Logistic models for familial association of binary traits

Many diseases show clear familial aggregation but the pattern of inheritance does not correspond to Mendelian segregation. A genetic component responsible for familial similarity may be suspected for a trait which is also known to be strongly influenced by environmental factors. In such cases, when the trait has dichotomous expression (i.e. a disease phenotype must be classified as sick or well), a binary data analogue of variance components estimation in the linear mixed model is needed to summarize the data, describe patterns of association and suggest disease etiology. Two modifications of the logistic regression model are proposed in this dissertation for application to family studies of binary such traits.; The first model is a logistic regression for the marginal probability that each family member will be affected given covariates on that person. This is the standard logistic regression model except that the family covariance matrix is a product of diagonal matrix terms and a family correlation matrix. We describe application of this model, which was proposed by Liang and Zeger (1986) to improve estimates of the regression coefficients in longitudinal studies, to the problem of estimating correlation. The asymptotic distribution of the correlation estimator is derived.; The second model is a logistic regression for the probability that each family member is affected conditional on that person's covariates and the disease outcome for each other family member. We define a family of distributions for the vector of family outcomes which has interactions of the first order and no higher order effects. In this model, the conditional probability that an individual is affected, given disease status of relatives, depends only on the number of relatives affected. Estimating equations are derived which have a much simpler form than the maximum likelihood equations for this distribution. The solution to these estimating equations, which maximizes the product of the conditional probabilities, is shown to have high asymptotic relative efficiency compared to the maximum likelihood estimators. In an example data analysis the estimates by both methods are similar. Conditional models for pedigree data are described and a data analysis example demonstrates applications and properties of the marginal and conditional models.