# How are robust standard errors calculated in the case of logistic regression?

I mean: the Huber/White/sandwich estimator of standard errors. It seems to me that, in the case of continuous outcomes, robust estimators of standard errors are rather simple, given that variance of residuals for each observation is calculated as the squared (estimated) residuals from the regression. But I can't figure out how this apply to a binary outcome: after estimating a probability for each observation, how can I estimate the variance of residuals for that observation, given it will necessarily have an outcome of 0 or 1, corresponding, in the logit scale, to either $+\infty$ or $-\infty$? In particular, I'm thinking at the case where I have continuous predictors (so I just have one occurence for each value of my set of regressors).

PS: I read some criticisms about the use of robust standard errors for logistic regression, because, if the estimates of variances are biased, then also the parameter estimates themselves are (given average and variance are linked, in the binomial case). However, I'm not wondering about whether robust standard errors make sense here, but simply how they are calculated.

• Do you have a reference for their calculation? Or are you looking for a reference? – David Smith Jun 6 '17 at 17:31
• Thank you. Yes, a reference would be great. This could be a reference for robust SEs in the case of regression for a continuous outcome (at page 3): stata.com/manuals13/rregress.pdf – Federico Tedeschi Jun 6 '17 at 20:27

## 1 Answer

The definition is completely analogous if you use the so-called working residuals and if regressors and residuals are weighted appropriately. The working weights are part of the usual iteratively weighted least squares (IWLS) algorithm employed for generalized linear models (including logistic regression).

The main idea is to use derivative of the log-likelihood with respect to the mean or the corresponding linear predictor as a measure of deviation. Alternatively, you can use the so-called scores or estimating functions as a measure of deviation, i.e., the derivative of the log-likelihood with respect to the coefficients of the linear predictor. All these ideas are employed in our sandwich package for R to enable object-oriented implementations of the different sandwich estimators. See Zeileis (2006, Object-Oriented Computation of Sandwich Estimators, doi:10.18637/jss.v016.i09) for the details.

A worked illustration in R using a logistic regression for labor market participation is:

data("SwissLabor", package = "AER")
m <- glm(participation ~ ., data = SwissLabor, family = binomial)


The classical Eicker/Huber/White sandwich covariance matrix can be computed using our sandwich package:

library("sandwich")
vcov_sandwich <- sandwich(m)


And by hand you can easily extract the working residuals and regressors, appropriately weighted with the working weights:

r <- residuals(m, "working") * sqrt(weights(m, "working"))
X <- model.matrix(m) * sqrt(weights(m, "working"))


Then, the formula for the covariance matrix is just the same as in the linear regression model

vcov_byhand <- solve(t(X) %*% X) %*% t(X) %*% diag(r^2) %*% X %*% solve(t(X) %*% X)
all.equal(vcov_sandwich, vcov_byhand)
##  TRUE


Two caveats: (1) Of course, one shouldn't compute the sandwich like this but more efficient and numerically more stable matrix computations can be employed. (2) For data with independent binary responses, these sandwich covariances are not robust against anything. They are consistent, that's ok, but there is no way to misspecify the likelihood without misspecifying the model equation.

• Thank you. Yes, of course with logistic regression there is problem that mean and variance are intertwined, however I am satisfied with having sensible confidence intervals. If I well understand, logistic regression always uses weights given the problem of variance depending on estimated probabilities, and the process is iterated until convergence is reached. In this case, why do working residuals guarantee more robustness? Is it just because they lead to higher standard errors than, for example, Pearson residuals (given $p*(1-p)<\sqrt{p*(1-p)}$) or there is something I'm missing? – Federico Tedeschi Jun 12 '17 at 15:51
• If you just want standard errors, why not just use the usual ones (based on the Hessian, i.e., second order derivatives only). As for the weighting: You can either understand the estimation process as a maximum likelihood problem or, equivalently, as an iterated least squares problem. The latter is useful of you want to use the same kind of formulae as in the linear regression case, as indicated in your question. – Achim Zeileis Jun 12 '17 at 16:01
• Thank you. I confess: I asked for robust standard errors because I want to explore the differences between separated and seemingly unrelated regressions. Given, with SUREG, robust standard errors are used, I wanted to understand what it means in the context of logistic regression. – Federico Tedeschi Jun 13 '17 at 9:13