# Estimating a single proportion from a marginal or conditional model

I had a single sample of a binary outcomes (success / failure), and I wanted to estimate the population proportion with a point estimate and a confidence interval. The problem was that some subjects contributed 2 samples, while others just 1, so my observations, or at least some of them, weren't independent.

To solve this problem, I ran both a GEE model and a GLMM model. I used the binary outcome as a dependent variable, and a vector of 1's as an independent variable, making the intercept my parameter of interest. I thought that it would give me the proportion I look for. The vector of 1's was declared in SAS as a classification variable.

And now for the problem. When I used the logit link function, I did not get the correct results. When I used the log link function, I did. I thought that the log link function suits the Poisson model, so I am confused. I suspect it might have something to do with the difference between proportion and predicted probabilities, but I'm not sure. I wanted to ask if someone can explain to me (and maybe please attach a formula) why the log link function is what I need here? To be more specific, if $b$ is the estimate of the intercept, I did at first $e^b/(1+e^b)$, which was wrong. When I did $e^b$, it was correct. I need to know why...

My SAS code was:

PROC GENMOD DATA=data descending;
CLASS ID D * D is vector of 1's;
MODEL Y  = D / dist = bin  link=log cl;
REPEATED subject= ID;
LSMEANS D/ cl exp;
RUN;

• What model do you have in mind here for how the observed values for the same individual are related? Or put another way, what makes you think you can't just consider multiple observations from the same user as additional independent observed values from the same bernoullibdistribution. – DavidR Mar 23 '14 at 14:05
• Observations from same subjects are correlated, it is confirmed. I already did the analysis and got the expected results. What I need is an estimate of the mean, which is identical to the one I would get with independent observations (correlation doesn't affect the mean), and an adjusted variance. I got this with the log link, and didn't with logit, I need to know why, the maths. Why the log link gives me the right outcome and logit doesn't ? – user42413 Mar 23 '14 at 14:09