A frailty-model based approach to estimating the age-dependent penetrance function of candidate genes using population-based case-control study designs: An application to data on the BRCA1 gene

Summary

The population-based case-control study design is perhaps one of, if not the most, commonly used designs for investigating the genetic and environmental contributions to disease risk in epidemiologic studies. Ages at onset and disease status of family members are routinely and systematically collected from the participants in this design. Considering age at onset in relatives as an outcome, this paper is focused on using the family history information to obtain the hazard function, i.e., age-dependent penetrance function, of candidate genes from case-control studies. A frailty-model based approach is proposed to accommodate the shared risk among family members that is not accounted for by observed risk factors. This approach is further extended to accommodate missing genotypes in family members and a two-phase case-control sampling design. Simulation results show that the proposed method performs well in realistic settings. Finally, a population-based two-phase case-control breast cancer study of the BRCA1 gene is used to illustrate the method.

1. Introduction

The population-based case-control study design is commonly employed in epidemiologic studies of chronic disease etiology and recently has been used in studying genetic association with disease risk. The odds ratios associated with mutations in these genes are estimated in the same approach as that for environmental exposures by using the logistic regression model as if the cases and controls were prospectively collected (Prentice and Pyke, 1979). When disease prevalence is low, the odds ratio approximates the relative but not the absolute risk. The absolute risk, unfortunately, is not estimable directly from case-control data because the proportion of cases in the sample is fixed artificially by design. Whittemore (1995) proposed the use of family data to estimate population-based baseline disease probability along with odds ratios and correlation coefficients of disease status among family members under the case-control study design. This approach requires factors of interest to be known for all family members, an assumption which is not satisfied in a typical case-control study because while family history of disease data are systematically collected for these cases and controls, relatives’ exposure information is not. Fortunately this issue is less of a problem for studies of candidate genes than environmental exposures as the carrier status for relatives can be inferred probabilistically from the genotypic status of cases and controls by Mendel’s law. This paper is thus focused on using family data gleaned through a case-control study design to estimate the disease risk of candidate genes.

Age is an important risk factor for many chronic diseases and investigators typically collect onset ages in relatives as part of disease family history information. This allows us to treat onset age as a censored survival outcome and to take the advantage of recent methodologic developments in multivariate survival analysis for estimating the cumulative risk, which is also called the age-dependent penetrance function in the genetic epidemiologic literature.

Wacholder et al. (1998) first proposed to use relatives’ age at onset data in population-based candidate gene studies in the context of volunteer-based studies. This study design goes by several names including the kin-cohort design and the genotyped-proband design (Gail et al., 1999). Methods of moments (Wacholder et al., 1998) and nonparametric maximum likelihood (Chatterjee and Wacholder, 2001) methods have been developed for obtaining hazard functions for individuals who do and do not carry high risk genotypes. However, these methods tend to yield biased hazard function estimators for case-control studies because of (i) the over-representation of cases due to the nature of the sampling and (ii) the residual dependency among relatives even after accounting for candidate gene effects. Therefore, methods that can both utilize relatives’ outcome and account for case-control sampling are necessary for analyzing case-control data with family history information.

To circumvent the sampling and residual dependency issues, Chatterjee et al. (2006) proposed to model the joint distribution of failure times of family members by a copula model. In this approach, the Cox proportional hazards model (Cox, 1972) was used for the marginal hazard function which represents population-averaged disease risks for carriers and non-carriers. Such estimates are important particularly in terms of public health impact. However, an individual is often more interested in the risk given his/her specific family background. The latter naturally leads to a conditional hazard function formulation, in that a frailty is assumed to represent the common unobserved risk for the family and it acts on the hazard function in a multiplicative fashion.

The purpose of this paper is to present a newly developed frailty-model-based method for estimating the hazard function from two-phase case-control data with family history information. The baseline hazard function in the frailty model is left unspecified, allowing for any arbitrary failure time distribution (age-dependent penetrance function). In Section 2, we first consider the situation where genotypes are observed for all family members and then extend the approach to accommodate unobserved genotypes in relatives which is the more typical situation. We then examine the performance of the proposed methods by a simulation study in Section 3. To demonstrate the potential insights that could be gleaned from population-based studies with this method, we apply our approach to data on the BRCA1 gene in Section 4. We conclude the paper with some final remarks.

2. Methods

2.1 Notation, Data structure, and Model Framework

Consider a two-phase case-control study. In the first phase a pool of cases and controls, here termed as probands, are randomly sampled from the population and an array of risk factors is collected from them. Stratified by certain aspects of variables collected at the first phase, in the second phase a random subset of cases and controls from each stratum are selected for collecting more detailed risk factor information, here, genotyping. The sampling fraction π for a genotyped proband is therefore the fraction of individuals being randomly selected for genotyping in the stratum to which the proband belongs. The sampling fraction is strictly positive π > 0, i.e., all strata have representative samples, to ensure a consistent estimation of odds ratios from two-phase data. When π = 1 for all strata, the two-phase sampling design becomes the usual standard case-control design. The two-phase sampling scheme is useful particularly when available resources constrain data collection for each individual. By a careful choice of stratification variables and π, the two-phase design can improve the efficiency of parameter estimates compared with a conventional case-control design with the same number of subjects. This design has been used in many genetic epidemiologic studies including the CARE study described further in Section 4 (Malone et al., 2006).

Below we introduce the notation and the model. Assume that there are n case-control probands indexed by i = 1, …, n with sampling fraction π₁, …, π_n in a two-phase case-control study. The relatives of each proband are also ascertained and we assume that members from different families are non-overlapping. Specifically, in the ith family, let the proband be indexed by subscript i0 and the n_i relatives be indexed by subscript ij for j = 1, …, n_i. Furthermore let (X_ij, δ_ij, Z_ij) be the observational time, disease status, and a vector of covariates for the jth individual in the ith family; δ_ij is a disease indicator which is 1 if the disease occurs at or before the censoring time and 0 otherwise, and X_ij is the failure time if δ_ij = 1 and the censoring time if δ_ij = 0. The total number of diseased in the ith family is denoted by ${\mathrm{\Delta }}_i={\sum }_{j=0}^{n_i}{\delta }_{ij}$. The observational time and disease status can also be equivalently represented by the counting process notation. The right censored counting process N_ij(t) is defined as N_ij(t) = I(X_ij ≤ t, δ_ij = 1), where I(·) is the indicator function. The at-risk process Y_ij(t) is defined as Y_ij(t) = I(X_ij ≥ t).

We use a shared gamma frailty with the conditional proportional hazards model to describe the dependent failure outcomes of family members. The unobserved frailty for the ith family, represented by ω_i, induces dependence among family members on their failure times. The family-specific frailty ω_i is assumed to be iid gamma(1/θ, θ) distributed with density ${\theta }^{-1/\theta }\mathrm{\Gamma }{(1/\theta )}^{-1}{\omega }_i^{1/\theta -1}exp(-{\omega }_i/\theta )$ and mean 1. The parameter θ measures the strength of dependence among failure times from family members, with a larger value of θ implying a stronger dependence. The individual hazard function conditional on frailty and covariates is given by

$${\lambda }_{ij}(t\mid Z_{ij},{\omega }_i)={\omega }_i{\lambda }_0(t)exp({\beta }^{\prime }{Z}_{ij}),j=0,\dots ,n_i,i=1,\dots ,n,$$ (1)

where λ₀(t) is the conditional baseline hazard function assumed common for all individuals and β is a vector of regression coefficients. Conditional on the frailty and the covariates, the censoring time is assumed to be independent of the failure time and non-informative of the frailty. In addition, we assume that the frailty is independent of the observed covariates. These assumptions are generally required by frailty models to allow for the distribution of frailty being separated from that of other variables. Our model specifies a parametric frailty distribution and a parametric form for the effect of the risk factors, but leaves the form of λ₀(t) unspecified.

2.2 A likelihood function for the frailty model when all covariates are observed

To obtain valid estimates for {β, Λ₀(t), θ}, we need to account for the case-control retrospective sampling. As relatives are sampled because of the proband’s disease outcomes and sometimes ages at onset, a natural consideration for a likelihood-based approach in this setting would be to let the joint likelihood of the relatives be conditional on the data from the proband. In this likelihood, we do not make parametric assumptions on the form of Λ₀(t). To do this, we treat the jump size at each observed failure time as a parameter, following the same idea as in the nonparametric maximum likelihood estimator (NPMLE) (Zeng and Lin, 2007). However, direct maximization of our likelihood with respect to the jump sizes is difficult because no closed form solution is available. We will show below that the frailty formulation in conjunction with an EM-based algorithm provide a closed form maximum likelihood solution to estimating Λ₀(t).

In a shared frailty model, the latent frailty is typically viewed as missing data. A standard approach for estimating parameters in the presence of missingness is to apply an EM algorithm (Dempster et al., 1977). We will use a variation of this approach, the expectation-conditional-maximization (ECM) algorithm (Meng and Rubin, 1993). The ECM algorithm differs from the conventional EM algorithm in the maximization step, where the estimates for multiple parameters are updated sequentially in an ECM algorithm rather than simultaneously as in the single M-step of an EM algorithm. It is particularly helpful when simultaneous maximization with respect to all parameters is difficult.

To carry out an ECM algorithm, we first construct the complete likelihood assuming the frailty were known. Under the assumptions given in Section 2.1, the complete likelihood can be decomposed into the product of several terms with only two of them, L_i1 and L_i2, involving the parameters of interest {β, Λ₀(t), θ}. The first term L_i1 is the likelihood for failure outcomes of the relatives conditional on the frailty and their covariates. It is the product of contributions from each relative because of the conditional independence of the relatives given the frailty and the covariates, and can be written as

$$L_{i1}=\prod _{j=1}^{n_i}{\{{\omega }_i{\lambda }_0(t)exp({\beta }^{\prime }{Z}_{ij})\}}^{{\delta }_{ij}}exp\{-{\omega }_i{\mathrm{\Lambda }}_0(X_{ij})exp({\beta }^{\prime }{Z}_{ij})\}.$$

The second term L_i2 is the likelihood of the frailty conditional on proband data. Since the gamma distribution is a conjugate prior, the posterior distribution of ω_i has a convenient mathematical form

$$L_{i2}=\frac{{\omega }_i^{1/\theta +{\delta }_{i0}-1}exp\{-{\omega }_i/\theta -{\omega }_i{\mathrm{\Lambda }}_0(X_{i0})exp({\beta }^{\prime }{Z}_{i0})\}}{\mathrm{\Gamma }(1/\theta +{\delta }_{i0}){\{1/\theta +{\mathrm{\Lambda }}_0(X_{i0})exp({\beta }^{\prime }{Z}_{i0})\}}^{-1/\theta -{\delta }_{i0}}}.$$

Interestingly the likelihood function does not involve λ₀(t) even when the proband is diseased, but the involvement of Λ₀(t) suggests that Λ̂₀(t) may take jumps at the probands’ failure times. We performed a simulation study assuming ω known for all probands who all had the same age at onset t₀ (for controls it would be age at last examination). We found that Λ₀(t₀) and β were estimable from the data and the estimates appeared to be unbiased. In other words, if Λ₀(t₀−) were known or estimated from the relatives’ data, the jump size at the proband’s failure times t₀ would be estimable despite L_i2 does not involve λ₀(t). However, the two estimates, Λ̂₀(t) and β, were highly collinear and the correlation coefficient was about −0.90. For practical purposes it seems reasonable to allow Λ₀(t) taking jumps only at the relatives’ failure times but not the probands’, as there will be little information from the probands for Λ₀(t) taking jumps at probands’ failure times.

When the probands come from a two-phase sampling study design where each proband is associated with a sampling fraction from the original stratum, the contribution of each family needs to be adjusted accordingly. Research in the analysis of studies with two-phase sampling design is very active, in the last two decades, see, e.g., Breslow and Chatterjee (1999). We adopt perhaps the simplest approach to handle the two-phase sampling, the weighted likelihood approach (Flanders and Greenland, 1991), where each selected proband is weighed by the inverse selection probability 1/π_i. Therefore the parameters are estimated by maximizing

$$L_w=\prod _{i=1}^n{(L_{i1}\times L_{i2})}^{\frac{1}{{\pi }_i}}.$$

In the E-step, we estimate the expectation of the weighted log complete likelihood at current parameter estimates. From the expression of log(L_w), it is easy to see that we would only need to calculate the posterior expectations of ω_i and ln ω_i, $\{1+\theta {\mathrm{\Delta }}_i\}/\{1+\theta {\sum }_{j=0}^{n_i}{\mathrm{\Lambda }}_0(X_{ij})exp({\beta }^{\prime }{Z}_{ij})\}$ and $\phi (1/\theta +{\mathrm{\Delta }}_i)-ln\{1/\theta +{\sum }_{j=0}^{n_i}{\mathrm{\Lambda }}_0(X_{ij})exp({\beta }^{\prime }{Z}_{ij})\}$, respectively, where φ(·) is the digamma function (Hougaard 2000, pp501).

In the CM-step, we update the parameter estimates sequentially between the finite dimensional vector of (β, θ) and infinite dimensional vector of Λ₀(t) by maximizing the expected weighted log completed likelihood. Given the current estimate Λ̂₀(t), β̂ and θ̂ can be updated by maximizing the likelihood function via solving the score equations from taking the partial derivative of the log-likelihood function with respect to (β, θ) using, for example, the Newton-Raphson algorithm.

The maximization of the expected weighted log complete likelihood over Λ₀(t) is more complex. Fixing β̂ and θ̂ at their current values, we obtain a closed-form expression for

$${\widehat{\mathrm{\Lambda }}}_0(t)=\sum _{i=1}^n\sum _{j=1}^{n_i}{\int }_0^t\frac{1}{S(u;\widehat{\beta })}{dN}_{ij}(u),$$ (2)

where S(u; β̂) is given by ${\sum }_{i=1}^n{\sum }_{j=1}^{n_i}\frac{1}{{\pi }_i}{\overline{\omega }}_i{Y}_{ij}(u)exp({\widehat{\beta }}^{\prime }{Z}_{ij})+{\sum }_{i=1}^n\frac{1}{{\pi }_i}\{{\overline{\omega }}_i-\overline{{\omega }_{i0}}\}{Y}_{i0}(u)exp({\widehat{\beta }}^{\prime }{Z}_{i0})$. Here $\overline{{\omega }_{i0}}$ equals (1 + θδ_i0)/{1 + θΛ₀(X_i0) exp(β′Z_i0)}, which is actually the expectation of the frailty conditional on proband data. It is worth noting that the second term in S(u, β̂) has an expectation of zero, suggesting that an alternative estimator for Λ₀(t) could have the same form as (2) with $S(u,\widehat{\beta })={\sum }_{i=1}^n{\sum }_{j=1}^{n_i}\frac{1}{{\pi }_i}{\overline{\omega }}_i{Y}_{ij}(u)exp({\widehat{\beta }}^{\prime }{Z}_{ij})$. This is the estimator that we proposed in earlier work (Hsu et al., 2004) using a heuristic argument that is based on the likelihood for the relatives’ data only, i.e., ${\prod }_{i=1}^n{L}_{i1}$, with the addition of the weights. See Web Appendix A for the derivation of (2).

The ECM algorithm described above is built upon formulation and maximization of a weighted log likelihood. Since the baseline hazard function is estimated at the observed failure times of the relatives, the dimension of the parameters involved in Λ̂₀(t) increases with sample size. The standard maximum likelihood theory for finite or fixed dimensional parameters does not apply here. One may consider using the nonparametric information matrix weighted by the inverse of sampling fractions to estimate the variance of the estimators following the idea of Andersen et al. (1997) for cohort data under frailty models. The procedure involves calculating and inverting a high dimensional matrix, which is quite complicated in both analytical derivation and numerical implementation. As an alternative, we use bootstrap to obtain the variance estimators. We would generate a fixed number, say, 100, of bootstrap data sets, each consisted of the same number of case and control families resampled with replacement from the original data set. Parameter estimates can then be obtained for each bootstrap data set and the variance of the estimates over these bootstrap samples would be the variance estimator of the parameter estimates from the original data set.

2.3 Extension for Relatives with Missing Genotypes

The ECM approach introduced in the previous section allows for handling the missing frailty and now the missing genotypes in a unified fashion. Let g and Z be the genotype and the observed covariates, and β and γ be the corresponding log-hazard ratios. Gene g is genotyped for the probands, but not for the relatives and Z is assumed observed for each individual including the probands and the relatives. For simplicity we assume g is a binary variable, which is 1 if an individual carries one or two high risk alleles and 0 otherwise. This is the dominant transmission model. The method can be easily generalized to other genetic transmission models such as additive, recessive or an unrestricted general model which allows for a separate hazard ratio for each different genotype.

Missing data now include the shared frailty and the genotypes for the relatives. To carry out the ECM algorithm, we need to calculate the joint distribution for the shared frailty and the genotypes for the relatives conditional on the observed data. This requires obtaining the joint distribution of the genotypes for multiple relatives conditional on the genotype of the proband. Unfortunately such joint distribution for genotype becomes very complex when there are more than 2 relatives, as it depends on the joint familial relationship among all family members. Incorporating such a complex joint distribution of the genotypes into the joint distribution of frailty and missing genotypes given other observed data would make the algebra rather complicated. Therefore, we consider a composite likelihood approach. That is, instead of treating a family with multiple relatives and one proband as a unit, we consider relative-proband pairs, viewing each relative-proband pair as a unit, and construct a composite-likelihood by taking the product of the likelihoods from these units as if they were independent even though there are multiple relative-proband pairs from the same family. In this approach, the joint distribution of more than two family members does not need to be explicitly worked out. This technique was first introduced in the longitudinal data setting by Liang and Zeger (1986) under the name of generalized estimating equations and has also been applied to family studies by, e.g., Chatterjee et al. (2006).

For the E-step of this ECM algorithm, we evaluate the expectation of ω_i, ln ω_i, g_ij, and ω_i exp(βg_ij) conditional on the observed data from the proband-relative pairs. The calculation of these expectations is in principle straightforward although it involves some algebra. We denote those expectations by Ẽ(ω_i), Ẽ(ln ω_i), Ẽ(g_ij), and Ẽ{ω_i exp(βg_ij)}. See Web Appendix B for their detailed experssions.

For CM-step, we sequentially maximize the expression below with respect to β and γ:

$$\sum _{i=1}^n\frac{1}{{\pi }_i}\sum _{j=1}^{n_i}[{\delta }_{ij}\{{\gamma }^{\prime }{Z}_{ij}+\beta \stackrel{\sim }{E}(g_{ij})\}-{\mathrm{\Lambda }}_0(X_{i0})\stackrel{\sim }{E}({\omega }_i)exp({\gamma }^{\prime }{Z}_{i0}+{\beta g}_{i0})-{\mathrm{\Lambda }}_0(X_{ij})exp({\gamma }^{\prime }{Z}_{ij})\stackrel{\sim }{E}\{{\omega }_{i}exp(\beta g_{ij})\}+(\frac{1}{\theta }+{\delta }_{i0})ln\{\frac{1}{\theta }+{\mathrm{\Lambda }}_0(X_{i0})exp({\gamma }^{\prime }{Z}_{i0}+{\beta }_{gi0})\}],$$

and maximize the following expression with respect to θ:

$$\sum _{i=1}^n\frac{1}{{\pi }_i}\sum _{j=1}^{n_i}\left[(\frac{1}{\theta }+{\delta }_{i0})ln\{\frac{1}{\theta }+{\mathrm{\Lambda }}_0(X_{i0})exp({\gamma }^{\prime }{Z}_{i0}+\beta g_{i0})\}+\frac{1}{\theta }\{\stackrel{\sim }{E}(ln{\omega }_i)-\stackrel{\sim }{E}({\omega }_i)\}-ln\{\mathrm{\Gamma }(\frac{1}{\theta }+{\delta }_{i0})\}\right].$$

Note that the break down of a family with n_i relatives into relative-proband pair can be treated as if there were n_i families and the proband is replicated n_i times.

For updating Λ̂₀(t), we obtain the following estimating equation:

$${\widehat{\mathrm{\Lambda }}}_0(t)=\sum _{i=1}^n\frac{1}{{\pi }_i}\left\{\sum _{j=1}^{n_i}{\int }_0^t\frac{1}{S_0(u;\widehat{\beta },\widehat{\gamma })}{dN}_{ij}(u)\right\},$$ (3)

where $S_0(u;\widehat{\beta },\widehat{\gamma })={\sum }_{i=1}^n\frac{1}{{\pi }_i}{\sum }_{j=1}^{n_i}[Y_{ij}(u)exp({\widehat{\gamma }}^{\prime }{Z}_{ij})\stackrel{\sim }{E}\{{\omega }_{i}exp(\widehat{\beta }{g}_{ij})\}+Y_{i0}(u)exp({\widehat{\gamma }}^{\prime }{Z}_{i0}+\widehat{\beta }{g}_{i0})\{\stackrel{\sim }{E}({\omega }_i)-\overline{{\omega }_{i0}}\}]$.

From the expressions it is clear that the ECM algorithm for data with missing genotypes in the relatives follows closely to the ECM algorithm in Section 2.2, except that a generalized likelihood rather than the full likelihood was used as the basis for the estimation. This simplification was mainly to make computation manageable though at the price of a potential efficiency loss and an invalid likelihood-based variance estimator. For the latter problem, we again use re-sampling techniques as described in Section 2.2 to obtain variance estimators for the parameter estimates.

2.4 Inclusion of Proband Data

The likelihood function of the proband data P(Z_i0, g_i0|X_i0, δ_i0) is a retrospective likelihood for the usual case-control data. If cases and controls are matched on age, the likelihood function can be replaced by the conditional likelihood for estimation under the frailty model by adding an offset term, which is a posterior expectation of ω_i conditional on {X_i0, δ_i0 = 1, Z_i0, g_i0} (Hsu et al., 2004). This approach, however, does not work for the two-phase design, where cases and controls are each sampled based on their own stratum and the matched case-control mechanism is not preserved. An alternative approach is to rewrite

$$P(Z_{i0},g_{i0}\mid X_{i0},{\delta }_{i0})=\frac{P(X_{i0},{\delta }_{i0}\mid Z_{i0},g_{i0})f(Z_{i0})f(g_{i0})}{{\int }_{z^{\ast }}{\sum }_{g^{\ast }}P(X_{i0},{\delta }_{i0}\mid z^{\ast },g^{\ast })f(g^{\ast })f(z^{\ast }){dz}^{\ast }},$$

assuming the independence of Z and g in the population. The distribution f(Z) and the allele frequency in f(g) need to be estimated from the data. The proband data alone do not allow us to uniquely identify these population parameters and {β, γ, Λ₀(t)}. However, in conjunction with the relatives’ data, they are identifiable, as {β, γ, Λ₀(t)} can be estimated from the relatives. Since f(Z) is not of main interest, we propose to take an additional condition given Z_i0, that is, P(g_i0|X_i0, δ_i0, Z_i0). Under the two-phase sampling design, the inverse-weighted log-likelihood function of the probands is given by

$$log(L)=\sum _{i=1}^n\frac{1}{{\pi }_i}log\frac{{\{1+\theta {\mathrm{\Lambda }}_0(X_{i0})exp({\beta }^{\prime }{g}_{i0}+{\gamma }^{\prime }{Z}_{i0})\}}^{-1/\theta -{\delta }_{i0}}exp({\beta }^{\prime }{g}_{i0}{\delta }_{i0})f(g_{i0})}{{\sum }_{g^{\ast }}{\{1+\theta {\mathrm{\Lambda }}_0(X_{i0})exp({\beta }^{\prime }{g}^{\ast }+{\gamma }^{\prime }{Z}_{i0})\}}^{-1/\theta -{\delta }_{i0}}exp({\beta }^{\prime }{g}^{\ast }{\delta }_{i0})f(g^{\ast })},$$

Although the expression involves both θ and Λ₀(t), intuitively the proband likelihood cannot provide direct information on them. We therefore use the proband likelihood log(L) only for estimating β along with the relatives’ data and q. Note that in the absence of covariates Z_i0 and under the conventional case-control design, this likelihood function was proposed by Chatterjee et al. (2006) for estimating allele frequency q, an important quantity that investigators often are interested in estimating.

3. Simulation Studies

We performed a simulation study to evaluate the performance of the proposed method under several realistic settings. In these simulations, we generated a candidate gene g and a continuous risk factor Z for each individual, where g followed Mendelian transmission with an autosomal dominant model and Z followed a N(0, 1). We also generated a family-specific frailty following gamma ( $\frac{1}{\theta }$, θ). The conditional baseline hazard function λ₀(t) was a Weibull distribution with p = 4.6 and λ = 0.01. The failure time for each individual was then generated according to the frailty model (1). Each family consisted of the proband, the mother, and a sister. The censoring distribution was N(60, 15), yielding censoring percentage about 80% – 85%.

We considered four different sampling scenarios for the simulation: a) no stratified sampling of cases and controls; b) stratified sampling of cases only but not controls; c) stratified sampling of controls only but not cases; and d) stratified sampling of both cases and controls. For each of these scenarios, we started by randomly selecting a pool of 1800 case families and a pool of 1800 control families from the population as the first phase sampling. The second phase sampling of the families varies for the four scenarios. For scenario a, we randomly chose 200 case families and 200 control families from each pool. For scenario b, we randomly selected 200 control families and performed a stratified sampling of 200 case families based on the number of diseased relatives that they had. If both the mother and the sister were diseased, the proband was always selected for genotyping. Among the families with only one diseased relative, we randomly selected 100 probands for genotyping. We then randomly sampled from the remaining families in which neither of the relatives was diseased to reach a total number of 200 case families. We employed a similar sampling strategy for selecting control probands in scenario c and selecting case and control probands separately in scenario d. The selection fraction is one, about one in three, and one in six to seven for families with two, one, and no diseased relatives, respectively.

We considered two situations for the candidate gene effect: 1) rare allele (q = 0.05) but relatively high penetrance (β = log(5) = 1.609) (Table 1), and 2) common allele (q = 0.2) with relatively low hazard ratio (β = log(2) = 0.693) (Table 2). There were about 46–60 carriers among 200 cases and 15–19 carriers in 200 controls under the rare allele with high penetrance model for scenarios a–d, and about 96–100 carriers in cases and 67–72 carriers in controls under the common allele with moderate penetrance model. The number of carriers generated under the rare allele high penetrance model is comparable to that in the BRCA1 data set that we will analyze in Section 4. In addition, θ takes two values: 0.5 for moderate dependence and 1.5 for strong dependence. For each parameter setting, a total of 500 data sets were simulated to assess the performance of the proposed method.

Table 1.

Biases and empirical standard errors (SEs) for the ECM approach for regression coefficients, dependence parameter and cumulative baseline hazard function at selected ages for a rare (q = 0.05) but high risk (β = log 5) gene. The data consist of (stratified) cases and controls and their mothers and sisters. The genotypes are missing by design in the relatives.

Moderate dependence				Strong dependence
Par.	True	Bias	SE	Par.	True	Bias	SE
a) no stratified sampling
θ	0.5	0.004	0.247	θ	1.5	0.006	0.479
β	1.609	−0.021	0.234	β	1.609	−0.026	0.284
γ	0.405	−0.002	0.099	γ	0.405	0.005	0.100
q	0.05	0.0011	0.0074	q	0.05	0.0011	0.0079
Λ(40)	0.0148	0.0002	0.0038	Λ(40)	0.0148	0.0000	0.0036
Λ(50)	0.0412	0.0003	0.0076	Λ(50)	0.0412	0.0006	0.0081
Λ(60)	0.0954	0.0007	0.0146	Λ(60)	0.0954	0.0011	0.0163
Λ(70)	0.1938	0.0009	0.0265	Λ(70)	0.1938	0.0031	0.0309
b) stratified sampling cases
θ	0.5	0.030	0.284	θ	1.5	0.049	0.536
β	1.609	−0.007	0.275	β	1.609	0.007	0.320
γ	0.405	0.006	0.099	γ	0.405	0.005	0.110
q	0.05	0.0009	0.0101	q	0.05	0.0011	0.0112
Λ(40)	0.0148	0.0000	0.0033	Λ(40)	0.0148	0.0002	0.0036
Λ(50)	0.0412	0.0003	0.0080	Λ(50)	0.0412	0.0004	0.0083
Λ(60)	0.0954	0.0008	0.0154	Λ(60)	0.0954	0.0010	0.0160
Λ(70)	0.1938	0.0012	0.0294	Λ(70)	0.1938	0.0023	0.0315
c) stratified sampling controls
θ	0.5	−0.041	0.259	θ	1.5	−0.093	0.427
β	1.609	0.004	0.268	β	1.609	0.002	0.360
γ	0.405	0.000	0.096	γ	0.405	0.002	0.098
q	0.05	−0.0002	0.0083	q	0.05	0.0001	0.0103
Λ(40)	0.0148	0.0001	0.0031	Λ(40)	0.0148	0.0002	0.0030
Λ(50)	0.0412	0.0002	0.0064	Λ(50)	0.0412	0.0007	0.0067
Λ(60)	0.0954	0.0009	0.0119	Λ(60)	0.0954	0.0008	0.0130
Λ(70)	0.1938	0.0012	0.0223	Λ(70)	0.1938	−0.0001	0.0248
d) stratified sampling both case & control
θ	0.5	−0.027	0.214	θ	1.5	0.024	0.374
β	1.609	−0.012	0.248	β	1.609	−0.014	0.291
γ	0.405	0.006	0.089	γ	0.405	0.010	0.097
q	0.05	0.0005	0.0089	q	0.05	0.0010	0.0101
Λ(40)	0.0148	0.0002	0.0029	Λ(40)	0.0148	0.0000	0.0028
Λ(50)	0.0412	0.0005	0.0060	Λ(50)	0.0412	0.0002	0.0063
Λ(60)	0.0954	0.0012	0.0114	Λ(60)	0.0954	0.0002	0.0121
Λ(70)	0.1938	0.0023	0.0228	Λ(70)	0.1938	0.0002	0.0224

Table 2.

Biases and empirical standard errors (SEs) for the ECM approach for regression coefficients, dependence parameter and cumulative baseline hazard function at selected ages for a common (q = 0.2) and low risk (β = log 2) gene. The data consist of (stratified) cases and controls and their mothers and sisters. The genotypes are missing by design in the relatives.

Moderate dependence				Strong dependence
Par	True	Bias	SE	Par.	True	Bias	SE
a) no stratified sampling
θ	0.5	−0.003	0.257	θ	1.5	0.030	0.454
β	0.693	−0.008	0.194	β	0.693	−0.012	0.216
γ	0.405	−0.001	0.100	γ	0.405	−0.004	0.096
q	0.20	0.0017	0.0167	q	0.20	0.0020	0.0175
Λ(40)	0.0148	0.0001	0.0039	Λ(40)	0.0148	0.0002	0.0041
Λ(50)	0.0412	0.0007	0.0081	Λ(50)	0.0412	0.0004	0.0091
Λ(60)	0.0954	0.0013	0.0161	Λ(60)	0.0954	0.0015	0.0187
Λ(70)	0.1938	0.0009	0.0314	Λ(70)	0.1938	0.0027	0.0363
b) stratified sampling cases
θ	0.5	0.006	0.251	θ	1.5	0.028	0.496
β	0.693	0.017	0.195	β	0.693	−0.015	0.216
γ	0.405	0.008	0.099	γ	0.405	0.013	0.105
q	0.20	0.0001	0.0229	q	0.20	0.0019	0.0240
Λ(40)	0.0148	0.0000	0.0038	Λ(40)	0.0148	0.0001	0.0035
Λ(50)	0.0412	−0.0003	0.0080	Λ(50)	0.0412	0.0003	0.0083
Λ(60)	0.0954	0.0000	0.0159	Λ(60)	0.0954	0.0015	0.0177
Λ(70)	0.1938	0.0019	0.0298	Λ(70)	0.1938	0.0043	0.0355
c) stratified sampling controls
θ	0.5	−0.051	0.244	θ	1.5	−0.095	0.397
β	0.693	−0.002	0.217	β	0.693	−0.022	0.265
γ	0.405	0.000	0.095	γ	0.405	−0.004	0.093
q	0.20	0.0001	0.0202	q	0.20	0.0007	0.0222
Λ(40)	0.0148	0.0002	0.0032	Λ(40)	0.0148	0.0003	0.0035
Λ(50)	0.0412	0.0005	0.0072	Λ(50)	0.0412	0.0012	0.0080
Λ(60)	0.0954	0.0007	0.0144	Λ(60)	0.0954	0.0026	0.0165
Λ(70)	0.1938	0.0014	0.0274	Λ(70)	0.1938	0.0049	0.0331
d) stratified sampling both case & control
θ	0.5	−0.005	0.211	θ	1.5	0.022	0.359
β	0.693	−0.002	0.212	β	0.693	0.004	0.255
γ	0.405	0.006	0.083	γ	0.405	0.000	0.095
q	0.20	0.0005	0.0210	q	0.20	−0.0001	0.0228
Λ(40)	0.0148	0.0001	0.0029	Λ(40)	0.0148	0.0001	0.0033
Λ(50)	0.0412	0.0005	0.0067	Λ(50)	0.0412	0.0003	0.0075
Λ(60)	0.0954	0.0006	0.0132	Λ(60)	0.0954	0.0009	0.0159
Λ(70)	0.1938	0.0028	0.0254	Λ(70)	0.1938	0.0031	0.0312

The biases appear to be small in all simulation situations. There is about 15–30% reduction in the standard errors (SEs) of θ̂ and Λ̂₀(t) when both cases and controls are sampled by stratification compared to the other three sampling schemes. However, there is no efficiency gain in β̂ for the genetic effect. If only cases or controls but not both are sampled based on stratification, the SEs of β̂ tend to be inflated compared to the sampling scheme without any stratification. This loss of efficiency may be due to an additional variation induced by the weights and/or the fact that a positive family history in our simulation setup is caused not only by or the risk factors under study but also by the shared frailty. It seems that stratifying based on positive family history with an intention of increasing the frequency of high risk allele carriers may not necessarily translate into an efficiency gain for β̂ for candidate genes. In these simulation settings, the inclusion of proband likelihood in the estimation of β greatly increases the efficiency compared with the estimator without including proband likelihood for β, the extent of which depends on the specific sampling schemes used in the simulations (results not shown). This implies that the proband likelihood can provide substantial information on β once other parameters are identified. Such an efficiency gain was also observed in simulation studies in Chatterjee et al. (2006).

We used the bootstrap method to estimate the se of the ECM estimators. Here we present the bootstrap results under scenario d, as it is the most likely approach for a real study and it is also the most comprehensive design. Table 3 shows empirical SE, bootstrap-based SE, and bootstrap-based coverage probabilities (CP)of 95% confidence intervals for 200 simulated data sets with 100 bootstrap samples from each simulated data set. Compared with the SDs of parameter estimates over simulated data sets, the bootstrap-based SEs appear working well for {β̂, γ̂} and q̂, but may over-estimate the SEs of θ̂ and Λ̂₀(t). The CPs appear to maintain 95% nominal level for β and q, but tend to be higher than nominal level for θ and the later time points of Λ₀(t) and lower for γ. The under- or over- estimation of 95% coverage probabilities are, to some extent, due to the small sample size. For θ, the overestimation could also be due to the fact that our bootstrap sampling unit is by family. The dependence parameter estimates are influenced by the number of families with more than one affected relatives and the number for such families is usually small. So when one or a few such families are in (or out of) the bootstrap samples, the estimates may be affected greatly.

Table 3.

Summary statistics of empirical standard errors (SE), bootstrap SEs (Bt SE), coverage probabilities of 95% bootstrap confidence intervals (95% CP) when both cases and controls are sampled according to the family history.

Moderate dependence					Strong dependence
Par.	True	SE	Bt SE	95% CP	Par	True	SE	Bt SE	95% CP
allele frequency=0.05
θ	0.500	0.210	0.252	0.96	θ	1.500	0.361	0.457	0.98
β	1.609	0.245	0.252	0.95	β	1.609	0.305	0.319	0.94
γ	0.405	0.095	0.089	0.91	γ	0.405	0.098	0.099	0.91
q	0.0500	0.009	0.009	0.92	q	0.050	0.010	0.010	0.92
Λ(40)	0.015	0.003	0.003	0.94	Λ(40)	0.015	0.003	0.003	0.94
Λ(50)	0.041	0.006	0.006	0.96	Λ(50)	0.041	0.006	0.007	0.96
Λ(60)	0.095	0.010	0.013	0.98	Λ(60)	0.095	0.011	0.014	1.00
Λ(70)	0.194	0.021	0.025	0.96	Λ(70)	0.194	0.024	0.030	0.955
allele frequency=0.2
θ	0.500	0.185	0.244	0.99	θ	1.500	0.362	0.452	0.97
β	0.693	0.206	0.213	0.94	β	0.693	0.232	0.246	0.94
γ	0.405	0.085	0.087	0.93	γ	0.405	0.101	0.095	0.90
q	0.200	0.019	0.021	0.94	q	0.200	0.022	0.023	0.95
Λ(40)	0.015	0.003	0.003	0.95	Λ(40)	0.015	0.003	0.003	0.93
Λ(50)	0.041	0.006	0.007	0.97	Λ(50)	0.041	0.007	0.008	0.97
Λ(60)	0.095	0.012	0.015	0.97	Λ(60)	0.095	0.014	0.017	0.98
Λ(70)	0.194	0.026	0.029	0.97	Λ(70)	0.194	0.029	0.034	0.97

4. Application to a Breast Cancer Dataset

As an illustration, we applied the proposed methods to a breast cancer dataset to estimate the hazard functions of developing breast cancer (BC) for BRCA genetic mutations. This population-based case-control study was conducted within the National Institute of Child Health and Human Development’s Women’s Contraceptive and Reproductive Experiences (CARE) study (Marchbanks et al., 2002). Due to funding constraints, the study could only collect blood from approximately 33% of the interviewed women and thus a second phase sampling design was developed, where women were stratified sampled according to their case-control status, study site, race, family history, and age. Within each stratum, a sampling fraction ranging from 0.05 to 1 was assigned so as to achieve a uniform distribution across strata. A study of the BRCA1/2 genes was conducted to evaluate their contribution to breast cancer risk (Malone et al., 2006).

For illustrative purpose, in this analysis we focus on only the BRCA1 gene and included only White probands and their blood-related first degree relatives. BRCA2 carriers were considered as non-carriers of BRCA1 mutations. The first degree relatives were restricted to those age 18 or older and had known breast cancer status at the time of study. Among the 1603 White probands with known BRCA1 mutation status, 1144 (71%) are cases. There are 42 (3.8%) BRCA1 mutation carriers in cases and only 1 in controls. A total of 4568 first degree relatives were included, and among them 634 (13.9%) had developed breast cancer. We applied the method proposed in section 2.3 and included the likelihood of the proband data in Section 2.4 for estimating the allele frequency. In this analysis we did not include the proband likelihood in estimating relative risk because we observed a collinearity problem between estimating β and allele frequency in the proband data. The collinearity is likely due to the rare mutational frequency of the BRCA1 gene. Such rare allele frequency, on the other hand, makes the estimation of β using relatives quite insensitive to changes in q̂, as the carrier probability of a relative given the proband’s carrier status changes little with the allele frequency when the frequency is low.

Figure 1 shows the estimated population cumulative cancer-free probabilities for BRCA1 mutation carriers and noncarriers and point-wise bootstrap confidence intervals. We present here the marginalized probabilities integrating over the frailty distribution mainly for the comparison of our result with those that have been published on other datasets (Table 4). Bootstrap confidence intervals were obtained based on 200 bootstrap samples. Mutation carriers had an estimated 45.4% or 54.9% chance of developing BC by age 70 or 80, in comparison to noncarriers who had an estimated chance of 7.2% or 10.2% at the same age. The BC risk estimates fall within the range, though at the lower end of current estimates reviewed by e.g. Chen and Parmigiani (2007). The estimates (β̂, $\widehat{\mathit{theta}}$, q̂) are 2.354 (95%CI: 1.374–3.534), 0.948 (95%CI: 0.505–1.444), and 0.181% (95%CI: 0.080%–0.401%), respectively. The high value $\widehat{\mathit{theta}}$ suggests there remains substantial residual dependency of ages at onset among family members, which Begg et al. (2008) observed as well. It is also worth noting that even though we assume β constant in the frailty model (1), the resulting population averaged hazard ratio decreases over the age and they are 8.822, 7.677, 6.371, and 5.479 at age 50, 60, 70, and 80 years old, respectively.

Figure 1.

Estimated population averaged probabilities of developing breast cancer for BRCA1 mutation carriers and noncarriers with 95% bootstrap confidence interval.

Table 4.

Cumulative probabilities of developing breast cancer by age (bootstrap 95% CI) for carriers and noncarriers of BRCA1 mutations.

	Probability of Developing Breast Cancer
	Noncarrier	Carrier
Age 50	0.021(0.017, 0.027)	0.188(0.082, 0.423)
Age 60	0.041(0.034, 0.050)	0.313(0.143, 0.612)
Age 70	0.072(0.060, 0.090)	0.454(0.227, 0.743)
Age 80	0.102(0.084, 0.125)	0.549(0.304, 0.814)

An advantage of the frailty model approach is to provide a woman with an individualized risk estimate. For example, a 40 year old woman who had a mother with BC diagnosed at age 38 and carried a BRCA1 mutation would have a 86.4% cumulative probability of developing BC by age 80. The estimated frailty value in this case is 1.77. If she were not a mutation carrier, with all other information the same her probability would be 18.7% with ω̂ = 1.93. Note that ω̂ is slighted lower if the woman is a carrier than if she is a non-carrier, because the BRCA1 mutation partly explains the aggregation of BC in a family. On the other hand, if the same age woman did not carry a BRCA1 mutation and her mother were still disease free at age 65, she would have only 9.6% chance of developing BC before age 80 (ω̂ = 0.94). As a bench mark, the average risk for a carrier and a non-carrier woman to develop breast cancer regardless of family history is 54.9% and 10.2%, respectively.

We also estimated the penetrance function assuming that the breast cancer risk for family members is independent given BRCA1 genotype, in other words, BRCA1 genotypes among family members explain completely the dependence of ages at onset among family members and the failure time follows a marginal proportional hazards model. As we can see, ignoring the residual dependence among family members over-estimated the population breast cancer risk for both carriers and noncarriers considerably (Figure 2).

Figure 2.

Estimated population averaged probabilities of developing breast cancer by age for BRCA1 mutation carriers and noncarriers, allowing for frailty or residual dependence and assuming independence of ages at onset among family members.

5. Final Remarks

In this paper we developed a weighted likelihood approach to estimating regression coefficients, nonparametric baseline hazard function, and dependence parameter under a shared frailty model from two-phase case-control data with family history information. While the work is focused on the estimation of penetrance function for the BRCA1 gene, the method is generalizable to other candidate genes. The family history information, which is typically collected in epidemiologic studies, if of sufficiently high quality, can provide population estimates that are not available from using case-control data alone. Moreover, the residual dependency estimates can shed light on whether one or more candidate genes or other shared environmental risk factors may contribute to diseases. Finally, the proposed likelihood, which conditions on the probands’ survival time and allows for residual dependency via a frailty, is robust against ascertainment biases that are often major issues in studies of gene penetrance.

Frailty models are useful especially when the goal is to make inference about individual families, e.g., in the situation of genetic counseling. As genetic data are increasingly available, public interests in genetic counseling are more intense than ever. It is thus critical to provide an accurate estimate of carrier probability and individualized disease probability for a counselee. Frailty model-based approaches that incorporate both the measured risk factors and those that are unknown or unmeasured by a shared frailty are appealing in this situation. One caveat, however, is that the estimate of individual risk depends on the frailty distribution and a misspecification could bias the estimate even though the estimate for the population-averaged hazard function is fairly robust (Hsu et al., 2007).

While nonparametric modeling puts no constraint in baseline hazard function, fitting may become unstable especially when the gene is rare and disease incidence low. Weakly parametric modeling, e.g., a three-parameter Weibull model proposed by Gail et al. (1999) may be useful in regularizing the potentially high-dimensional baseline hazard function. In principle the proposed approach should apply with much reduced computation. It would be useful to provide this as an alternative to the nonparametric modeling.