1. University of Wisconsin School of Medicine and Public Health
An Introduction to Bayesian GLM Methods for Cost-Effectiveness Analysis of Primary Data
Dave Vanness, Ph.D.
University of Wisconsin School of Medicine and Public Health

2. Introduction
In cost-effectiveness analysis conducted alongside clinical trials, it is common to simply calculate the Incremental Cost-Effectiveness Ratio (ICER) or Net Benefit (NB) directly from average costs and outcomes observed in treatment groups.
In a classical (“frequentist”) inference, we would construct confidence intervals for the ICER or NB.
“95% confidence” does NOT express the probability that an unknown (e.g., the "true" NB) lies within a specific confidence interval.
Rather, it represents the likelihood that your procedure of calculating such intervals would capture the true parameter value about 95% of the time when an experiment is repeated over and over.

3. Is Inference Irrelevant in CEA?
“If the objective is to maximise health gain for a given budget, then programmes should be selected based on the posterior mean net benefit irrespective of whether any differences are regarded as statistically significant or fall outside a Bayesian range of equivalence. This is because one of the mutually exclusive alternatives must be chosen and this decision cannot be deferred. The opportunity costs of failing to make the correct decision based on the mean are symmetrical and the historical accident that dictates which of the alternatives is regarded as current practice is irrelevant.”
-- Karl Claxton, 1999 p

4. Overview/Learning Objectives
Introduction to Bayesian Analysis
Probability as Uncertainty
Bayes’ Rule
Markov Chain Monte Carlo
Analytical Example
Hypothetical CEA alongside a trial (simulated dataset)
Generalized Linear Models (GLM) of:
Cure and Adverse Events - Bernoulli with Probit Link
Cost - Gamma with Log Link
QALY - Beta with Logistic Link
Decision-Theoretic Analysis of Results
Goal: The probability that a treatment is cost-effective
Posterior ICER, expected net benefit and acceptability

5. Introduction to Bayesian Analysis

6. Probability as Uncertainty
1654 – Pascal and Fermat (The “Points Problem”) – how to divide winnings when a sequence of games is interrupted.
stochastic 1662, "pertaining to conjecture," from Gk. stokhastikos "able to guess, conjecturing," from stokhazesthai "guess," from stokhos "a guess, aim, target, mark," lit. "pointed stick set up for archers to shoot at" (see sting). [
1701?-1761, Rev. Thomas Bayes

7. Philos. Trans. R. Soc. London, 1763, 53: 370-418.

8. Bayes’ Rule
Posterior
Likelihood
Prior
Normalizing Constant

9. Is proportional to…

10. A Simple Bayesian Hierarchical Model
Let Yi = 1 if a treatment is successful for individual i, Yi = 0 otherwise.
Yi ~ Bernoulli (θ)
θ = P(Yi = 1) for all i.
All individuals are “exchangeable” members of the population with unknown probability of success, θ.
As Bayesians, we represent uncertainty about θ with a probability distribution:
θ ~ P(θ) (prior: before observing data)
θ ~ P(θ|Y) (posterior: after observing data)

11. P(θ) Pr(Yi = 1) = E[Yi] = θ θ 1
Prior beliefs for θ (must integrate to 1) θ 1

12. Suppose we observe one individual…
Suppose Y1 = 0.
What is P(Y2 = 1 | Y1 = 0)?
It’s still θ – but do we now have the same beliefs about θ that we had before?
No! We apply Bayes’ Rule to update our prior beliefs.

13. The Likelihood Function
L(Y1 = 0 | θ) 1 θ 1

14. Applying Bayes’ Rule
P(θ)
L(Y1 = 1 | θ) 1 1 X θ θ 1 1

15. P(θ|Y1 = 1) 1 = θ 1
This doesn’t integrate to 1.

16. Just a normalizing constant: P(Y)

17. L(Y1 = 1 | θ) 1 θ 1

18. 1 ÷
P(θ|Y1) = θ θ 1
Now it integrates to 1

19. Now, we observe one more…
Suppose Y2 = 1.
Now, what is P(Y3 = 1 | Y2 = 1)?
It’s still θ.
This time, we brought some information with us.
Our posterior P(θ|Y1) becomes our new prior, and we apply Bayes’ Rule again.

20. P(θ)
L(Y1 = 1 |θ) 2 1 X θ θ θ 1 1

21. 1
P(θ|Y2) α θ 1

22. Conjugate Analysis
It happens to be that the prior and posterior distributions you just saw are Beta distributions.
The likelihood function was Bernoulli (binomial if we observed multiple individuals at once).
If you multiply a Beta prior distribution by a Bernoulli/Binomial Likelihood function, you get back a Beta posterior distribution.
Posterior(θ|Data) = Beta(NS + a, NF + b)
NS = number of observed successes (say, 6)
NF = number of observed failures (say, 3)
Where prior is Beta(a, b) [a = b = 1 in our example]

23. Posterior(θ) = Beta(7,4)
Prior(θ) = Beta(1,1)

24. Markov Chain Monte Carlo
The Metropolis-Hastings algorithm uses distributions we already know to generate random draws from a Markov Chain whose stationary distribution is 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.).

25. P(θj|θ-j,X) = L(X|θj,θ-j) P(θj|θ-j)
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|θ-j,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.

26. θ2
θ02 1 5 4 2 3 θ1

27. Heterogeneity
Our first example was a very simple and 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.

28. Modeling Heterogeneity
Usually, we assume there is a systematic relationship that explains some of the heterogeneity in observed outcomes.
The classical normal regression model (which we usually estimate with Ordinary Least Squares) can be thought of as a hierarchical 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
θ = {β, σ2} ~ P(θ)
P(θ) summarizes our knowledge about the joint distribution of unknown parameters.
β is probably multidimensional; σ2 is a variance term – has to be positive; this is probably a weird mixture of distributions. Yikes!

30. Analytical Example: Cost-Effectiveness Analysis

31. Using MCMC to Conduct CEA
We can use Markov Chain Monte Carlo in WinBUGS to estimate models of virtually any type.
Draws from the posterior distributions can be used to conduct inference, test simple hypotheses – or can become inputs for policy-relevant simulations.
Using the flexibility of WinBUGS, we can also explore the relationships among treatments, covariates, costs and health outcomes using regression analysis with Generalized Linear Models (GLM) and perform probabilistic sensitivity analysis at the same time.

32. Simulated CEA Dataset
We use a combination of real-world variables and simulated relationships.
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)

33. Simulated CEA Dataset Treatment (T = 1) Ti ~ Bernoulli (0.5)
Adverse events (AE = 1)
AEi ~ Bernoulli (PiAE)
PiAE = *Ti*Classi
“Cure” (S = 1)
Si ~ Bernoulli (PiS)
PiS = 0.8*Ti + 0.1*(1-Ti)

34. CiX ~ Gamma(4,1/exp(XiβC))
Simulated CEA Dataset
Costs were simulated from Gamma-distributions as follows:
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.

35. Simulated Costs by Treatment Group

36. Simulated CEA Dataset QALYs were simulated from Beta distributions:
Qi ~ Beta(αi,βi)
αi = βi exp(XiβQ)
βi = *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

37. Simulated QALYs by Treatment Group

38. -> t = 0
Variable | Obs Mean Std. Dev Min Max
c |
q |
t |
age |
male |
ae |
s |
bmi |
class |
-> t = 1
c |
q |
t |
age |
male |
ae |
s |
bmi |
class |

39. 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

40. BMI by Class

41. GLM Cure and Adverse Event Models (Bernoulli with Probit Link)
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 through meta-analysis.

42. What the BUGS Code Looks Like
for ( i in 1 : 800 ){
S[i] ~ dbern(p_S[i])
p_S[i] <- phi(arg.S[i])
arg.S[i] <- max(min(bS[1] + bS[2]*st_AGE[i] + bs[3]*MALE[i] + bS[4]*st_BMI[i] + bS[5]*T[i]
+ bS[6]*T[i]*st_AGE[i] + bS[7]*T[i]*MALE[i]
+ bS[8]*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(bAE[1] + bAE[2]*st_AGE[i]
+ bAE[3]*MALE[i] + bAE[4]*st_BMI[i] +
bAE[5]*T[i] + bAE[6]*T[i]*st_AGE[i] +
bAE[7]*T[i]*MALE[i] +
bAE[8] *T[i]*st_BMI[i],5),-5) }
Note: For complete code,

43. GLM Cost Model (Gamma with Log Link)
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,scaleC1)
Ci0 ~ Gamma(shapeC0,scaleC0)
Using log function to link mean cost to shape and scale parameters.
Ln[mean(C)] = Xβ
exp(Ln[mean(C)]) = exp(Xβ)
mean(C) = exp(Xβ)
shape/scale = exp(Xβ)
scale = r/exp(Xβ)

44. GLM QALY Model (Beta with Logit Link)
Rescale Q to [0,1] interval by dividing by maximum possible Q (follow-up time)
Qi = Ti*Qi1 + (1-Ti)*Qi0
Qi1 ~ Beta(aQ1, bQ1)
Qi0 ~ Beta(aQ0, bQ0)
We use the logit function to link mean QALYs to the Beta parameters.
mean(Q) = exp(Xβ)/(1+exp(Xβ))
mean(Q) = a/(a + b)
a + a exp(Xβ) = a exp(Xβ) + b exp(Xβ)
a = b exp(Xβ)

45. Posterior Credible Interval Posterior Standard Deviation
Posterior Inference
Posterior Credible Interval
Node
Posterior Mean
Posterior Standard Deviation
Monte Carlo Error
2.50%
median
97.50%
Adverse Events (Bernoulli - Probit)
AGE
0.0094
0.0783
0.0025
0.0090
0.1647
BMI
0.0029
0.1339
CONSTANT
0.1181
0.0054
MALE
0.1690
0.0080
0.3178 T
0.6391
0.1487
0.0069
0.3501
0.6405
0.9348
TxAGE
0.0608
0.1023
0.0034
0.0620
0.2571
TxBMI
0.2338
0.1043
0.0037
0.0342
0.2353
0.4395
TxMALE
0.2211
0.0104
0.4152
Cure (Bernoulli - Probit)
0.0885
0.0031
0.1693
0.0859
0.0002
0.1633
0.1214
0.0055
0.1391
0.1687
0.0077
0.1384
0.4846
2.2460
0.1519
0.0068
1.9600
2.2480
2.5600
0.0326
0.1156
0.0039
0.0336
0.2552
0.1131
0.0040
0.0684
0.2189
0.0097
0.2165

49. Posterior Credible Interval Posterior Standard Deviation
Posterior Inference
Posterior Credible Interval
Node
Posterior Mean
Posterior Standard Deviation
Monte Carlo Error
2.50%
median
97.50%
Cost w/o Treatment (Gamma-Log) AE
1.5190
0.0737
0.0015
1.3770
1.6690
AGE
0.1552
0.0240
0.0004
0.1098
0.1550
0.2037
BMI
0.0031
0.0230
0.0032
0.0477
CONSTANT
0.0358
0.0012
MALE
0.0277
0.0490
0.0289
0.1231 S
0.0775
0.0016
SHAPE
4.5590
0.3130
0.0029
3.9700
4.5530
5.1890
Cost w/Treatment (Gamma - Log)
0.7035
0.0253
0.0006
0.6544
0.7544
0.0746
0.0114
0.0002
0.0522
0.0747
0.0971
0.0147
0.0121
0.0144
0.0390
0.2002
0.0303
0.1382
0.2013
0.2563
0.0229
0.0472
0.0299
0.0014
1.2890
0.0119

50. Posterior Credible Interval Posterior Standard Deviation
Posterior Inference
Posterior Credible Interval
Node
Posterior Mean
Posterior Standard Deviation
Monte Carlo Error
2.50%
median
97.50%
QALY w/o Treatment (Beta - Logit) AE
0.1578
0.0030
AGE
0.0493
0.0009
0.0795
BMI
0.0479
0.0010
0.0927
CONSTANT
0.5294
0.0733
0.0023
0.3849
0.5300
0.6753
MALE
0.1366
0.0963
0.1368
0.3198 S
1.3100
0.1615
0.9928
1.3160
1.6240 β
1.1600
0.0756
0.0008
1.0170
1.1580
1.3110
QALY w/Treatment (Beta - Logit)
0.0677
0.0015
0.0300
0.0006
0.0484
0.0865
0.0305
0.0298
0.0862
0.1478
0.4160
0.0797
0.0041
0.2625
0.4129
0.5812
0.2812
0.0577
0.0017
0.1621
0.2827
0.3906
1.4720
0.0799
0.0040
1.3120
1.4760
1.6230
3.4980
0.2322
0.0022
3.0580
3.4950
3.9740

51. Subgroup Simulations
Within WinBUGS, we take the draws from the parameter posteriors and, for a hypothetical individual (X profile):
Assign to No Treatment (T = 0)
Simulate cure or no cure
Simulate adverse event
Simulate cost and QALY, given simulated cure and adverse event status
Assign to Treatment (T = 1)
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}

52. Posterior ICERs by BMI z-score

53. Posterior Acceptability (BMI = -2) (WTP from $0 to $200,000)

54. Posterior Acceptability (BMI = 2) (WTP from $0 to $200,000)

55. Posterior Net Benefit (WTP from $0 to $200,000)
BMI = -2
BMI = -1
BMI = 0
BMI = 1
BMI = 2

56. References
George Woodworth’s book “Biostatistics: A Bayesian Introduction” (Wiley-Interscience, 2004 ISBN: has an excellent WinBUGS tutorial (Appendix B), the text of which may be found here:
Carlin BP, TA Louis. Bayes and Empirical Bayes Methods for Data Analysis. 2nd Edition, London: Chapman & Hall, 2000.
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Vanness DJ and Kim WR. “Empirical Modeling, Simulation and Uncertainty Analysis Using Markov Chain Monte Carlo: Ganciclovir Prophylaxis in Liver Transplantation,” Health Economics, 2002: 11(6),