Cure rate estimation is an important issue in clinical trials for diseases such as lymphoma and breast cancer and mixture models are the main statistical methods. In the last decade, mixture models under different distributions, such as exponential, Weibull, log-normal and Gompertz, have been discussed and used. However, these models involve stronger distributional assumptions than is desirable and inferences may not be robust to departures from these assumptions. In this paper, a mixture model is proposed using the generalized F distribution family. Although this family is seldom used because of computational difficulties, it has the advantage of being very flexible and including many commonly used distributions as special cases.
The generalised F mixture model can relax the usual stronger distributional assumptions and allow the analyst to uncover structure in the data that might otherwise have been missed. This is illustrated by fitting the model to data from large-scale clinical trials with long follow-up of lymphoma patients. Computational problems with the model and model selection methods are discussed. Comparison of maximum likelihood estimates with those obtained from mixture models under other distributions are included.
1998 John Wiley & Sons, Ltd.
Gamel et al. studied data from patients with intraocular melanoma by a mixture model of the log-normal distribution. The background distribution was a degenerate one and the cure rate and mean log survival time were estimated as functions of tumour size. The logrank test was used to test the difference between the cure rates and median survival times under treatment and control groups. Denham et al. employed the model to estimate cure rates in lymphoma patients.
Jones et al. employed the exponential distribution for uncured patients in their mixture model, a simulation study of which was carried out by Goldman. This method was studied further by Ghitany et al. and Ghitany and Maller. Farewell proposed the Weibull distribution instead of the exponential distribution to model the failure time of uncured patients.
A Gompertz distribution is also used for this purpose by Gordon in breast cancer. However, he assumed that the failure time of cured patients also follows a Gompertz distribution.
McLachlan and McGiffin studied a similar model but assumed a degenerate distribution for the failure time of cured patients. They also reviewed related parametric models.
All of the above research was carried out using a specific distribution. As remarked by Taylor and Kim, one disadvantage of these models is that they are too restrictedly parametric and may involve unsuitably strong distributional assumptions, and hence conclusions from the models may not be sufficiently robust to departures from the assumptions. Actual survival data can come from almost any positive valued continuous distribution. A typical example where these models may not be appropriate is given by Taylor and Kim. Therefore, it is desirable to consider other approaches to achieve greater robustness. A similar problem also exists with accelerated failure time models, for which Kalbfleisch and Prentice proposed using the generalized F distribution. This method was not exploited completely because of its computational difficulties and is seldom mentioned in the statistical literature.
In this paper, we investigate the application of the generalized F distribution to a mixture model for cure rate estimation. The generalized F distribution has more parameters than most commonly used distributions; it is also a flexible distribution and contains other distributions as special cases. The generalized F mixture model provides great flexibility to model the survival time distribution of uncured patients as well as covariate effects on the cure rate. It can potentially uncover structure in survival data which otherwise might be missed using other parametric mixture models. Section 2 gives a description of the data from lymphoma patients which will be analysed throughout the paper. Section 3.1 gives the definition of the generalized F distribution and some basic properties. The generalized F mixture model for cure rate estimation is described in Section 3.2 and the computational problems and their solution are discussed in Section 3.3. Some model selection methods involved in the generalized F mixture model are given in Section 3.4. In Section 4, the goodness-of-fit of the model to the lymphoma data is investigated. A comparison is drawn between maximum likelihood estimates under the generalized F mixture model and those obtained from mixture models using other distributions. Finally, there is a brief discussion in Section 5.
NON-HODGKIN'S LYMPHOMA
Patients with follicular non-Hodgkin's lymphoma entered clinical trials conducted by the British National Lymphoma Investigation (BNLI) between 1974 and 1980. The time to relapse (in years, from the date of starting treatment to the time that relapse was first recorded in patients who responded completely to therapy), together with other demographic and pathological factors, was recorded for each patient. Complete details are presented elsewhere. Of 398 patients enrolled, 164 failed to respond completely and data from them are not used.