14.3: Standardized Regression Coefficients

In most cases, the various IVs in a model are represented on different measurement scales. For example, ideology ranges from 1 to 7, while age ranges from 18 to over 90 years old. These different scales make comparing the effects of the various IVs difficult. If we want to directly compare the magnitudes of the effects of ideology and age on levels of environmental concern, we would need to standardize the variables.

One way to standardized variables is to create a ZZ-score based on each variable. Variables are standardized in this way as follows:

Zi=Xi−¯Xsx(14.1)(14.1)Zi=Xi−X¯sx

where sxsx is the s.d. of XX. Standardizing the variables by creating ZZ-scores re-scales them so that each variables has a mean of 00 and a s.d. of 11. Therefore, all variables have the same mean and s.d. It is important to realize (and it is somewhat counter-intuitive) that the standardized variables retain all of the variation that was in the original measure.

A second way to standardize variables converts the unstandardized BB, into a standardized B′B′.

B′k=BksksY(14.2)(14.2)Bk′=BksksY

where BkBk is the unstandardized coefficient of XkXk, sksk is the s.d. of XkXk, and sysy is the s.d. of YY. Standardized regression coefficients, also known as beta weights or “betas”, are those we would get if we regress a standardized YY onto standardized XX’s.

Interpreting Standardized Betas

• The standard deviation change in YY for a one-standard deviation change in XX
• All XX’ss on an equal footing, so one can compare the strength of the effects of the XX’s
• Cannot be used for comparisons across samples
• Variances will differ across different samples

We can use the scale function in R to calculate a ZZ score for each of our variables, and then re-run our model.

stan.ds <- ds.temp %>%
  dplyr::select(glbcc_risk, age, education, income, ideol, gender) %>% 
  scale %>%
  data.frame()

ols3 <- lm(glbcc_risk ~ age + education + income + ideol + gender, data = stan.ds)
summary(ols3)

## 
## Call:
## lm(formula = glbcc_risk ~ age + education + income + ideol + 
##     gender, data = stan.ds)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.92180 -0.54357  0.06509  0.48646  2.20164 
## 
## Coefficients:
##                           Estimate             Std. Error t value
## (Intercept)  0.0000000000000001685  0.0167531785616065292   0.000
## age         -0.0187675384877126518  0.0169621356203379960  -1.106
## education    0.0395657731919867237  0.0178239180606745221   2.220
## income      -0.0466922668201090602  0.0178816880127353542  -2.611
## ideol       -0.5882792369403809785  0.0170882328807871603 -34.426
## gender      -0.0359158695199312886  0.0170016561132237121  -2.112
##                         Pr(>|t|)    
## (Intercept)              1.00000    
## age                      0.26865    
## education                0.02653 *  
## income                   0.00908 ** 
## ideol       < 0.0000000000000002 ***
## gender                   0.03475 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7984 on 2265 degrees of freedom
## Multiple R-squared:  0.364,  Adjusted R-squared:  0.3626 
## F-statistic: 259.3 on 5 and 2265 DF,  p-value: < 0.00000000000000022

In addition, we can convert the original unstandardized coefficient for ideology, to a standardized coefficient.

sdX <- sd(ds.temp$ideol, na.rm=TRUE)
sdY <- sd(ds.temp$glbcc_risk, na.rm=TRUE)
ideology.prime <- ols1$coef[5]*(sdX/sdY)
ideology.prime

##      ideol 
## -0.5882792

Using either approach, standardized coefficients allow us to compare the magnitudes of the effects of each of the IVs on YY.