# A combined gamma frailty and normal random-effects model for repeated, overdispersed time-to-event data

TL;DR: This paper presents, extends, and studies a model for repeated, overdispersed time-to-event outcomes, subject to censoring, and two estimation methods are presented.

Abstract: This paper presents, extends, and studies a model for repeated, overdispersed time-to-event outcomes, subject to censoring. Building upon work by Molenberghs, Verbeke, and Demetrio (2007) and Molenberghs et al. (2010), gamma and normal random effects are included in a Weibull model, to account for overdispersion and between-subject effects, respectively. Unlike these authors, censoring is allowed for, and two estimation methods are presented. The partial marginalization approach to full maximum likelihood of Molenberghs et al. (2010) is contrasted with pseudo-likelihood estimation. A limited simulation study is conducted to examine the relative merits of these estimation methods. The modeling framework is employed to analyze data on recurrent asthma attacks in children on the one hand and on survival in cancer patients on the other.

## Summary (3 min read)

### 1 Introduction

- Time-to-event data are prominent in contemporary statistical analysis, not only for univariate outcomes but also in hierarchical settings.
- The timeto-event case is but one of the applications of their framework.
- Basic ingredients for their modeling framework, standard generalized linear models, extensions for overdispersion, and the generalized linear mixed model are the subject of Section 3.
- Avenues for parameter estimation and ensuing inferences are explored in Section 5, with particular emphasis on so-called partial marginalization and pseudo-likelihood estimation.

### 2.1 Recurrent asthma attacks in children

- These data have been studied in Duchateau and Janssen.
- A prevention trial is set up with such children randomized to placebo or drug, and the asthma events that developed over time are recorded in a diary.
- The different events are thus clustered within a patient and ordered in time.
- The data are presented in calendar time format, where the time at risk for a particular event is the time from the end of the previous event (asthma attack) to the start of the next event (start of the next asthma attack).
- Data for the first two patients are listed in Table 1.

### 2.2 Survival in cancer patients

- Hand et al.11 presented data on patients with advanced cancer of the stomach, bronchus, colon, ovary, or breast, who were treated, in addition to standard treatment, with ascorbate.
- The outcome of interest, survival time in days, is recorded to address the question as to whether survival times differ with the organ being affected.
- Individual-patient data are listed in Table 2.
- There are no censored observations in this case.

### 3 Background

- The authors model is based upon the generalized linear model and two of its extensions, the first one to accommodate overdispersion, and the second one to account for data hierarchies, such as in longitudinal data.
- Þ and cð , Þ. Often, and are termed ‘‘natural parameter’’ (or ‘‘canonical parameter’’) and ‘‘dispersion parameter’’, respectively.
- When the standard exponential-family models constrain the mean–variance relationship, so-called overdispersion is introduced.
- In their exponential and Weibull cases, it is in line with the data range to assume such a random effect to follow a gamma distribution, giving rise to the exponential-gamma and Weibull-gamma models.
- The model elements are listed in Table 3.

### 4.1 General model formulation

- The authors now need two different notations, ij and ij, to refer to the linear predictor and/or the natural parameter.
- It is convenient, but not strictly necessary, to assume that the two sets of random effects, hi and bi, are independent of each other.
- Obviously, parameterization (12) allows for random effects ij capturing overdispersion, and formulated directly at mean scale, whereas ij could be considered the GLMM component.
- They considered conjugacy, conditional upon the normally-distributed random effect bi.
- Fortunately, the Weibull and exponential cases satisfy this property, with gamma random effects.

### 4.2 Weibull- and exponential-type models for time-to-event data

- The general Weibull model for repeated measures, with both gamma and normal random effects can be expressed as f ð yijhi, biÞ ¼ First, it is implicit that the gamma random effects are independent.
- This need not be the case and, like in the Poisson case, extension via multivariate gamma distributions is possible.
- Third, it is evident that the classical gamma frailty model (i.e., no normal random effects) and the Weibull-based GLMM (i.e., no gamma random effects) follow as special cases.
- This is typically considered for the exponential model, but it holds for the Weibull model too, by observing that the Weibull model is nothing but an exponential model for the random variable Y ij.
- Fifth, the above expressions are derived for a two-parameter gamma density.

### 5 Estimation

- The key problem in maximizing (23) is the presence of N integrals over the random effects bi and hi.
- The authors consider so-called partial marginalization, in agreement with Molenberghs et al.
- 5 but, unlike these authors, also allowing for censorship.

### 5.1 Partial marginalization

- While closed-form expressions, can be used to implement maximum likelihood estimation, with numerical accuracy governed by the number of terms included in the Taylor series, one can also proceed by what Molenberghs et al.5 termed partial marginalization.
- By this the authors refer to integrating the conditional density over the gamma random effects only, leaving the normal random effects untouched.
- Now, in the survival case it is evidently very likely that censoring occurs.
- Focusing on rightcensored data, it is then necessary to integrate the marginal density over the survival time within the interval ½0,Ci .
- Recall that, while expressions of the type (16) appear to be for the univariate case, they extend without problem to the longitudinal setting as well.

### 5.2 Pseudo-likelihood

- Pseudo-likelihood,10,18 as generalized estimating equations19 are useful when the computational burden of full likelihood becomes burdensome and/or when robustness against misspecification of higher order moments is desirable.
- Essentially then, the joint distribution is replaced with a product of factors of marginal and/or conditional distributions of lower dimensions.
- Let us define pseudo-likelihood in general and formally, after which the authors turn to the special case of pairwise likelihood.
- Grouping the outcomes for subject i into a vector.

### 6 Marginal distributions and moments

- Along the lines of Molenberghs et al.5 and Molenberghs and Verbeke,29 the marginal density and moments are derived.
- Molenberghs and Verbeke29 showed that only a finite number of moments is finite.
- This holds not only for the combined model, but as soon as gamma random effects are combined with Weibull outcomes, i.e., it also applies to the conventional Weibullgamma model.
- Because it is possible that even the second and first moments may be infinite, it is wise to check the number of finite moments.

### 7.1 Recurrent asthma attacks in children

- The authors will analyze the times-to-event, introduced in Section 2.1.
- The treatment effect 1 is stably identifiable in all four models.
- As a result, overdispersion now disappears, given that the standard error values are more trustworthy.
- Re-fitted results for all four models in this way are reported in Table 7.
- A related finding was reported in Geys et al.26 where excessive computational requirements could be avoided when using pseudo-likelihood.

### 7.2 Survival in cancer patients

- For the generalized Cauchy model, predictor function ’ is set equal to instead.
- Parameter estimates are presented in Table 9.
- The key scientific question is directed toward the difference in survival across cancer types.
- Thus, their analysis illustrates the occurrence, in real life, of distributions without finite moments, with all moments finite, and with a finite number of finite moments.
- The generalized Cauchy model has a finite mean and finite variance and provides a parsimonious description, unlike the generalized logistic model, in spite of its full series of finite moments.

### 8 Simulation study

- The authors aim to evaluate the performance of the combined model, Weibull model with gamma frailties and normal random effects, under full likelihood and pseudo-likelihood.
- The design of the simulation study was carried out under different settings, to investigate the impact of sample size, censoring percentage, and method of estimation.
- The authors used two sets of true parameters, similar in spirit to the ones in Table 7, without and with censoring (full likelihood).
- The Mahalanobis distance has the advance of taking the variance-covariance structure into account.
- While with increasing proportion of censored observations, within the same sample size, the relative distance seems to be stable when using pseudo-likelihood estimation method.

### 9 Concluding remarks

- Building upon work by Molenberghs et al.,5 the authors have studied the combination of normal and nonnormal random effects in the time-to-event case.
- The statistical loss of efficiency of pseudo-likelihood is relatively small, although the consistency behavior for the maximum-likelihood case is better.
- The gamma and normal random effects play distinct roles.
- The model can be extended further and/or adapted to specific cases.

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