We previously proposed using the probability that a set of distantly related affected subjects share a rare variant to test the null hypothesis of a complete absence of linkage and association to a disease, without requiring a control group to estimate variant frequency. Within-family heterogeneity, the fact that different rare variants, common variants or non-genetic causes may influence the trait in different subsets of members from large extended families may result in causal rare variants being shared by only a subset of affected relatives, and be missed by a test considering only variants shared by all affected relatives. Here we introduce an extension of our algorithm to compute exact probabilities of sharing patterns where a subset of affected subjects share a rare variant instead of all affected family members. We define an exact test statistic based on these pattern probabilities, and study the factors influencing its power analytically and by simulation on extended families. Under single-locus and heterogeneity models, power is mostly driven by a causal variant relative risk of disease. The test incorporating sharing by a subset of affected relatives proved moderately more powerful than the test requiring all relatives to share, under single locus models and one form of genetic heterogeneity model.