Comparison of Bayesian Network Meta-Analysis Models for Survival Data Purvi Prajapati James Stamey John Seaman May 22, 2019 1
Table of Contents 1. Introduction 2. Bayesian Network Meta-Analysis Models for Survival Data 3. Simulation Study 4. Application 5. Conclusion 6. Appendix 2
Introduction
Introduction • In the health-care field, decisions are often made using meta-analysis or network meta-analysis to allow for direct or indirect comparisons of treatments. • Survival data is a crucial endpoint in the pharmaceutical field. • Meta-analysis for survival data is commonly based on reported hazard ratios. • Rather than basing the treatment effects on hazard ratios, the models in consideration base the treatment effects on parameters used to model the log hazard rate over time. • The purpose of this work was to study via simulation the models available in the literature. 3
Survival Analysis • Survival analysis is used to analyze time to event data. • There are parametric and non-parametric approaches to analyze this type of data. • Some issues with survival data include the proportional hazards assumption and censoring. − Censoring is inevitable in time-to-event data. − Proportional hazards assumption is often needed for the Cox proportional hazards model. 4
Survival Distributions • Exponential Distribution • f ( t ) = 1 λ exp � − t � . λ • S ( t ) = exp � � − t . λ • h ( t ) = 1 λ . • Weibull Distribution � t � φ − 1 exp � � φ . • f ( t ) = φ − t λ λ λ − � t � � φ � • S ( t ) = exp . λ • h ( t ) = φ t φ − 1 λ φ . • Gompertz Distribution λ exp � t exp � φ 1 − exp � t • f ( t ) = 1 � � ��� . φ λ φ exp � t • S ( t ) = exp � − φ � � − 1 �� . λ φ λ exp � t • h ( t ) = 1 � . φ 5
Meta-Analysis/Network Meta-Analysis Meta-Analysis • Allows for the pooling of information from clinical trials that have a common outcome for a given disease. • Summary data is then pooled from the selected literature to make comparisons between treatments. Network Meta-Analysis • Network meta-analysis is an extension which allows indirect comparisons as well as direct comparisons. • This makes estimation more precise for all comparisons. • Exchangability is a standard assumption of hierarchical models, and meta-analysis. Network meta-analysis adds another assumption, consistency. 6
Meta-Analysis vs. Network Meta-Analysis Figure 1: Example of a Treatment Network. Here d bk is, on average, the “direct” effect between b and k . 7
Bayesian Network Meta-Analysis Models for Survival Data
Model Basis • The literature contains summary data, such as means and confidence intervals. • For survival literature, the summaries are usually given in the form of hazard ratios, Figure 2: Example of a and often include Kaplan-Meier Curve. Kaplan-Meier curves. Ouwens et al, 2010. 8
Model Basis • The likelihood for each of the models is r jkt ∼ Bin( n jkt , p jkt ) . − r jkt : Observed number of events in interval [ t , t + △ t ] for treatment k in study j . − n jkt : Number of subjects that have not experienced an event at time t for treatment k in study j . − p jkt : Observed cumulative incidence of events in interval [ t , t + △ t ]. • The relationship between p jkt and h jkt is ln(1 − p jkt ) h jkt = − lim . △ t △ t → 0 9
Ouwens Model • This model is based on the parameters of the parametric survival curves. • For a given shape, φ , and scale, λ the reparameterization used by this model are � φ � ν = ln − ( φ − 1) ln( λ ) λ and, θ = φ − 1 . • These reparameterizations are then used to model the log hazard rates which are − Exponential: ln( h ( t )) = ν , − Weibull: ln( h ( t )) = ν + θ ln( t ), − Gompertz: ln( h ( t )) = ν + θ t , • The parameters of interest will be the treatments effect, d 1 k (1) and d 1 k (2) , for all treatments k . − The d 1 k (1) is related to the differences in the value of the ν for each treatment. − The d 1 k (2) is related to the differences in the value of the θ for each treatment. 10
Jansen Model • This method utilizes fractional polynomials to model the log hazard rates in the survival network meta-analysis model. • Fractional polynomials are an alternate to regular polynomials. They have the form y = β 0 + β 1 x p 1 + β 2 x p 2 + · · · , where the exponents p 1 , p 2 , . . . are restricted to be from {− 2 , − 1 , − 0 . 5 , 0 , 0 . 5 , 1 , 2 , 3 } . • The first-order fractional polynomial is ln( h kt ) = β 0 k + β 1 k t p , with t 0 = ln( t ). − For the exponential distribution β 1 k = 0 for all treatments, k . − For the Weibull distribution β 1 k � = 0 and p = 0 for all treatments, k . − For the Gompertz distribution β 1 k � = 0 and p = 1 for all treatments, k . • The parameters of interest will be the treatments effect, d 1 k (1) d 1 k (2) , for all treatments k . − The d 1 k (1) is related to the differences in the value of the β 0 k for each treatment. − The d 1 k (2) is related to the differences in the value of the β 1 k for each treatment. 11
Simulation Study
Data Generation • Binomial counts were generated using the hazard rate function for a given distribution. Table 1: Sample of generated data • Scale and shape parameters were assigned for the Study r n t Time dt b Arm distribution for each treatment 1 6 300 1 1 1 1 1 k in study j . 1 8 294 1 2 1 1 1 • The hazard rates were then 1 3 286 1 3 1 1 1 1 7 283 1 4 1 1 1 used to calculate probabilities 1 11 276 1 5 1 1 1 using 1 12 265 1 6 1 1 1 1 6 253 1 7 1 1 1 1 6 247 1 8 1 1 1 p jkt = 1 − exp( − h jkt △ t ) . 1 13 241 1 9 1 1 1 1 5 228 1 10 1 1 1 • Then the binomial counts are . . . . . . . . . . . . . . . . . . . . . . . . sampled using r jkt ∼ Bin( n jkt , p jkt ) . 12
Gompertz Simulation Table 2: Parameters used for Gompertz data generation • 100 datasets • 10 studies Treatment Scale Shape • 5 treatments 1 75.000 10.0 • 300 initial subjects 2 75.067 10.1 3 75.133 10.2 • 30 time points 4 75.200 10.3 5 75.267 10.4 13
Gompertz Simulation Table 3: Average bias and standard deviation for the treatment effects in the Ouwens models Ouwens Models Exponential Weibull Gompertz d 11(1) 0.0000 (0.0000) 0.0000 (0.0000) 0.0000 (0.0000) d 12(1) − 0 . 0013 (0.0255) 0.0270 (0.2396) 0.0091 (0.0999) 0.0014 (0.0387) − 0 . 0180 (0.3964) − 0 . 0060 (0.1611) d 13(1) − 0 . 0013 (0.0346) − 0 . 0148 (0.3279) − 0 . 0081 (0.1311) d 14(1) d 15(1) − 0 . 0045 (0.0365) − 0 . 0237 (0.3514) − 0 . 0104 (0.1291) d 11(2) — 0.0000 (0.0000) 0.0000 (0.0000) d 12(2) — − 0 . 0103 (0.0860) 0.0002 (0.0054) d 13(2) — 0.0093 (0.1433) 0.0025 (0.0088) d 14(2) — 0.0069 (0.1188) 0.0032 (0.0072) — 0.0097 (0.1266) 0.0039 (0.0069) d 15(2) 14
Gompertz Simulation Table 4: Average bias and standard deviation for the treatment effects in the Jansen models Jansen Models Exponential Weibull Gompertz d 11(1) 0.0000 (0.0000) 0.0000 (0.0000) 0.0000 (0.0000) d 12(1) − 0 . 0013 (0.0256) 0.0267 (0.2399) 0.0070 (0.0987) 0.0015 (0.0386) − 0 . 0207 (0.3963) − 0 . 0057 (0.1600) d 13(1) − 0 . 0013 (0.0345) − 0 . 0158 (0.3296) − 0 . 0077 (0.1320) d 14(1) d 15(1) − 0 . 0045 (0.0365) − 0 . 0237 (0.3526) − 0 . 0127 (0.1321) d 11(2) — 0.0000 (0.0000) 0.0000 (0.0000) d 12(2) — − 0 . 0102 (0.0861) 0.0003 (0.0054) d 13(2) — 0.0103 (0.1432) 0.0024 (0.0087) d 14(2) — 0.0073 (0.1195) 0.0031 (0.0072) — 0.0097 (0.1271) 0.0041 (0.0070) d 15(2) 15
Gompertz Simulation Table 5: Coverage for the treatment effects Models Ouwens Jansen Exponential Weibull Gompertz Exponential Weibull Gompertz d 11(1) 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.93 0.97 1.00 0.93 0.97 d 12(1) d 13(1) 1.00 0.87 0.95 1.00 0.87 0.93 d 14(1) 1.00 0.91 0.98 1.00 0.91 0.97 1.00 0.91 0.97 1.00 0.91 0.97 d 15(1) d 11(2) — 1.00 1.00 — 1.00 1.00 d 12(2) — 0.93 0.94 — 0.93 0.94 — 0.87 0.91 — 0.87 0.90 d 13(2) d 14(2) — 0.91 0.93 — 0.90 0.93 d 15(2) — 0.93 0.98 — 0.92 0.96 16
Gompertz Simulation Figure 3: Estimated hazard rates for the proportional Gompertz simulation 17
Gompertz Simulation Figure 4: Estimated hazard rates for Treatment 1 in Study 1 18
Application
Application • 4 Treatments • 7 Studies • Looking at overall survival. Figure 5: Network for the application example 19
Application Results Table 6: Summary results for the Ouwens models Ouwens Models Exponential Weibull Gompertz d 11(1) 0.0000 (0.0000) 0.0000 (0.0000) 0.0000 (0.0000) d 12(1) 0.2364 (0.1881) − 0.1258 (0.4371) − 0.1135 (0.3490) d 13(1) − 0.3163 (0.1580) − 0.4946 (0.3627) − 0.4534 (0.2955) d 14(1) 0.1905 (0.1369) 0.0565 (0.3054) 0.1397 (0.2373) — 0.0000 (0.0000) 0.0000 (0.0000) d 11(2) — 0.2594 (0.2519) 0.0671 (0.0485) d 12(2) — 0.1147 (0.2091) 0.0248 (0.0414) d 13(2) — 0.0941 (0.1645) 0.0090 (0.0293) d 14(2) DIC 1545.272 1527.678 1548.400 20
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