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An Introduction to Bayesian GLM Methods for CEA with Individual Data Dave Vanness, PhD Research Scientist Center for Health Economics & Science Policy Overview/Learning Objectives Introduction to Bayesian Analysis Bayes’ Rule Markov Chain Monte Carlo Analytical Example Hypothetical CEA alongside a trial (simulated dataset based on MEPS data) Bayesian Generalized Linear Models (GLM) of: Potential latent class of individuals with different response to treatment Cure and Adverse Events - Bernoulli with Probit Link Cost - Gamma with Log Link QALY - Beta with Logistic Link Counter-factual Simulation Analysis of Results Goal: The probability that a treatment is cost-effective Posterior ICER, expected net benefit and acceptability Basics of Bayesian Analysis Basics of Bayesian Analysis Y : Data – events we observe θ : Parameter – probability of event Bayes’ Rule Likelihood Posterior P(θ | Y ) = Prior L(Y | θ ) × P (θ ) θ θ θ L Y P d ( | ) × ( ) ∫ Normalizing Constant is proportional to … P(θ | Y ) α L(Y | θ ) × P(θ ) What you know now times what you knew before (PRIOR) (POSTERIOR) the likelihood of what you just observed A simple binomial example Consider a treatment that can either succeed (Yi=1) or fail (Yi=0) for individual i. Let θ be the unknown average success rate for that treatment. No Prior Knowledge Suppose you know absolutely nothing about the treatment success rate before observing a new set of data (yet to be collected). What might your (lack of) prior knowledge look like if you plotted it out? Suppose instead that you had a lot of knowledge about the treatment success rate θ and were almost certain treatment is between 60% and 70% successful. What might that look like? How do we get from this … to this? With data and a model… Suppose we observe one patient who has undergone treatment. Suppose that patient failed (rats!): Y1 = 0 The Likelihood Function (model) L(Y1 = 0 | θ) 1 0 0 1 θ Applying Bayes’ Rule P(θ) L(Y1 = 0 |θ) 1 1 X 0 0 1 θ 0 0 1 θ The Posterior P(θ|Y1=0) 2 0 0 1 θ If at first you don’t succeed … This time, we observe patient 2, whose treatment is a success (yay!) Y2 = 1 Now, we bring information with us. Our posterior P(θ|Y1=0) becomes our new prior. P(θ) L(Y2 = 1 |θ) 1 2 X 0 0 1 Our non-flat prior… θ θ 0 0 1 The likelihood of observing a success. P(θ|Y2=1,Y1=0) 1 0 0 1 θ The Bayesian Learning Process Heterogeneity This was a very simple (homogenous) model: every individual’s outcome is drawn from the same distribution. The extreme in the opposite direction (complete heterogeneity) is also fairly simple: Yi ~ Bernoulli(θi) But it’s pretty difficult to extrapolate (make predictions) when there is no systematic variation. Modeling Heterogeneity Usually, we assume there is a relationship that explains some heterogeneity in observed outcomes. The classical normal regression model: Yi = Xiβ + εi Xi is a row vector of individual covariates β is a column vector of parameters εi ~ N(0,σ2) We can also write this as: Yi ~ N(µi,,σ2), µi = Xiβ [likelihood] θ = (β, σ2) ~ P(θ) [prior] The posterior we are interested in is the product of the likelihood and prior. We will not be able to calculate this analytically Markov Chain Monte Carlo (MCMC) Good text is Markov Chain Monte Carlo in Practice by Gilks, Richardson and Spiegelhalter (Chapman & Hall) Goal is to construct an algorithm that generates random draws from a distribution that converges to P(θ|X). We then collect those draws and analyze them (take their mean, median, etc., or run them through as parameters of a cost-effectiveness simulation, etc.). Gibbs Sampling When θ is multidimensional, it can be useful to break down the joint distribution P(θ|X) into a sequence of “full conditional distributions” P(θj|θ-j,X) = L(X|θj,θ-j) P(θj|θ-j) where “-j” signifies all elements of θ other than j. We can then specify a starting vector θ-j0 and, if P(θj|θj,X) is not from a known type of distribution, we can use the Metropolis algorithm to sample from it. Running from j = 1 to M gives one full sample of θ. θ2 θ1 θ2 θ02 θ1 θ2 θ02 θ1 θ2 θ02 0 θ01 θ1 θ2 θ02 0 θ01 θ1 θ2 θ02 0 θ12 θ01 θ1 θ2 θ02 0 θ12 θ01 θ1 θ2 θ02 θ12 0 1 θ11 θ01 θ1 θ2 θ02 0 4 3 1 2 θ1 http://www.mrc-bsu.cam.ac.uk/bugs/welcome.shtml Using MCMC to Conduct CEA Using the flexibility of WinBUGS, we can explore the relationships among treatments, covariates, costs and health outcomes using Generalized Linear Models (GLM) and perform probabilistic sensitivity analysis at the same time. Obtain Markov Chain Monte Carlo samples from the posterior distribution of unknown GLM model parameters, given our data, model and prior beliefs. These "draws" can be used not simply to conduct inference (the "credible interval") but also to perform uncertainty analysis. For an introduction to the general approach, see O'Hagan A and Stevens JW. A framework for cost-effectiveness analysis from clinical trial data. Health Economics, 2001; 10(4): 303-315. Simulated CEA Dataset 800 individuals selected at random from 2,452 individuals who self-reported hypertension in the 2005 MEPS-HC Covariates were: Age Sex (Male = 1) BMI We created a latent class that equals 1 if an individual self-reported diabetes; 0 otherwise. The class variable was excluded from all analysis (assumed to be unobservable) Simulated CEA Dataset Treatment (T = 1) assigned randomly (no treatment selection bias) Ti ~ Bernoulli (0.5) Adverse events (AE = 1) : can occur in anyone, but will occur with probability 1 for individuals in the latent class who receive treatment. AEi ~ Bernoulli (PiAE) PiAE = 0.1 + 0.9*Ti*Classi “Cure” (S = 1) : 8 times more likely with treatment than without and does not depend on class. Si ~ Bernoulli (PiS) PiS = 0.8*Ti + 0.1*(1-Ti) Simulated CEA Dataset Costs are increasing in age and are higher for the treated and people with AEs, lower if treatment is successful: Ci = CiT*Ti + CiX CiT ~ Gamma(25,1/400) CiX ~ Gamma(4,1/exp(XiβC)) where Xi is a row vector consisting of: 1~Agei~Sexi~AEi~Si and βC is a column vector of parameters: 7|.03|0|1.5|-.5. Simulated Costs Simulated CEA Dataset QALYs are decreasing in age and are lower for males and people with AEs and higher for people with successful treatment: Qi ~ Beta(αi,βi) αi = βi exp(XiβQ) βi = 1.2 + 2*Ti where Xi is a row vector consisting of: 1~Agei~Sexi~AEi~Si and βQ is a column vector of parameters: 1|-.01|.25|-1|1.5 Simulated QALYs A simple CEA Sample (n=800) ICER: $74,357/QALY Population (n=2452) ICER: $79,933/QALY Population ICER by (unobserved class): Class 0 (no diabetes): $43,519/QALY Class 1 (diabetes): $589,954/QALY Latent Class and BMI Estimating Bayesian GLM Models S ~ Bernoulli(Pis) Pis = Φ(Xisβs) Xis = 1~Agei~BMIi~Sexi~Ti~Ti*(Agei~BMIi~Sexi) AE ~ Bernoulli(PiAE) PiAE = Φ(XiAEβAE) XiAE = 1~Agei~BMIi~Sexi~Ti~Ti*(Agei~BMIi~Sexi) Note: we are using non-informative (“flat”) priors, which give results comparable to Maximum Likelihood. But we could bring outside information into the prior (from other related trial or observational data, meta analysis, expert opinion, etc). BUGS Code (Probits) #Treatment Success and Adverse Event Model for ( i in 1 : 800 ){ S[i] ~ dbern(p_S[i]) p_S[i] <- phi(arg.S[i]) arg.S[i] <- max(min(S_CONSTANT + S_AGE*st_AGE[i] + S_MALE*MALE[i] + S_BMI*st_BMI[i] + S_T*T[i] + S_TxAGE*T[i]*st_AGE[i] + S_TxMALE*T[i]*MALE[i] + S_TxBMI*T[i]*st_BMI[i],5),-5) AE[i] ~ dbern(p_AE[i]) p_AE[i] <- phi(arg.AE[i]) arg.AE[i] <- max(min(AE_CONSTANT + AE_AGE*st_AGE[i] + AE_MALE*MALE[i] + AE_BMI*st_BMI[i] + AE_T*T[i] + AE_TxAGE*T[i]*st_AGE[i] + AE_TxMALE*T[i]*MALE[i] + AE_TxBMI*T[i]*st_BMI[i],5),-5) } BUGS Code (Probits) # Priors # Success coefficients S_CONSTANT ~ dnorm(0,.0001) S_AGE ~ dnorm(0,.0001) S_MALE ~ dnorm(0,.0001) S_BMI ~ dnorm(0,.0001) S_T ~ dnorm(0,.0001) S_TxAGE ~ dnorm(0,.0001) S_TxMALE ~ dnorm(0,.0001) S_TxBMI ~ dnorm(0,.0001) # Adverse event coefficients AE_CONSTANT ~ dnorm(0,.0001) AE_AGE ~ dnorm(0,.0001) AE_MALE ~ dnorm(0,.0001) AE_BMI ~ dnorm(0,.0001) AE_T ~ dnorm(0,.0001) AE_TxAGE ~ dnorm(0,.0001) AE_TxMALE ~ dnorm(0,.0001) AE_TxBMI ~ dnorm(0,.0001) Draws from the MCMC sampler for treatment Adverse Event Parameter Draws from the MCMC sampler for Treatment x BMI Adverse Event parameter Another way of looking at the draws: Posterior Density Posterior Credible Interval Node Adverse Events (Bernoulli - Probit) AGE BMI CONSTANT MALE T TxAGE TxBMI TxMALE Success (Bernoulli Probit) AGE BMI CONSTANT MALE T TxAGE TxBMI TxMALE Posterior Mean Posterior Standard Deviatio n Monte Carlo Error 2.50% median 97.50% 0.0094 -0.0119 -1.1630 -0.0121 0.6391 0.0608 0.2338 -0.0047 0.0783 0.0783 0.1181 0.1690 0.1487 0.1023 0.1043 0.2211 0.0025 0.0029 0.0054 0.0080 0.0069 0.0034 0.0037 0.0104 -0.1363 -0.1665 -1.4030 -0.3376 0.3501 -0.1457 0.0342 -0.4334 0.0090 -0.0095 -1.1610 -0.0135 0.6405 0.0620 0.2353 -0.0014 0.1647 0.1339 -0.9381 0.3178 0.9348 0.2571 0.4395 0.4152 -0.0022 -0.0012 -1.3500 0.1391 2.2460 0.0326 -0.1557 -0.2274 0.0885 0.0859 0.1214 0.1687 0.1519 0.1156 0.1131 0.2189 0.0031 0.0031 0.0055 0.0077 0.0068 0.0039 0.0040 0.0097 -0.1753 -0.1764 -1.5990 -0.1796 1.9600 -0.1936 -0.3741 -0.6529 -0.0013 0.0002 -1.3490 0.1384 2.2480 0.0336 -0.1560 -0.2242 0.1693 0.1633 -1.1180 0.4846 2.5600 0.2552 0.0684 0.2165 GLM Cost Model We model cost as a mixture of Gammas (separate distributions of cost with and without treatment). Ci = Ti*Ci1 + (1-Ti)*Ci0 Ci1 ~ Gamma(shapeC1,scaleiC1) Ci0 ~ Gamma(shapeC0,scaleiC0) Using log function to link mean cost to shape and scale parameters. Key: mean of a gamma distribution = shape/scale… Ln[mean(Ci)] = Xiβ exp(Ln[mean(Ci)]) = exp(Xiβ) mean(Ci) = exp(Xiβ) shape/scalei = exp(Xiβ) scalei = shape/exp(Xiβ) Posterior Credible Interval Posterior Mean Posterior Standard Dev. Monte Carlo Error 2.50% median 97.50% AE 1.5190 0.0737 0.0015 1.3770 1.5190 1.6690 AGE 0.1552 0.0240 0.0004 0.1098 0.1550 0.2037 BMI 0.0031 0.0230 0.0004 -0.0424 0.0032 0.0477 CONSTANT -0.8600 0.0358 0.0012 -0.9289 -0.8600 -0.7893 MALE 0.0277 0.0490 0.0015 -0.0671 0.0289 0.1231 S -0.5440 0.0775 0.0016 -0.6897 -0.5430 -0.3867 SHAPE 4.5590 0.3130 0.0029 3.9700 4.5530 5.1890 AE 0.7035 0.0253 0.0006 0.6544 0.7035 0.7544 AGE 0.0746 0.0114 0.0002 0.0522 0.0747 0.0971 BMI 0.0147 0.0121 0.0002 -0.0080 0.0144 0.0390 CONSTANT 0.2002 0.0303 0.0015 0.1382 0.2013 0.2563 MALE 0.0016 0.0229 0.0006 -0.0420 0.0015 0.0472 S -0.1725 0.0299 0.0014 -0.2294 -0.1736 -0.1133 SHAPE 18.3500 1.2890 0.0119 15.9000 18.3300 20.9600 Node Cost w/o Treatment (GammaLog) Cost w/Treatment (Gamma Log) GLM QALY Model Rescale Q to [0,1] interval by dividing by maximum possible Q (follow-up time) Qi = Ti*Qi1 + (1-Ti)*Qi0 Qi1 ~ Beta(aiQ1, bQ1) Qi0 ~ Beta(aiQ0, bQ0) We use the logit function to link mean QALYs to shape and scale parameters. Key: mean of a Beta distribution = a/(a+b) mean(Qi) = exp(Xiβ)/(1+exp(Xiβ)) mean(Qi) = ai/(ai + b) ai + b exp(Xβ) = ai exp(Xiβ) + b exp(Xiβ) ai = b exp(Xiβ) Posterior Credible Interval Node Posterior Posterior Standard Mean Dev. Monte Carlo Error 2.50% median 97.50% QALY w/o Treatment (Beta - Logit) AE AGE BMI CONSTANT MALE S β -1.1760 -0.0193 -0.0015 0.5294 0.1366 1.3100 1.1600 0.1578 0.0493 0.0479 0.0733 0.0963 0.1615 0.0756 0.0030 0.0009 0.0010 0.0023 0.0030 0.0030 0.0008 -1.4950 -0.1144 -0.0954 0.3849 -0.0565 0.9928 1.0170 -1.1740 -0.0205 -0.0017 0.5300 0.1368 1.3160 1.1580 -0.8751 0.0795 0.0927 0.6753 0.3198 1.6240 1.3110 QALY w/Treatment (Beta Logit) AE AGE BMI CONSTANT MALE S β -0.9542 -0.0092 0.0865 0.4160 0.2812 1.4720 3.4980 0.0677 0.0300 0.0305 0.0797 0.0577 0.0799 0.2322 0.0015 0.0006 0.0006 0.0041 0.0017 0.0040 0.0022 -1.0880 -0.0673 0.0298 0.2625 0.1621 1.3120 3.0580 -0.9524 -0.0088 0.0862 0.4129 0.2827 1.4760 3.4950 -0.8262 0.0484 0.1478 0.5812 0.3906 1.6230 3.9740 Counterfactual Simulations Within WinBUGS, we take the draws from the parameter posteriors and, for a hypothetical individual (X profile): Assign to No Treatment (T = 0) Assign to Treatment (T = 1) Simulate cure or no cure Simulate adverse event Simulate cost and QALY, given simulated cure and adverse event status Repeat cure, adverse event, cost and QALY simulations Repeat, say, 1,000 times and calculate average incremental costs and QALYs. The following ICERs apply to a female (Sexi = 0) of average age (z-transformed Agei = 0) at 5 different levels of z-transformed BMIi = {-2,-1,0,1,2} Posterior ICERs $50K BMI z-score -2.0 -1.0 0.0 1.0 2.0 Posterior Acceptability (BMI = -2) (WTP from $0 to $200,000) Posterior Acceptability (BMI = 2) (WTP from $0 to $200,000) Posterior Expected Net Benefit (WTP from $0 to $200,000) BMI z-score -2.0 BMI = -2 -1.0 0.0 1.0 2.0 BMI = -1 BMI = 0 BMI = 1 BMI = 2 WTP Thresholds $50K $60K $70K $90K $120K Conclusion Advantages to Bayesian Approach Statistical analysis is coherent with rational decision-making under uncertainty Inference is on parameters, not future data Therefore, can hypothesize about arbitrary “value functions” of parameters – like cost-effectiveness or net benefit Posterior draws are ready-made for inclusion in decision-analytic models MCMC is a great tool for estimating complicated models Probabilistic sensitivity analysis parameters will have correct interparameter correlation if model is correctly specified Very flexible and robust in WinBUGS Many convergence diagnostics, including nice visualization Can incorporate “prior information” from previous related studies, metaanalysis, etc. Comes under the general topic of “evidence-synthesis.” Conclusion Drawbacks to Bayesian approach Still controversial: resistance among some journals and reviewers. However, Bayesian methods are playing a key role in submissions to NICE (e.g., mixed treatment comparison meta-analysis) Choice of prior is not always clear Informative or “non-informative” may matter if observed data are sparse Some models are sensitive to what seem to be trivial differences in non-informative priors (e.g., rare events models) WinBUGS has a steep learning curve – “classical” MLE methods are easier to implement, say, in STATA References George Woodworth’s book “Biostatistics: A Bayesian Introduction” (Wiley-Interscience, 2004 ISBN: 0471468428 9780471468424 has an excellent WinBUGS tutorial (Appendix B), the text of which may be found here: http://www.stat.uiowa.edu/~gwoodwor/BBIText/AppendixBWinbugs.pdf Carlin BP, TA Louis. Bayes and Empirical Bayes Methods for Data Analysis. 2nd Edition, London: Chapman & Hall, 2000. Claxton K. The irrelevance of inference: a decision-making approach to the stochastic evaluation of health care technologies. 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