Inference from incomplete data
Lecture 4 : meta analysis and publication bias
semi-parametric modelparametric 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 = ˜
Meta analysis of 15 clinical trials on the effectiveness of intravenous magnesium in acute myocardial infarction θ} = .58(.46, .73) P-value 2 × 106 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)dzya(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 × 105 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) = Φ {1 + β2(1 + τ22)}12 P (select)P (σ|select) P (select)Eσ{P (select)1|select} {1 + β2(1 + τ22)}12 L(α, β, θ, τ ) = 1 log(τ 2 + σ2) 1 {1 + β2(1 + τ22)}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.



UNPUBLISHED No. 07-4602 Appeal from the United States District Court for the EasternDistrict of North Carolina, at Raleigh. Malcolm J. Howard, SeniorDistrict Judge. (5:06-cr-00007-H)Before KING, Circuit Judge, HAMILTON, Senior Circuit Judge, andHenry F. FLOYD, United States District Judge for the District ofSouth Carolina, sitting by designation. Affirmed by unpublished per curiam opinion.

#2050 the yom kippur war and the abomination of desolation – the post-world war ii u.s. waxing great toward the south and toward the east as a second syria/antiochus iv epiphanes, part 309, nuremberg day of judgment, (xii), julius streicher and the second feast of purim

#2050 The Yom Kippur War and the Abomination of Desolation – The post-World War II U.S. waxing great toward the South and toward the East as a second Syria/Antiochus IV Epiphanes, part 309, Nuremberg Day of Judgment, (xii), Julius Streicher and the second Feast of Purim Comment [KM1]: This Unsealing is repeated in Unsealing #2089. Julius Streicher. Julius Streicher (Fe

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