Example 8.14: generating standardized regression coefficients

Standardized (or beta) coefficients from a linear regression model are the parameter estimates obtained when the predictors and outcomes have been standardized to have variance = 1. Alternatively, the regression model can be fit and then standardized post-hoc based on the appropriate standard deviations. The parameters are thus interpreted as change in the outcome, in standard deviations, per standard deviation change in the predictors. However they’re calculated, standardized coefficients facilitate an assessment of which variables have the greatest association with the outcome (or response) variable, though such an assessment ignores the confidence limits associated with each pairwise association.

It’s straightforward to calculate these quantities in SAS and R. We’ll demonstrate with data from the HELP study, modeling PCS as a function of MCS and homelessness among female subjects.

SAS

In SAS, standardized coefficients are available as the stb option for the model statement in proc reg.

proc reg data="c:\book\help";
   where female eq 1;
   model pcs = mcs homeless / stb;
run;
                       The REG Procedure
                         Model: MODEL1
                    Dependent Variable: PCS

                      Parameter Estimates
                   Parameter      Standard
Variable    DF      Estimate         Error   t Value   Pr > |t|

Intercept    1      39.62619       2.49830     15.86     

R
In R we demonstrate the use of the lm.beta() function in the QuantPsyc package (due to Thomas D. Fletcher of State Farm). The function is short and sweet, and takes a linear model object as argument:

>lm.beta
function (MOD) 
{
    b <- summary(MOD)$coef[-1, 1]
    sx <- sd(MOD$model[-1])
    sy <- sd(MOD$model[1])
    beta <- b * sx/sy
    return(beta)
}

Here we apply the function to data from the HELP study.

ds = read.csv("http://www.math.smith.edu/r/data/help.csv")
female = subset(ds, female==1)
lm1 = lm(pcs ~ mcs + homeless, data=female)

The results, in terms of unstandardized regression parameters are the same as in SAS:

> summary(lm1)

Call:
lm(formula = pcs ~ mcs + homeless, data = female)

Residuals:
    Min      1Q  Median      3Q     Max 
-28.163  -5.821  -1.017   6.775  29.979 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 39.62619    2.49830  15.861  < 2e-16 ***
mcs          0.21945    0.07644   2.871  0.00496 ** 
homeless    -2.56907    1.95079  -1.317  0.19075    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Residual standard error: 9.761 on 104 degrees of freedom
Multiple R-squared: 0.0862, Adjusted R-squared: 0.06862 
F-statistic: 4.905 on 2 and 104 DF,  p-value: 0.009212 

To generate the standardized parameter estimates, we use the lm.beta() function.

library(QuantPsyc)
lm.beta(lm1)

This generates the following output:

       mcs   homeless 
 0.2691888 -0.1234776 

A change in 1 standard deviation of MCS has more than twice the impact on PCS than a 1 standard deviation change in the HOMELESS variable. This example points up another potential weakness of standardized regression coefficients, however, in that the homeless variable can take on values of 0 or 1, and a 1 standard deviation change is hard to interpret.