poisson regression for binary outcome: genmod and Zou's variance estimator

[pmbrown]

i would normally use genmod with a log link to get risk estimates, but i notice others reporting poisson regression with Zou's sandwich variance estimator

The robust error estimate is commonly used to deal with variance underestimation in correlated data analysis. I have applied this approach here to deal with variance overestimation when Poisson regression is applied to binary data.

you can see this used in practice here, for example: ucla.edu_relative-risk-in-sas-using-proc-genmod 

Zou suggests using a modified Poisson approach to estimate the relative risk and confidence intervals by using robust error variances. Using a Poisson model without robust error variances will result in a confidence interval that is too wide. The robust error variances can be estimated by using the repeated statement and the subject identifier ... [on the class statement], even if there is only one observation per subject, as Zou cleverly points out. ... [T]he repeated statement (with an unstructured correlation matrix), telling proc genmod to calculate the robust errors

however, I can't find anything in the sas documentation that confirms Zou's estimator is implied by the use of type=un (unstructured) on the repeated statement (Zou is not listed among the references for the details of genmod: https://documentation.sas.com/doc/en/pgmsascdc/9.4_3.5/statug/statug_genmod_references.htm). I have seen others use type =ind (independent) and claim to be using Zou's estimator. The difference between the correlation matrix is found in the documentation:

can someone point me to something that confirms how the repeated statement should be specified to implement Zou's method? Note, the 'repeated' statement is a misnomer, it is a single obs per patient, and I wonder if the unstructured and independent correlation matrices reduce the same equation under the circumstances and then the user can specify un or ind.

[StatDave]

This note discusses various methods of estimating relative risks including Zou's method. With only single observations per subject, all of the correlation structures are the same, so it makes no difference which is selected. 

[pmbrown]

it would be nice to see it worked thru numerically. That link simply says: Zou says so, as if it was not known this was a feature of their code (perhaps that is why it is not indicated in the documentation). And Zou states it as a fact and then uses simulations to evaluate it. I'd like to see that the correction -(1/n0+1/n1) falls out when it is a single obs per patient, but i would have to remember the equations. Without knowing the detail it may be surprising that genmod would even run when it is expecting multiple obs per patient and the user feeds it a single obs.