Building on Beta: Indicators of Predictor Importance

Researchers are rarely interested in knowing only whether a variable exhibits a significant relationship with another variable. The more interesting research questions, and answers, are often about the relative importance of a variable in predicting an outcome. Indicators of relative importance may rank order predictors’ contributions to an overall regression effect or partition the variance explained by individual predictors into unique and shared variance. Relative importance of predictors is important for theoretical development because it encourages parsimony, but also for practicality as researchers and practitioners are faced with limited resources.

A brief review of psychology publications that report multiple regression (MR) analyses would show that beta coefficients are clearly the most popular indicator of relative importance reported and interpreted. However, an exclusive reliance on beta coefficients to interpret MR results, and predictor importance in particular, is often misguided and we will briefly review additional indicators that can provide useful information.

Beta Coefficients

In multiple regression, beta coefficients (i.e., standardized regression coefficients) represent the expected change in the dependent variable (in standard deviation units) per standard deviation increase in the independent variable, holding all other independent variables constant. When independent variables are perfectly uncorrelated, one can estimate variable importance by squaring the beta weights. However, this becomes problematic when independent variables are correlated, as is often the case. In these situations, a given beta weight can reflect the explained variance it shares with other variables in the model, and it becomes difficult to disentangle a variable’s unique importance from the “extra credit” it gets from shared variance. As such, beta weights should be relied on as an easily computed but preliminary indicator of a predictor’s contribution to a regression coefficient.

Pratt Product Measure

The Pratt product measure is a relatively simple method of partitioning a model’s explained variance (R²) into non-overlapping segments. The Pratt measure multiplies a variable’s zero-order correlation with a dependent variable by its beta weight. The correlation measure captures the direct effect of the predictor in isolation from other predictors while the beta weight is an indicator of a predictor’s contribution accounting for all other predictors in the model.  If a given beta weight is inflated by shared variance, multiplying the value by its corresponding (smaller) zero order correlation will correct for the beta’s inflation. Summing the Pratt product measures for all the variables in a model generally results in the R² except in some cases where a negative product is yielded (often a signal of suppression), allowing for straightforward partitioning and ranking of variables.

Commonality Coefficients

Commonality analysis partitions R² into variance that is unique to each independent variable (unique effects; e.g., X1…X3) and variance that is shared by all possible subsets of predictors (common effects; e.g., X1X2, X2X3, etc.). Partitioning variance in this way produces non-overlapping values that can be compared easily. Common effects, in particular, provide a great deal of information about the overlap between independent variables and how this overlap contributes to predicting the dependent variable. However, common effects can be difficult to interpret, especially as the number of variables increases in a model and commonalities reflect the combination of more than two variables.

Relative Weights

Relative weight analysis (RWA) is a slightly more complex method of partitioning R² between independent variables. When predictors in a MR are correlated, RWA uses principal components analysis to generate principal components that are the most highly correlated with the dependent variable while being uncorrelated with one another. The dependent variable is regressed on these components in one analysis and a second analysis is conducted in which the original independent variables are regressed on the components. A given relative weight is equal to the product of the squared regression coefficient from the first analysis and the squared regression coefficient from the second analysis. Dividing relative weights by R² then allows for ranking of individual predictors’ contributions. Essentially, relative weights are an indicator of a predictor’s contribution to explaining the dependent variable as a joint function of how highly related independent variables are to the principal components and how highly related the principal components are to the dependent variable. As such, RWA partitions R² while minimizing the influence of multicollinearity among predictors, a major strength of the method.

Dominance Analysis

Dominance analysis compares the unique variance contributed by pairs of independent variables across all possible subsets of predictors. To assess complete dominance, the unique effect of a given independent variable when entered last in multiple regression equation is compared to another independent variable across all subsets of a model. Complete dominance occurs when the unique effect is greater across all the models. Conditional dominance is determined by calculating the averages of all independent variables’ contributions to all possible model subsets and comparing those averages. Conditional dominance is shown when an independent variable contributes more to predicting the dependent variable, on average across all possible models, compared to another independent variable.  

References

Azen, R., & Budescu, D. V. (2003). The dominance analysis approach for comparing predictors in multiple regression. Psychological Methods, 8, 129-148.

Johnson, J. W. (2000). A heuristic method for estimating the relative weight of predictor variables in multiple regression. Multivariate Behavioral Research, 35, 1-19. 

Nimon, K. F. & Oswald, F. L. (In Press). Understanding the results of multiple linear regression: Beyond standardized regression coefficients. Organizational Research Methods.  

Nimon, K. F., & Reio, T. (2011). Regression commonality analysis: A technique for quantitative theory building. Human Resource Development Review, 10, 329-340.