# Survey weights in logistic regression models

We are interested in fitting a multiple logistic regression model using data obtained from a national survey of children with special health care needs. The data has an accompanying weight variable intended to standardize children to the national population in which we intend to make inference. This weight variable does not sum to 1 nor are the weights integral (they may take values such as 23.2). This model is being fit using SAS v9.2. In consulting the documentation for the logistic procedure, I notice in the syntax description the following statement:

Caution: PROC LOGISTIC does not compute the proper variance estimators if you are analyzing survey data and specifying the sampling weights through the WEIGHT statement. The SURVEYLOGISTIC procedure is designed to perform the necessary, and correct, computations.

I don't understand why this should be an issue. If model based standard errors are being computed, then the weighted maximum likelihood estimator should give standard errors which are correct for the population of interest. Is this correct? What likelihood function is SAS's logistic regression solver optimizing if the above statement is correct?

Most stats software that is not built specifically for survey weights-to-population interprets weights as either a frequency variable or as a correction for heteroscedasticity and fits likelihood functions accordingly. Estimates of parameters will be correct but standard errors will be wildly wrong (usually way too small ie claiming excess precision).

• Can you enlighten me on the difference between survey weights and frequency weights? If I understand correctly, the point estimates for odds ratios should be the same but the confidence intervals are computed differently. How is this done, exactly? And what is the intuitive reason. I understand that "magnifying" the sample size should "magnify" the uncertainty in survey weights (relative to frequency weights). Feb 16 '12 at 19:41
• Frequency weights are when the row of data actually represents w observations in the sample; whereas survey weights mean one observation in the sample representing w entities in the population. I don't know about SAS, but SPSS thinks of weights as frequencies and hence concludes "ah ha, this weight of 23 means they saw 23 people in the sample with exactly the same score, that gives me lots of confidence about the result..." So apart from anything else it uses the population size (which the weights will sum to, in a weighted-to-population sample) as the sample size. Feb 16 '12 at 19:46

In order to compute standard error correctly when dealing with complex sampling design (like e.g. stratified, cluster or stratified cluster or even multistage stratified cluster sampling) one needs to take into account two things:
1. Probabilities of inclusion for every observation.
2. Sampling design, defining the stratas, clusters and stages of sampling design that each observation belong to and size of those stratas (or at least their share in population).

First point is enough only for computing unbiased estimator of the mean of the variable. To adequately compute unbiased estimator of the standard error of the mean one needs also to know the second. Otherwise it is impossible to compute it, no matter if you would like to use traditional survey weights (which requires individually estimating standard error in every strata, and thus knowing the design), or if you would like use some resampling methods, like replicate weights or bootstrap, where resampling must account for the stratas and clusters.

The SAS's PROC LOGISTIC use only first information and although the coefficients estimates are correct, the estimate of the size of their standard error is biased and incorrect.

I don't understand why this should be an issue. If model based standard errors are being computed, then the weighted maximum likelihood estimator should give standard errors which are correct for the population of interest.

The issue is, that formulas SAS is using for computing standard error are simplified versions of those formulas, that are much more easy to compute and do not require information about sampling design. These formulas can be applied only to simple sampling design with replacement, which is as you said, not the case.