An Application of a Theorem of Johnstone and Forrester to Testing for Familial Aggregation

In family studies, genetic epidemiologists are interested in choosing traits, or combinations of traits, of high degree of familial aggregation to use in the linkage analysis. Principal components analysis of heritability similar to canonical correlation analysis, estimates the linear combination with the largest heritability. However, the first and foremost task is to test if there is any trait, or combination of traits, of familial aggregation. To this aim, a type of permutation test is proposed. Since the permutation test is time-consuming, a theorem of Johnstone and Forrester (2004) can be used to obtain the approximate p-value.