Pseudo-likelihood estimates of the cumulative risk of an autosomal dominant disease from a kin-cohort study

Wacholder et al. [1998: Am J Epidemiol 148:623-629] and Struewing et al. [1997: N Engl J Med 336:1401-1408] have recently proposed a design called the kin-cohort design to estimate the probability of developing disease (penetrance) associated with an autosomal dominant gene. In this design, volunteers (probands) agree to be genotyped and one also determines the disease history (phenotype) of first-degree relatives of the proband. They used this design to estimate that the chance of developing breast cancer by age 70 in Ashkenazi Jewish women who carried mutations of the genes BRCA1 or BRCA2 was 0.56, a figure that was lower than previously estimated from highly affected families. The method that they used to estimate the cumulative risk of breast cancer, while asymptotically correct, does not necessarily produce monotone estimates in small samples. To obtain monotone, weakly parametric estimates, we consider separate piece-wise exponential models for carders and non-carriers. As the number of intervals on which constant hazards are assumed increases, however, the maximum likelihood score equations become unstable and difficult to solve. We, therefore, developed alternative pseudo-likelihood procedures that are readily solvable for piece-wise exponential models with many intervals. We study these techniques through simulations and a re-analysis of a portion of the data used by Struewing et al. [1997] and discuss possible extensions.