gms | German Medical Science

Introduction: In modelling occurrence of chronic diseases, death as a competing risk must be accounted for. In particular, the “semi-competing” character of the data has to be acknowledged: Disease occurrence (the “non-terminal” event) can be observed before death (the “terminal” event), but not vice versa [1]. Consequently, the observed ages at the two events within the same individual are correlated. As suggested by Lee et. al. [2], statistical approaches in this setting can be divided in at least three groups: 1) Copula based models [3], [4], 2) illness-death models, and 3) causal inference motivated methods. A common approach in the first group is to model the dependency between the two events by a bivariate copula with two marginal distributions for age at disease onset and age at death. However, such approaches cannot distinguish between the two different modes of mortality, the one with and the one without the disease.

Methodology: We here propose a “double copula” model that estimates the three marginal distributions given in the semi-competing risks framework: 1) age at disease onset, 2) age at death for individuals with the disease and 3) age at death for individuals without the disease. The model uses two bivariate copulas modeling the joint distribution of age at disease onset with each of the two ages at death separately. Further, different parametric copulas are implemented, allowing flexible modelling of two distinct copulas for each pair of distributions. We assume the marginals of the two mortalities to be Gompertz distributed [5], whereas for the disease onset a Weibull distribution is adopted. Model parameters are estimated by maximum likelihood, accounting for the complex censoring and truncation mechanisms in a cohort study. The likelihood function incorporates left truncation for both terminal and non-terminal events, reflecting delayed entry into the study. As in cohort studies the exact age at disease onset is only known to lie between two intermittent follow-up visits, we additionally consider interval censoring for the age of disease occurrence.

Results: The model is illustrated using data from the Paquid study, a large cohort study on mental and physical aging with age at dementia onset describing the non-terminal event. Our model leads to plausible results, indicating that people with dementia diagnosis die earlier compared with people without dementia.

Conclusion: We propose a new flexible, copula-based approach for modelling semi-competing risks data. Two copulas are incorporated in the likelihood function, allowing the comparison of three distributions: the age at disease onset as well as the two underlying age at death distributions, the one with and the one without the disease of interest. Thereby, we aim not only to estimate the disease distribution, but to quantify the disease impact on life expectancy.

The authors declare that they have no competing interests.

The authors declare that an ethics committee vote is not required.