Age at Onset and Liability to Disorder: Estimating Covariances in Censored Populations

Studies of resemblance for disorders and other traits measured at the binary (yes/no) level between relatives frequently contain individuals who are currently in the negative category but who will become positive in future. For example, a 10-year-old may develop depression in the future, but is as yet unaffected. Such censoring can substantially bias estimates of correlation between relatives. To overcome this problem we develop a model for the association between liability to a disorder, and its age at onset. The model is designed for data from pairs of relatives to enable estimation of the correlation between an individuals liability to disorder and their age at onset. Usually, such information is not available at the individual level, because age at onset is uniquely available when onset has occurred. Lacking variation in disorder status, data from non-related persons cannot estimate the covariance between liability and age at onset. Data from relatives can resolve this issue when there is a correlation in liability between the relatives, because different age at onset distributions would be expected in concordant vs. discordant pairs of relatives. Greater severity and worse outcomes are often observed among those with earlier onset, so a correlation between disorder liability and age at onset seems likely in many cases. In this article we present the basic theory of the model, implemented as a mixture distribution, and an application to cannabis use in a Virginia Twin Study of Adolescent Behavioral Development. A negative association of (-.212) between age at onset an liability was found, with confidence intervals of -.263 to -.152, which do not cross zero. The method contrasts with Cox Proportional Hazards, in which disorder liability and onset timing are treated as a single dimension.