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1. **Interactional Dominance** - This is the incremental R<sup>2</sup> contribution of the predictor to the complete model. Hence, the Interactional Dominance of a particular predictor 'X' will be the diffrence between the R<sup>2</sup> of the complete model and the R<sup>2</sup> of the model with all other predictors except the particular predictor 'X'. <br>
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<img src='images/formulas.JPG'>
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*This package is designed to determine relative importance of predictors for both regression and classification models. The determination of relative importance depends on how one defines importance; Budescu (1993) and Azen and Budescu (2003) proposed using dominance analysis (DA) because it invokes a general and intuitive definition of "relative importance" that is based on the additional contribution of a predictor in all subset models. The purpose of determining predictor importance in the context of DA is not model selection but rather uncovering the individual contributions of the predictors.*
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Based on these criteria, the following four R<sup>2</sup> analogues were chosen that satisfied at least three of these four properties:
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2. **Individual Dominance -** The individual dominance of a predictor is the R<sup>2</sup> of the model between the dependent variable and the predictor. So, the individual dominanace of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub></sub>. <br>
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* **Conditional Dominance** - If a predictor’s averaged additional contribution within each model size is greater than that of another predictor, then the first predictor is said to conditionally dominate the latter. Here, the model size is indicated by the number of predictors included in a given model. If a predictor’s averaged additional contribution is greater for some model sizes but not for all, then conditional dominance between the two predictors cannot be established.
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Regression - <a href="https://www.kaggle.com/iamsajanbhagat/dominance-analysis-on-boston-house-prices-dataset" target="_blank">Dominane Analysis on Boston House Price Data</a> &
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Classification- <a href="https://www.kaggle.com/iamsajanbhagat/dominance-analysis-on-breast-cancer-dataset" target="_blank"> Dominance Analysis on Breast Cancer Dataset</a>*
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## Important Parameters
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<img src="https://latex.codecogs.com/svg.latex?\Large&space;R_{Estrella}^{2}=1-\{\frac{LL_{full}}{LL_{null}}\}^{\frac{-2}{N}*LL_{null}}" title="\Large R_{Estrella}^{2}=1-\}\frac{LL_{full}}{LL_{null}}\}^{\frac{2}{N}*LL_{null}}" />
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Hence, individual dominance can be interpreted as the variability explained by the predictor alone or the quantum of impact that a predictor will have in absence of all other predictors.
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3. **Average Partial Dominance -** This is average of average incremental R<sup>2</sup> contributions of the predictor to all subset models except complete model and bi-variate (when only one predcitor is present) model. <br>
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[![Downloads](https://static.pepy.tech/badge/dominance-analysis)](https://pepy.tech/project/dominance-analysis)
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# Dominance-Analysis : A Python Library for Accurate and Intuitive Relative Importance of Predictors
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```
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* data : Complete Dataset, should be a Pandas DataFrame.
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```
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in the context of all 2<sup>(p−2)</sup> models that contain some subset of the other predictors. So, if we have a total of 'p' predictors, we will build 2<sup>p</sup>-1 models (all possible subset models) and compute the incremental R<sup>2</sup> contribution of each predictor to the subset model of all other predictors. The additional contribution of a given predictor is measured by the increase in R<sup>2</sup> that results from adding that predictor to the regression model.
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<img src="https://latex.codecogs.com/svg.latex?\Large&space;R_{Cox\&Snell}^{2}=1-\{\frac{L_{null}}{L_{full}}\}^{2/N}" title="\Large R_{Cox\&Snell}^{2}=1-\{\frac{L_{null}}{L_{full}}\}^{2/N}" />
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It can bee seen that the Percentage Relative Importance of predictors has been computed by dividing the Overall Average Incremental R<sup>2</sup> contribution of predictors by the R<sup>2</sup> of the complete model. This explains the intuitive nature of Dominance Analysis wherein the overall R<sup>2</sup> of the model can be attributed to individual predictors within the model.
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4. **Total Dominance -** The last measure of dominance summarizes the additional contributions of each predictor to all subset models by averaging all the conditional values. In the example on table 3, this consists of averaging the four averaged entries in each column. <br>
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1. Azen, R. (2000). Inference for predictor comparisons:Dominance analysis and the distribution of R<sup>2</sup> differences. Dissertation Abstracts International B, 61/10, 5616.
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3. Azen, R., Budescu, D. V. (2003). The Dominance Analysis Approach for Comparing Predictors in Multiple Regression. Psychological Methods, 2003, Vol. 8, No. 2, 129–148. https://doi.org/10.1037/1082-989X.8.2.129
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* data_format: It can take value 0, 1 or 2. 0 is for raw data, 1 is when correlation matrix (correlation of predictors with the target variable) is being passed, 2 is when covraiance matrix (covariance of predictors with the the traget variable) is being passed. By default, the package will run for raw data (data_format=0). This parameter is not needed in case of classification.
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Consider a scenario when we have Y as the dependent variable and four predictors X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>, let R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>2</sub> ;
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4. Azen, R., Budescu, D. V. (2006). Comparing Predictors in Multivariate Regression Models: An Extension of Dominance Analysis. Journal of Educational and Behavioral Statistics Summer 2006, Vol. 31, No. 2, pp. 157-180. https://doi.org/10.3102/10769986031002157
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2. Azen, R., Budescu, D. V., & Reiser, B. (2001). Criticality of predictors in multiple regression. British Journal of Mathematical and Statistical Psychology, 54, 201–225.
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<br> If the additional contributions are inconsistent in favoring the same predictor across all subset models, then complete dominance is undetermined while weaker levels of dominance may still be achieved.
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<img src="https://latex.codecogs.com/svg.latex?\Large&space;R_{Nagelkerke}^{2}=\frac{1-\{\frac{L_{null}}{L_{full}}\}^{2/N}}{1-L_{null}^{2/N}}" title="\Large R_{Nagelkerke}^{2}=\frac{1-\{\frac{L_{null}}{L_{full}}\}^{2/N}}{1-L_{null}^{2/N}}" />
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This measure satisfies three of the four properties and doesn't satisfy the property of Interpretability.
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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* objective : It can take value either 0 or 1. 0 for Classification and 1 for Regression. By default, the package will run for Regression.
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Dominance Analysis, according to Azen and Budescu meets three important criteria for measuring relative importance. First, the technique should be defined in terms of its ability to reduce error in predicting the outcome variable. Next, it should permit direct comparison of measures within a model (that is, X<sub>1</sub> is twice as important as X<sub>2</sub>). Finally, the technique should permit inferences concerning an attribute's direct effect (that is, when considered by itself), total effect (that is, when considered with other attributes) and partial effect (that is, when considered with various combinations of other predictors). Hence, Dominance analysis is both robust and intuitive and its interpretation is also very straightforward.
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3. **Average Partial Dominance -** This is average of average incremental R<sup>2</sup> contributions of the predictor to all subset models except complete model and bi-variate (when only one predcitor is present) model. <br>
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Hence, this can be interpreted as the average impact that a predictor has when it is available in all possible combinations with other predictors except the combination when all predcitors are available.
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The following three levels of dominance can be achieved between each pair of predictors in Dominance Analysis:
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* top_k : No. of features to choose from all available features. By default, the package will run for top 15 features.
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1. Boundedness: The measure should vary between a minimum of zero, indicating complete lack of fit, and a maximum of one, indicating perfect fit.
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4. Intuitive Interpretability: The measure of fit is intuitively interpretable, in that it agrees with the scale of the linear case for intermediate values.
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Note: Since, Dominance Analysis is computationally intensive as it builds all subset model (2<sup>p</sup>-1 models), we have provided the user the flexibility to choose number of top predictors that they want to compute relative importance for. For regression, Top K features are selected based on F-regression and for classification it is based on Chi-Squared statistic. Dominance Analysis can be used in combination with Principal Component Analysis (PCA) or Factor Analysis or any other feature reduction algorithm for getting accurate and intutive importance of predictors.
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Hence, interactional dominance can be interpreted as the incremental impact or incremental variability explained by the predictor in presence of all other predictors.
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hold; that is, general dominance does not imply conditional dominance and conditional dominance does not necessarily imply complete dominance.
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*Complete code for below examples is available in example folder or the following public kernels on Kaggle:
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5. Azen, R., Traxel, N. (2009). Using Dominance Analysis to Determine Predictor Importance in Logistic Regression. Journal of Educational and Behavioral Statistics September 2009, Vol. 34, No. 3, pp. 319-347. https://doi.org/10.3102/1076998609332754
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7. Luo, W., & Azen, R. (2013). Determining Predictor Importance in Hierarchical Linear Models Using Dominance Analysis. Journal of Educational and Behavioral Statistics, 38(1), 3-31. https://doi.org/10.3102/1076998612458319
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```
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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The beauty of the math of Dominance Analysis is that the sum of the overall average incremental R<sup>2</sup> of all predictors is equal to the R<sup>2</sup> of the complete model (model with all predictors). Hence, the total R<sup>2</sup> can be attributed to each predictor in the model. Below is an illustration of Dominance Analysis in the Population for Hypothetical example with four predictors:
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4. Azen, R., Budescu, D. V. (2006). Comparing Predictors in Multivariate Regression Models: An Extension of Dominance Analysis. Journal of Educational and Behavioral Statistics Summer 2006, Vol. 31, No. 2, pp. 157-180. https://doi.org/10.3102/10769986031002157
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6. Budescu, D. V. (1993). Dominance analysis: A new approach to the problem of relative importance of predictors in multiple regression. Psychological Bulletin, 114(3), 542-551. https://doi.org/10.1037/0033-2909.114.3.542
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```
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Dominance Analysis is unique as it measures relative importance in a pairwise fashion, and the two predictors are compared
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<br> If the additional contributions are inconsistent in favoring the same predictor across all subset models, then complete dominance is undetermined while weaker levels of dominance may still be achieved.
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# Dominance-Analysis : A Python Library for Accurate and Intuitive Relative Importance of Predictors
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in the context of all 2<sup>(p−2)</sup> models that contain some subset of the other predictors. So, if we have a total of 'p' predictors, we will build 2<sup>p</sup>-1 models (all possible subset models) and compute the incremental R<sup>2</sup> contribution of each predictor to the subset model of all other predictors. The additional contribution of a given predictor is measured by the increase in R<sup>2</sup> that results from adding that predictor to the regression model.
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# Markdown header
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Inline HTML Open
Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Inline HTML Open
Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Inline HTML Open
Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Trailing spaces Open
```
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MD009 - Trailing spaces
Tags: whitespace
Aliases: no-trailing-spaces
Parameters: br_spaces (number; default: 0)
This rule is triggered on any lines that end with whitespace. To fix this, find the line that is triggered and remove any trailing spaces from the end.
The brspaces parameter allows an exception to this rule for a specific amount of trailing spaces used to insert an explicit line break/br element. For example, set brspaces to 2 to allow exactly 2 spaces at the end of a line.
Note: you have to set brspaces to 2 or higher for this exception to take effect - you can't insert a br element with just a single trailing space, so if you set brspaces to 1, the exception will be disabled, just as if it was set to the default of 0.
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<a href="https://github.com/dominance-analysis/dominance-analysis/blob/master/LICENSE"><img alt="GitHub license" src="https://img.shields.io/github/license/dominance-analysis/dominance-analysis"></a>
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MD013 - Line length
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do this, set the code_blocks
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Code blocks are included in this rule by default since it is often a requirement for document readability, and tentatively compatible with code rules. Still, some languages do not lend themselves to short lines.
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*In case the target is a continuous variable, the package determines the dominance of one predictor over another by comparing their incremental R-squared contribution across all subset models. In case the target variable is binary, the package determines the dominance over another by comparing their incremental Pseudo R-Squared contribution across all subset models.*
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MD013 - Line length
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This rule is triggered when there are lines that are longer than the configured line length (default: 80 characters). To fix this, split the line up into multiple lines.
This rule has an exception where there is no whitespace beyond the configured line length. This allows you to still include items such as long URLs without being forced to break them in the middle.
You also have the option to exclude this rule for code blocks and tables. To
do this, set the code_blocks
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Code blocks are included in this rule by default since it is often a requirement for document readability, and tentatively compatible with code rules. Still, some languages do not lend themselves to short lines.
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<img src="https://latex.codecogs.com/svg.latex?\Large&space;R_{McFadden}^{2}=1-\frac{log(L_{full})}{log(L_{null})}" title="\Large R_{McFadden}^{2}=1-\frac{log(L_{full})}{log(L_{null})}" />
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MD013 - Line length
Tags: line_length
Aliases: line-length Parameters: linelength, codeblocks, tables (number; default 80, boolean; default true)
This rule is triggered when there are lines that are longer than the configured line length (default: 80 characters). To fix this, split the line up into multiple lines.
This rule has an exception where there is no whitespace beyond the configured line length. This allows you to still include items such as long URLs without being forced to break them in the middle.
You also have the option to exclude this rule for code blocks and tables. To
do this, set the code_blocks
and/or tables
parameters to false.
Code blocks are included in this rule by default since it is often a requirement for document readability, and tentatively compatible with code rules. Still, some languages do not lend themselves to short lines.
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* **General Dominance** - If overall averaged additional R<sup>2</sup> contribution of one predictor is greater than the other then that predictor is said to generally dominate the other. <br>
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MD013 - Line length
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Aliases: line-length Parameters: linelength, codeblocks, tables (number; default 80, boolean; default true)
This rule is triggered when there are lines that are longer than the configured line length (default: 80 characters). To fix this, split the line up into multiple lines.
This rule has an exception where there is no whitespace beyond the configured line length. This allows you to still include items such as long URLs without being forced to break them in the middle.
You also have the option to exclude this rule for code blocks and tables. To
do this, set the code_blocks
and/or tables
parameters to false.
Code blocks are included in this rule by default since it is often a requirement for document readability, and tentatively compatible with code rules. Still, some languages do not lend themselves to short lines.
Ordered list item prefix Open
2. Azen, R., Budescu, D. V., & Reiser, B. (2001). Criticality of predictors in multiple regression. British Journal of Mathematical and Statistical Psychology, 54, 201–225.
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MD029 - Ordered list item prefix
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Aliases: ol-prefix
Parameters: style ("one", "ordered"; default "one")
This rule is triggered on ordered lists that do not either start with '1.' or do not have a prefix that increases in numerical order (depending on the configured style, which defaults to 'one').
Example valid list if the style is configured as 'one':
1. Do this.
1. Do that.
1. Done.
Example valid list if the style is configured as 'ordered':
1. Do this.
2. Do that.
3. Done.
Inconsistent indentation for list items at the same level Open
4. **Total Dominance -** The last measure of dominance summarizes the additional contributions of each predictor to all subset models by averaging all the conditional values. In the example on table 3, this consists of averaging the four averaged entries in each column. <br>
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MD005 - Inconsistent indentation for list items at the same level
Tags: bullet, ul, indentation
Aliases: list-indent
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* Item 1
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* A misaligned item
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* Item 1
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* Nested Item 3
Fenced code blocks should be surrounded by blank lines Open
```
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MD031 - Fenced code blocks should be surrounded by blank lines
Tags: code, blank_lines
Aliases: blanks-around-fences
This rule is triggered when fenced code blocks are either not preceded or not followed by a blank line:
Some text
```
Code block
```
```
Another code block
```
Some more text
To fix this, ensure that all fenced code blocks have a blank line both before and after (except where the block is at the beginning or end of the document):
Some text
```
Code block
```
```
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```
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Rationale: Aside from aesthetic reasons, some parsers, including kramdown, will not parse fenced code blocks that don't have blank lines before and after them.
Inconsistent indentation for list items at the same level Open
1. Azen, R. (2000). Inference for predictor comparisons:Dominance analysis and the distribution of R<sup>2</sup> differences. Dissertation Abstracts International B, 61/10, 5616.
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MD005 - Inconsistent indentation for list items at the same level
Tags: bullet, ul, indentation
Aliases: list-indent
This rule is triggered when list items are parsed as being at the same level, but don't have the same indentation:
* Item 1
* Nested Item 1
* Nested Item 2
* A misaligned item
Usually this rule will be triggered because of a typo. Correct the indentation for the list to fix it:
* Item 1
* Nested Item 1
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Fenced code blocks should be surrounded by blank lines Open
```
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MD031 - Fenced code blocks should be surrounded by blank lines
Tags: code, blank_lines
Aliases: blanks-around-fences
This rule is triggered when fenced code blocks are either not preceded or not followed by a blank line:
Some text
```
Code block
```
```
Another code block
```
Some more text
To fix this, ensure that all fenced code blocks have a blank line both before and after (except where the block is at the beginning or end of the document):
Some text
```
Code block
```
```
Another code block
```
Some more text
Rationale: Aside from aesthetic reasons, some parsers, including kramdown, will not parse fenced code blocks that don't have blank lines before and after them.
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5. Azen, R., Traxel, N. (2009). Using Dominance Analysis to Determine Predictor Importance in Logistic Regression. Journal of Educational and Behavioral Statistics September 2009, Vol. 34, No. 3, pp. 319-347. https://doi.org/10.3102/1076998609332754
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MD005 - Inconsistent indentation for list items at the same level
Tags: bullet, ul, indentation
Aliases: list-indent
This rule is triggered when list items are parsed as being at the same level, but don't have the same indentation:
* Item 1
* Nested Item 1
* Nested Item 2
* A misaligned item
Usually this rule will be triggered because of a typo. Correct the indentation for the list to fix it:
* Item 1
* Nested Item 1
* Nested Item 2
* Nested Item 3
Fenced code blocks should be surrounded by blank lines Open
```
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MD031 - Fenced code blocks should be surrounded by blank lines
Tags: code, blank_lines
Aliases: blanks-around-fences
This rule is triggered when fenced code blocks are either not preceded or not followed by a blank line:
Some text
```
Code block
```
```
Another code block
```
Some more text
To fix this, ensure that all fenced code blocks have a blank line both before and after (except where the block is at the beginning or end of the document):
Some text
```
Code block
```
```
Another code block
```
Some more text
Rationale: Aside from aesthetic reasons, some parsers, including kramdown, will not parse fenced code blocks that don't have blank lines before and after them.
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1. Boundedness: The measure should vary between a minimum of zero, indicating complete lack of fit, and a maximum of one, indicating perfect fit.
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MD009 - Trailing spaces
Tags: whitespace
Aliases: no-trailing-spaces
Parameters: br_spaces (number; default: 0)
This rule is triggered on any lines that end with whitespace. To fix this, find the line that is triggered and remove any trailing spaces from the end.
The brspaces parameter allows an exception to this rule for a specific amount of trailing spaces used to insert an explicit line break/br element. For example, set brspaces to 2 to allow exactly 2 spaces at the end of a line.
Note: you have to set brspaces to 2 or higher for this exception to take effect - you can't insert a br element with just a single trailing space, so if you set brspaces to 1, the exception will be disabled, just as if it was set to the default of 0.
Fenced code blocks should be surrounded by blank lines Open
```
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MD031 - Fenced code blocks should be surrounded by blank lines
Tags: code, blank_lines
Aliases: blanks-around-fences
This rule is triggered when fenced code blocks are either not preceded or not followed by a blank line:
Some text
```
Code block
```
```
Another code block
```
Some more text
To fix this, ensure that all fenced code blocks have a blank line both before and after (except where the block is at the beginning or end of the document):
Some text
```
Code block
```
```
Another code block
```
Some more text
Rationale: Aside from aesthetic reasons, some parsers, including kramdown, will not parse fenced code blocks that don't have blank lines before and after them.
Trailing spaces Open
2. Linear invariance: The measure should be invariant to nonsingular linear transformations of the variables (Ys and Xs).
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MD009 - Trailing spaces
Tags: whitespace
Aliases: no-trailing-spaces
Parameters: br_spaces (number; default: 0)
This rule is triggered on any lines that end with whitespace. To fix this, find the line that is triggered and remove any trailing spaces from the end.
The brspaces parameter allows an exception to this rule for a specific amount of trailing spaces used to insert an explicit line break/br element. For example, set brspaces to 2 to allow exactly 2 spaces at the end of a line.
Note: you have to set brspaces to 2 or higher for this exception to take effect - you can't insert a br element with just a single trailing space, so if you set brspaces to 1, the exception will be disabled, just as if it was set to the default of 0.
Inline HTML Open
<a href="https://github.com/dominance-analysis/dominance-analysis/blob/master/LICENSE"><img alt="GitHub license" src="https://img.shields.io/github/license/dominance-analysis/dominance-analysis"></a>
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
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# Dominance-Analysis : A Python Library for Accurate and Intuitive Relative Importance of Predictors
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MD013 - Line length
Tags: line_length
Aliases: line-length Parameters: linelength, codeblocks, tables (number; default 80, boolean; default true)
This rule is triggered when there are lines that are longer than the configured line length (default: 80 characters). To fix this, split the line up into multiple lines.
This rule has an exception where there is no whitespace beyond the configured line length. This allows you to still include items such as long URLs without being forced to break them in the middle.
You also have the option to exclude this rule for code blocks and tables. To
do this, set the code_blocks
and/or tables
parameters to false.
Code blocks are included in this rule by default since it is often a requirement for document readability, and tentatively compatible with code rules. Still, some languages do not lend themselves to short lines.
Inline HTML Open
<a href="https://github.com/dominance-analysis/dominance-analysis/blob/master/LICENSE"><img alt="GitHub license" src="https://img.shields.io/github/license/dominance-analysis/dominance-analysis"></a>
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Line length Open
In below table, we have illustrated the calculation used to arrive at the four measures of dominance.
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MD013 - Line length
Tags: line_length
Aliases: line-length Parameters: linelength, codeblocks, tables (number; default 80, boolean; default true)
This rule is triggered when there are lines that are longer than the configured line length (default: 80 characters). To fix this, split the line up into multiple lines.
This rule has an exception where there is no whitespace beyond the configured line length. This allows you to still include items such as long URLs without being forced to break them in the middle.
You also have the option to exclude this rule for code blocks and tables. To
do this, set the code_blocks
and/or tables
parameters to false.
Code blocks are included in this rule by default since it is often a requirement for document readability, and tentatively compatible with code rules. Still, some languages do not lend themselves to short lines.
Inline HTML Open
<a href="https://github.com/dominance-analysis/dominance-analysis/stargazers">
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Inline HTML Open
<hr>
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Line length Open
The following three levels of dominance can be achieved between each pair of predictors in Dominance Analysis:
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MD013 - Line length
Tags: line_length
Aliases: line-length Parameters: linelength, codeblocks, tables (number; default 80, boolean; default true)
This rule is triggered when there are lines that are longer than the configured line length (default: 80 characters). To fix this, split the line up into multiple lines.
This rule has an exception where there is no whitespace beyond the configured line length. This allows you to still include items such as long URLs without being forced to break them in the middle.
You also have the option to exclude this rule for code blocks and tables. To
do this, set the code_blocks
and/or tables
parameters to false.
Code blocks are included in this rule by default since it is often a requirement for document readability, and tentatively compatible with code rules. Still, some languages do not lend themselves to short lines.
Inline HTML Open
Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD033 - Inline HTML
Tags: html
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# Markdown header
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Headers should be surrounded by blank lines Open
### References
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MD022 - Headers should be surrounded by blank lines
Tags: headers, blank_lines
Aliases: blanks-around-headers
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# Header 1
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# Header 1
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Inline HTML Open
Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD033 - Inline HTML
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This rule is triggered whenever raw HTML is used in a markdown document:
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Ordered list item prefix Open
5. Azen, R., Traxel, N. (2009). Using Dominance Analysis to Determine Predictor Importance in Logistic Regression. Journal of Educational and Behavioral Statistics September 2009, Vol. 34, No. 3, pp. 319-347. https://doi.org/10.3102/1076998609332754
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MD029 - Ordered list item prefix
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Aliases: ol-prefix
Parameters: style ("one", "ordered"; default "one")
This rule is triggered on ordered lists that do not either start with '1.' or do not have a prefix that increases in numerical order (depending on the configured style, which defaults to 'one').
Example valid list if the style is configured as 'one':
1. Do this.
1. Do that.
1. Done.
Example valid list if the style is configured as 'ordered':
1. Do this.
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3. Done.
Inline HTML Open
Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
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# Markdown header
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Fenced code blocks should be surrounded by blank lines Open
```
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MD031 - Fenced code blocks should be surrounded by blank lines
Tags: code, blank_lines
Aliases: blanks-around-fences
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Some text
```
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```
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```
Some more text
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```
Code block
```
```
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<img src="https://latex.codecogs.com/svg.latex?\Large&space;R_{Nagelkerke}^{2}=\frac{1-\{\frac{L_{null}}{L_{full}}\}^{2/N}}{1-L_{null}^{2/N}}" title="\Large R_{Nagelkerke}^{2}=\frac{1-\{\frac{L_{null}}{L_{full}}\}^{2/N}}{1-L_{null}^{2/N}}" />
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
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# Markdown header
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Lists should be surrounded by blank lines Open
1. Boundedness: The measure should vary between a minimum of zero, indicating complete lack of fit, and a maximum of one, indicating perfect fit.
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MD032 - Lists should be surrounded by blank lines
Tags: bullet, ul, ol, blank_lines
Aliases: blanks-around-lists
This rule is triggered when lists (of any kind) are either not preceded or not followed by a blank line:
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* Some
* List
1. Some
2. List
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* List
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Note: List items without hanging indents are a violation of this rule; list items with hanging indents are okay:
* This is
not okay
* This is
okay
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<img src="https://latex.codecogs.com/svg.latex?\Large&space;R_{Cox\&Snell}^{2}=1-\{\frac{L_{null}}{L_{full}}\}^{2/N}" title="\Large R_{Cox\&Snell}^{2}=1-\{\frac{L_{null}}{L_{full}}\}^{2/N}" />
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
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# Markdown header
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Lists should be surrounded by blank lines Open
* **Conditional Dominance** - If a predictor’s averaged additional contribution within each model size is greater than that of another predictor, then the first predictor is said to conditionally dominate the latter. Here, the model size is indicated by the number of predictors included in a given model. If a predictor’s averaged additional contribution is greater for some model sizes but not for all, then conditional dominance between the two predictors cannot be established.
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MD032 - Lists should be surrounded by blank lines
Tags: bullet, ul, ol, blank_lines
Aliases: blanks-around-lists
This rule is triggered when lists (of any kind) are either not preceded or not followed by a blank line:
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* List
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* List
1. Some
2. List
Some text
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Note: List items without hanging indents are a violation of this rule; list items with hanging indents are okay:
* This is
not okay
* This is
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Inline HTML Open
in the context of all 2<sup>(p−2)</sup> models that contain some subset of the other predictors. So, if we have a total of 'p' predictors, we will build 2<sup>p</sup>-1 models (all possible subset models) and compute the incremental R<sup>2</sup> contribution of each predictor to the subset model of all other predictors. The additional contribution of a given predictor is measured by the increase in R<sup>2</sup> that results from adding that predictor to the regression model.
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
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# Markdown header
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Inline HTML Open
Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
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# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Inline HTML Open
Using each of these four R<sup>2</sup> analogues, the additional contribution of a given predictor to a specific logistic model can be measured as the change (i.e., increase) in the R<sup>2</sup> analogues when the predictor is added to the model. Even though, all the four measures will give similar results, we recommend using either Estrella's (1998) model fit measure or McFadden's (1974) measure for conducting dominance analysis in logistic regression. We have a slight preference for McFadden's measure (and that is what the package will compute by default) because it is computationally simpler, but both McFadden's and Estrella’s measures satisfy the minimum requirements for an R<sup>2</sup> analogues.
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
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# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Line length Open
Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD013 - Line length
Tags: line_length
Aliases: line-length Parameters: linelength, codeblocks, tables (number; default 80, boolean; default true)
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Code blocks are included in this rule by default since it is often a requirement for document readability, and tentatively compatible with code rules. Still, some languages do not lend themselves to short lines.
Line length Open
6. Budescu, D. V. (1993). Dominance analysis: A new approach to the problem of relative importance of predictors in multiple regression. Psychological Bulletin, 114(3), 542-551. https://doi.org/10.1037/0033-2909.114.3.542
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MD013 - Line length
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Aliases: line-length Parameters: linelength, codeblocks, tables (number; default 80, boolean; default true)
This rule is triggered when there are lines that are longer than the configured line length (default: 80 characters). To fix this, split the line up into multiple lines.
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parameters to false.
Code blocks are included in this rule by default since it is often a requirement for document readability, and tentatively compatible with code rules. Still, some languages do not lend themselves to short lines.
Inconsistent indentation for list items at the same level Open
7. Luo, W., & Azen, R. (2013). Determining Predictor Importance in Hierarchical Linear Models Using Dominance Analysis. Journal of Educational and Behavioral Statistics, 38(1), 3-31. https://doi.org/10.3102/1076998612458319
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MD005 - Inconsistent indentation for list items at the same level
Tags: bullet, ul, indentation
Aliases: list-indent
This rule is triggered when list items are parsed as being at the same level, but don't have the same indentation:
* Item 1
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* A misaligned item
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* Item 1
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Trailing spaces Open
```
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MD009 - Trailing spaces
Tags: whitespace
Aliases: no-trailing-spaces
Parameters: br_spaces (number; default: 0)
This rule is triggered on any lines that end with whitespace. To fix this, find the line that is triggered and remove any trailing spaces from the end.
The brspaces parameter allows an exception to this rule for a specific amount of trailing spaces used to insert an explicit line break/br element. For example, set brspaces to 2 to allow exactly 2 spaces at the end of a line.
Note: you have to set brspaces to 2 or higher for this exception to take effect - you can't insert a br element with just a single trailing space, so if you set brspaces to 1, the exception will be disabled, just as if it was set to the default of 0.
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[![Downloads](https://static.pepy.tech/badge/dominance-analysis)](https://pepy.tech/project/dominance-analysis)
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MD013 - Line length
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This rule is triggered when there are lines that are longer than the configured line length (default: 80 characters). To fix this, split the line up into multiple lines.
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2. Linear invariance: The measure should be invariant to nonsingular linear transformations of the variables (Ys and Xs).
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MD013 - Line length
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Code blocks are included in this rule by default since it is often a requirement for document readability, and tentatively compatible with code rules. Still, some languages do not lend themselves to short lines.
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Using each of these four R<sup>2</sup> analogues, the additional contribution of a given predictor to a specific logistic model can be measured as the change (i.e., increase) in the R<sup>2</sup> analogues when the predictor is added to the model. Even though, all the four measures will give similar results, we recommend using either Estrella's (1998) model fit measure or McFadden's (1974) measure for conducting dominance analysis in logistic regression. We have a slight preference for McFadden's measure (and that is what the package will compute by default) because it is computationally simpler, but both McFadden's and Estrella’s measures satisfy the minimum requirements for an R<sup>2</sup> analogues.
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MD013 - Line length
Tags: line_length
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This rule is triggered when there are lines that are longer than the configured line length (default: 80 characters). To fix this, split the line up into multiple lines.
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You also have the option to exclude this rule for code blocks and tables. To
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Code blocks are included in this rule by default since it is often a requirement for document readability, and tentatively compatible with code rules. Still, some languages do not lend themselves to short lines.
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As described earlier, a relative importance measure should be able to describe a predictor's direct, total and partial effect, therefore in the Dominance Statistics, we have come up with four different types of Dominance measures. **These measures have been conceptualized, defined and formulated by us and are unique to this library**. Below are the definitions and interpretations of the measures:
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MD013 - Line length
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The three levels of dominance (complete, conditional,and general) are related to each other in a hierarchical fashion: Complete dominance implies conditional dominance, which, in turn, implies general dominance. However, for p > 3 the converse may not
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MD013 - Line length
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Aliases: line-length Parameters: linelength, codeblocks, tables (number; default 80, boolean; default true)
This rule is triggered when there are lines that are longer than the configured line length (default: 80 characters). To fix this, split the line up into multiple lines.
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parameters to false.
Code blocks are included in this rule by default since it is often a requirement for document readability, and tentatively compatible with code rules. Still, some languages do not lend themselves to short lines.
Inconsistent indentation for list items at the same level Open
2. **Individual Dominance -** The individual dominance of a predictor is the R<sup>2</sup> of the model between the dependent variable and the predictor. So, the individual dominanace of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub></sub>. <br>
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MD005 - Inconsistent indentation for list items at the same level
Tags: bullet, ul, indentation
Aliases: list-indent
This rule is triggered when list items are parsed as being at the same level, but don't have the same indentation:
* Item 1
* Nested Item 1
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* A misaligned item
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* Nested Item 1
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* Nested Item 3
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3. Azen, R., Budescu, D. V. (2003). The Dominance Analysis Approach for Comparing Predictors in Multiple Regression. Psychological Methods, 2003, Vol. 8, No. 2, 129–148. https://doi.org/10.1037/1082-989X.8.2.129
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MD005 - Inconsistent indentation for list items at the same level
Tags: bullet, ul, indentation
Aliases: list-indent
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* Item 1
* Nested Item 1
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```
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MD009 - Trailing spaces
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Parameters: br_spaces (number; default: 0)
This rule is triggered on any lines that end with whitespace. To fix this, find the line that is triggered and remove any trailing spaces from the end.
The brspaces parameter allows an exception to this rule for a specific amount of trailing spaces used to insert an explicit line break/br element. For example, set brspaces to 2 to allow exactly 2 spaces at the end of a line.
Note: you have to set brspaces to 2 or higher for this exception to take effect - you can't insert a br element with just a single trailing space, so if you set brspaces to 1, the exception will be disabled, just as if it was set to the default of 0.
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MD012 - Multiple consecutive blank lines
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The beauty of the math of Dominance Analysis is that the sum of the overall average incremental R<sup>2</sup> of all predictors is equal to the R<sup>2</sup> of the complete model (model with all predictors). Hence, the total R<sup>2</sup> can be attributed to each predictor in the model. Below is an illustration of Dominance Analysis in the Population for Hypothetical example with four predictors:
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MD013 - Line length
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Measures of fit in logistic regression can be classified by those based on sums of squares and those based on maximum likelihood statistics. Reviews of a variety of measures of fit proposed for logistic regression can be found in Amemiya (1981), Menard (2000), Mittlbock and Schemper (1996) and Zheng and Agresti (2000). Given the large number of proposed measures, criteria for defining appropriate R<sup>2</sup> analogues need to he determined. The following criteria, which are also found in the linear regression literature (e.g., Kvilseth. 1985: Van den Burg & Lewis, 1988), were used to select R<sup>2</sup> analogues for logistic regression:
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1. **Interactional Dominance** - This is the incremental R<sup>2</sup> contribution of the predictor to the complete model. Hence, the Interactional Dominance of a particular predictor 'X' will be the diffrence between the R<sup>2</sup> of the complete model and the R<sup>2</sup> of the model with all other predictors except the particular predictor 'X'. <br>
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The Dominance Analysis package is based on the concept developed by Azen and Budescu (see references). This package is released under a <a href="https://github.com/bhagatsajan0073/dominance_analysis/blob/master/LICENSE"> MIT License</a>. Dominance Analysis Python package has been developed by <a href="https://github.com/quintshekhar"> Shashank Shekhar</a>, <a href="https://github.com/bhagatsajan0073"> Sajan Bhagat</a>, <a href="https://github.com/Vibish"> Kunjithapatham Sivakumar</a> and <a href="https://github.com/balakolluri"> Bala Koteshwar Kolluri</a> . Pull requests submitted to the GitHub Repo are highly encouraged!
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2. Azen, R., Budescu, D. V., & Reiser, B. (2001). Criticality of predictors in multiple regression. British Journal of Mathematical and Statistical Psychology, 54, 201–225.
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### Installation
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* pseudo_r2 : It can take one of the Pseudo R-Squared measures - "mcfadden","nagelkerke", "cox_and_snell" or "estrella", where default="mcfadden". It's not needed in case of regression (objective=1).
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Below is the illustration of formulas used to compute the averaged additional contributions of X<sub>1</sub> and X<sub>2</sub> within model size in the poupulation with four predictors (We use the notation <img src='images/formula1.JPG'> to represent the proportion of variance in Y that is accounted for by the predictors in the model X. For example,<img src='images/formula2.JPG'> represents the proportion of variance in Y that is accounted for by the model consisting of X<sub>1</sub> and X<sub>3</sub>. The additional contribution of a given predictor is measured by the increase in proportion of variance that results from adding that predictor to the regression model):
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The measure for proportion of variance that we have used for regression is R<sup>2</sup> but since we don't have R<sup>2</sup> in logsitic regression/classification models, we have used Pseudo R<sup>2</sup>.
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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If we calculate the four measures of dominance from the above example, we will get the following values:
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* data : Complete Dataset, should be a Pandas DataFrame.
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3. **Average Partial Dominance -** This is average of average incremental R<sup>2</sup> contributions of the predictor to all subset models except complete model and bi-variate (when only one predcitor is present) model. <br>
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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The measure for proportion of variance that we have used for regression is R<sup>2</sup> but since we don't have R<sup>2</sup> in logsitic regression/classification models, we have used Pseudo R<sup>2</sup>.
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Based on these criteria, the following four R<sup>2</sup> analogues were chosen that satisfied at least three of these four properties:
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Using each of these four R<sup>2</sup> analogues, the additional contribution of a given predictor to a specific logistic model can be measured as the change (i.e., increase) in the R<sup>2</sup> analogues when the predictor is added to the model. Even though, all the four measures will give similar results, we recommend using either Estrella's (1998) model fit measure or McFadden's (1974) measure for conducting dominance analysis in logistic regression. We have a slight preference for McFadden's measure (and that is what the package will compute by default) because it is computationally simpler, but both McFadden's and Estrella’s measures satisfy the minimum requirements for an R<sup>2</sup> analogues.
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2. **Individual Dominance -** The individual dominance of a predictor is the R<sup>2</sup> of the model between the dependent variable and the predictor. So, the individual dominanace of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub></sub>. <br>
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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* **General Dominance** - If overall averaged additional R<sup>2</sup> contribution of one predictor is greater than the other then that predictor is said to generally dominate the other. <br>
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**Breast Cancer Wisconsin (Diagnostic) dataset downloaded from: https://goo.gl/U2Uwz2**
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7. Luo, W., & Azen, R. (2013). Determining Predictor Importance in Hierarchical Linear Models Using Dominance Analysis. Journal of Educational and Behavioral Statistics, 38(1), 3-31. https://doi.org/10.3102/1076998612458319
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<img src="https://img.shields.io/github/issues/dominance-analysis/dominance-analysis.svg">
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2. **Individual Dominance -** The individual dominance of a predictor is the R<sup>2</sup> of the model between the dependent variable and the predictor. So, the individual dominanace of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub></sub>. <br>
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Below is the illustration of formulas used to compute the averaged additional contributions of X<sub>1</sub> and X<sub>2</sub> within model size in the poupulation with four predictors (We use the notation <img src='images/formula1.JPG'> to represent the proportion of variance in Y that is accounted for by the predictors in the model X. For example,<img src='images/formula2.JPG'> represents the proportion of variance in Y that is accounted for by the model consisting of X<sub>1</sub> and X<sub>3</sub>. The additional contribution of a given predictor is measured by the increase in proportion of variance that results from adding that predictor to the regression model):
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It can bee seen that the Percentage Relative Importance of predictors has been computed by dividing the Overall Average Incremental R<sup>2</sup> contribution of predictors by the R<sup>2</sup> of the complete model. This explains the intuitive nature of Dominance Analysis wherein the overall R<sup>2</sup> of the model can be attributed to individual predictors within the model.
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Consider a scenario when we have Y as the dependent variable and four predictors X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>, let R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>2</sub> ;
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Consider a scenario when we have Y as the dependent variable and four predictors X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>, let R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>2</sub> ;
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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<img src='images/Dom Stat.jpg'>
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Dominance Analysis, according to Azen and Budescu meets three important criteria for measuring relative importance. First, the technique should be defined in terms of its ability to reduce error in predicting the outcome variable. Next, it should permit direct comparison of measures within a model (that is, X<sub>1</sub> is twice as important as X<sub>2</sub>). Finally, the technique should permit inferences concerning an attribute's direct effect (that is, when considered by itself), total effect (that is, when considered with other attributes) and partial effect (that is, when considered with various combinations of other predictors). Hence, Dominance analysis is both robust and intuitive and its interpretation is also very straightforward.
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* **General Dominance** - If overall averaged additional R<sup>2</sup> contribution of one predictor is greater than the other then that predictor is said to generally dominate the other. <br>
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Regression - <a href="https://www.kaggle.com/iamsajanbhagat/dominance-analysis-on-boston-house-prices-dataset" target="_blank">Dominane Analysis on Boston House Price Data</a> &
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<img src='images/breast_cancer_data_dominance_stats.JPG'>
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Below is the illustration of formulas used to compute the averaged additional contributions of X<sub>1</sub> and X<sub>2</sub> within model size in the poupulation with four predictors (We use the notation <img src='images/formula1.JPG'> to represent the proportion of variance in Y that is accounted for by the predictors in the model X. For example,<img src='images/formula2.JPG'> represents the proportion of variance in Y that is accounted for by the model consisting of X<sub>1</sub> and X<sub>3</sub>. The additional contribution of a given predictor is measured by the increase in proportion of variance that results from adding that predictor to the regression model):
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**Using Boston Housing Dataset downloaded from: https://www.cs.toronto.edu/~delve/data/boston/bostonDetail.html**
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The measure for proportion of variance that we have used for regression is R<sup>2</sup> but since we don't have R<sup>2</sup> in logsitic regression/classification models, we have used Pseudo R<sup>2</sup>.
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<img src='images/PercentRel.jpg'>
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<p align="center"> Table 2</p>
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It can bee seen that the Percentage Relative Importance of predictors has been computed by dividing the Overall Average Incremental R<sup>2</sup> contribution of predictors by the R<sup>2</sup> of the complete model. This explains the intuitive nature of Dominance Analysis wherein the overall R<sup>2</sup> of the model can be attributed to individual predictors within the model.
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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4. Azen, R., Budescu, D. V. (2006). Comparing Predictors in Multivariate Regression Models: An Extension of Dominance Analysis. Journal of Educational and Behavioral Statistics Summer 2006, Vol. 31, No. 2, pp. 157-180. https://doi.org/10.3102/10769986031002157
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## Important Parameters
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### Dominance Analysis - The Significance!
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### Dominance Analysis - The Math!
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3. **Average Partial Dominance -** This is average of average incremental R<sup>2</sup> contributions of the predictor to all subset models except complete model and bi-variate (when only one predcitor is present) model. <br>
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3. Azen, R., Budescu, D. V. (2003). The Dominance Analysis Approach for Comparing Predictors in Multiple Regression. Psychological Methods, 2003, Vol. 8, No. 2, 129–148. https://doi.org/10.1037/1082-989X.8.2.129
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2. **Individual Dominance -** The individual dominance of a predictor is the R<sup>2</sup> of the model between the dependent variable and the predictor. So, the individual dominanace of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub></sub>. <br>
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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It can bee seen that the Percentage Relative Importance of predictors has been computed by dividing the Overall Average Incremental R<sup>2</sup> contribution of predictors by the R<sup>2</sup> of the complete model. This explains the intuitive nature of Dominance Analysis wherein the overall R<sup>2</sup> of the model can be attributed to individual predictors within the model.
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Consider a scenario when we have Y as the dependent variable and four predictors X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>, let R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>2</sub> ;
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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### Authors & License
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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4. **Total Dominance -** The last measure of dominance summarizes the additional contributions of each predictor to all subset models by averaging all the conditional values. In the example on table 3, this consists of averaging the four averaged entries in each column. <br>
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2. **Individual Dominance -** The individual dominance of a predictor is the R<sup>2</sup> of the model between the dependent variable and the predictor. So, the individual dominanace of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub></sub>. <br>
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
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To fix this, use 'pure' markdown instead of including raw HTML:
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The Dominance Analysis package is based on the concept developed by Azen and Budescu (see references). This package is released under a <a href="https://github.com/bhagatsajan0073/dominance_analysis/blob/master/LICENSE"> MIT License</a>. Dominance Analysis Python package has been developed by <a href="https://github.com/quintshekhar"> Shashank Shekhar</a>, <a href="https://github.com/bhagatsajan0073"> Sajan Bhagat</a>, <a href="https://github.com/Vibish"> Kunjithapatham Sivakumar</a> and <a href="https://github.com/balakolluri"> Bala Koteshwar Kolluri</a> . Pull requests submitted to the GitHub Repo are highly encouraged!
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MD033 - Inline HTML
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Aliases: no-inline-html
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To fix this, use 'pure' markdown instead of including raw HTML:
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Ordered list item prefix Open
6. Budescu, D. V. (1993). Dominance analysis: A new approach to the problem of relative importance of predictors in multiple regression. Psychological Bulletin, 114(3), 542-551. https://doi.org/10.1037/0033-2909.114.3.542
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MD029 - Ordered list item prefix
Tags: ol
Aliases: ol-prefix
Parameters: style ("one", "ordered"; default "one")
This rule is triggered on ordered lists that do not either start with '1.' or do not have a prefix that increases in numerical order (depending on the configured style, which defaults to 'one').
Example valid list if the style is configured as 'one':
1. Do this.
1. Do that.
1. Done.
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1. Do this.
2. Do that.
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Bare URL used Open
**Breast Cancer Wisconsin (Diagnostic) dataset downloaded from: https://goo.gl/U2Uwz2**
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MD034 - Bare URL used
Tags: links, url
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Rationale: Without angle brackets, the URL isn't converted into a link in many markdown parsers.
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`http://www.example.com`
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD033 - Inline HTML
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Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
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# Markdown header
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Emphasis used instead of a header Open
**3. Cox and Snell R-Squared**
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MD036 - Emphasis used instead of a header
Tags: headers, emphasis
Parameters: punctuation (string; default ".,;:!?")
Aliases: no-emphasis-as-header
This check looks for instances where emphasized (i.e. bold or italic) text is used to separate sections, where a header should be used instead:
**My document**
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# My document
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Note: this rule looks for single line paragraphs that consist entirely of emphasized text. It won't fire on emphasis used within regular text, multi-line emphasized paragraphs, and paragraphs ending in punctuation. Similarly to rule MD026, you can configure what characters are recognized as punctuation.
Inline HTML Open
Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
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To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Fenced code blocks should be surrounded by blank lines Open
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MD031 - Fenced code blocks should be surrounded by blank lines
Tags: code, blank_lines
Aliases: blanks-around-fences
This rule is triggered when fenced code blocks are either not preceded or not followed by a blank line:
Some text
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```
Another code block
```
Some more text
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Rationale: Aside from aesthetic reasons, some parsers, including kramdown, will not parse fenced code blocks that don't have blank lines before and after them.
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Below is the illustration of formulas used to compute the averaged additional contributions of X<sub>1</sub> and X<sub>2</sub> within model size in the poupulation with four predictors (We use the notation <img src='images/formula1.JPG'> to represent the proportion of variance in Y that is accounted for by the predictors in the model X. For example,<img src='images/formula2.JPG'> represents the proportion of variance in Y that is accounted for by the model consisting of X<sub>1</sub> and X<sub>3</sub>. The additional contribution of a given predictor is measured by the increase in proportion of variance that results from adding that predictor to the regression model):
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
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To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
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Below is the illustration of formulas used to compute the averaged additional contributions of X<sub>1</sub> and X<sub>2</sub> within model size in the poupulation with four predictors (We use the notation <img src='images/formula1.JPG'> to represent the proportion of variance in Y that is accounted for by the predictors in the model X. For example,<img src='images/formula2.JPG'> represents the proportion of variance in Y that is accounted for by the model consisting of X<sub>1</sub> and X<sub>3</sub>. The additional contribution of a given predictor is measured by the increase in proportion of variance that results from adding that predictor to the regression model):
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
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To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Fenced code blocks should be surrounded by blank lines Open
```
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MD031 - Fenced code blocks should be surrounded by blank lines
Tags: code, blank_lines
Aliases: blanks-around-fences
This rule is triggered when fenced code blocks are either not preceded or not followed by a blank line:
Some text
```
Code block
```
```
Another code block
```
Some more text
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Some text
```
Code block
```
```
Another code block
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Rationale: Aside from aesthetic reasons, some parsers, including kramdown, will not parse fenced code blocks that don't have blank lines before and after them.
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Fenced code blocks should be surrounded by blank lines Open
```
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MD031 - Fenced code blocks should be surrounded by blank lines
Tags: code, blank_lines
Aliases: blanks-around-fences
This rule is triggered when fenced code blocks are either not preceded or not followed by a blank line:
Some text
```
Code block
```
```
Another code block
```
Some more text
To fix this, ensure that all fenced code blocks have a blank line both before and after (except where the block is at the beginning or end of the document):
Some text
```
Code block
```
```
Another code block
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Rationale: Aside from aesthetic reasons, some parsers, including kramdown, will not parse fenced code blocks that don't have blank lines before and after them.
Inline HTML Open
R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Fenced code blocks should be surrounded by blank lines Open
```
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MD031 - Fenced code blocks should be surrounded by blank lines
Tags: code, blank_lines
Aliases: blanks-around-fences
This rule is triggered when fenced code blocks are either not preceded or not followed by a blank line:
Some text
```
Code block
```
```
Another code block
```
Some more text
To fix this, ensure that all fenced code blocks have a blank line both before and after (except where the block is at the beginning or end of the document):
Some text
```
Code block
```
```
Another code block
```
Some more text
Rationale: Aside from aesthetic reasons, some parsers, including kramdown, will not parse fenced code blocks that don't have blank lines before and after them.
Inline HTML Open
R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Inline HTML Open
<a href="https://github.com/dominance-analysis/dominance-analysis/issues">
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Inline HTML Open
R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Inline HTML Open
Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
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To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Inline HTML Open
2. **Individual Dominance -** The individual dominance of a predictor is the R<sup>2</sup> of the model between the dependent variable and the predictor. So, the individual dominanace of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub></sub>. <br>
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Inline HTML Open
Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Inline HTML Open
<img src='images/Dominance_Analysis.JPG'>
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
Inline HTML Open
Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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MD033 - Inline HTML
Tags: html
Aliases: no-inline-html
This rule is triggered whenever raw HTML is used in a markdown document:
Inline HTML header
To fix this, use 'pure' markdown instead of including raw HTML:
# Markdown header
Rationale: Raw HTML is allowed in markdown, but this rule is included for those who want their documents to only include "pure" markdown, or for those who are rendering markdown documents in something other than HTML.
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<hr>
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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<img src='images/Housing_Data_Dominance.JPG'>
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Below is the illustration of formulas used to compute the averaged additional contributions of X<sub>1</sub> and X<sub>2</sub> within model size in the poupulation with four predictors (We use the notation <img src='images/formula1.JPG'> to represent the proportion of variance in Y that is accounted for by the predictors in the model X. For example,<img src='images/formula2.JPG'> represents the proportion of variance in Y that is accounted for by the model consisting of X<sub>1</sub> and X<sub>3</sub>. The additional contribution of a given predictor is measured by the increase in proportion of variance that results from adding that predictor to the regression model):
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<hr>
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<img src="https://latex.codecogs.com/svg.latex?\Large&space;R_{McFadden}^{2}=1-\frac{log(L_{full})}{log(L_{null})}" title="\Large R_{McFadden}^{2}=1-\frac{log(L_{full})}{log(L_{null})}" />
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1. **Interactional Dominance** - This is the incremental R<sup>2</sup> contribution of the predictor to the complete model. Hence, the Interactional Dominance of a particular predictor 'X' will be the diffrence between the R<sup>2</sup> of the complete model and the R<sup>2</sup> of the model with all other predictors except the particular predictor 'X'. <br>
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Consider a scenario when we have Y as the dependent variable and four predictors X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>, let R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>2</sub> ;
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**Using Boston Housing Dataset downloaded from: https://www.cs.toronto.edu/~delve/data/boston/bostonDetail.html**
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**Dominance Level**
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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Classification- <a href="https://www.kaggle.com/iamsajanbhagat/dominance-analysis-on-breast-cancer-dataset" target="_blank"> Dominance Analysis on Breast Cancer Dataset</a>*
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**Selecting top K features and getting R<sup>2</sup> of the Complete Model**
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<hr>
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**Getting R<sup>2</sup> of the Complete Model**
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Fenced code blocks should be surrounded by blank lines Open
```
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Dominance Analysis, according to Azen and Budescu meets three important criteria for measuring relative importance. First, the technique should be defined in terms of its ability to reduce error in predicting the outcome variable. Next, it should permit direct comparison of measures within a model (that is, X<sub>1</sub> is twice as important as X<sub>2</sub>). Finally, the technique should permit inferences concerning an attribute's direct effect (that is, when considered by itself), total effect (that is, when considered with other attributes) and partial effect (that is, when considered with various combinations of other predictors). Hence, Dominance analysis is both robust and intuitive and its interpretation is also very straightforward.
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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The measure for proportion of variance that we have used for regression is R<sup>2</sup> but since we don't have R<sup>2</sup> in logsitic regression/classification models, we have used Pseudo R<sup>2</sup>.
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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<img src='images/housing_data_dominance_stats.JPG'>
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<img src='images/Dominance_Levels_Breast_Cancer.JPG'>
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The Dominance Analysis package is based on the concept developed by Azen and Budescu (see references). This package is released under a <a href="https://github.com/bhagatsajan0073/dominance_analysis/blob/master/LICENSE"> MIT License</a>. Dominance Analysis Python package has been developed by <a href="https://github.com/quintshekhar"> Shashank Shekhar</a>, <a href="https://github.com/bhagatsajan0073"> Sajan Bhagat</a>, <a href="https://github.com/Vibish"> Kunjithapatham Sivakumar</a> and <a href="https://github.com/balakolluri"> Bala Koteshwar Kolluri</a> . Pull requests submitted to the GitHub Repo are highly encouraged!
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The Dominance Analysis package is based on the concept developed by Azen and Budescu (see references). This package is released under a <a href="https://github.com/bhagatsajan0073/dominance_analysis/blob/master/LICENSE"> MIT License</a>. Dominance Analysis Python package has been developed by <a href="https://github.com/quintshekhar"> Shashank Shekhar</a>, <a href="https://github.com/bhagatsajan0073"> Sajan Bhagat</a>, <a href="https://github.com/Vibish"> Kunjithapatham Sivakumar</a> and <a href="https://github.com/balakolluri"> Bala Koteshwar Kolluri</a> . Pull requests submitted to the GitHub Repo are highly encouraged!
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6. Budescu, D. V. (1993). Dominance analysis: A new approach to the problem of relative importance of predictors in multiple regression. Psychological Bulletin, 114(3), 542-551. https://doi.org/10.1037/0033-2909.114.3.542
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**Dominance Level**
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Consider a scenario when we have Y as the dependent variable and four predictors X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>, let R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>2</sub> ;
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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<img src='images/pie_boston_house_price.png'>
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1. **Interactional Dominance** - This is the incremental R<sup>2</sup> contribution of the predictor to the complete model. Hence, the Interactional Dominance of a particular predictor 'X' will be the diffrence between the R<sup>2</sup> of the complete model and the R<sup>2</sup> of the model with all other predictors except the particular predictor 'X'. <br>
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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<img src='images/pie_boston_house_price_corr.png'>
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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<img src='images/bar_breast_cancer_latest.png'>
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<img src='images/housing_data_corr_dominance_stats.PNG'>
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The Dominance Analysis package is based on the concept developed by Azen and Budescu (see references). This package is released under a <a href="https://github.com/bhagatsajan0073/dominance_analysis/blob/master/LICENSE"> MIT License</a>. Dominance Analysis Python package has been developed by <a href="https://github.com/quintshekhar"> Shashank Shekhar</a>, <a href="https://github.com/bhagatsajan0073"> Sajan Bhagat</a>, <a href="https://github.com/Vibish"> Kunjithapatham Sivakumar</a> and <a href="https://github.com/balakolluri"> Bala Koteshwar Kolluri</a> . Pull requests submitted to the GitHub Repo are highly encouraged!
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**Using Boston Housing Dataset downloaded from: https://www.cs.toronto.edu/~delve/data/boston/bostonDetail.html**
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**4. Estrella R-Squared**
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**Using Boston Housing Dataset downloaded from: https://www.cs.toronto.edu/~delve/data/boston/bostonDetail.html**
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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**Selecting top K features and getting Pseudo R<sup>2</sup> of the Complete Model**
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<img src='images/pie_boston_house_price_cov.png'>
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3. Azen, R., Budescu, D. V. (2003). The Dominance Analysis Approach for Comparing Predictors in Multiple Regression. Psychological Methods, 2003, Vol. 8, No. 2, 129–148. https://doi.org/10.1037/1082-989X.8.2.129
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**1. McFadden's Pseudo-R Squared**
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MD036 - Emphasis used instead of a header
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**My document**
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1. **Interactional Dominance** - This is the incremental R<sup>2</sup> contribution of the predictor to the complete model. Hence, the Interactional Dominance of a particular predictor 'X' will be the diffrence between the R<sup>2</sup> of the complete model and the R<sup>2</sup> of the model with all other predictors except the particular predictor 'X'. <br>
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Consider a scenario when we have Y as the dependent variable and four predictors X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>, let R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>2</sub> ;
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Consider a scenario when we have Y as the dependent variable and four predictors X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>, let R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>2</sub> ;
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**Dominance Level**
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1. **Interactional Dominance** - This is the incremental R<sup>2</sup> contribution of the predictor to the complete model. Hence, the Interactional Dominance of a particular predictor 'X' will be the diffrence between the R<sup>2</sup> of the complete model and the R<sup>2</sup> of the model with all other predictors except the particular predictor 'X'. <br>
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<a href="https://github.com/dominance-analysis/dominance-analysis/network/members">
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Below is the illustration of formulas used to compute the averaged additional contributions of X<sub>1</sub> and X<sub>2</sub> within model size in the poupulation with four predictors (We use the notation <img src='images/formula1.JPG'> to represent the proportion of variance in Y that is accounted for by the predictors in the model X. For example,<img src='images/formula2.JPG'> represents the proportion of variance in Y that is accounted for by the model consisting of X<sub>1</sub> and X<sub>3</sub>. The additional contribution of a given predictor is measured by the increase in proportion of variance that results from adding that predictor to the regression model):
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Consider a scenario when we have Y as the dependent variable and four predictors X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>, let R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>2</sub> ;
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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2. **Individual Dominance -** The individual dominance of a predictor is the R<sup>2</sup> of the model between the dependent variable and the predictor. So, the individual dominanace of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub></sub>. <br>
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3. **Average Partial Dominance -** This is average of average incremental R<sup>2</sup> contributions of the predictor to all subset models except complete model and bi-variate (when only one predcitor is present) model. <br>
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4. **Total Dominance -** The last measure of dominance summarizes the additional contributions of each predictor to all subset models by averaging all the conditional values. In the example on table 3, this consists of averaging the four averaged entries in each column. <br>
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**Using Boston Housing Dataset downloaded from: https://www.cs.toronto.edu/~delve/data/boston/bostonDetail.html**
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4. Azen, R., Budescu, D. V. (2006). Comparing Predictors in Multivariate Regression Models: An Extension of Dominance Analysis. Journal of Educational and Behavioral Statistics Summer 2006, Vol. 31, No. 2, pp. 157-180. https://doi.org/10.3102/10769986031002157
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in the context of all 2<sup>(p−2)</sup> models that contain some subset of the other predictors. So, if we have a total of 'p' predictors, we will build 2<sup>p</sup>-1 models (all possible subset models) and compute the incremental R<sup>2</sup> contribution of each predictor to the subset model of all other predictors. The additional contribution of a given predictor is measured by the increase in R<sup>2</sup> that results from adding that predictor to the regression model.
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Measures of fit in logistic regression can be classified by those based on sums of squares and those based on maximum likelihood statistics. Reviews of a variety of measures of fit proposed for logistic regression can be found in Amemiya (1981), Menard (2000), Mittlbock and Schemper (1996) and Zheng and Agresti (2000). Given the large number of proposed measures, criteria for defining appropriate R<sup>2</sup> analogues need to he determined. The following criteria, which are also found in the linear regression literature (e.g., Kvilseth. 1985: Van den Burg & Lewis, 1988), were used to select R<sup>2</sup> analogues for logistic regression:
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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<br> If the additional contributions are inconsistent in favoring the same predictor across all subset models, then complete dominance is undetermined while weaker levels of dominance may still be achieved.
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<hr>
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<img src='images/Breast_Cancer_Data_Model_Training_Latest.JPG'>
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<img src='images/Housing_Data_Corr_Dominance.PNG'>
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The Dominance Analysis package is based on the concept developed by Azen and Budescu (see references). This package is released under a <a href="https://github.com/bhagatsajan0073/dominance_analysis/blob/master/LICENSE"> MIT License</a>. Dominance Analysis Python package has been developed by <a href="https://github.com/quintshekhar"> Shashank Shekhar</a>, <a href="https://github.com/bhagatsajan0073"> Sajan Bhagat</a>, <a href="https://github.com/Vibish"> Kunjithapatham Sivakumar</a> and <a href="https://github.com/balakolluri"> Bala Koteshwar Kolluri</a> . Pull requests submitted to the GitHub Repo are highly encouraged!
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1. Azen, R. (2000). Inference for predictor comparisons:Dominance analysis and the distribution of R<sup>2</sup> differences. Dissertation Abstracts International B, 61/10, 5616.
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**2. Nagelkerke Pseudo-R Squared**
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**Getting R<sup>2</sup> of the Complete Model**
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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<img src="https://img.shields.io/github/stars/dominance-analysis/dominance-analysis.svg">
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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The beauty of the math of Dominance Analysis is that the sum of the overall average incremental R<sup>2</sup> of all predictors is equal to the R<sup>2</sup> of the complete model (model with all predictors). Hence, the total R<sup>2</sup> can be attributed to each predictor in the model. Below is an illustration of Dominance Analysis in the Population for Hypothetical example with four predictors:
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Measures of fit in logistic regression can be classified by those based on sums of squares and those based on maximum likelihood statistics. Reviews of a variety of measures of fit proposed for logistic regression can be found in Amemiya (1981), Menard (2000), Mittlbock and Schemper (1996) and Zheng and Agresti (2000). Given the large number of proposed measures, criteria for defining appropriate R<sup>2</sup> analogues need to he determined. The following criteria, which are also found in the linear regression literature (e.g., Kvilseth. 1985: Van den Burg & Lewis, 1988), were used to select R<sup>2</sup> analogues for logistic regression:
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<img src="https://latex.codecogs.com/svg.latex?\Large&space;R_{Estrella}^{2}=1-\{\frac{LL_{full}}{LL_{null}}\}^{\frac{-2}{N}*LL_{null}}" title="\Large R_{Estrella}^{2}=1-\}\frac{LL_{full}}{LL_{null}}\}^{\frac{2}{N}*LL_{null}}" />
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### Dominance Levels
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Consider a scenario when we have Y as the dependent variable and four predictors X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>, let R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>2</sub> ;
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Consider a scenario when we have Y as the dependent variable and four predictors X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>, let R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>2</sub> ;
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<p align="center"> Table 1</p>
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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1. Azen, R. (2000). Inference for predictor comparisons:Dominance analysis and the distribution of R<sup>2</sup> differences. Dissertation Abstracts International B, 61/10, 5616.
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Using each of these four R<sup>2</sup> analogues, the additional contribution of a given predictor to a specific logistic model can be measured as the change (i.e., increase) in the R<sup>2</sup> analogues when the predictor is added to the model. Even though, all the four measures will give similar results, we recommend using either Estrella's (1998) model fit measure or McFadden's (1974) measure for conducting dominance analysis in logistic regression. We have a slight preference for McFadden's measure (and that is what the package will compute by default) because it is computationally simpler, but both McFadden's and Estrella’s measures satisfy the minimum requirements for an R<sup>2</sup> analogues.
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in the context of all 2<sup>(p−2)</sup> models that contain some subset of the other predictors. So, if we have a total of 'p' predictors, we will build 2<sup>p</sup>-1 models (all possible subset models) and compute the incremental R<sup>2</sup> contribution of each predictor to the subset model of all other predictors. The additional contribution of a given predictor is measured by the increase in R<sup>2</sup> that results from adding that predictor to the regression model.
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<img src='images/WaterFall_Boston.png'>
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Note: Since, Dominance Analysis is computationally intensive as it builds all subset model (2<sup>p</sup>-1 models), we have provided the user the flexibility to choose number of top predictors that they want to compute relative importance for. For regression, Top K features are selected based on F-regression and for classification it is based on Chi-Squared statistic. Dominance Analysis can be used in combination with Principal Component Analysis (PCA) or Factor Analysis or any other feature reduction algorithm for getting accurate and intutive importance of predictors.
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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<img src='images/Dominance_Level_Boston.JPG'>
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<p align="center"> Table 3</p>
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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<img src='images/bar_boston_house_price.png'>
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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Let's consider a scenario when we have 4 predictors; X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>. We will have to build a total of 2<sup>4</sup>-1 models i.e. 15 models- <sup>4</sup>C<sub>1</sub> = 4 models with only one predictor, <sup>4</sup>C<sub>2</sub> = 6 models with two predictors each, <sup>4</sup>C<sub>3</sub> = 4 models with three predictors each and 1 (<sup>4</sup>C<sub>4</sub>) complete model with all 4 predictors. Thus, the additional contributions of X<sub>1</sub> are computed as the increases in the proportion of variance accounted for when X<sub>1</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>2</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>2</sub>X<sub>3</sub>}, {X<sub>2</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>2</sub>X<sub>3</sub>X<sub>4</sub>}). Similarly, the additional contributions of X<sub>2</sub> are the increases in the proportion of variance accounted for when X<sub>2</sub> is added to each subset of the remaining predictors (i.e., the null subset {.}, {X<sub>1</sub>}, {X<sub>3</sub>}, {X<sub>4</sub>}, {X<sub>1</sub>X<sub>3</sub>}, {X<sub>1</sub>X<sub>4</sub>}, {X<sub>3</sub>X<sub>4</sub>} and {X<sub>1</sub>X<sub>3</sub>X<sub>4</sub>})
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<img src='images/WaterFall_Breast_Cancer.png'>
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**Selecting top K features and getting R<sup>2</sup> of the Complete Model**
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The beauty of the math of Dominance Analysis is that the sum of the overall average incremental R<sup>2</sup> of all predictors is equal to the R<sup>2</sup> of the complete model (model with all predictors). Hence, the total R<sup>2</sup> can be attributed to each predictor in the model. Below is an illustration of Dominance Analysis in the Population for Hypothetical example with four predictors:
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<img src='images/bar_boston_house_price_corr.png'>
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**Using Boston Housing Dataset downloaded from: https://www.cs.toronto.edu/~delve/data/boston/bostonDetail.html**
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1. **Interactional Dominance** - This is the incremental R<sup>2</sup> contribution of the predictor to the complete model. Hence, the Interactional Dominance of a particular predictor 'X' will be the diffrence between the R<sup>2</sup> of the complete model and the R<sup>2</sup> of the model with all other predictors except the particular predictor 'X'. <br>
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<img src='images/Dominance_Level_Boston_Correlation.PNG'>
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Consider a scenario when we have Y as the dependent variable and four predictors X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub> and X<sub>4</sub>, let R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>2</sub> ;
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R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>3</sub></sub> be the R<sup>2</sup> of the model between Y and X<sub>1</sub>, X<sub>3</sub> so on and so forth. In this case, the interactional dominance of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub>,X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub> - R<sup>2</sup><sub>Y.X<sub>2</sub>,X<sub>3</sub>,X<sub>4</sub></sub>. <br>
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2. **Individual Dominance -** The individual dominance of a predictor is the R<sup>2</sup> of the model between the dependent variable and the predictor. So, the individual dominanace of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub></sub>. <br>
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5. Azen, R., Traxel, N. (2009). Using Dominance Analysis to Determine Predictor Importance in Logistic Regression. Journal of Educational and Behavioral Statistics September 2009, Vol. 34, No. 3, pp. 319-347. https://doi.org/10.3102/1076998609332754
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2. **Individual Dominance -** The individual dominance of a predictor is the R<sup>2</sup> of the model between the dependent variable and the predictor. So, the individual dominanace of predictor X<sub>1</sub> will be R<sup>2</sup><sub>Y.X<sub>1</sub></sub>. <br>
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**Getting R<sup>2</sup> of the Complete Model**
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3. **Average Partial Dominance -** This is average of average incremental R<sup>2</sup> contributions of the predictor to all subset models except complete model and bi-variate (when only one predcitor is present) model. <br>
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<p align="center"> Table 4</p>
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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* **Complete Dominance** - One predictor is said to completely dominate another predictor if its dominance holds across all possible subset models (that do not include the two predictors under comparison). Back to the four-predictor model, for example, complete dominance of X<sub>1</sub> over X<sub>2</sub> is achieved if the additional R<sup>2</sup> contribution of X<sub>1</sub> is more than that of X<sub>2</sub> to the null model, the model consisting of X<sub>3</sub>, the model consisting of X<sub>4</sub>, and the model consisting of both X<sub>3</sub> and X<sub>4</sub>. In Table 3, we can see that incremental R<sup>2</sup> of X<sub>1</sub> is greater than that of X<sub>2</sub> for all subset models and hence X<sub>1</sub> completely dominates X<sub>2</sub>.
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**Getting R<sup>2</sup> of the Complete Model**
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7. Luo, W., & Azen, R. (2013). Determining Predictor Importance in Hierarchical Linear Models Using Dominance Analysis. Journal of Educational and Behavioral Statistics, 38(1), 3-31. https://doi.org/10.3102/1076998612458319
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