Logistic regression is widely used in statistics and machine learning for modeling binary outcome variables. However, standard errors in logistic regression can be sensitive to violations of model assumptions, such as heteroscedasticity or clustering of observations. Robust standard errors provide a way to mitigate these issues and produce more reliable inferences. This article will explore how to compute robust standard errors for logistic regression in both Stata and R, focusing on different types of robust standard errors, including heteroscedasticity-consistent (HC), cluster-robust, and bootstrapped standard errors.

Robust Standard Errors in Stata

Stata is a powerful statistical software package with extensive capabilities for estimating robust standard errors. Below, we will demonstrate how to compute robust standard errors for a logistic regression model.

1. Heteroscedasticity-Consistent Standard Errors

To estimate a logistic regression with heteroscedasticity-consistent standard errors in Stata, you can use the logit command with the vce(robust) option:

logit outcome predictor1 predictor2, vce(robust)

2. Cluster-Robust Standard Errors

If your data have a clustered structure (e.g., individuals nested within groups), you can compute cluster-robust standard errors using the vce(cluster clustvar) option:

logit outcome predictor1 predictor2, vce(cluster group)

Here, group is the variable indicating the clustering structure.

3. Bootstrapped Standard Errors

Bootstrapping is a resampling technique that can be used to estimate robust standard errors. In Stata, you can perform bootstrapping using the bootstrap command:

bootstrap, reps(1000): logit outcome predictor1 predictor2

The reps(1000) option specifies the number of bootstrap replications.

Robust Standard Errors in R

R provides a variety of packages and functions for estimating robust standard errors in logistic regression models. Here, we will demonstrate how to compute heteroscedasticity-consistent, cluster-robust, and bootstrapped standard errors in R Programming Language.

1. Heteroscedasticity-Consistent Standard Errors

The sandwich package in R can be used to compute heteroscedasticity-consistent standard errors. First, you need to fit a logistic regression model using the glm function, and then use the vcovHC function from the sandwich package:

# Load necessary libraries
library(sandwich)
library(lmtest)

# Set seed for reproducibility
set.seed(123)

# Generate a sample dataset
n <- 100  # Number of observations
your_data <- data.frame(
  predictor1 = rnorm(n, mean = 5, sd = 2),
  predictor2 = rnorm(n, mean = 10, sd = 3),
  outcome = rbinom(n, size = 1, prob = 0.5)
)


model <- glm(outcome ~ predictor1 + predictor2, family = binomial, data = your_data)
robust_se <- coeftest(model, vcov = vcovHC(model, type = "HC0"))
print(robust_se)

Output:

z test of coefficients:

             Estimate Std. Error z value Pr(>|z|)
(Intercept)  0.511441   0.937666  0.5454   0.5855
predictor1  -0.167804   0.113103 -1.4836   0.1379
predictor2   0.019991   0.070286  0.2844   0.7761

2. Cluster-Robust Standard Errors

The clubSandwich package in R provides tools for computing cluster-robust standard errors. You can use the vcovCL function:

library(clubSandwich)

robust_se_cluster <- coeftest(model, vcov = vcovCL(model, cluster = your_data$group))
print(robust_se_cluster)

Output:

z test of coefficients:

             Estimate Std. Error z value Pr(>|z|)
(Intercept)  0.511441   0.942390  0.5427   0.5873
predictor1  -0.167804   0.113672 -1.4762   0.1399
predictor2   0.019991   0.070641  0.2830   0.7772

3. Bootstrapped Standard Errors

The boot package in R allows for bootstrapping. You can use the boot function to perform bootstrapping on the logistic regression model:

library(boot)

boot_fn <- function(data, indices) {
  d <- data[indices,]
  model <- glm(outcome ~ predictor1 + predictor2, family = binomial, data = d)
  return(coef(model))
}

boot_results <- boot(data = your_data, statistic = boot_fn, R = 1000)
print(boot_results)

Output:

ORDINARY NONPARAMETRIC BOOTSTRAP


Call:
boot(data = your_data, statistic = boot_fn, R = 1000)


Bootstrap Statistics :
       original       bias    std. error
t1*  0.51144113 -0.011828375  0.96259223
t2* -0.16780441 -0.004944558  0.12078082
t3*  0.01999114  0.002874668  0.07508054

Conclusion

Robust standard errors are essential for reliable inference in logistic regression, especially when model assumptions are violated. Both Stata and R offer various methods to compute these robust standard errors. In Stata, you can use vce(robust), vce(cluster), and the bootstrap command. In R, the sandwich, clubSandwich, and boot packages provide tools to estimate heteroscedasticity-consistent, cluster-robust, and bootstrapped standard errors, respectively. By using these robust methods, researchers can ensure more accurate and reliable results in their logistic regression analyses.