Logistic regression of family data from retrospective study designs.

We wish to study the effects of genetic and environmental factors on disease risk, using data from families ascertained because they contain multiple cases of the disease. To do so, we must account for the way participants were ascertained, and for within-family correlations in both disease occurrences and covariates. We model the joint probability distribution of the covariates of ascertained family members, given family disease occurrence and pedigree structure. We describe two such covariate models: the random effects model and the marginal model. Both models assume a logistic form for the distribution of one person's covariates that involves a vector beta of regression parameters. The components of beta in the two models have different interpretations, and they differ in magnitude when the covariates are correlated within families. We describe ascertainment assumptions needed to estimate consistently the parameters beta(RE) in the random effects model and the parameters beta(M) in the marginal model. Under the ascertainment assumptions for the random effects model, we show that conditional logistic regression (CLR) of matched family data gives a consistent estimate beta(RE) for beta(RE) and a consistent estimate for the covariance matrix of beta(RE). Under the ascertainment assumptions for the marginal model, we show that unconditional logistic regression (ULR) gives a consistent estimate for beta(M), and we give a consistent estimator for its covariance matrix. The random effects/CLR approach is simple to use and to interpret, but it can use data only from families containing both affected and unaffected members. The marginal/ULR approach uses data from all individuals, but its variance estimates require special computations. A C program to compute these variance estimates is available at http://www.stanford.edu/dept/HRP/epidemiology. We illustrate these pros and cons by application to data on the effects of parity on ovarian cancer risk in mother/daughter pairs, and use simulations to study the performance of the estimates.