# When to use robust standard errors in Poisson regression?

I am using a Poisson regression model for count data and am wondering whether there are reasons not to use the robust standard error for the parameter estimates? I am particularly concerned as some of my estimates without robust are not significant (e.g., p=0.13) but with robust are significant (p<0.01).

In SAS this is available by using the repeated statement in proc genmod (e.g., repeated subject=patid;). I've been using http://www.ats.ucla.edu/stat/sas/dae/poissonreg.htm as an example which cites a paper by Cameron and Trivedi (2009) in support of using robust standard errors.

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In general if you have any suspicion that your errors are heteroskedastic, you should use robust standard errors. The fact that your estimates become non-significant when you don't use robust SEs suggests (but does not prove) the need for robust SEs! These SEs are "robust" to the bias that heteroskedasticity can cause in a generalized linear model.

This situation is a little different, though, in that you're layering them on top of Poisson regression.

Poisson has a well known property that it forces the dispersion to be equal to the mean, whether or not the data supports that. Before considering robust standard errors, I would try a Negative Binomial regression, which does not suffer from this problem. There is a test (see the comment) to help determine whether the resultant change in standard errors is significant.

I do not know for sure whether the change you're seeing (moving to robust SEs narrows the CI) implies under-dispersion, but it seems likely. Take a look at the appropriate model (I think negative binomial, but a quick googling also suggests quasi-Poisson for under-dispersion?) and see what you get in that setting.

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Nice answer! Typically in OLS heteroskedasticity does not cause the parameters to be unbiased (merely inefficient). This is not true though for generalized linear models, see this post by Dave Giles about it for references. I don't think I've seen the Vuong test recommend for this though (for comparisons of non-nested zero inflated models I have seen it suggested). Poisson is nested within the Neg. Binomial model, so one can use a likelihood ratio test for the dispersion parameter. –  Andy W Nov 2 '12 at 12:42
Thank for your answer. I did try Negative Binomial regression but ran into the warning: "The relative Hessian convergence criterion of 0.0046138565 is greater than the limit of 0.0001. The convergence is questionable." Note that my response variable is a count with values ranging from 0 to 4. Is there a transformation of the dependent or independent variable that would help convergence? Or what does one do in this case? –  kara Nov 2 '12 at 13:36
Also, related to the non-robust SEs being smaller - in my analysis I see that it is the robust SEs that are smaller and this is where the significance lies (not in the non-robust results). This is why I want to be careful about whether or not to report the robust results - I don't want to choose this method just because of significant pvalues! Thanks again! –  kara Nov 2 '12 at 13:42
@AndyW I checked my notes and Vuong is indeed for ZI vs Poisson. Updated post. kara I missed the reversal. You could have under-dispersed data, in which case the NBD is also potentially the solution :-) –  Ari B. Friedman Nov 2 '12 at 13:58
@kara Hard to diagnose your non-convergence problem in comments. I'd try a new question just on that with as much information as you can provide. –  Ari B. Friedman Nov 2 '12 at 14:24

I'll differentiate analyses using model based versus robust standard errors by referring to the latter as "GEEs" which is in fact an exchangeable definition. In addition to Scortchi's fantastic explanation:

GEEs can be "biased" in small samples, i.e. 10-50 subjects: (Lipsitz, Laird, and Harrington, 1990; Emrich and Piedmonte, 1992; Sharples and Breslow, 1992; Lipsitz et al., 1994; Qu, Piedmonte, and Williams, 1994; Gunsolley, Getchell, and Chinchilli, 1995; Sherman and le Cessie, 1997.) When I say that GEEs are biased what I mean is that the standard error estimate can be either conservative or anticonservative due to small or zero cell counts, depending upon which fitted values exhibit this behavior and how consistent they are with the overall trend of the regression model.

In general, when the parametric model is correctly specified, you still get correct standard error estimates from the model based CIs, but the whole point of using GEE is to accommodate that very big "if". GEEs allow the statistician to merely specify a working probability model for the data, and the parameters (instead of being interpreted in the strictly parametric framework) are considered a type of "sieve" that can generate reproducible values regardless of the underlying, unknown data generating mechanism. This is the heart and soul of semi-parametric analysis, which a GEE is an example of.

GEEs also handle unmeasured sources of covariation in the data, even with specification of an independent correlation matrix. This is because of the use of empirical rather than model based covariance matrix. In Poisson modeling, for instance, you might be interested in fertility rates of salmon sampled from various streams. The ova harvested from female fish might have an underlying Poisson distribution, but genetic variation that comprise of shared heretibility and available resources in specific streams might make fish within those streams more similar than among other streams. The GEE will give correct population standard error estimates as long as the sampling rate is consistent with their population proportion (or is in other ways stratified).

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