Estimating the parameters of multi-state models with time-dependent covariates through likelihood decomposition

Highlights

• Currently, various successive treatments are offered to people with ESRD.

• The demand increase requires modelling these treatments for provisional purposes.

• Modeling should allow for several treatments as for time-dependent covariates.

• We present and test an approach that allows for an unlimited number of treatments.

• This approach may be extended to allow for various other management constraints.

Abstract

Background

Multi-state models become complex when the number of states is large, when back and forth transitions between states are allowed, and when time-dependent covariates are inevitable. However, these conditions are sometimes necessary in the context of medical issues. For instance, they were needed for modelling the future treatments of patients with end-stage renal disease according to age and to various treatments.

Methods

The available modelling tools do not allow an easy handling of all issues; we designed thus a specific multi-state model that takes into account the complexity of the research question. Parameter estimation relied on decomposition of the likelihood and separate maximisations of the resulting likelihoods. This was possible because there were no interactions between patient treatment courses and because all exact times of transition from any state to another were known. Poisson likelihoods were calculated using the time spent at risk in each state and the observed transitions between each state and all others. The likelihoods were calculated on short time intervals during which age was considered as constant.

Results

The method was not limited by the number of parameters to estimate; it could be applied to a multi-state model with 10 renal replacement therapies. Supposing the parameters of the model constant over each of seven time intervals, this method was able to estimate one hundred age-dependent transitions.

Conclusions

The method is easy to adapt to any disease with numerous states or grades as long as the disease does not imply interactions between patient courses.

Introduction

Multi-state models have already been widely used in medicine. These models are of great interest in studying chronic diseases with several grades and/or treatments [1]. When the number of states is large, when back and forth transitions between states are allowed, and when time-dependent covariates (such as age) have to be taken into account, the resulting model can become quite complex. We faced all these difficulties when we designed a model to describe the successive treatments offered to patients with end-stage renal disease (ESRD).

ESRD is a life-threatening condition but, in the sixties, renal replacement therapies (RRTs) radically changed patients׳ prognoses and led to longer survivals, especially in renal transplant patients. Currently, ESRD patients may experience several switches between various RRTs [2], [3], and, in several countries, registries have been established to collect data on these switches [4], [5], [6], [7]. These switches can be described using the rates of transition between different RRTs.

Various solutions have been proposed to solve the specific problems of multi-state models in medical research [1], [8], [9]. These solutions are extensions of survival data analyses applied to settings with multiple transitions and competing risks [10], [11], [12], [13], [14], [15], [16]. In fact, several authors succeeded in using these solutions to analyse ESRD datasets, but their models used small numbers of transitions (only the main RRTs were studied), and they considered only age as a categorical variable (i.e., the estimates of the parameters were made for age classes) [17], [18], [19].

In France, 10 different RRT modalities are used and all of them should be considered to model the successive treatments a patient may undergo; this requires the estimation of a high number of transition rates. Moreover, the transitions to renal transplantation or death depend on the patients׳ age which should be integrated into the model.

The data recorded in the French ESRD registry [4] have two features that helped us overcome these difficulties. First, the patients׳ treatment courses are independent (that is, the switch of a patient from one treatment to another does not depend on the treatments the other patients are undergoing). Second, the exact times of all transitions from any treatment to another are known. These two features make it possible to decompose a multi-state model into sub-models whose likelihoods can be separately maximised [8], [10].

In this paper, we explain the way we solved a complex problem with a simple implementation of likelihood decomposition. This decomposition greatly facilitates the theoretical analysis and the numerical solution. Afterwards, considering short time intervals during which the time-dependent covariates are supposed constant led to the estimation of these covariate effects on the transition rates.