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)dyθ|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 aP (θ ∈ (La, Ua)|studies selected) = 1 − αL(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.

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 Comment [KM1]: This Unsealing is repeated in Unsealing #2089. Julius Streicher. Julius Streicher (Fe