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Adjusted Likelihoods for Synthesizing Empirical Evidence from Studies that Differ in Quality and Design: Effects of Environmental Tobacco Smoke. Kerrie Mengersen, Newcastle Aust Robert Wolpert, Duke USA. Is ETS associated with lung cancer?.
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Adjusted Likelihoods for Synthesizing Empirical Evidence from Studies that Differ in Quality and Design: Effects of Environmental Tobacco Smoke Kerrie Mengersen, Newcastle Aust Robert Wolpert, Duke USA
Is ETS associated with lung cancer? 1981: First studies published (Hirayama, Garfinkel)1986: First meta-analyses (Wald, NRC)1992: EPA Review of 29 studies NHMRC Review (Australia)1994: OSHA Review2002: Boffetta review of 51 studies
Individual studies Exposed Unexposed Case Control N
A common measure: log odds ratio Cohort Studies Variety of ways to extract information about e amid nuisance parameters. Bayesian approach? Eg, Jeffrey’s prior Be(1/2,1/2) on pc|e,pc|e Then eLOR ~ N(m,s2)
A common measure: log odds ratio Case Control Studies • Only have indirect evidence about pc|e, pc|e • If we know pc and pe in target population, then • pc|e=pe|c pc / pe and similarly for pc|e • Also if the outcome is ‘rare’, the relative risk and the odds ratio are approximately equal, so which does not depend on collateral quantities
Exchangeable combination of evidence • Simple pooling: same cond’l exposure probs Cohort studies Case control studieseLOR = 0.70 ± 0.12 eLOR = 0.17 ± 0.05 • Fixed effects: same ei=e for each study iIeLOR = 0.21 ± 0.05 (RR=1.23) • Random effects: same overall eeLOR = 0.28 ± 0.20 (RR=1.32) • Boffetta (2002): Overall RR 1.25 (1.15-1.37)
Exchangeable hierarchical models • Model variation in study-specific parameters pi explicitly • Exchangeable implies cond’ly independent random vectors given the hyperparameter q • Write e as a function of q • Then the joint posterior for e, q and all pi can be factored as p(de) p(dq|e) • The marginal likelihood is thusLEHM = [ Pi Li(pi) p(dpi|q) ] p(dq|e)
Exchangeable hierarchical model likelihood Reduces to • RE for q=(m,t,s) and normal logistic conditional distributions p(dpi|q) • FE for q=e and beta conditional distributions p(dpic|e|q),p(dpie|c|q) • Simple pooling for q=(pce,pce,pce,pce) with unit point masses p(dpi|q) at pi=q.
Partially exchangeable hierarchical model Studies are exchangeable within known groups but the G groups may differ systematically among themselves e ~ p(de); q ~ p(dq|e);qg ~ p(dqg|q); qi~p(dqi|qg) Normal priors: e~N(0,t2), eg~N(e,sg2), ei~N(eg,si2)
1992 EPA results Country Gp All Studies Tiers 1-2Greece 2.01 (1.42,2.84) 1.92 (1.13,3.23)Hong Kong 1.48 (1.21,1.81) 1.61 (1.25,2.07)Japan 1.41 (1.18,1.69) 1.39 (1.16,1.66)USA 1.19 (1.04,1.35) 1.23 (1.04,1.42)W. Europe 1.17 (0.84,1.62) 1.17 (0.85,1.64)China 0.95 (0.81,1.12)
Synthesizing heterogeneous evidence • Threshold exclusion • Weighted likelihood functions • Block mixtures • Mixtures • Hierarchical models • Systematic adjustment of likelihoods
Adjusting the likelihood • Parametric adjustment:LiAdj(q) = Li(f(q,ai)eg, Shift in binomial probability parameter q by setting f(q,ai)=q+ai • Uncertain adjustment:ai is uncertain so has (informative) prior pia(dai|q), soLiAdj(q) = Li(f(q,ai) pia(dai|q)
Adjustment in the ETS studies • True fraction pijkl of ith population with case status i, exposure status j, elibility status k [eg, pices is true fraction of cases, exposed, smokers] • True classification probabilitiesqi=(qice,qice,qice,qice) • Apparent classification probabilities(qice,qice,qice,qice)
Eligibility violation Change of notation: c,C; e,E; s,S Suppress superscript i for each study qceS = aS|ces pces + aceS|ceS pceS + ae|cES pcES + ac|CeS pCeS (Similarly for qcES, qCeS ,qCES ) So we want (pces, pcEs, pCes, pCEs)
Gathering the evidence • EPA Review gives us ps for each study • Get pe|s from pe|S and K=pespES / peSpEs (3, Lee) • Take pc|es = pc|Es = Rspc|S(Rs is RR lung cancer among active smokers)Write pc=pc|s ps + pc|s pS = pc|S(psRs+pS)Thus pc|s = pcRs/psRs+1-ps)and pc|s = pc/(psRs+1-ps ) • This gives us (pis,pie|s,pic|es,pic|Es) so we can find all four required probabilities for each study
Still on eligibility violation: gathering the evidence • We still need p`S’|sLee, EPA: ~ 5% of eversmokers deny smoking.Mixed evidence of dependene on case status; we ignore this.Little evidence of dependence on exposure status; we ignore this. • EPA assigns ‘penalty points’ Ai ranging from –0.5 (bonus) to +1.0 for each study’s control of this bias. • So we takeaiS|ces= aiS|cEs = aiS|Ces = aiS|CEs = 0.05 2Ai
Misclassification of exposure • We want p`e’|E and p`E’|e • Some studies report the other probs eg pE|`e’so use Bayes theorem to invert • Friedman’83: 47% of currently nonsmoking wives have <1 hr/day exposure at home. 40-50% women with nonsmoking spouses have significant ETS exposure outside the home.Lee’92: ‘not a major issue’Jarvis et al’01: good surrogate • We take pE|`e’ = 0.25, pe|`e’=0.10
Misclassification of lung cancer • We need p`c’|c, p`C’|c • No evidence that these differ w.r.t. ETS exposure among nonsmokers • Lee: 30-40% lung cancers seen at autopsy are missed clinically • Thus we take p`c’|c=0.35. • EPA: ‘penalty points’ Bi from –0.5 to +2.5 for each study’s control of this bias. • So we take aiE|ceS = aiE|CeS = .19 2Bi -2.5aie|CES = aie|CES = .14 2Bi-2.5
Misclassification of exposure (cont.) • Lee: 22.5% report exposure: p`e’ = 0.225EPA: rates from 15% to 87% in studies • We take p`e’ = 0.36 • Thus we calculate p`E’|e = 0.19, p`e’|E=0.14 • Reduce by fraction Ci of histologically verified cases, given by the EPA • So we take aic|CeS = aic|CES = 0 aiC|ceS = aiC|cES = .35 (1-Ci)
ei Study-specific LOR Adjusted conditional ‘true’ probabilities pc|e, pc|E, pC|e, pC|E a qce, qcE, qCe, qCE Apparent classificationprobabilities nce , ncE, nCe, nCE Observed data
Hierarchical prior distribution qi can be recovered from qic=qice+qicE, qie=qice+qiCe and eiLOR so we construct a joint distribution for q from that of qc, qe, e. • log(qc/qe) ~ N(mg, 0.5) log(mg) ~ N(m 25/105, 0.5)low precision, little prior opinion on nonsmokers’ cancer rates • log(qie/qiE)~N(me=log(.36/.64)=-0.57,0.842) p`e’ 0.36; reported apparent exposure rates 15%-87%; calculate variance so that P(0.1<qie<0.75) 0.90
A tale of two studies Tier 4 case control study (Chan, Hong Kong)- evidence about qe but not qc Tier 2 cohort study (Hirayama, Japan)- evidence about qc but not qe
Conclusions • Flexible meta-analysis method that directly adjusts the likelihoods • Requires specific, explicit account of factors for which adjustment is made • Allows quantification and introspection about the impact of quality issues • Allows detailed interpretation