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

Weibull model

In this section, we define the Weibull model for analysing survival of patients in the context of human health. We confine ourselves to survival times that are the difference between a nominated start time and a declared failure (uncensored data) or a nominated end time (censored time). Let T be a nonnegative random variable for a person’s survival time and t be a realisation of the random variable T. Kleinbaum and Klein (2005) give some reasons for the occurrence of right censoring in survival studies, including termination of the study, drop outs, or loss to follow-up. For the censored observations, one could impute the missing survival times or assume that they are event-free. The former is often difficult, especially if the censoring proportion is large, and extreme imputation assumptions (such as all censored cases fail right after the time of censoring) may distort inferences (Leung et al. 1997; Stajduhar et al. 2009). In this study, we treat all censored cases as event-free regardless of observation time.

for α > 0 and γ > 0, where α is a shape parameter and γ is a scale parameter (Ibrahim et al. 2001).

where t > 0, α > 0 and γ>0.

and we allow Σ to be diagonal with elements ${\sigma }_j^2,j=1,2,\dots ,p$.

Diffuse priors are represented by large positive values for σ ², and small positive values for u _α  and v _α .

MCMC analysis is performed by sampling from the conditional distributions of the parameters. The conditional distribution of α does not have an explicit form but can be sampled from MCMC algorithms such as Metropolis Hastings or slice sampling (Gilks et al. 1996).

Weibull mixture model

where α = (α ₁,…, α _K ), γ = (γ ₁,…, γ _K ) are the parameters of each Weibull distribution and w = (w ₁,…, w _K ) is a vector of nonnegative weights which sum to one.

We now assume that we observe possibly right-censored data for n patients; y = (y ₁,…, y _n ) where y _i  = (t _i , δ _i ) and δ _i  is an indicator function as described in Section “Weibull model”.

where x _i  = (x _{i 1},…, x _ip ), γ _m  = (γ _1m ,…, γ _pm ) and β _m  = (β _1m ,…, β _pm ), for i = 1,2,…, n and m = 1,2,…, K.

Here, the incomplete information is modelled via the survivor function, which reflects the probability that the patient was alive for duration greater than t _i .

and we allow Σ to be diagonal with elements ${\sigma }_j^2,j=1,2,\dots ,p$. Again, we express a vaguely informative prior by setting a large positive value for ${\sigma }_j^2$. The diagonal matrices were used here but this changed recently (Bhadra and Mallick 2013), so one may argue that a non-diagonal variace-covariance matrix may be more appropriate.

The model described in this section can be fitted using MCMC sampling with latent values Z _i  to indicate component membership of the i ^th  observation (Diebolt and Robert 1994; Robert and Casella 2000). Since w _m  = P r(Z _i  = m), we can write Z _i  ∼ M(w ₁,…, w _K ). In this scheme, the Z _i  are sampled by computing posterior probabilities of membership, and the other parameters are sampled from their full conditional distributions. This was implemented in the WinBUGS software package (Spiegelhalter et al. 2002).

The WinBUGS software (Lunn et al. 2000; Ntzoufras 2009; Spiegelhalter et al. 2002) is an interactive Windows version of the BUGS program for Bayesian analysis of complex statistical models using MCMC techniques.

Label switching, caused by non-identifiability of the mixture components, was dealt with post-MCMC using the reordering algorithm of Marin et al. (2005b). The algorithm proceeded by selecting the permutation of components at each iteration that minimised the vector dot product with the so-called “pivot”, a high density point from the posterior distribution. The MCMC output was then reordered according to each selected permutation. In this paper, the approximate maximum a posteriori (MAP) (i.e. the realization of parameters corresponding to the MCMC iterate that maximised the unnormalised posterior) was chosen as the pivot.

Cure model

The commonly used parametric distributions include Exponential and Weibull for S ^∗(t).

A corresponding cure fraction in model (8) is ${lim}_{t\to \infty }{S}_{\mathit{\text{pop}}}\left(t\right)=\text{exp}(-\theta )>0$. We also know from (8) that the cure fraction is given by S _pop (∞) = P(N = 0) = exp(-θ). As θ → ∞, the cure fraction tends to 0, whereas as θ→0, the cure fraction tends to 1. Corresponding population density and hazard functions are $f_{\mathit{\text{pop}}}\left(t\right)=-\frac{d}{\mathit{\text{dt}}}{S}_{\mathit{\text{pop}}}\left(t\right)=\theta f\left(t\right)\text{exp}(-\theta FF(t\left)\right)$ and h _pop (t) = θ f(t), respectively.

where $S^{\ast }\left(t\right)=\frac{\text{exp}(-\theta F(t\left)\right)-\text{exp}(-\theta )}{1-\text{exp}(-\theta )}$, and $f^{\ast }\left(t\right)=\frac{\text{exp}(-\theta F(t\left)\right)}{1-\text{exp}(-\theta )}\theta f\left(t\right)$.

Following Chen et al. (1999) and Ibrahim et al. (2001), we construct the likelihood function. Suppose we have n subjects and we assume that the N _i  are i.i.d with Poisson distributions with means θ _i , i = 1,…, n. Let Z _{i 1},…, Z _iN  denote the times for the N _i  competing causes, which are unobserved, and which have a cumulative distribution function, F(.). In this section, we will specify a parametric form for F(.) that is a Weibull distribution. Let ψ = (α, λ)^′, where α is the shape parameter and λ is the scale parameter. We incorporate covariates for the cure rate model through the cure parameter θ and we have a different cure rate parameter, θ _i , for each subject.

Again, we assume independent priors for β and ψ, where α∼G amma(a _α , b _α ), λ ∼ N(μ _λ , Σ _λ ) and β ∼ N(μ _β , Σ _β ). We also assume p(α, λ) = p(α ∣ δ ₀, τ ₀)p(λ), $p(\alpha \mid {\delta }_0,{\tau }_0)\propto {\alpha }^{{\delta }_0-1}\text{exp}(-{\tau }_0\alpha )$, and the hyperparameters (δ ₀, τ ₀) are specified (Chen et al. 1999; Ibrahim et al. 2001).

The joint posterior density of (α, λ, β) in equation (10) is analytically intractable because the integration of the joint posterior density is not easy to perform. Hence, inferences are based on MCMC simulation methods. We can use, for example, the Metropolis-Hastings algorithms or slice sampling to simulate samples of α, λ and β. MCMC computations were implemented using the WinBUGS system (Spiegelhalter et al. 2002).

DLBCL dataset

We applied the proposed method of model averaging across the three candidate survival models to a dataset containing gene expression of Diffuse Large B-cell Lymphoma (DLBCL). The dataset comprises gene expression measurements and survival times of patients with DLBCL (Rosenwald et al. 2002). DLBCL (Lenz et al. 2008) is a type of cancer of the lymphatic system in adults which can be cured by anthracycline-based chemotherapy in only 35 to 40 percent of patients (Rosenwald et al. 2002). In general, types of this disease are very diverse and their biological properties are largely unknown, meaning that this is a relatively difficult cancer to cure and prevent. Rosenwald et al. (2002) proposed that there are three phenotypes subgroups of patients of DLBCL: activated B-like DLBCL, germinal centre (GC)-B like and type III DLBCL. The GC B-like DLBCL is less dangerous than the others in the progression of the tumour; the activated B-like DLBCL is more active than the others and the type III DLBCL is the most dangerous in the progression of tumour (Alizadeh et al. 2000). These groups were defined using microarray experiments and hierarchical clustering. The authors showed that these phenotypes subgroups were differentiated from each other by distinct gene expressions of hundreds of different genes and had different survival time patterns. This dataset contains 219 patients with DLBCL, including 138 patient deaths during follow-up. Patients with missing values for a particular microarray element were excluded from all analyses involving that element.

Based on patterns of gene expression in biopsy specimens of the lymphoma, Rosenwald et al. (2002) analysed this dataset to predict the likelihood of patients’ survival after chemotherapy for DLBCL. By using a Cox proportional-hazards model, Rosenwald et al. (2002) identified five individual gene expressions which correlated with the survival after chemotherapy. These gene expressions are germinal center B-cell (GC-B), lymphoma node, proliferation, BMP6 and MHC. In this study, these five gene expressions are used as covariates for estimating survival times based on the three competing models in Section “Models”.

Results

As discussed in Section “Methods”, to account for model uncertainty, the model averaging technique which combines estimates from different survival models was carried out. This was accomplished through a weighted average of the survival considered in the analysis. First, we calculated the Kaplan-Meier estimates of overall survival according to the gene expression and the relation between the gene expression score and the subgroups phenotype of DLBCL. We confirmed that these phenotypes had different survival time patterns (Figure 1). Following this, we fitted the three models to all gene expression data and to the three phenotype subgroups. We then applied the BMA approach described in Section “Methods”. For each model, we ran the corresponding MCMC algorithm for 100 000 iterations, discarding the first 10 000 iterations as burn-in.

Table 1 shows the estimated posterior mean of the parameters, the 95% credible intervals (CI), the BIC values and the BMA weights for each of the fitted models for the whole dataset. The BMA weights reflect the relative posterior probability of the models. As can be seen from Table 1, for the Weibull model, there are three genes that substantially describe patients’ survival times, namely GC-B (β ₁), lymphoma node (β ₂) and MHC (β ₅). These three genes have a negative effect on the expected survival time. For the mixture model, GC-B (β ₁), lymphoma node (β ₂) and proliferation (β ₃) accounted for patients’ survival times in the first component. In the second component, GC-B (β ₁), lymphoma node (β ₂) and MHC signature (β ₅) substantially explained patients’ survival times. All these genes have negative effects on the expected survival time for their respective component. For the cure model, four of these genes substantially describe patients’ survival times, namely GC-B (β ₁), lymphoma node (β ₂), BMP6 (β ₄) and MHC (β ₅) signature. Three of these, namely GC-B (β ₁), lymphoma node (β ₂) and MHC signature (β ₅), have a negative effect on the expected survival time. Under the cure model, approximately 33.8% of the patients are cured of DLBCL (Figure 2).

Model	Parameter	Mean	95% CI	BIC	Weight
Weibull	α	0.7305	(0.626,0.840)	687.0953	0.0009
	β 0	-1.578	(-1.84, -1.33)
	β 1	-0.3446	(-0.516, -0.172)
	β 2	-0.2844	(-0.454, -0.116)
	β 3	0.2097	(-0.049, 0.468)
	β 4	0.3292	(0.115, 0.537)
	β 5	-0.3019	(-0.488, -0.112)
Mixture	α 1	4.029	(2.411, 6.631)	734.0054	≈0
	α 2	0.7707	(0.662, 0.885)
	β 01	6.857	(5.479, 8.205)
	β 02	-1.724	(-2.007, -1.457)
	β 11	-11.62	(-12.88, -10.35)
	β 12	-0.3956	(-0.575, -0.216)
	β 21	-2.087	(-3.54, -0.689)
	β 22	-0.3172	(-0.495, -0.143)
	β 31	-2.241	(-3.425, -1.059)
	β 32	0.1972	(-0.064, 0.461)
	β 41	-0.2849	(-1.434, 0.854)
	β 42	0.3594	(0.141, 0.574)
	β 51	-0.7928	(-2.107, 0.477)
	β 52	-0.3102	(-0.500, -0.115)
	π 1	0.01992	(0.002, 0.053)
	π 2	0.9801	(0.946, 0.997)
Cure	α	0.9884	(0.828, 1.145)	673.1359	0.9991
	β 0	0.1611	(-0.124, 0.560)
	β 1	-0.3151	(-0.484, -0.144)
	β 2	-0.2821	(-0.451, -0.115)
	β 3	0.189	(-0.070, 0.442)
	β 4	0.3303	(0.118, 0.539)
	β 5	-0.3039	(-0.490, -0.112)

This is clearly exhibited in Table 1, which shows that the cure model has the largest posterior model probability (or BMA weight). To evaluate the model fit, a comparison of predicted values under the models and of the observed data was carried out.

From Tables 1 and 2, we can see that the Weibull model is better than a two-component Weibull mixture model.

As can be seen in Figure 3, in the full DLBCL dataset, the predicted curve for the cure model is quite close to the observed data, suggesting a good fit of the data. Specifically, in this model, 94.3% of observed survival times in the dataset fall in the corresponding 95% posterior prediction intervals. As expected, this is quite similar to the result obtained from model averaging (91.9%) (Table 3).

Furthermore, in the GCB phenotype, the genes corresponding to the BMP6 (β ₄) and MHC signature (β ₅) in the Weibull model and MHC signature (β ₅) in the cure model substantially affect patients’ survival time. In the ABC phenotype, in the Weibull model, with the exception of proliferation (β ₃), all genes were involved substantially in the description of patients’ survival and lymphoma node (β ₂), BMP6 (β ₄) and MHC signature (β ₅) are potentially important prognostic factors for predicting survival in the cure model. For the type III phenotype, the GC-B gene (β ₁) in both models and only the BMP6 gene (β ₄) in the cure model are substantial in explaining the survival times of the patients.

Under the cure model, in the GCB phenotype, approximately 33.2% of the patients are estimated to be cured of DLBCL. In the ABC and type III phenotypes, the respective cure rates are approximately 26.6% and 18.7% (Figure 2).

The results of the posterior densities prediction for the individual models and the model averaged prediction based on these three phenotypes are presented in Figure 3. In comparison to other models, the mixture model fitted the data poorly for each phenotype. In detail, using model averaging, for the GCB phenotype, 94% of the observed survival times in the dataset lie in the respective 95% prediction intervals. For the other two phenotypes, namely the ABC and the type III, 93.4% and 92.8% of the observed survival times in the dataset are in the corresponding 95% prediction intervals, respectively (Table 3).