Comparison of Bayesian Network Meta-Analysis Models for Survival - - PowerPoint PPT Presentation

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
• r network meta-analysis to allow for direct or indirect comparisons
• f 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.

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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.

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Survival Distributions

• Exponential Distribution
• f (t) = 1

λ exp

− t

λ

• .
• S(t) = exp

− t

λ

• .
• h(t) = 1

λ.

• Weibull Distribution
• f (t) = φ

λ

t

λ

φ−1 exp

− t

λ

φ .

• S(t) = exp
• − t

λ

φ

.

• h(t) = φtφ−1

λφ .

• Gompertz Distribution
• f (t) = 1

λ exp t φ

• exp φ

λ

• 1 − exp t

φ

• .
• S(t) = exp

− φ

λ

• exp t

φ

• − 1

.

• h(t) = 1

λ exp t φ

• .

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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.

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Meta-Analysis vs. Network Meta-Analysis

Figure 1: Example of a Treatment Network. Here dbk is, on average, the “direct” effect between b and k.

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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, and often include Kaplan-Meier curves.

Figure 2: Example of a Kaplan-Meier Curve. Ouwens et al, 2010.

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Model Basis

• The likelihood for each of the models is

rjkt ∼ Bin(njkt, pjkt).

− rjkt: Observed number of events in interval [t, t + △t] for treatment k in study j. − njkt: Number of subjects that have not experienced an event at time t for treatment k in study j. − pjkt: Observed cumulative incidence of events in interval [t, t + △t].

• The relationship between pjkt and hjkt is

hjkt = − lim

△t→0

ln(1 − pjkt) △t .

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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 ν = ln φ λ

• − (φ − 1) ln(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 + θt,

• The parameters of interest will be the treatments effect, d1k(1) and d1k(2), for

all treatments k.

− The d1k(1) is related to the differences in the value of the ν for each treatment. − The d1k(2) is related to the differences in the value of the θ for each treatment.

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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 = β0 + β1xp1 + β2xp2 + · · · , where the exponents p1, p2, . . . are restricted to be from {−2, −1, −0.5, 0, 0.5, 1, 2, 3}.

• The first-order fractional polynomial is

ln(hkt) = β0k + β1ktp, with t0 = ln(t).

− For the exponential distribution β1k = 0 for all treatments, k. − For the Weibull distribution β1k = 0 and p = 0 for all treatments, k. − For the Gompertz distribution β1k = 0 and p = 1 for all treatments, k.

• The parameters of interest will be the treatments effect, d1k(1) d1k(2), for all treatments

k.

− The d1k(1) is related to the differences in the value of the β0k for each treatment. − The d1k(2) is related to the differences in the value of the β1k for each treatment. 11

Simulation Study

Data Generation

• Binomial counts were generated

using the hazard rate function for a given distribution.

• Scale and shape parameters

were assigned for the distribution for each treatment k in study j.

• The hazard rates were then

used to calculate probabilities using pjkt = 1 − exp(−hjkt △ t).

• Then the binomial counts are

sampled using rjkt ∼ Bin(njkt, pjkt).

Table 1: Sample of generated data

Study r n t Time dt b Arm 1 6 300 1 1 1 1 1 1 8 294 1 2 1 1 1 1 3 286 1 3 1 1 1 1 7 283 1 4 1 1 1 1 11 276 1 5 1 1 1 1 12 265 1 6 1 1 1 1 6 253 1 7 1 1 1 1 6 247 1 8 1 1 1 1 13 241 1 9 1 1 1 1 5 228 1 10 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . .

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Gompertz Simulation

• 100 datasets
• 10 studies
• 5 treatments
• 300 initial subjects
• 30 time points

Table 2: Parameters used for Gompertz data generation Treatment Scale Shape 1 75.000 10.0 2 75.067 10.1 3 75.133 10.2 4 75.200 10.3 5 75.267 10.4

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Gompertz Simulation

Table 3: Average bias and standard deviation for the treatment effects in the Ouwens models

Ouwens Models Exponential Weibull Gompertz d11(1) 0.0000 (0.0000) 0.0000 (0.0000) 0.0000 (0.0000) d12(1) −0.0013 (0.0255) 0.0270 (0.2396) 0.0091 (0.0999) d13(1) 0.0014 (0.0387) −0.0180 (0.3964) −0.0060 (0.1611) d14(1) −0.0013 (0.0346) −0.0148 (0.3279) −0.0081 (0.1311) d15(1) −0.0045 (0.0365) −0.0237 (0.3514) −0.0104 (0.1291) d11(2) — 0.0000 (0.0000) 0.0000 (0.0000) d12(2) — −0.0103 (0.0860) 0.0002 (0.0054) d13(2) — 0.0093 (0.1433) 0.0025 (0.0088) d14(2) — 0.0069 (0.1188) 0.0032 (0.0072) d15(2) — 0.0097 (0.1266) 0.0039 (0.0069)

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Gompertz Simulation

Table 4: Average bias and standard deviation for the treatment effects in the Jansen models

Jansen Models Exponential Weibull Gompertz d11(1) 0.0000 (0.0000) 0.0000 (0.0000) 0.0000 (0.0000) d12(1) −0.0013 (0.0256) 0.0267 (0.2399) 0.0070 (0.0987) d13(1) 0.0015 (0.0386) −0.0207 (0.3963) −0.0057 (0.1600) d14(1) −0.0013 (0.0345) −0.0158 (0.3296) −0.0077 (0.1320) d15(1) −0.0045 (0.0365) −0.0237 (0.3526) −0.0127 (0.1321) d11(2) — 0.0000 (0.0000) 0.0000 (0.0000) d12(2) — −0.0102 (0.0861) 0.0003 (0.0054) d13(2) — 0.0103 (0.1432) 0.0024 (0.0087) d14(2) — 0.0073 (0.1195) 0.0031 (0.0072) d15(2) — 0.0097 (0.1271) 0.0041 (0.0070)

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Gompertz Simulation

Table 5: Coverage for the treatment effects

Models Ouwens Jansen Exponential Weibull Gompertz Exponential Weibull Gompertz d11(1) 1.00 1.00 1.00 1.00 1.00 1.00 d12(1) 1.00 0.93 0.97 1.00 0.93 0.97 d13(1) 1.00 0.87 0.95 1.00 0.87 0.93 d14(1) 1.00 0.91 0.98 1.00 0.91 0.97 d15(1) 1.00 0.91 0.97 1.00 0.91 0.97 d11(2) — 1.00 1.00 — 1.00 1.00 d12(2) — 0.93 0.94 — 0.93 0.94 d13(2) — 0.87 0.91 — 0.87 0.90 d14(2) — 0.91 0.93 — 0.90 0.93 d15(2) — 0.93 0.98 — 0.92 0.96

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Gompertz Simulation

Figure 3: Estimated hazard rates for the proportional Gompertz simulation

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Gompertz Simulation

Figure 4: Estimated hazard rates for Treatment 1 in Study 1

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Application

Application

• 4 Treatments
• 7 Studies
• Looking at overall survival.

Figure 5: Network for the application example

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Application Results

Table 6: Summary results for the Ouwens models Ouwens Models Exponential Weibull Gompertz d11(1) 0.0000 (0.0000) 0.0000 (0.0000) 0.0000 (0.0000) d12(1) 0.2364 (0.1881) −0.1258 (0.4371) −0.1135 (0.3490) d13(1) −0.3163 (0.1580) −0.4946 (0.3627) −0.4534 (0.2955) d14(1) 0.1905 (0.1369) 0.0565 (0.3054) 0.1397 (0.2373) d11(2) — 0.0000 (0.0000) 0.0000 (0.0000) d12(2) — 0.2594 (0.2519) 0.0671 (0.0485) d13(2) — 0.1147 (0.2091) 0.0248 (0.0414) d14(2) — 0.0941 (0.1645) 0.0090 (0.0293) DIC 1545.272 1527.678 1548.400

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Application Results

Table 7: Summary results for the Jansen models Jansen Models Exponential Weibull Gompertz d11(1) 0.0000 (0.0000) 0.0000 (0.0000) 0.0000 (0.0000) d12(1) 0.2344 (0.1871) −0.1339 (0.4371) −0.1430 (0.3400) d13(1) −0.3146 (0.1597) −0.4943 (0.3633) −0.4459 (0.2930) d14(1) 0.1897 (0.1373) 0.0630 (0.3041) 0.1228 (0.2402) d11(2) — 0.0000 (0.0000) 0.0000 (0.0000) d12(2) — 0.2626 (0.2517) 0.0698 (0.0476) d13(2) — 0.1135 (0.2091) 0.0245 (0.0409) d14(2) — 0.0915 (0.1642) 0.0108 (0.0297) DIC 1545.420 1527.483 1548.295

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Application Results

Figure 6: Hazard ratios for the Weibull version of Ouwens model

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Application Results

Figure 7: Hazard ratios for the Weibull version of Jansen model

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Conclusion

Conclusion

• Take into consideration the requirements when using these models.
• Multiple studies per treatment.
• Large starting sample sizes in for the treatments in each study.
• Our work was limited to the fixed effects models, but there are

random effects versions available for the Ouwens and Jansen models.

• Each of the models had similar performance, but the Jansen model

allows for more flexibility due to the fractional polynomials.

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Acknowledgments

Special thanks to my advisors, Dr. Stamey and Dr. Seaman. Many thanks to our colleagues at Eli Lilly & Co., especially Michael Sonksen and Min-Hua Jen.

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References

• 1. J. W. Seaman, J. D. Stamey, D. Kahle, and S. Blair. An

Introduction to Bayesian Meta-Analysis and Network Meta-Analysis.

• 2. J. Seaman. Notes for Survival Analysis.
• 3. M. J. N. M. Ouwens, Z. Phillips, and J. P. Jansen. Network

meta-analysis of parametric survival curves. Research Synthesis Methods, 1(3-4):258-71, 2010.

• 4. J. P. Jansen. Network meta-analysis of survival data with fractional
• polynomials. BMC Medical Research Methodology, 11:61, 2011.
• 5. P. Royston and D. G. Altman. Regression Using Fractional

Polynomials of Continuous Covariates: Parsimonious Parametric

• Modelling. Journal of the Royal Statistical Society, Series C,

43(3):429-67, 1994.

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Appendix

Ouwens Model

27

Jansen Model

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Model Comparison Using DIC

Table 8: Average DIC and Standard Deviation for Ouwens Models Ouwens Models Exp Weibull Gompertz Proportional Exponential 2783.17 (36.29) 2796.79 (35.20) 2796.53 (34.60) Weibull (λ < 1) 2302.35 (30.74) 2307.37 (30.70) 2309.86 (30.79) Weibull (λ > 1) 2868.39 (37.86) 2790.44 (33.44) 2806.43 (35.69) Gompertz 6273.34 (119.70) 3392.77 (48.52) 2979.66 (35.15) Nonproportional Exp-Exp 2786.61 (37.36) 2799.57 (35.83) 2798.67 (35.07) Weibull-Weibull 2835.88 (35.63) 2792.93 (33.46) 2794.45 (35.46) Exp-Weibull 2712.83 (33.50) 2708.66 (31.47) 2693.09 (31.33) Weibull-Exp 2854.17 (31.82) 2801.30 (31.37) 2826.01 (33.23)

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Model Comparison Using DIC

Table 9: Average DIC and Standard Deviation for Jansen Models

Jansen Models Exponential Weibull Gompertz Proportional Exponential 2783.17 (35.28) 2796.83 (35.22) 2796.53 (34.60) Weibull (λ < 1) 2302.36 (30.75) 2307.38 (30.71) 2309.86 (30.79) Weibull (λ > 1) 2868.40 (37.86) 2790.42 (33.44) 2806.41 (35.77) Gompertz 6273.35 (119.70) 3392.77 (48.62) 2979.66 (35.15) Piecewise Proportional Exp-Exp 2786.62 (37.38) 2799.55 (35.90) 2798.67 (35.07) Weibull-Weibull 2835.86 (35.63) 2792.92 (33.49) 2794.45 (35.46) Exp-Weibull 2712.86 (33.49) 2708.69 (31.47) 2693.09 (31.33) Weibull-Exp 2854.14 (31.81) 2801.31 (31.36) 2826.02 (33.22)

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Model Comparison

Table 10: Model Selection Percentages using DIC

Models Ouwens Jansen Simulation Exp Weibull Gompertz Exp Weibull Gompertz Proportional Exponential 96% 3% 1% 97% 2% 1% Weibull (λ < 1) 69% 24% 7% 70% 23% 7% Weibull (λ > 1) 0% 98% 2% 0% 98% 2% Gompertz 0% 0% 100% 0% 0% 100% Piecewise Proportional Exp-Exp 94% 2% 4% 94% 2% 4% Weibull-Weibull 0% 57% 43% 0% 57% 43% Exp-Weibull 3% 0% 97% 3% 0% 97% Weibull-Exp 0% 100% 0% 0% 100% 0%

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