s209 m5 standardized coefficients

Alternative Standardizations of Regression Coefficients

Calculation of the Coefficients

Using SYSTAT the information needed to calculate alternative standardizations of regression coefficients can be obtained in the stats module with the stat/all command (standard deviations and ranges) and the stem command (calculate interquartile range IQR for each variable as IQR = upper hinge - lower hinge).

**Information Needed for Alternative Standardizations of Regression Coefficients**
Statistic (Symbol)	PAUP	OUTRATIO	PROPOLD	POP
Standard deviation (Std)	16.488	12.796	9.107	30.248
Min	23	5	81	68
Max	81	64	119	203
Range (Rng)	58	59	38	135
Lower hinge	37	18	93.5	97.5
Upper hinge	61	34	103	138.5
IQR (Iqr)	24	16	9.5	41

**Alternative Standardizations of Regression Coefficients**
Coefficient	Formula	OUTRATIO	PROPOLD	POP
Unstandardized coefficient	b_k	.752	.056	-.311
Standardized coefficient	b_k* = b_k(Std(X_k)/Std(Y))	.584	.031	-.570
Robust standardized coefficient	b_k(Iqr(X_k)/Iqr(Y))	.501	.022	-.531
Semi-standardized coefficient	b_k(Std(X_k))	9.623	.510	-9.407
Robust semi-standardized	b_k(Iqr(X_k))	12.032	.532	-12.751
Maximum impact (Y metric)	b_k(Rng(X_k))	44.368	2.128	-41.985
Maximum impact (proportion Y range)	b_k(Rng(X_k)/Rng(Y))	.765	.037	-.724

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).

Last modified 24 Feb 2002