A time‐dependent logistic hazard function for modeling variable age of onset in analysis of familial diseases

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Abstract
The paper presents an extension of the regressive logistic models proposed by Bonney [Biometrics42:611–625, 1986], to address the problems of variable age‐of‐onset and time‐dependent covariates in analysis of familial diseases. This goal is achieved by using failure time data analysis methods, and partitioning the time of follow up inKmutually exclusive intervals. The conditional probability of being affected within thekth interval (k= 1…K) given not affected before represents the hazard function in this discrete formulation. A logistic model is used to specify a regression relationship between this hazard function and a set of explanatory variables including genotype, phenotypes of ancestors, and other covariates which can be time dependent. The probability that a given person either becomes affected within thekth interval (i.e., intervalkincludes age of onset of the person) or remains unaffected by the end of thekth interval (i.e., intervalkincludes age at examination of the person) are derived from the general results of failure time data analysis and used for the likelihood formulation. This proposed approach can be used in any genetic segregation and linkage analysis in which a penetrance function needs to be defined. Application of the method to familial leprosy data leads to results consistent with our previous analysis performed using the unified mixed model [Abel and Demenais,Am J Hum Genet42:256–266, 1988], i.e., the presence of a recessive major gene controlling susceptibility to leprosy. Furthermore, a simulation study shows the capability of the new model to detect major gene effects and to provide accurate parameter estimates in a situation of complete ascertainment.

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