In a standard kin–cohort design, a volunteer (either affected or unaffected) agrees to be genotyped, and the phenotype information on the disease histories of his or her first-degree relatives is obtained through a questionnaire. When the information on both phenotype and genotype for relatives is available, alternative approaches are needed in order to take full advantage of all available data while correcting for bias due to the effects of ascertainment. In this paper, we assume that all probands are affected carriers, though the proposed approach can be extended to include unaffected probands carrying the mutation. Recently, Wang and others (2006) proposed a nonparametric method for estimating the penetrance of a rare mutation. Olschwang and others (2009) proposed an alternative parametric logistic regression model. Both approaches rely on the assumption that the penetrance of noncarriers is zero. This assumption might not be true for many genetic diseases. The penetrance estimate could be severely biased if this assumption was not valid in real applications.

In this paper, we focus on rare mutations and aim at developing a rigorous statistical inference framework for such case–family design. The main difference between this design and the standard kin–cohort design is that the former collects information on both phenotypes and genotypes of the probands’ relatives, while the latter simply collects the phenotypes of the relatives through a questionnaire. The assumption of zero penetrance for the noncarriers is not required for our approach. Furthermore, the proposed approach is based on a likelihood model conditioned on the phenotypes of all individuals; therefore, the derived estimate should not suffer from the biases mentioned for the kin–cohort design.

Some covariates such as gender and ethnicity can be incorporated easily in our approach. Multiple rare mutations can also be handled in the context of the proposed conditional likelihood framework. The derivation of the conditional likelihood functions requires minor assumptions. The maximum likelihood estimates (MLEs) can be obtained through standard optimization algorithm available in mathematical/statistical softwares. Statistical inferences, such as constructing confidence intervals and testing hypotheses for the parameters characterizing the penetrance, can be performed based on the standard large-sample theories.

First, we studied the age-independent penetrances. We assumed 2 independent disease related single-nucleotide polymorphisms (SNPs): one is the study mutation with minor allele frequency (MAF) 0.01 or 0.001 and the other one is unobserved with MAF 0.2. We assumed dominant mode of inheritance for both the SNPs. The disease and risk factors were related by a logistic regression model:where g (r) is 1 if the genotype of the study SNP (unobserved SNP) is of higher risk and 0 otherwise and OR is the odds ratio parameter for the unobserved SNP, which takes value 1 or 2. The marginal penetrance f₁ for carriers was fixed at 0.5 and the other penetrance f₀ was 0.03 or 0.1. The values of log-OR parameters a and b were determined by the other parameters. The genotypes of parents were generated under Hardy–Weinberg equilibrium and random mating, and the genotypes of offspring were independently generated given parental genotypes. From a large number of generated families with 3 offspring, we randomly selected 1 000 000 families with the first offspring being affected and carrying the study mutation and treated them as the source population from which the study sample was collected. A sample of size 200, 500, or 1000 was drawn from this population and simulation results based on 100 000 replications were produced. Reported in Table 1 are the Rbias of the estimates defined as the mean estimated penetrances divided by the true penetrance minus 1, empirical standard errors (SE) and mean estimated standard errors (SEE) of the estimates, and empirical coverage probability (ECP) of the penetrances.

Overall, the estimates have minor bias when the disease is rare (f₀ = 0.03) and the study mutation is rare (MAF = 0.001), with absolute relative biases no more than 1.2%. Common disease (f₀ = 0.1), increased MAF (0.01) of the study mutation, and positive effect of unobserved mutation (OR = 2) has small impact on the estimates, with Rbias − 7.9% ~ 3.4%. In all situations, the SEE are very close to the empirical ones. The relative bias tends to be stable and remain to be small when the sample size increases. We also estimated the penetrance of carriers using only the genotypes of unaffected relatives by assuming zero penetrance of noncarriers. The resulting Rbias is generally small when f₀ = 0.03 but it could become considerably large when f₀ = 0.1 (results not shown).

It is also seen from Table 1 that the Rbias for a mutation with MAF = 0.001 tends to be smaller than that observed for a mutation with MAF = 0.01. Additional simulation results show that the relative biases get larger when the MAF increases. For example, a MAF of 0.03 produces relative biases at the range of − 10.3% ~ − 23.7%, and an MAF of 0.1 produces relative biases at the range of − 41.9% ~ − 65.1%, with the other parameters the same as those in Table 1. It appears that the proposed approach is suitable for rare mutation with MAF ≤ 0.01.

Second, we studied the proposed approach when the penetrance is age dependent. We generated data from the following Cox model with Weibull baseline hazard function:where g and r are the same as those in (5.1) with the same MAFs. The OR was fixed at 1 or 2. The other parameters ξ, ψ, and β were determined by 3 cumulative risk probabilities: p_30,0 = P(T ≤ 30|g = 0), p_60,0 = P(T ≤ 60|g = 0), and p_60,1 = P(T ≤ 60|g = 1), where T is the age at onset. To mimic common disease, we set p_30,0 = 0.03 and p_60,0 = 0.09; to mimic rare disease, we set p_30,0 = 0.01 and p_60,0 = 0.03. In both situations, we set p_60,1 = 0.5. The ages of the relatives of a proband were generated from the uniform distribution in the interval (a − 5,a + 5), where a is the current age of the proband that is uniformly distributed in the interval (20, 70). The ages, genotypes, and disease status were generated for a large number of families similarly to the age-dependent situation. In each family, there were 2 parents and 3 offspring whose data were generated. Altogether, 1 000 000 families with 1 affected proband (the first offspring) carrying the mutation in each family were obtained. From these families, we sampled 400 or 1000 families and estimated ξ, ψ, and β in model (5.2) by ignoring the unobserved mutation. Substituting the estimated parameters gave the estimates of marginal survival functions of carriers and noncarriers. Based on 5000 replications, we calculated the mean estimated survival functions of both carriers and noncarriers and the 90% confidence intervals of the survival functions.

Presented in Figures 1 and ​and22 are the results for carriers and noncarriers, respectively, with sample size 1000 and OR = 1 (unobserved mutation does not play a role on the disease). We can see that the bias of the estimates reduces dramatically when the MAF of study mutation decreases from 0.01 to 0.001, showing that the approximation of the likelihood function works pretty good for relatively rare mutation. When the disease gets common, the proposed method using both affected and unaffected relatives does not produce extra bias. However, the method that uses only unaffected relatives has much larger bias for common disease. This extra bias is due to the improper assumption of zero penetrance function of noncarriers for common disease. Other results for sample size 400 or OR = 2 are presented in Figures s1–s6 of the supplementary material (available at Biostatistics online). In summary, the bias of the penetrance functions get smaller as the sample size increases. The positive effect of unobserved mutation (OR = 2) has only limited impact on the penetrance function estimates. In particular, the impact is minimal when the MAF of the study mutation is small and the disease is rare.

With pooled data, we assumed a Weibull survival function (t/e^ψ)^ξ for carriers. With gender or mutation type adjusted, we assumed a proportional hazards model with survival function (t/e^ψ)^ξe^β₁x₁ or (t/e^ψ)^ξe^β₂x₂. Here, x₁ = 1 if male and 0 if female and x₂ = 1 if MSH1 and 0 if MSH2. We obtained the MLEs of the unknown parameters (ψ, ξ, β₁, and β₂), estimated SE of the MLEs, and 95% confidence intervals of the parameters. The estimation and hypothesis-testing results are presented in Table 3. The estimated survival function together with its confidence interval curves based on 5000 bootstrappings (Efron and Tibshirani, 1993) are plotted in Figure s7 of of the supplementary material (available at Biostatistics online).

It is helpful to check the parametric assumption of the baseline hazard function. Because the design is retrospective and the observations are subject to censoring, rigorously checking the parametric assumption is a great challenge. In practice, one can try some commonly used parametric baselines and choose the one with the largest likelihood. This technique, however, could be misleading if the true baseline is very different from the selected ones. Instead, a nonparametric approach that does not assume any parametric baseline is much more desirable, although it involves some computational and theoretic issues. We will pursue this in the future research.

The proposed approach allows for unobserved risk factors that are correlated among family members, provided that there is no interaction effect between the study mutation and unobserved risk factors. When the interaction effect is present, the proposed approach can produce considerably large bias on the penetrance estimates. More advanced methods such as the frailty model could be helpful in resolving this problem, for example, Hsu and others (2004) and Hsu and Gorfine (2006). The development of an inference procedure is still under way.