Let T₁ and T₃ be the times from the healthy state to events 1 and 3, respectively and Y₁ = min{T₁, T₃}. Define ε₁ by ε₁ = 1 if T₁ < T₃ and ε₁ = 3, otherwise. Note that events 1 and 3 are competing risks so it is of interest to define the following cause-specific hazard functions

where G is the individual genotype information corresponding to the mutation carrier status (carrier=1, non-carrier=0) and X a set of measured covariates. By standard theory of competing risks,

are the hazard and survival functions associated with Y₁, respectively and

is the cause-specific cumulative incidence function of event 1.

Individuals satisfying ε₁ = 1 are afterwards at risk of observing either event 2 or event 4. Let T₂ and T₄ be the times from event 1 to events 2 and 4, respectively and Y₂ = min(T₂, T₄). Define ε₂ by ε₂ = 2 if T₂ < T₄ and ε₂ = 4, otherwise. Similarly, define the conditional cause-specific hazard functions given ε₁ = 1 by

The conditional hazard and survival functions associated with Y₂ given ε₁ = 1 are then, respectively,

We assume that the cause-specific hazard for event k, k = 1, 2, 3, 4, follows a proportional hazards regression model

where λ_k0 is the baseline hazard function and β_k and β_gk are the regression coefficients related to event k. Two approaches are considered in this paper: (i) a parametric approach where a parametric distribution is specified for each λ_k0, and (ii) a piecewise constant hazard approach where λ_k0 is assumed to be constant within each interval of a partition of [0, ∞). In both cases, we denote by θ_k the set of baseline distribution parameters and regression coefficients related to event k.

For individuals satisfying ε₁ = 1, we model the dependence in the pair (Y₁, Y₂) through a semi-survival copula, 𝒞_γ, (Lakhal-Chaieb et al., 2006; Zhao & Zhou, 2010; Ding, 2012) defined as follows:

where the parameter γ measures the conditional dependency in the pair (Y₁, Y₂) given ε₁ = 1 and p(G,X)=P(ε1=1|G,X)=limt→∞F11(t|G,X).

The model is completed by specifying P(ε₂ = 2|G, X, Y₁ = y₁, Y₂ = y₂, ε₁ = 1). This probability has to satisfy

where the expectation is taken with respect to Y₁. A natural and mathematically convenient strategy to ensure that (1) holds is to assume

When this condition is not met, we are in the presence of an additional aspect of the dependency between the successive competing risks. In Web Appendix A, we present a procedure to test equation (2). Applying this test to the LS families cancer data suggests that it is plausible to assume (2) in our case. Therefore, the developments presented throughout the rest of this paper are relying on this assumption.

The penetrance functions are defined as cause-specific cumulative incidence functions. The penetrance for event 1 is 𝒫₁(y₁; G, X) = F₁₁(y₁|G, X), which is the cumulative risk of developing event 1 by age y₁ in the presence of the competing event 3. The penetrance function for event 2 is the cause-specific cumulative incidence function conditional on the age at onset of event 1. When the assumption (2) is satisfied, we show in Web Appendix B that this penetrance function equals

where Cγij(u,υ)=∂i+jCγ(u,υ)/∂iu∂jυ. It is the probability of developing event 2 within y₂ since event 1 which has occurred at y₁. One is often interested in a 5-year or 10-year penetrance for second event.

In this section, we describe the observed data and derive an estimation procedure for the parameters {θ₁, θ₂, θ₃, θ₄, γ}. In the LS families, Y₁ is right-censored by the age of last follow-up a. The observed data related to the events 1 and 3 is then {a, Ỹ₁, ε̃₁}, where Ỹ₁ = min(Y₁, a) and ε̃₁ = ε₁ × I(Y₁ < a) ∈ {0,1, 3}. For those satisfying ε̃₁ = 1, we also observe Ỹ₂ = min(Y₂, a − Y₁) and ε̃₂ = ε₂ × I(Y₂ < a − Y₁) ∈ {0, 2, 4}.

The observations are clustered into I families. The data is then

where n_i is the size of the i^th family.

A family is included into the study if and only if the first examined person or proband has observed either event 1 or event 3 by age a. We assume a unique proband per family, whom we index by the subscript j = 1. Close relatives of this proband for whom some genotype and cancer history information are available from the corresponding family unit. As this data collection protocol induces a selection bias, an ascertainment correction is required. To this end, we employ a conditional likelihood approach where the contribution of each family is corrected for its probability of being ascertained. For parameter estimation, we consider a two-stage estimation procedure. In the first stage, we estimate the parameters related to events 1 and 3 by maximizing the conditional log-likelihood function

where

is the standard contribution of an individual to the log-likelihood function and

is the familial ascertainment correction term. This log-likelihood function is derived under the assumption of conditional independence of ages at onset of cancer of family members given their mutation carrier statuses. This assumption is plausible in our case given the strong association between the genotype and the risk of developing cancer.

At the second stage, we estimate the parameters related to events 2 and 4 as well as the copula parameter γ by maximizing the log-likelihood function

where

and θ̂₁ and θ̂₃ are obtained from the first stage.

In this section, we modify the estimation procedure derived above in order to include the individuals whose genotype information is missing in the analysis. In what follows, we assume that the genotypes are missing at random and that the probands' genotypes are known. Let G̃_ij = G_ij if G_ij is observed and − 1 otherwise.

We consider the following two-stage procedure. At the first stage, we estimate the parameters related to events 1 and 3 via an Expectation-Maximization (EM) algorithm. After m iterations, the E-step and the M-step of this algorithm are

E-step: For i = 1, ⋯, I, j = 1, ⋯, n_i, if G̃_ij = − 1, compute

where p_ij = P(G_ij = 1|G_i1) depends only on the relationship between the individual j and the proband in family i. In this paper, these probabilities are estimated empirically from the subset of data with observed genotypes.

M-step: Compute θ^1(m+1) and θ^3(m+1) by maximizing

We iterate between these steps until convergence to obtain θ̂₁ and θ̂₃.

At the second stage, we estimate θ₂, θ₄ and γ by maximizing the weighted loglikelihood function

where wij(∞) are the conditional probabilities computed at the E-step of the EM-algorithm evaluated at convergence.

The estimation procedure derived in this paper simultaneously involves several inference techniques including a two-stage estimation setting and an EM-algorithm to handle missing genotypes. Therefore, it may not be straightforward to derive explicit formulae for the variances of the obtained estimators. In this work, we propose to estimate the variances using a nonparametric bootstrap procedure. At each bootstrap iteration, we resample I families with replacement from the original data in order to obtain a bootstrapped sample. Afterwards, we apply the iterative estimation procedure described above to each bootstrapped sample. Finally, the variances are computed empirically from B bootstrapped samples. Applying the complete estimation procedure to each bootstrapped sample insures the validity of this variance estimation procedure.

We conducted a simulation study to evaluate the performance of our proposed successive competing risks model by examining the accuracy and precision of the estimates of the model parameters and penetrance functions. We simulated samples of 781 families with family structures and inclusion criteria similar to those of the Lynch Syndrome families from the Colon CFR. For each family member, the times to the first and second events of interest were generated in the presence of competing events based on the proposed model assuming Weibull baseline hazard functions and a Clayton copula, with parameters estimated from the Colon CFR's data in order to mimic realistic disease risks. We considered 0% (no missing), 50% and 80% of missing genotypes among family members of the probands for studying the impact of missing genotypes. For each genotype missing rate, we generated 1000 samples and for each generated sample, we estimated the parameters of the model and deduced plug-in estimators for the penetrance functions for the first and second cancers. We fitted the simulated data assuming various forms for the baseline hazard functions: parametric Weibull, log-logistic, and gamma distributions and piecewise constant hazards, where λ₀₁ and λ₀₃ were assumed to be constant within the intervals (0, 5], (5,10], ⋯, (60, ∞) and λ₀₂ and λ₀₄ within (0, 5], ⋯, (30, ∞).

The EM-algorithm derived in Section 3.2 takes a very long time to converge with the piecewise constant hazards approach. Therefore, we consider this approach only when no genotypes are missing.

Our interest lies on the log cause-specific relative risks for gender and mutation status β_1sex, β_1gene for the first cancer, β_2sex, β_2gene for the second cancer, the copula parameter log(γ), the gender-specific penetrance among mutation carriers by age 70 for the first cancer 𝒫₁(70; G = 1, X) and the 10-year penetrance for the second event given the first cancer occurred at ages 40 and 50, 𝒫₂(10; 40, G = 1, X) and 𝒫₂(10;50, G = 1, X), respectively. For each of these quantities of interest, we computed the average bias, the empirical standard deviation (SE) and the root mean square error (RMSE). The results are summarized in Tables 1 (first cancer) and 2 (second cancer). From Table 1, the bias values for β_1sex estimates are small across the different baseline distributions even when data involves high proportion of missing genotypes. On the other hand, β_1gene estimates are almost unbiased when 0% and 50% of the genotypes are missing; however, they are slightly underestimated when 80% of the genotypes are missing. This Table also suggests that the biases of the penetrance estimates for the first cancer are generally small regardless the proportions of missing genotypes and the choice of the baseline distributions, although penetrance estimates for female carriers are slightly more biased and more variable compared to those for male carriers. For most of the estimates, as we expected, the SEs and RMSEs increase slightly when the proportion of missing genotypes increases.

From Table 2, the biases in β_2sex and β_2gene estimates are small when the true baseline distribution, Weibull, is assumed; however, larger biases are observed when the baseline hazard functions are misspecified. Considering the true values of β_2sex and β_2gene are set close to zero, the model misspecification provides relatively large bias in those estimates. Despite of the biased parameter estimates, the penetrance estimates for the second cancer are generally unbiased, even in the presence of missing genotypes. We found that the misspecification of the baseline distribution can lead to biased penetrance estimates; gamma baseline distribution underestimes the penetrance while the log-logistic baselines provide almost unbiased penetrance estimates. Finally, the piecewise constant hazards provide penetrance estimates as accurate as using the true parametric baseline distribution. However, it only applies to the situation with no missing genotypes.

The Colon CFR is an international consortium regrouping six institutes in North America and Australia and formed as a resource to support studies on the etiology, prevention, and clinical management of CRC. Details of recruitment methods for each centre of the Colon CFR have been published previously (Newcomb et al., 2007) and can be found at http://coloncfr.org/. The Colon CFR includes lifestyle, medical history, and family history data collected from more than 41,000 men and women from 14,500 families with and without CRC. The Colon CFR recruited families between 1997 to 2012 and all participants were followed-up approximately every 5 years to update personal and family histories and expand recruitment if new cases have occurred since baseline. A total of 781 Lynch Syndrome (LS) families, defined as families in which at least one member is affected by CRC and carrying a mutation in one of the following genes: MLH1, MSH2, MSH6, MSP2 and EPCAM, has been identified through the Colon CFR. The risk of developing a first CRC in LS people has been well evaluated (Dowty et al., 2013), however the risk of developing a second CRC following a first CRC is not well known.

In this study, our goal is based on LS families from the Colon CFR, to estimate the cumulative risks (penetrances) of developing a first CRC and developing a second CRC following a first CRC for people who carry germline mutations in the five genes listed above, in males and females separately. Here, competing risks refer to death related to LS cancer and only families whose probands have observed the first CRC cancer are included in the sample.

The number of CRCs and competing events observed by mutation status and gender are given in Web Figure 1 and Web Table 1. For each family, we considered three generations including the probands, their children, spouses, parents, siblings, nephews and nieces. The sample considered consists of 781 LS families including a total of 7703 individuals. We observed 1501 individuals who developed a first primary CRC and 89 who died from other LS related cancers. Among the 1501 individuals who developed a first CRC, 276 developed a second primary CRC following the first one and 163 died from other LS related cancers. Deaths from other LS cancers were considered as competing events for both the first and second CRCs. Unknown mutation status was inferred as outlined in section 3.2.

We analysed the LS families data using the methodology presented in Sections 2 and 3. We considered different survival models for λ_k, k = 1, ⋯,4: parametric Weibull and log-logistic models and piecewise constant baseline hazards model. Proportional hazards are assumed in the Weibull and piecewise constant baseline hazards models, whereas hazards ratios are not constant over time in the log-logistic model, which in fact assumes proportional odds. The same model was used for all four events (i.e. the first and second CRC and the two competing events). We also specified a Clayton copula to model the dependence between the first and second CRC times. We tested the proportional hazards assumption under the Weibull specification using the goodness-of-fit test described in Web Appendix C and obtained p-values equal to 0.24 and 0.59 for events 1 and 3, respectively. Therefore, the proportional hazards assumption seems plausible for our data. Furthermore, we tested the partial independence assumption given by equation (2), as outlined in Web Appendix A, and obtained a p-value equal to 0.92, which leads us to conduct the analysis under this assumption.

In our application, families whose proband was dead before observing the first CRC cancer were not identified by the data collection protocol. Therefore, we replaced the familial ascertainment term given by equation (5) by

in the estimation process.

In addition, we analysed the data using a naive approach that ignores competing risks and treats LS related deaths as right-censored observations. This approach, whose details are given in Web Appendix D, is referred to as “No competing risks model” henceforth.

The log-likelihood for the first step analysis (i.e. parameter estimates related to events 1 and 3) was –7533.21 for the log-logistic model, –7648.97 for the Weibull model and –7758.97 for the piecewise constant hazard model. Table 3 summarizes the estimates of model parameters and penetrance for the first and second CRCs from the three models with and without competing risks taken into account.

Our results showed that mutation carriers of any of the five MMR genes had a very high risk of developing a first CRC with a corresponding log hazard ratio (HR), β_1gene, of 3.22 for the Weibull model and 2.73 for the piecewise constant hazard model. For the log-logistic model, the log HR varied with age, being in males 3.62 at 30 years, 3.53 at 40 years, 3.36 at 50 years and in females 3.63 at 30 years, 3.57 at 40 years and 3.46 at 50 years. The gender effect was highly significant in all the three models with substantial increased risks in males than in females. The cumulative probability of developing a first CRC (i.e. penetrance) by age 70 was among male carriers 55.9% with the Weibull model, 50.2% with the piecewise constant hazard model and 54.6% with the log-logistic model, and among female carriers 42.1%, 39.7% and 43.3%, respectively (see Web Figure 2). When no competing risks were considered, the Weibull and log-logistic models provided estimates of the genetic effect, β_1gene, of the mutation, equal to 3.48 and 3.16 respectively, which corresponds to cumulative penetrances of 54.2% and 42.9% in male and female carriers for the log-logistic model and 55.0% and 40.9%, respectively, for the Weibull model.

We also examined the risk of first CRCs for different types of MMR gene mutations (see Table 4). In 278 MLH1 carrier families, we observed 592 first CRCs (345 carriers, 6 non-carriers). The penetrance of the first cancer by age 70 was 72.2% in males and 52.3% in females. For 342 MSH2 carrier families, we observed 690 first CRCs (381 carriers, 11 non-carriers). The first cancer penetrance was 57.7% in males and 52.8% in females. Finally, for 101 MSH6 carrier families, we observed 135 first CRCs (76 carriers, 2 non-carriers). The penetrance for the first cancer was 30.5% in males and 15.8% in females.

For the second step analysis (i.e. parameter estimates related to events 2 and 4), the log-likelihood of the model was –2246.78 for the log-logistic model, –2249.35 for the Weibull PH model and –2336.60 for the piecewise constant hazard model. Table 3 shows significant correlations between the two CRC events measured by the copula parameter. They correspond to a Kendall's tau of 0.082 (p<0.001) for the log-logistic and Weibull model and 0.062 (p=0.002) for the piecewise constant hazard model. These correlations are relatively small but highly significant, indicating that the gap time between the two CRCs depends significantly on the age at the first CRC. Among gene carriers, the 10-year risk of developing a second CRC after a first CRC under the log-logistic model was about 13.8% in males and 12.8% in females when the first CRC occurred at 40 years and it was close to 15.4% in males and 14.5% in females when the first CRC occurred at 50 years (Figure 1). Interestingly, the effect of the gene mutation on the second CRC was not significant for any of three models considered, nor the gender effect. When competing risks were ignored, the 10-year risk of developing a second CRC among gene carriers was slightly smaller with the log-logistic model.

We also assessed the effect of the type of surgery after a first CRC on the risk of a second CRC using the log-logistic regression models. Among 788 individuals who had a first CRC and have had surgery recorded between the first and second CRCs, 6 had complete bowel removal and 170 partial removal. The rate of second CRCs (after exclusion of competing events) was 0/6 among individuals with complete bowel removal and 38/170 among those with partial removal (all of them being mutation carriers). Among mutation carriers, the 10-year risk of developing a second CRC after having partial surgery was close to 16.9% in males and 14.5% in females when a first CRC occurred at 40 years. Those rates were about 22.1% and 19.1% when the first CRC occurred at 50 years. The correlation between the times of first and second CRC corresponds to a Kendall's tau of 0.096 (SE^b=0.077), where SE^b is a bootstrap SE obtained from 1000 bootstrapped samples of the families.

Finally, we examined the risk of second CRCs for different types of MMR gene mutations and the dependence between the times to first and second CRCs. The results are summarized in Table 4. In 278 MLH1 carrier families, we observed 122 second CRCs (80 carriers, 42 unknown genotypes) among 592 first CRCs. The 10-year risk of developing a second CRC among carriers was 16.7% in males and 12.5% in females when a first CRC occurred at 40 years and 19.1% and 14.9% when the first CRC occurred at 50 years. For 342 MSH2 carrier families, we observed 139 second CRCs (94 carriers, 45 unknown genotypes) among 690 first CRCs. The 10-year risk of developing a second CRC among carriers was 13.1% in males and 15.1% in females when a first CRC occurred at 40 years and 13.8% and 15.9% when the first CRC occurred at 50 years. Finally, for 101 MSH6 carrier families, we observed 13 second CRCs (7 carriers, 6 unknown genotypes) among 135 first CRCs. The 10-year risk of developing a second CRC among carriers was 4.9% in males and 9.3% in females when a first CRC occurred at 40 years and 5.0% and 9.5% when the first CRC occurred at 50 years. Interestingly, the dependence between the times to first and second CRCs varied according to the mutation type, with a Kendall's tau of 0.109 (SE^b=0.033), 0.037 (SE^b=0.022), and 0.008 (SE^b=0.071) for MLH1, MSH2 and MSH6 mutations, respectively.