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When studying familial aggregation of a disease, the following two-stage design is often used: first select index subjects (cases and controls); then record data on their relatives. The likelihood corresponding to this design is derived and a score test of homogeneity is proposed for testing the hypothesis of no-aggregation. This test takes into account the selection procedure and allows adjustment to be made for explanatory variables. It appears as the sum of three terms: a pure test of homogeneity, a test of comparison of observed minus expected cases in the two groups, and a term which adjusts for the possible unequal probabilities of disease of the index subjects. Asymptotic efficiency and a simulation study show that the proposed test is superior to either the pure homogeneity test or tests based on the comparison of numbers of affected in the two groups. The test statistic, which has an asymptotically standard normal distribution, is applied to a study of familial aggregation of early-onset Alzheimer's disease for which a highly significant value (9.46) is obtained: this is the highest value among the three tests compared, in agreement with the simulation study. A logistic normal model is fitted to the data, taking account of the selection procedure: it allows to estimate the regression parameters and the variance of the random effect; the likelihood ratio test for familial aggregation seems less powerful than the score test.


Journal article



Publication Date





542 - 551


Alzheimer Disease, Case-Control Studies, Family, Humans, Longevity, Mathematics, Models, Statistical, Probability, Risk Factors