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## Math.nankai.edu.cn

**Meta Analysis**
**Zhezhen Jin**
A statistical technique for summarizing the results of several
Karl Pearson (1904) Averaged correlations for studies of the
eﬀectiveness of inoculation for typhoid fever
R. A. Fisher (1944) When a number of quite independent tests of
signiﬁcance have been made, it sometimes happens that although
few or none can be claimed individually as signiﬁcant, yet the
aggregate gives an impression that the probabilities are on the
whole lower than would often have been obtained by chance Source
of the idea of cumulating probability values
W. G. Cochran (1953) Discusses a method of averaging means
across independent studies Laid-out much of the statistical
foundation that modern meta-analysis is built upon (e.g., inverse
variance weighting and homogeneity testing)
Cochrane Collaboration started in 1993 (Evidence based health
Team work: based on protocol and use summary statistical
“The Cochrane Collaboration is an international network of more
than 28,000 dedicated people from over 100 countries. We work
together to help healthcare practitioners, policy-makers, patients,
their advocates and carers, make well-informed decisions about
health care, by preparing, updating, and promoting the
accessibility of Cochrane Reviews over 5,000 so far, published
online in the Cochrane Database of Systematic Reviews, part of
The Cochrane Library. We also prepare the largest collection of
records of randomised controlled trials in the world, called
CENTRAL, published as part of The Cochrane Library.”
Driven by the evidence-based medicine movement and the
Resolves controversy between conﬂicting ﬁndings
Provides reliable basis for decision making
4. Study selection (inclusion/exclusion criteria)

**Example: Marfan’s syndrome**
It is genetic disorder of the connective tissues
Usually tall, with long limb, leg, thin ﬁngers
Major causes of morbidity and mortality: cardiovascular
complication of aortic dissection and rupture
In late 1960s, blood pressure lowing medication improves survival
of general patients with acute dissection of aortic aneurysms.

With the observation: Blood pressure lowering drug used to treat
patients with aortic root dilatation related to Marfan’s syndrome.

Beta blocker therapy: make heart beats more slowly and with less
force, thereby reducing blood pressure, and also help blood vessels
Several small studies available, using echocardiography to measure
Clinically, beta-blocker therapy is used routinely for Marfan’s
Problem: No convincing evidence of long-term outcome.

*• *Observational studies ((e.g. case control, non-randomized
cohorts, cross-sectional prevalence studies, etc.)

*• *Combination of randomized and observational studies

*• *Individual patient data studies

**Outcome Measures**
Create a summary statistic that is comparable across all studies:
1. Binary data: alive/dead, diseased/non-diseased,
Risk diﬀerence, relative risk (risk ratio), odds ratio
2. Continuous data: weight loss, blood pressure
Mean diﬀerence, standardized mean diﬀerence,

*z*-statistic,
3. Survival data: time to death, time to recurrence, time to
4. Ordinal data (ordered categorical data): disease severity,

**Binary data**
Risk diﬀerence, relative risk (risk ratio), odds ratio
where

*qT *= 1

*− p *,

*n*
treated and control patients, and

*a, b, c, d *denote the number of
Relative risk and odds ratio both use logarithmic scales

**Continuous data**
Required for each group: mean, standard deviation, sample size
(

*nT − *1)

*s*2 + (

*nC − *1)

*s*2
2. Eﬀect size (standardized mean diﬀerence)
Diﬀerence of means divided by the variability of the measures

**Survival data**
Time to event arise whenever subjects were followed over time until
Problem: not everyone has an event, censoring
HR: ratio of the risk of having an event at any given time in
treatment group over the the risk of an event in the control group.

Analysis with Cox proportional hazards models
Estimates from diﬀerent studies are diﬀerent
Two possibilities: sampling error (homogeneous), true variation
Cochran’s Chi-square test and

*I*2-test, plots
Suppose there are

*K *studies with summary statistics

*θi*, astatistical test for the homogeneity

*H*0 :

*θ*1 =

*θ*2 =

*· · · *=

*θK *=

*θ*
*H*1 : At least one

*θi *is diﬀerent
, the

*χ*2 distribution with

*K − *1 degrees of freedom.

Some comments on Cochran’s

*Q *statistic
Power of the test might be very low due to small number of studies
When sample sizes in each study are very large, then

*H*0 may be
rejected even when the individual eﬀect size estimates do not
The likelihood of design ﬂaws in primary studies and publication
biases makes the interpretation of test complex.

**Higgins and Thompson’s ***I*2

*I*2 = 100(

*Q − *(

*K − *1))

*/Q*
the proportion of total variability explained by heterogeneity
Values

*< *25% be thought to be ‘low’
The eﬀect size: has two values, estimate and its standard error
In usual statistics: only one measure is available
Notation: (

*yi, si*), var(

*yi*)=

*s*2

*i*
Fixed eﬀects model and random eﬀects model
All studies share common true eﬀect size
Factors that could impact on the true eﬀect size are the same
Observed eﬀect sizes vary among studies only because a random
The random sampling error can be estimated
where

*ϵi ∼ N *(0

*, s*2) for

*i *= 1

*, *2

*, · · · , K*.

Approximate 100(1

*− α*)% conﬁdence interval for

*θ *is:

**Random Eﬀects Model (DerSimonian and Laird model)**
Studies are a random sample of a hypothetical population of
Two sources of variation: the between and within study variance.

*yi *=

*θ *+

*δi *+

*ϵi*
where

*δi ∼ N *(0

*, τ *2) and

*ϵi ∼ N *(0

*, s*2) for

*i *= 1

*, *2

*, · · · , K*. The

*τ *2 is
The random eﬀects model will usually generate a conﬁdence
interval as wide or wider than that using the ﬁxed eﬀect model.

Results from a random eﬀects model will usually be more
conservative (there are exceptions).

If

*τ *2 is known, the pooled estimate of

*θ *is given by
where

*Wi*(

*τ *) = 1

*/*(

*s*2 +

*τ *2).

Approximate 100(1

*− α*)% conﬁdence interval for

*θ *is:
As we said,

*τ *2 is unknown, and need to be estimated.

Under normal error assumptions, maximum likelihood estimate or
restricted maximum likelihood estimate can be used.

Random eﬀects models account for heterogeneity between studies,
Two approaches can address the issue: subgroup analyses and
Subgroup analyses: focus factors that are possibly diﬀerent across
studies, such as patient characteristics, study conduct
Meta-regression: include covariates to the ﬁxed and mixed eﬀects
models. Used to estimate the impact/inﬂuence of categorical
and/or continuous covariates (moderators) on eﬀect sizes or to
predict eﬀect sizes in studies with speciﬁc characteristics. A ratio
of 10:1 (studies to covariates) is recommended

**Publication Bias**
One major concern of meta-analysis is publication bias:
1. If missing studies are random, failure to include these studies
will result in less information, wider conﬁdence intervals, and
2. If missing studies are systematically diﬀerent from available
1. Large studies are likely to be published regardless of statistical
2. Moderately-sized studies are at risk for being lost, only those
with signiﬁcant results might be published
3. Small studies are at the greatest risk of being lost, studies with
small sample sizes only very large eﬀects are likely to be
signiﬁcant and those with small and moderate eﬀects are likely
How to assess publication bias and how to adjust it?
Funnel plot: is a scatter plot of sample size or other measure of
precision on the

*y*-axis versus the estimated eﬀect size on the

*x*-axis.

In the absence of publication bias the studies will be distributed
symmetrically about the combined eﬀect size
In the presence of bias, the plot may become skewed, the bottom
of the plot would tend to show a higher concentration of
studies on one side of the mean than the other
This would reﬂect the fact that smaller studies (which appear
toward the bottom) are more likely to be published if they have
larger than average eﬀects, which makes them more likely to
meet the criterion for statistical signiﬁcance
It is informal visual method, and a useful funnel plot needs a range
Diﬀerent people might interpret the same plot diﬀerently
Skewed funnel plot might be caused by factors other than

**Formal tests**
Rank correlation test (Begg and Mazumdar):

*yw*) =

*s*2

*− *1

*/*
Then the rank correlation test is based on statistic
Eggers linear regression method, quantiﬁes the bias captured by
the funnel plot using the actual values of the eﬀect sizes and their
In the Egger test, the standardized eﬀect (eﬀect size divided by
standard error) is regressed on precision (inverse of standard error)
Then ﬁt weighted linear regression either with weight 1

*/si *orunweighted and the equation

*α *is used to measure asymmetry.

If it is signiﬁcantly diﬀerent from 0, then it is concluded that there
A negative

*α *indicates that smaller studies are associated with
Small studies generally have a precision close to zero, due to their
large standard error In the absence of bias such studies would be
associated with small standardized eﬀects and large studies
associated with large standardized eﬀects
This would create a regression line whose intercept approaches the
If the intercept deviates from this expectation, publication bias
This would occur when small studies are disproportionately
If the publication bias is suspected, may model the selection
process into the model for bias correction. One possibility is to
view the problem as a missing data problem and assume that the
studies are missing with probabilities that are a function of their
lack of statistical signiﬁcance. For example,
0

*, *if

*z ≤ *1

*.*96

*pi*(

*z*) = 1

*, *if

*z > *1

*.*96
Treat

*pi*(

*z*) as missing probability and carry out analysis.

**Meta regression**
Two types of regression models are possible: ﬁxed eﬀects
meta-regression and random eﬀects mete-regression model

*yi *=

*θ *+

*XT β *+

*ϵi*
*yi *=

*θ *+

*XT β *+

*δi *+

*ϵi*
where

*δi ∼ N *(0

*, τ *2) and

*ϵi ∼ N *(0

*, s*2).

Challenging issues Studies do not report same measures. For
example, the measure for variation might be
‘Not signiﬁcant’ or ‘

*P < *0

*.*05’
Individual-patients meta-analysis (recent and still ongoing)
Example: Prevention of fractures after organ transplantation
Organ transplantation: Heart, kidney, lung, liver
Treatment: bisphosphonates, active metabolites of vitamin D
Available studies: small, no deﬁnitive clinical trial
To assess diﬀerences in bone fracture among treated and untreated

**Data sources and Searches**
Cochrane Controlled Clinical Trials Register
Unpublished abstracts for various meetings
Keywords: transplant, osteoporosis, bone loss, fracture, transplant
and calcitriol, transplant and bisphosphonates, transplant and

**Study selection**
Randomized clinical trials, treatment and control groups, fracture
Eligible treatment: oral or IV bisphosphonates (alendronate,
risedronate, pamidronate, ibandronate, zoledronic acid) or active
vitamin D analogs (calcitriol, calcidiol, 1

*α*-hydroxyvitamin D)
No restriction on sample size or specific dose of bisphosphonate or
Studies with historical controls were excluded.

Bone marrow transplants were excluded.

Treatments for bone loss prevention: hormone replacement therapy,
calcitonin, or resistence exercise were excluded
Study design, methods, subjects, interventions, fracture, and bone
Primary outcome: vertebral or nonvertebral fracture sustained
Radiographs of the thoracic and lumbar spine (LS) at baseline and
Secondary outcome: BMD measured by dual-energy x-ray
absorptiometry in grams per square centimeter at the LS and
Method of randomization, presence/absence of double-blinding,
Studies were scored between 0 and 5, with 5 as the highest quality.

685 abstracts: 607 eliminated, 42 duplicate
36 remained, and among them, 28 published
9 were not adequately randomized, 8 were excluded due to delay on
treatment and no response from the authors
Included: 11 studies with 780 participants
Prophylactic use of lidocaine after heart attack
Source: Hine, L.K., Laird, N., Hewitt, P. and Chalmers, T.C.

(1989) Meta-analytic evidence against prophylactic use of lidocaine
in Myocardial Infarction,

*Archives of Internal Medicine*,

**149**,

Sodium fuoride (NaF) with sodium monoﬂouorophosphate (SMFP)
dentrifrices for the purpose of reducing dental decay.

The outcome: decayed, missing or ﬁlled teeth score (DMFS)
Source: Johnson, M.F. (1993) Comparative eﬃcacy of NaF and
SMFP dentrifrices in caries prevention: a meta-analytic overview,

*Caries Res.*,

**27**, 328-336.

Source: http://www.math.nankai.edu.cn/~mqliu/teach/JinZhezhen-nankai2013-meta626.pdf

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