When this assumption is violated, we say that heteroscedasticity is present in a regression model.
When this occurs, the standard errors for the regression coefficients in the model become untrustworthy.
To account for this, we can calculate robust standard errors, which are “robust” against heteroscedasticity and can give us a better idea of the true standard error values for the regression coefficients.
The following example shows how to calculate robust standard errors for a regression model in R.
Example: Calculating Robust Standard Errors in R
Suppose we have the following data frame in R that contains information on the hours studied and exam score received by 20 students in some class:
#create data frame df <- data.frame(hours=c(1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 8), score=c(67, 68, 74, 70, 71, 75, 80, 70, 84, 72, 88, 75, 95, 75, 99, 78, 99, 65, 96, 70)) #view head of data frame head(df) hours score 1 1 67 2 1 68 3 1 74 4 1 70 5 2 71 6 2 75
We can use the lm() function to fit a regression model in R that uses hours as the predictor variable and score as the response variable:
#fit regression model fit <- lm(score ~ hours, data=df) #view summary of model summary(fit) Call: lm(formula = score ~ hours, data = df) Residuals: Min 1Q Median 3Q Max -19.775 -5.298 -3.521 7.520 18.116 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 71.158 4.708 15.11 1.14e-11 *** hours 1.945 1.075 1.81 0.087 . --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 10.48 on 18 degrees of freedom Multiple R-squared: 0.154, Adjusted R-squared: 0.107 F-statistic: 3.278 on 1 and 18 DF, p-value: 0.08696
The easiest way to visually check if heteroscedasticity is a problem in the regression model is to create a residual plot:
#create residual vs. fitted plot plot(fitted(fit), resid(fit)) #add a horizontal line at y=0 abline(0,0)
The x-axis shows the fitted values of the response variable and the y-axis shows the corresponding residuals.
From the plot we can see that the variance in the residuals increases as the fitted values increase.
This is an indication that heteroscedasticity is likely a problem in the regression model and the standard errors from the model summary are untrustworthy.
To calculate robust standard errors, we can use the coeftest() function from the lmtest package and the vcovHC() function from the sandwich package as follows:
library(lmtest) library(sandwich) #calculate robust standard errors for model coefficients coeftest(fit, vcov = vcovHC(fit, type = 'HC0')) t test of coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 71.1576 3.3072 21.5160 2.719e-14 *** hours 1.9454 1.2072 1.6115 0.1245 --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Notice that the standard error for the hours predictor variable increased from 1.075 in the previous model summary to 1.2072 in this model summary.
Since heteroscedasticity is present in the original regression model, this estimate for the standard error is more trustworthy and should be used when calculating a confidence interval for the hours predictor variable.
Note: The most common type of estimate to calculate in the vcovHC() function is ‘HC0’, but you can refer to the documentation to find other estimate types.
The following tutorials explain how to perform other common tasks in R: