For many complex disorders, genetically relevant disease definition is still unclear. For this reason, researchers tend to collect large numbers of items related directly or indirectly to the disease diagnostic. Since the measured traits may not be all influenced by genetic factors, researchers are faced with the problem of choosing which traits or combinations of traits to consider in linkage analysis. To combine items, one can subject the data to a principal component analysis. However, when family date are collected, principal component analysis does not take family structure into account. In order to deal with these issues, Ott & Rabinowitz (1999) introduced the principal components of heritability (PCH), which capture the familial information across traits by calculating linear combinations of traits that maximize heritability. The calculation of the PCHs is based on the estimation of the genetic and the environmental components of variance. In the genetic context, the standard estimators of the variance components are Lange's maximumudlikelihood estimators, which require complex numerical calculations. The objectives of this paper are the following: i) to review some standard strategies available in the literature to estimate variance components for unbalanced data in mixed models; ii) to propose an ANOVA method for a genetic random effect model to estimate the variance components, which can be applied to general pedigrees and high dimensional family data within the PCH framework; iii) to elucidate the connection between PCH analysis and Linear Discriminant Analysis. We use computer simulations to show that the proposed method has similar asymptotic properties as Lange's method when the number of traits is small, and we study the efficiency of our method when the number of traits is large. A data analysis involving schizophrenia and bipolar quantitative traits is finally presented to illustrate the PCH methodology.
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