Modelling survival data to account for model uncertainty: a single model or model averaging?

This study considered the problem of predicting survival, based on three alternative models: a single Weibull, a mixture of Weibulls and a cure model. Instead of the common procedure of choosing a single “best” model, where “best” is defined in terms of goodness of fit to the data, a Bayesian model averaging (BMA) approach was adopted to account for model uncertainty. This was illustrated using a case study in which the aim was the description of lymphoma cancer survival with covariates given by phenotypes and gene expression. The results of this study indicate that if the sample size is sufficiently large, one of the three models emerge as having highest probability given the data, as indicated by the goodness of fit measure; the Bayesian information criterion (BIC). However, when the sample size was reduced, no single model was revealed as “best”, suggesting that a BMA approach would be appropriate. Although a BMA approach can compromise on goodness of fit to the data (when compared to the true model), it can provide robust predictions and facilitate more detailed investigation of the relationships between gene expression and patient survival. Keywords: Bayesian modelling; Bayesian model averaging; Cure model; Markov Chain Monte Carlo; Mixture model; Survival analysis; Weibull distribution

Afatinib versus cisplatin-based chemotherapy for EGFR mutation-positive lung adenocarcinoma (LUX-Lung 3 and LUX-Lung 6): analysis of overall survival data from two randomised, phase 3 trials

Background We aimed to assess the effect of afatinib on overall survival of patients with EGFR mutation-positive lung adenocarcinoma through an analysis of data from two open-label, randomised, phase 3 trials. Methods Previously untreated patients with EGFR mutation-positive stage IIIB or IV lung adenocarcinoma were enrolled in LUX-Lung 3 (n=345) and LUX-Lung 6 (n=364). These patients were randomly assigned in a 2:1 ratio to receive afatinib or chemotherapy (pemetrexed-cisplatin [LUX-Lung 3] or gemcitabine-cisplatin [LUX-Lung 6]), stratified by EGFR mutation (exon 19 deletion [del19], Leu858Arg, or other) and ethnic origin (LUX-Lung 3 only). We planned analyses of mature overall survival data in the intention-to-treat population after 209 (LUX-Lung 3) and 237 (LUX-Lung 6) deaths. These ongoing studies are registered with ClinicalTrials.gov, numbers NCT00949650 and NCT01121393. Findings Median follow-up in LUX-Lung 3 was 41 months (IQR 35–44); 213 (62%) of 345 patients had died. Median follow-up in LUX-Lung 6 was 33 months (IQR 31–37); 246 (68%) of 364 patients had died. In LUX-Lung 3, median overall survival was 28·2 months (95% CI 24·6–33·6) in the afatinib group and 28·2 months (20·7–33·2) in the pemetrexed-cisplatin group (HR 0·88, 95% CI 0·66–1·17, p=0·39). In LUX-Lung 6, median overall survival was 23·1 months (95% CI 20·4–27·3) in the afatinib group and 23·5 months (18·0–25·6) in the gemcitabine-cisplatin group (HR 0·93, 95% CI 0·72–1·22, p=0·61). However, in preplanned analyses, overall survival was significantly longer for patients with del19-positive tumours in the afatinib group than in the chemotherapy group in both trials: in LUX-Lung 3, median overall survival was 33·3 months (95% CI 26·8–41·5) in the afatinib group versus 21·1 months (16·3–30·7) in the chemotherapy group (HR 0·54, 95% CI 0·36–0·79, p=0·0015); in LUX-Lung 6, it was 31·4 months (95% CI 24·2–35·3) versus 18·4 months (14·6–25·6), respectively (HR 0·64, 95% CI 0·44–0·94, p=0·023). By contrast, there were no significant differences by treatment group for patients with EGFR Leu858Arg-positive tumours in either trial: in LUX-Lung 3, median overall survival was 27·6 months (19·8–41·7) in the afatinib group versus 40·3 months (24·3–not estimable) in the chemotherapy group (HR 1·30, 95% CI 0·80–2·11, p=0·29); in LUX-Lung 6, it was 19·6 months (95% CI 17·0–22·1) versus 24·3 months (19·0–27·0), respectively (HR 1·22, 95% CI 0·81–1·83, p=0·34). In both trials, the most common afatinib-related grade 3–4 adverse events were rash or acne (37 [16%] of 229 patients in LUX-Lung 3 and 35 [15%] of 239 patients in LUX-Lung 6), diarrhoea (33 [14%] and 13 [5%]), paronychia (26 [11%] in LUX-Lung 3 only), and stomatitis or mucositis (13 [5%] in LUX-Lung 6 only). In LUX-Lung 3, neutropenia (20 [18%] of 111 patients), fatigue (14 [13%]) and leucopenia (nine [8%]) were the most common chemotherapy-related grade 3–4 adverse events, while in LUX-Lung 6, the most common chemotherapy-related grade 3–4 adverse events were neutropenia (30 [27%] of 113 patients), vomiting (22 [19%]), and leucopenia (17 [15%]). Interpretation Although afatinib did not improve overall survival in the whole population of either trial, overall survival was improved with the drug for patients with del19 EGFR mutations. The absence of an effect in patients with Leu858Arg EGFR mutations suggests that EGFR del19-positive disease might be distinct from Leu858Arg-positive disease and that these subgroups should be analysed separately in future trials.

Regression models for bivariate survival data

Multivariate lifetime data arise in various forms including recurrent event data when individuals are followed to observe the sequence of occurrences of a certain type of event; correlated lifetime when an individual is followed for the occurrence of two or more types of events, or when distinct individuals have dependent event times. In most studies there are covariates such as treatments, group indicators, individual characteristics, or environmental conditions, whose relationship to lifetime is of interest. This leads to a consideration of regression models.The well known Cox proportional hazards model and its variations, using the marginal hazard functions employed for the analysis of multivariate survival data in literature are not sufficient to explain the complete dependence structure of pair of lifetimes on the covariate vector. Motivated by this, in Chapter 2, we introduced a bivariate proportional hazards model using vector hazard function of Johnson and Kotz (1975), in which the covariates under study have different effect on two components of the vector hazard function. The proposed model is useful in real life situations to study the dependence structure of pair of lifetimes on the covariate vector . The well known partial likelihood approach is used for the estimation of parameter vectors. We then introduced a bivariate proportional hazards model for gap times of recurrent events in Chapter 3. The model incorporates both marginal and joint dependence of the distribution of gap times on the covariate vector . In many fields of application, mean residual life function is considered superior concept than the hazard function. Motivated by this, in Chapter 4, we considered a new semi-parametric model, bivariate proportional mean residual life time model, to assess the relationship between mean residual life and covariates for gap time of recurrent events. The counting process approach is used for the inference procedures of the gap time of recurrent events. In many survival studies, the distribution of lifetime may depend on the distribution of censoring time. In Chapter 5, we introduced a proportional hazards model for duration times and developed inference procedures under dependent (informative) censoring. In Chapter 6, we introduced a bivariate proportional hazards model for competing risks data under right censoring. The asymptotic properties of the estimators of the parameters of different models developed in previous chapters, were studied. The proposed models were applied to various real life situations.

Estimation of age- and stage-specific Catalan breast cancer survival functions using US and Catalan survival data

During the last part of the 1990s the chance of surviving breast cancer increased. Changes in survival functions reflect a mixture of effects. Both, the introduction of adjuvant treatments and early screening with mammography played a role in the decline in mortality. Evaluating the contribution of these interventions using mathematical models requires survival functions before and after their introduction. Furthermore, required survival functions may be different by age groups and are related to disease stage at diagnosis. Sometimes detailed information is not available, as was the case for the region of Catalonia (Spain). Then one may derive the functions using information from other geographical areas. This work presents the methodology used to estimate age- and stage-specific Catalan breast cancer survival functions from scarce Catalan survival data by adapting the age- and stage-specific US functions. Methods: Cubic splines were used to smooth data and obtain continuous hazard rate functions. After, we fitted a Poisson model to derive hazard ratios. The model included time as a covariate. Then the hazard ratios were applied to US survival functions detailed by age and stage to obtain Catalan estimations. Results: We started estimating the hazard ratios for Catalonia versus the USA before and after the introduction of screening. The hazard ratios were then multiplied by the age- and stage-specific breast cancer hazard rates from the USA to obtain the Catalan hazard rates. We also compared breast cancer survival in Catalonia and the USA in two time periods, before cancer control interventions (USA 1975–79, Catalonia 1980–89) and after (USA and Catalonia 1990–2001). Survival in Catalonia in the 1980–89 period was worse than in the USA during 1975–79, but the differences disappeared in 1990–2001. Conclusion: Our results suggest that access to better treatments and quality of care contributed to large improvements in survival in Catalonia. On the other hand, we obtained detailed breast cancer survival functions that will be used for modeling the effect of screening and adjuvant treatments in Catalonia

A practical comparison of blinded methods for sample size reviews in survival data clinical trials.

This paper presents practical approaches to the problem of sample size re-estimation in the case of clinical trials with survival data when proportional hazards can be assumed. When data are readily available at the time of the review, on a full range of survival experiences across the recruited patients, it is shown that, as expected, performing a blinded re-estimation procedure is straightforward and can help to maintain the trial's pre-specified error rates. Two alternative methods for dealing with the situation where limited survival experiences are available at the time of the sample size review are then presented and compared. In this instance, extrapolation is required in order to undertake the sample size re-estimation. Worked examples, together with results from a simulation study are described. It is concluded that, as in the standard case, use of either extrapolation approach successfully protects the trial error rates. Copyright © 2012 John Wiley & Sons, Ltd.

A Bayesian Analysis in the Presence of Covariates for Multivariate Survival Data: An example of Application

In this paper, we introduce a Bayesian analysis for survival multivariate data in the presence of a covariate vector and censored observations. Different ""frailties"" or latent variables are considered to capture the correlation among the survival times for the same individual. We assume Weibull or generalized Gamma distributions considering right censored lifetime data. We develop the Bayesian analysis using Markov Chain Monte Carlo (MCMC) methods.

A flexible model for survival data with a cure rate: a Bayesian approach

In this paper we deal with a Bayesian analysis for right-censored survival data suitable for populations with a cure rate. We consider a cure rate model based on the negative binomial distribution, encompassing as a special case the promotion time cure model. Bayesian analysis is based on Markov chain Monte Carlo (MCMC) methods. We also present some discussion on model selection and an illustration with a real dataset.

A bivariate regression model for matched paired survival data: local influence and residual analysis

The use of bivariate distributions plays a fundamental role in survival and reliability studies. In this paper, we consider a location scale model for bivariate survival times based on the proposal of a copula to model the dependence of bivariate survival data. For the proposed model, we consider inferential procedures based on maximum likelihood. Gains in efficiency from bivariate models are also examined in the censored data setting. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and compared to the performance of the bivariate regression model for matched paired survival data. Sensitivity analysis methods such as local and total influence are presented and derived under three perturbation schemes. The martingale marginal and the deviance marginal residual measures are used to check the adequacy of the model. Furthermore, we propose a new measure which we call modified deviance component residual. The methodology in the paper is illustrated on a lifetime data set for kidney patients.

Tests of proportional hazards and proportional odds models for grouped survival data

In this paper, we derive score test statistics to discriminate between proportional hazards and proportional odds models for grouped survival data. These models are embedded within a power family transformation in order to obtain the score tests. In simple cases, some small-sample results are obtained for the score statistics using Monte Carlo simulations. Score statistics have distributions well approximated by the chi-squared distribution. Real examples illustrate the proposed tests.

Modeling grouped survival data with time-dependent covariates

In this article, proportional hazards and logistic models for grouped survival data were extended to incorporate time-dependent covariates. The extension was motivated by a forestry experiment designed to compare five different water stresses in Eucalyptus grandis seedlings. The response was the seedling lifetime. The data set was grouped since there were just three occasions in which the seedlings was visited by the researcher. In each of these occasions also the shoot height was measured and therefore it is a time-dependent covariate. Both extended models were used in this example, and the results were very similar.

Likelihood approximations and discrete models for tied survival data

Ties among event times are often recorded in survival studies. For example, in a two week laboratory study where event times are measured in days, ties are very likely to occur. The proportional hazards model might be used in this setting using an approximated partial likelihood function. This approximation works well when the number of ties is small. on the other hand, discrete regression models are suggested when the data are heavily tied. However, in many situations it is not clear which approach should be used in practice. In this work, empirical guidelines based on Monte Carlo simulations are provided. These recommendations are based on a measure of the amount of tied data present and the mean square error. An example illustrates the proposed criterion.

A Generalized Log-Normal Model for Grouped Survival Data

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)