Whole-genome sequencing (WGS) and whole-exome sequencing studies have become increasingly available and are being used to identify rare genetic variants associated with health and disease outcomes. Investigators routinely use mixed models to account for genetic relatedness or other clustering variables (e.g., family or household) when testing genetic associations. However, no existing tests of the association of a rare variant with a binary outcome in the presence of correlated data control the type 1 error where there are (1) few individuals harboring the rare allele, (2) a small proportion of cases relative to controls, and (3) covariates to adjust for. Here, we address all three issues in developing a framework for testing rare variant association with a binary trait in individuals harboring at least one risk allele. In this framework, we estimate outcome probabilities under the null hypothesis and then use them, within the individuals with at least one risk allele, to test variant associations. We extend the BinomiRare test, which was previously proposed for independent observations, and develop the Conway-Maxwell-Poisson (CMP) test and study their properties in simulations. We show that the BinomiRare test always controls the type 1 error, while the CMP test sometimes does not. We then use the BinomiRare test to test the association of rare genetic variants in target genes with small-vessel disease (SVD) stroke, short sleep, and venous thromboembolism (VTE), in whole-genome sequence data from the Trans-Omics for Precision Medicine (TOPMed) program.