The generalized score (GS) statistic is widely used to test hypotheses about mean model parameters in the generalized estimating equation (GEE) framework. However, even in the simple problem of comparing two proportions with paired and independent observations, GS has room for improvement. Kosinski (2013, Stat Med) showed that GS does not reduce to the score statistic when applied to independent data, and that GS can have imprecise type I error control when the two comparison groups have extremely unbalanced sample sizes. The weighted generalized score (WGS) statistic, introduced in the same paper, resolves both of these issues. This talk covers recent further results on WGS and its properties in comparison with GS, with applications to diagnostic testing and other biomedical topics. New work to be discussed includes the optimal choices of weights under certain optimality criteria, as well as extensions of WGS beyond two correlated proportions to broader comparisons of correlated means.