I carried out a GEE Poisson regression on my dependent variable, number of days, and my independent variables are binary categories, including a high/low treatment indicator of interest. I obtained the following 95% confidence intervals for a high treatment group and low treatment group (the other binary covariates are held fixed):
High Treatment Group Point Estimate, (95% CI): 10.7099, (10.2836, 11.1540)
Low Treatment Group Point Estimate, (95% CI): 8.9673, (6.3788, 12.6062)
I noted that the two confidence intervals overlap, which leads me to believe that the two treatment groups means are not statistically significantly different.
However, I then calculated an estimate of the predicted group differences and the corresponding confidence interval (e.g. $\hat{\mu_{high}}-\hat{\mu_{low}}$), using the Delta method, and I got the following:
1.7426, (0.1714438, 3.313756).
Now, I noted that the point estimate makes perfect sense as it is $10.7099-8.9673=1.7426$, but the confidence interval does not contain zero, which implies that the difference in means is statistically significantly different from zero.
Why are the two conclusions different? Is the confidence made smaller and therefore significant in the mean difference calculations because maybe, I'm taking advantage of the covariances between the estimates or is something else maybe going on here? Which one of the conclusions should I believe?
UPDATE: I've found some articles that talk about how overlapping confidence intervals don't necessarily imply that the differences between estimated means are not significant in a t-test. They made reference to the sample size being larger (and hence smaller confidence intervals) in the two-sided t-test. That doesn't seem to be going on here since I'm using regression. Can anyone help?
