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Inference from incomplete data
Lecture 4 : meta analysis and publication bias
• semi-parametric model• parametric model • Literature search and systematic review of relevant • Statistical summary of each study – Study estimates ˆ
– Within-study variances σ2i
• Combining summary statistics into an overall inference – fixed effects model
– MLE = ˜
– Var{˜
Meta analysis of 15 clinical trials on the effectiveness of intravenous magnesium in acute myocardial infarction θ} = .58(.46, .73) P-value ≈ 2 × 10−6 Published conclusion: ”magnesium is an effective, safe, simple and inexpensive intervention that should be introduced into clinical practice without delay” This was soon contradicted by ISIS-4 (1995), a very large multi-centre randomized clinical trial which gave mortality • Relative risk = 1.06(0.99, 1.13) • P-value ≈ 0.09 Conclusion: there is no significant difference,
Selection Model for publication bias
• There is a population of studies (ˆθ, σ2) from which the n observed studies are a (possibly non-random) θobs θ = 0, all studies have been selected} only if the selection is random
Conjecture : the probability that a study is selected is a ⇒ P(selected | study with ˆθ, σ2) = a(y), • we don’t know the function a(y) • but if we did know a(y) then we could work out the Under the null hypothesis H0 : θ = 0, for each study a(y)ϕ(y)dz ∫ ya(y)ϕ(y)dy ∫ (y − µa)2a(y)ϕ(y)dy θ|study selected) = µaσ ∝ µa(sample size)− 12 θ|study selected) for different values of µa studies have been selected is, under H0, Hence the approximate bias-corrected P-value is: θobs|studies selected, H0) • The bias corrected P-value depends on the selection • If a(y) = 1 then pa = 1 and Pa is the usual P-value • If pa = p < 1 then Pa may be larger than the crude For a ‘worst case’ sensitivity analysis, plot P (p) against p.
(Hemni, Copas and Eguchi, 2007, Biometrics) Example: Hackshaw et al. (1997), BMJ Meta analysis of 37 case-control studies on passive smoking Exposure defined as prolonged exposure to other people’s θ = log P(lung cancer|not exposed) θ = 0.217 , CI = (.120, .326) , P-value = 2.5 × 10−5 This can be extended to confidence intervals: For given a(y), let (La, Ua) be the bias-corrected α-levelconfidence interval for θ. Then for given a P (θ ∈ (La, Ua)|studies selected) = 1 − α L(p) = inf{La|pa = p} U (p) = sup{Ua|pa = p} P (θ ∈ (L(p), U (p))|studies selected) ≥ 1 − α for all possible selection function a(y) with pa = p.
For a sensitivity analysis: plot (L(p), U (p)) against p Passive Smoking: Worst Case Confidence Intervals Probit random effects selection model
∼ N(θ, σ2 + τ2) P (select|y) = Φ(α + βy) P (select|σ) = Φ {1 + β2(1 + τ2/σ2)}12 P (select)P (σ|select) P (select)Eσ{P (select|σ)−1|select} {1 + β2(1 + τ2/σ2)}12 L(α, β, θ, τ ) = − 1 log(τ 2 + σ2) − 1 {1 + β2(1 + τ2/σ2)}12 Fix pa(α, β, θ, τ ) = p and find the profile likelihood for θ: This gives, for any p, the 95% likelihood ratio confidence Lp(θ(L)) = L Estimated log relative risk and 95% confidence limits θ|select, pa = p) for p = 1, 0.9, 0.5, 0.1 General comments
• it is impossible to adjust for publication bias unless we make some assumptions about the selection mechanism • ‘selection by P-value’ ⇒ a(y) • a(y) known ⇒ OK • a(y) cannot be estimated • a(y) unknown ⇒ test of H0 is possible but can have • Proposed sensitivity analysis : ‘worst case’ for given • see Hemni et al (2007) for a more general version Meta analysis of passive smoking studies
• standard analysis give strongly significant evidence of • but largest studies give no evidence of risk at all (trend • publication bias means that risk is exaggerated • a(y) selection model explains funnel plot trend • ‘a(y) unknown’ kills all evidence of risk • sensitivity analysis suggests that evidence is significant only when p > 0.7 i.e. if there are less than about 16 References
accumulated evidence on lung cancer and environmental tobacco smoke. British Medical Journal, 315, 980-988.
Hemni M, Copas JB, Eguchi S. (2007) Confidence intervals and P-values for meta-analysis with publication bias.
Biometrics, 63, 475-482.
ISIS-4 Collaborative Group (1995) A randomized factorial trial assessing early oral captopril, oral mononitrate and intravenous magnesium sulphate in 58,050 patients with suspected myocardial infarction. Lancet, 345, 669-685.
Yusuf S, Koon T, Woods K. (1993) Intravenuos magnesium in acute myocardial infarction: an effective, safe, simple and inexpensive intervention. Circulation, 87, 2043-2046.

Source: http://snovit.math.umu.se/Aktuellt/vinterkonf/v-konf-11/Copas4.pdf