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

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

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.

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. 347-8.

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

Introduction to Bayesian Analysis

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). [http://www.etymonline.com/index.php?search=stochastic&searchmode=none]

• 1701?-1761, Rev. Thomas Bayes

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

dPYL

PYLYP

)()|(

)()|()|(

Posterior PriorLikelihood

Normalizing Constant

Bayes’ Rule

)()|()|( PYLYP

Is proportional to…

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)

0 1

0

1

P(θ)

θ

Prior beliefs for θ (must integrate to 1)

Pr(Yi = 1) = E[Yi] = θ

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.

)()|()|( PYLYP

0 1

0

1

L(Y1 = 0 | θ)

θ

The Likelihood Function

0 1

0

1

P(θ)

θ

X

0 1

0

1

L(Y1 = 1 | θ)

θ

Applying Bayes’ Rule

=

0 1

0

1

θ

This doesn’t integrate to 1.

P(θ|Y1 = 1)

dYL

PYLYP

)|(

)()|()|(

Just a normalizing constant: P(Y)

0 1

0

1

θ

2

1)()|( dPYL

L(Y1 = 1 | θ)

=

0 1

0

1

θθ

÷ 2

1

Now it integrates to 1

P(θ|Y1)

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.

0 1

0

2

θθ

P(θ)

X

0 1

0

1

L(Y1 = 1 |θ)

θ

αP(θ|Y2)

0 1

0

1

θ

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 + , NF + )– NS = number of observed successes (say, 6)– NF = number of observed failures (say, 3)– Where prior is Beta(, ) [in our

example]

Prior(θ) = Beta(1,1)

Posterior(θ) = Beta(7,4)

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.).

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.

θ1

θ02

θ2

1

2

3

4

5

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.

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!

http://www.mrc-bsu.cam.ac.uk/bugs/welcome.shtml

Analytical Example: Cost-Effectiveness Analysis

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.

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)

Simulated CEA Dataset

• Treatment (T = 1)

Ti ~ Bernoulli (0.5)• Adverse events (AE = 1)

AEi ~ Bernoulli (PiAE)

PiAE = 0.1 + 0.9*Ti*Classi

• “Cure” (S = 1)

Si ~ Bernoulli (PiS)

PiS = 0.8*Ti + 0.1*(1-Ti)

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.

Simulated Costs by Treatment Group

Simulated CEA Dataset

• QALYs were simulated from Beta distributions:

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 by Treatment Group

-> t = 0

Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- c | 390 7671.389 7850.924 477.4742 65191 q | 390 .6354654 .2557378 .0053504 .9961702 t | 390 0 0 0 0 age | 390 54.7641 5.513782 45 64 male | 390 .4871795 .5004777 0 1 ae | 390 .1230769 .3289475 0 1 s | 390 .1025641 .3037784 0 1 bmi | 390 31.78385 6.994469 16 57.4 class | 390 .2512821 .4343075 0 1

-> t = 1

Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- c | 410 18385.52 8746.241 6958.342 55654.68 q | 410 .7849 .163569 .1765754 .9979415 t | 410 1 0 1 1 age | 410 54.7561 5.482493 45 64 male | 410 .4414634 .4971683 0 1 ae | 410 .297561 .4577439 0 1 s | 410 .802439 .3986455 0 1 bmi | 410 30.90756 6.540621 17.2 56.5 class | 410 .2268293 .4192929 0 1

• 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

BMI by Class

GLM Cure and Adverse Event Models(Bernoulli with Probit Link)

S ~ Bernoulli(Pis)

Pis = Φ(Xi

sβs)Xi

s = 1~Agei~BMIi~Sexi~Ti~Ti*(Agei~BMIi~Sexi)

AE ~ Bernoulli(PiAE)

PiAE = Φ(Xi

AEβAE)Xi

AE = 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.

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, email [email protected]

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)*Ci

0

Ci1 ~ Gamma(shapeC1,scaleC1)

Ci0 ~ Gamma(shapeC0,scaleC0)

• Using log function to link mean cost to shape and scale parameters.

Ln[mean(C)] = Xβexp(Ln[mean(C)]) = exp(Xβ)

mean(C) = exp(Xβ)shape/scale = exp(Xβ)

scale = r/exp(Xβ)

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)*Qi

0

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)

        Posterior Credible Interval

NodePosterior

Mean

Posterior Standard Deviation

Monte Carlo Error 2.50% median 97.50%

Adverse Events (Bernoulli - Probit)

AGE 0.0094 0.0783 0.0025 -0.1363 0.0090 0.1647

BMI -0.0119 0.0783 0.0029 -0.1665 -0.0095 0.1339

CONSTANT -1.1630 0.1181 0.0054 -1.4030 -1.1610 -0.9381

MALE -0.0121 0.1690 0.0080 -0.3376 -0.0135 0.3178

T 0.6391 0.1487 0.0069 0.3501 0.6405 0.9348

TxAGE 0.0608 0.1023 0.0034 -0.1457 0.0620 0.2571

TxBMI 0.2338 0.1043 0.0037 0.0342 0.2353 0.4395

TxMALE -0.0047 0.2211 0.0104 -0.4334 -0.0014 0.4152

Cure (Bernoulli - Probit)

AGE -0.0022 0.0885 0.0031 -0.1753 -0.0013 0.1693

BMI -0.0012 0.0859 0.0031 -0.1764 0.0002 0.1633

CONSTANT -1.3500 0.1214 0.0055 -1.5990 -1.3490 -1.1180

MALE 0.1391 0.1687 0.0077 -0.1796 0.1384 0.4846

T 2.2460 0.1519 0.0068 1.9600 2.2480 2.5600

TxAGE 0.0326 0.1156 0.0039 -0.1936 0.0336 0.2552

TxBMI -0.1557 0.1131 0.0040 -0.3741 -0.1560 0.0684

TxMALE -0.2274 0.2189 0.0097 -0.6529 -0.2242 0.2165

Posterior Inference

        Posterior Credible Interval

NodePosterior

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 1.5190 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

Cost w/Treatment (Gamma - Log)

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

Posterior Inference

        Posterior Credible Interval

NodePosterior

Mean

Posterior Standard Deviation

Monte Carlo Error 2.50% median 97.50%

QALY w/o Treatment (Beta - Logit)

AE -1.1760 0.1578 0.0030 -1.4950 -1.1740 -0.8751

AGE -0.0193 0.0493 0.0009 -0.1144 -0.0205 0.0795

BMI -0.0015 0.0479 0.0010 -0.0954 -0.0017 0.0927

CONSTANT 0.5294 0.0733 0.0023 0.3849 0.5300 0.6753

MALE 0.1366 0.0963 0.0030 -0.0565 0.1368 0.3198

S 1.3100 0.1615 0.0030 0.9928 1.3160 1.6240

β 1.1600 0.0756 0.0008 1.0170 1.1580 1.3110

QALY w/Treatment (Beta - Logit)

AE -0.9542 0.0677 0.0015 -1.0880 -0.9524 -0.8262

AGE -0.0092 0.0300 0.0006 -0.0673 -0.0088 0.0484

BMI 0.0865 0.0305 0.0006 0.0298 0.0862 0.1478

CONSTANT 0.4160 0.0797 0.0041 0.2625 0.4129 0.5812

MALE 0.2812 0.0577 0.0017 0.1621 0.2827 0.3906

S 1.4720 0.0799 0.0040 1.3120 1.4760 1.6230

β 3.4980 0.2322 0.0022 3.0580 3.4950 3.9740

Posterior Inference

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}

Posterior ICERs by BMI z-score

BMI = -2 BMI = -1 BMI = 0 BMI = 1 BMI = 2

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

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

Posterior Net Benefit(WTP from $0 to $200,000)

BMI = -2

BMI = -1BMI = 0

BMI = 1BMI = 2

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