## Snovit.math.umu.se

**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

*σ*2

*i*
*• *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 (ˆ

*θ, σ*2) 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 ˆ

*θ, σ*2) =

*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

*H*0 :

*θ *= 0, for each study

*a*(

*y*)

*ϕ*(

*y*)

*dz*
∫

*ya*(

*y*)

*ϕ*(

*y*)

*dy*
∫ (

*y − µa*)2

*a*(

*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

*H*0,
Hence the approximate bias-corrected P-value is:

*θobs|*studies selected

*, H*0)

*• *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, σ*2 +

*τ*2)

*P *(select

*|y*) = Φ(

*α *+

*βy*)

*P *(select

*|σ*) = Φ

*{*1 +

*β*2(1 +

*τ*2

*/σ*2)

*}*12

*P *(select)

*P *(

*σ|*select)

*P *(select)E

*σ{P *(select

*|σ*)

*−*1

*|*select

*}*
*{*1 +

*β*2(1 +

*τ*2

*/σ*2)

*}*12

*L*(

*α, β, θ, τ *) =

*− *1
log(

*τ *2 +

*σ*2)

*− *1

*{*1 +

*β*2(1 +

*τ*2

*/σ*2)

*}*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

*H*0 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

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