Summary

Using Generalized Estimating Equations, you have fit a repeated measures logistic regression to longitudinal binary data, and seen how the goodness of fit statistics can be used to compare models.

• Concerning the correlation structure, before settling on Unstructured, you should compare it to other models. Given the temporal nature of the repeated measurements, trying the AR(1) structure may be reasonable, though it seems unlikely to perform well since the size of the values in the working correlation matrix for the Unstructured structure do not decay with the distance between measurements as you would expect if the AR(1) structure is to perform well. Alternatively, given that the correlations all fall within a relatively narrow range, the Exchangeable structure, which estimates a single correlation for all cells in the matrix, may perform well.
• Concerning the model effects, you probably want to keep Smoker, even though it's not statistically significant, because it is of policy interest. It seems unlikely that removing Age would result in a better model; however, given near-equivalence of the parameter estimates for the first 3 levels of Age, you could consider computing a binary factor that takes value 0 when Age is 7, 8, or 9 and value 1 when Age is 10. This may improve your results slightly.