Causal Inference in Epidemiology UsingBayesian Methods: The Example of

Meta-Analysis of Statins and Fracture Risk

Lawrence McCandlesslmccandl@sfu.ca

Faculty of Health Sciences, Simon Fraser University, Vancouver Canada

Summer 2013

OverviewThis talk is about Bayesian Causal Inference

Bayesian Causal Inference

Meta−analysis Confounding

Meta-analysis of observational studies is a wonderful andneglected corner of causal inference.

How many people have heard of the drug called Statins?

What are Statins?

Statins are blockbuster cholesterol drug (sales of $1billion/year).

They reduce the risk of strokes and cardiovascular events by25%-30%.

There has been intensive study of the health benefits of statinsthat are not related to cardiovascular disease, such as reducingarthritis, cancer, and fractures (e.g hip fractures in the elderly).

Pharmaceutical companies can make a lot of $$$ fromsuch research.

This talk concerns a tale ofstatins and fracture risk

Scientific question: Do statins reduce the risk of fractures(e.g. hip fractures)???

FACTS:

1. In 2000, three observational studies in JAMA and Lancetshowed that statin use was associated with huge (>50%)reductions in fracture risk in elderly patients.

2. In the next 5 years, a torrent of observational studies werepublished that seemed to confirm the finding.

3. Statins increase bone formation in rats, which gives somebiological basis of causation.

HOWEVER:

1. Causal inference from observational studies is prone tobias

2. A 2007 re-analysis of 4 randomized trials of cardiovascularendpoints found NO reduction in fracture risk amongstatin users.

3. Controversy in the literature: Two epidemiologicanalyses of the same database came to oppositeconclusions about the health benefits of statins!

Study #1 Fracture OR=0.12 95% CI (0.04 - 0.41)Study #2 Fracture OR=0.59 95% CI (0.31 - 1.13).

de Vries (2006) Int J Epidemiol

The statins and fractures controversy is now a textbookexample of a "failure of epidemiology"

Postscript: Why were all the epidemiologicalstudies biased?

Widespread belief: residual unmeasured confounding

There are some confounders that they didn’t control for.

What confounders??

Measurements of a healthy lifestyle and health seekingbehaviours, including

• regular exercise• healthy diet• BMI• Smoking• Alcohol• Use of preventative therapies (Mammogram, prostate

cancer screening ...)

Unmeasured Confounding

Epidemiologists call this Health User Bias (Henessey 2000JAMA)

In fact, Statin users have lower risk of being in car crashes(Dormuth 2009, Circulation).

Research Question for Statistical Science

Suppose that there was a single binary confounder that wasunmeasured in all 17 studies.

Could confounding bias account for the discrepancy between17 observational studies versus the 4 RCTs?

Is that possible?

Sensitivity analysis in a meta-analysis

Objective: In this talk I will describe a new Bayesian procedureto adjust for bias in a meta-analysis (unmeasured confounding,and selection bias).

The procedure is an example of Bayesian bias modelling forobservational studies:

Greenland (2005) Multiple bias modelling for analysis of observational data JRoyal Stat Soc Ser A

Findings published in:

McCandless (2013) Statins and fracture risk: Can we quantify thehealthy-user effect? Epidemiology

McCandless (2012) Meta-analysis of observational studies with unmeasuredconfounders. Int J Biostatistics

The Model: How to Adjust for a MissingConfounder?

Data:

• Log relative risk estimates y1, y2, . . . , yk , with• Standard error estimates σ1, σ2, . . . , σk

for the association between a dichotomous exposure anddichotomous outcome in k different observational studies.

Studies could be case-control, cross-sectional or cohortstudies.

We assume that yj been adjusted for a set of confounders inthe j th study, which may depend on j .

The Model: How to Adjust for a MissingConfounder?

Pooling RRs, ORs and HRs:

The incidence of osteoporitic fractures is roughly 5 cases/1000person years.

When the disease incidence is low, then the the relative risk isapproximated by the odds ratio and by the hazard ratio (Jewell2000).

Thus we allow for the possiblity that some of the y1, . . . , yk areadjusted log ORs (from case-control/cross sectional studies viaunconditional logistic) or are adjusted HRs (from cohort studiesvia Cox proportional hazards models).

The Model: How to Adjust for a MissingConfounder?

Bayesian random effects meta-analysis (Carlin 1992,Gelman et al. 2003):

Assume

yj ∼ N(θ∗j , σ2j ). for j ∈ 1 : k

where σ2j is assumed to be known.

The quantity θ∗j :

The log relative risk for the association between statin use andfractures in study j , conditional the set of covariates that aremeasured in study j .

The Model: How to Adjust for a MissingConfounder?

Suppose that we assume that relative risks are estimated froma loglinear model. Following confounder model of Lin et al.(1997) Biometrics, we conceptualize

Reduced model

log P(Y = 1|X ,Cj) = α∗j + θ∗j X + ξ∗T

j Cj

Full model

log P(Y = 1|X ,Cj) = αj + θjX + ξTj Cj + λU

We can alternatively use logistic or Cox PH regression.

The Model: How to Adjust for a MissingConfounder?

When incidence of disease is low, then we have

θ∗j ≈ θj + Ω(RRj ,p1j ,p0j)

where

Ω(RRj ,p1j ,p0j) = logRRj × p1j + (1− p1j)

RRj × p0j + (1− p0j).

with bias parameters: (RRj ,p1j ,p0j)

Formula applies when θ∗j and θ∗j are obtained from logistic orCox regression models provided the disease incidence islow.

The Model: How to Adjust for a MissingConfounder?

This gives us the model:

yj ∼ N(θj + Ω(RRj ,p1j ,p0j), σ

2j

)for j ∈ 1 : k

with parameters:θ = (θ1, . . . , θk )

RR = (RR1, . . . ,RRk )

p1 = (p1,1, . . . ,p1,k )

p0 = (p0,1, . . . ,p0,k )

Not identifable because in study j we have 1 estimate yj andfour parameters (θj ,RRj ,p1,j ,p0,j)

Priors for θ and (RR , p1 , p0 )

The exposure effects θ :

θjIID∼ N(µ, τ2) for j ∈ 1 : k

µ ∼ N(0,103)

τ ∼ Uniform(0,103)

The quantities µ and τ are the usual parameters from a randomeffects meta-analysis.

Priors for θ and (RR , p1 , p0 )

The bias parameters (RR , p1 , p0 ):

log RRjIID∼ N(µRR, τ

2RR)

logit(pj0)IID∼ N(µp0τ

2p0

)

logit(pj1)IID∼ N(µp1τ

2p1

).

We require hyperparameters (µRR, τ2RR), (µp0τ

2p0

) and (µp1τ2p1

),which are taken from the litterature.

Hyperparameters (µRR, τ2RR), (µp0τ

2p0) and (µp1τ

2p1)

What prior distribution would describe the typical magnitudeconfounding from a single missing binary variable?

U =engaging in 3 or more prevantative medical practices

µRR = −0.62 and τRR = 0.27

µp0 = −0.476 and τp0 = 0.014

µp1 = −0.285 and τp1 = 0.0423

Based on Dormuth al. (2007).

→ Statin users are more likely to engage in prevantativemedical practices, and this reduces fracture risk. OR≈ 1.5 or 0.67.

MCMC Fitting and Computation

Update in 3 blocks. Two blocks are conditionally conjugate.

The posterior:

P(θ , RR , p1 , p0 , µ, τ2|y , σ) ∝

k∏j=1

exp[− 1

2σ2iyi − (θj + Ω(RRj , p1j , p0j ))2

P(θi |µ, τ)P(RRj )P(p1j )P(p0j )]×

P(µ)P(τ 2).

Three Blocks:

[θ |RR , p1 , p0 , µ, τ2|y , σ ] [µ, τ2|θ ] [RR , p1 , p0 |y , θ ],

MCMC Fitting and Computation

[θj |RRj ,p1j ,p0j , µ, τ2, yj , σj ] ∼ N

[τ2yi − Ω(RRj ,p1j ,p0j)+ σ2

j µ

σ2j + τ2

,

σ2j τ

2

σ2j + τ2

]

[µ, τ2|θ ]

→ Usual meta-analysis conditionally conjugate.

[RR , p1 , p0 |y , θ ]

→ Complicated dependence on Ω requires Metropolis steps.

Results

Guess what: Adjustment for a typical missing confounder hasno impact on the results.

Reverse Engineering the Prior distribution on BiasParameters

Suppose that statins do not reduce fracture risk, and theprotective associaiton is entirely due to a single unmeasuredconfounder.

What are the likely caracteristics of such a confounder?

We can answer this question by locking the bias-correctedpooled RR to be 1.0 and then looking at the posteriordistribution that is induced over the bias parameters.

log RR

01

23

45

6

−6 −5 −4 −3 −2 −1 0

log

OR

Posterior mean of bias parameters is (-1.65, 1.65). ORs andRRs roughly 5 or 1/5. Tremendous confounding is required todestroy association between statins and fractures.

Results

This investigation reveals that adjustment for a single missingvariable would not remove the discrepancy betweenobservational and randomized estimates.

This is important because much of the debate in the literaturecenters on the role of variabes covariates that may or may nothave been measured in different studies.

Conclusions & Going Forward

What are other hypothese for the failure of epidemiology in thecontext of statins and fractures?

• Multiple correlated unmeasued unconfounders (andextremely hard/impossible statistical problem).

• Publication bias• Selection bias because of prevalent users of statins.

McCandless (2013) Statins and fracture risk: Can we quantify thehealthy-user effect? Epidemiology (in press)

McCandless (2012) Meta-analysis of observational studies with unmeasuredconfounders. Int J of Biostatistics.