A multivariate logistic model (MLM) for analyzing binary family data

We consider modeling the familial correlation between 2 related individuals using a multiple logistic regressive model. It is shown that there is a discrepancy in the marginal probability of the second individual. We investigate the conditions under which this discrepancy can be minimized and show how it can have a direct effect on handling missing values and ascertainment. We derive a functional relationship between the parameters in the model that eliminates this discrepancy, hence solving the problems that can arise in the handling of missing values and ascertainment. Because this methodology fails when there are more than 2 related individuals, we present a new model based on a multivariate logistic distribution. Residual familial correlations can be directly related to the parameters of this model. The likelihood for family data under this model is independent of the order in which the family members enter the calculation. The marginal probabilities can be easily computed.