Alternative Standardizations of Regression Coefficients

Calculation of the Coefficients

Interpretation of the Coefficients

• The unstandardized regression coefficient b_k measures the increase in Y (in units of Y) associated with an increase in X_k of one unit (in units of X_k), keeping levels of the other independent variables constant.
• The standardized regression coefficient b_k* measures the increase in Y (in standard deviations of Y) associated with an increase in X_k of one standard deviation, keeping levels of the other independent variables constant.  EX: increasing OUTRATIO by one standard deviation (i.e., 12.796 units) is associated with an increase in PAUP of .584 standard deviation (i.e., .584*16.488 = 9.629 units of PAUP)
• The robust standardized regression coefficient measures the increase in Y (in IQR of Y) associated with an increase in X_k of one IQR, keeping levels of the other independent variables constant.  The robust standardized coefficient is similar to the standardized coefficient, but uses the interquartile range IQR instead of the standard deviation as a measure of dispersion.  Thus the robust standardized coefficient is also comparable across independent variables measured in different units while being less sensitive than the standardized coefficient to the presence of extreme values (outliers) in either X_k or Y.  For a published example see Nielsen and Alderson (2001).  In that article we estimated regression models of decennial change in income inequality in U.S. counties.  The models were estimated with a robust regression method because there were many outliers.  We used robust-standardized coefficients because of the presence of these outliers, because they would have affected sample standard deviations too.
• The semi-standardized regression coefficient measures the increase in Y (in units of Y) associated with an increase in X_k of one standard deviation, keeping levels of the other independent variables constant.  Semi-standardized coefficients are also comparable across independent variables in different metrics.  EX:  an increase of one standard deviation in OUTRATIO is associated with an increase in PAUP of 9.623 units; an increase of one standard deviaiton in POP in associated with a decrease in PAUP of 9.407 units.  So the same conclusion as for standardized coefficients can be drawn: OUTRATIO and POP have effects on PAUP of similar magnitude in opposite directions.
• The robust semi-standardized regression coefficient measures the increase in Y (in units of Y) associated with an increase in X_k of one IQR, keeping levels of the other independent variables constant.  The robust semi-standardized coefficient is similar to the semi-standardized coefficient, but uses the interquartile range instead of the standard deviation as a measure of dispersion.  Robust semi-standardized coefficients are also comparable across independent variables in different metrics.  EX: an increase of one IQR in OUTRATIO is associated with an increase in PAUP of 12.032 units; an increase of one IQR in POP is associated with a decrease in PAUP of 12.751 units.  The effects of OUTRATIO and POP on PAUP again appear of the same magnitude in opposite directions.
• The maximum impact coefficient measures the increase in Y (in units of Y) associated with an increase in X_k equal to the range of X_k, keeping levels of the other independent variables constant.  The maximum impact coefficient helps visualize the greatest possible effect of Xk on Y, given the range of variation in X_k in the sample.  EX: the maximum impact of OUTRATIO (corresponding to the difference of 59 units between the minimum and maximum values of OUTRATIO in the sample) is equal to 44.368 units of PAUP.  For a published example of the use of the maximum impact coefficient see Alderson and Nielsen (2002).  In that paper we are using time series of cross sections data.  We also use standardized coefficients that measure separately the maximum longitudinal and maximum cross-sectional impact of X_k on Y, using the maximum longitudinal and cross-sectional range of X_k, respectively. 
• The maximum impact (proportion Y range) coefficient measures the increase in Y (as a proportion of the range of Y) associated with an increase in X_k equal to the range of X_k, keeping levels of the other independent variables constant.  EX: the maximum impact of OUTRATIO on PAUP, which is equal to 44.368 units of PAUP, corresponds to a proportion .765 of the maximum range of PAUP (58 units).