Meta-analysis The EBM workshop A.A.Haghdoost, MD; PhD of Epidemiology [email protected]

Meta-analysis

The EBM workshopThe EBM workshop

A.A.Haghdoost, MD; PhD of EpidemiologyA.A.Haghdoost, MD; PhD of [email protected]@kmu.ac.ir

EBM workshopEBM workshop Meta analysisMeta analysis A.A.HaghdoostA.A.Haghdoost

Definition

Meta-analysis: a type of systemic review that uses statistical techniques to quantitatively combine and summarize results of previous research

A review of literature is a meta-analytic review only if it includes quantitative estimation of the magnitude of the effect and its uncertainty (confidence limits).


Function of Meta-Analysis(1)

1-Identify heterogeneity in effects among multiple studies and, where appropriate, provide summary measure

2-Increase statistical power and precision to detect an effect

3-Develop ,refine, and test hypothesis

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Function of Meta-Analysis(2)

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4-Reduce the subjectivity of study comparisons by using systematic and explicit comparison procedure

5-Identify data gap in the knowledge base and suggest direction for future research

6-Calculate sample size for future studies


Historical background

Ideas behind meta-analysis predate Glass’ work by several decades– 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” (p. 99).

• 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)


Basic concepts

The main outcome is the overall magnitude of the effect.

It's not a simple average of the magnitude in all the studies.

Meta-analysis gives more weight to studies with more precise estimates. – The weighting factor is 1/(standard error)2.


Main magnitude of effects

DescriptiveDescriptiveMeanPrevalence

AnalyticalAnalyticalAdditiveAdditive

Mean differenceStandardized mean differenceRisk, rate or hazard differenceCorrelation coefficient

MultiplicativeMultiplicativeOdds ratio, Risk, Rate or Hazard Ratio


Statistical concepts(1)

You can combine effects from different studies only when they are expressed in the same units.

Meta-analysis uses the magnitude of the effect and its precision from each study to produce a weighted mean.



The impact of fish oil consumption on Cardio-vascular diseases


Forest plot

the graphical display of results from individual studies on a common scale is a “Forest plot”.

In the forest plot each study is represented by a black square and a horizontal line (CI:95%).The area of the black square reflects the weight of the study in the meta-analysis.

A logarithmic scale should be used for plotting the Relative Risk.


Forest plot



There are two basic approach to Quantitative meta –analysis:

Weighted-sum

Fixed effect model

Random effect model

Meta-regression model


Fixed effect model

General Fixed effect model- the inverse variance – weighted method

Specific methods for combining odds ratio

Mantel- Haenszel method

Peto’s method

Maximum-Likelihood techniques

Exact methods of interval estimation


Fixed effect model

In this model, all of the observed difference between the studies is due to chance

Observed study effect=Fixed effect+ error

Xi= θ + ei ei is N (0,δ2 )

Xi = Observed study effect

θ = Fixed effect common to all studies


General Fixed effect model

Ť=∑ wiTi/ ∑ wi

The weights that minimize the variance of Ť are inversely proportional to the conditional variance in each study

Wi=1/vi

Var(Ť)=1/ ∑ wi


Mantel- Haenszel method

Each study is considered a strata.

Ť=∑ai di / ni / ∑bi ci /ni


Random effect model

The “random effect” model, assumes a different underlying effect for each study.

This model leads to relatively more weight being given to smaller studies and to wider confidence intervals than the fixed effects models.

The use of this model has been advocated if there is heterogeneity between study results.


Source of heterogeneity

Results of studies of similar interventions usually differ to some degree.

Differences may be due to:- inadequate sample size- different study design- different treatment protocols- different patient follow-up- different statistical analysis- different reporting- different patient response


An important controversy has arisen over whether the primary objective a meta-analysis should be the estimation of an overall summary or average effect across studies (a synthetic goal)

or the identification and estimation of differences among study-specific effects (analytic goal)


Test of Homogeneity

This is a test that observed scatter of study outcomes is consistent with all of them estimating the same underlying effect.

Q= X2homo=∑i=1

nwi (mi -M)2

df=n-1

wi =weight

M=meta analytic estimate of effect

mi =effect measure of each study


Dealing with statistical heterogeneity

The studies must be examined closely to see if the reason for their wide variation in effect. If it’s found the analysis can be stratified by that factor.

Subgroup analysisExclusion of studyChoose another scale

Random effect modelMeta-regression


Random effect model

Assume there are two component of variability:

1)Due to inherent differences of the effect being sought in the studies (e.g. different design, different populations, different treatments, different adjustments ,etc.) (Between study)

2)Due to sampling error (Within study)


Random effect model

There are two separable effects that can be measured

The effect that each study is estimating

The common effect that all studies are estimating

Observed study effect=study specific (random )effect + error


Random effect model

This model assumes that the study specific effect sizes come from a random distribution of effect sizes with a fixed mean and variance.

There are five approach for this model:Weighted least squares

Un-weighted least squares

Maximum likelihood

Restricted Maximum likelihood

Exact approach to random effects of binary data.


Random effect

Xi= θi + ei ei is N (0,δ2 )

Xi = Observed study effect

θi = Random effect specific to each study θi =U+di

U=Grand mean (common effect)

di is N (0, 2ד ) – Random term


Weighted least squares for Random Effect

Ŵ=∑wi/k

S2w=1/k-1(∑wi

2-k Ŵ2)

U=(k-1)(Ŵ-S2w/kŴ)

0=2ד if Q<k-1

2ד =(Q-(k-1))/U if Q>k-1

wi* = 1/var.+ 2ד var.=within study variances


Weighted least squares for Random Effect (WLS)

Ť.RND=∑ wi* Ti/ ∑ wi*

Var(Ť.RND)=1/ ∑ wi*

Where Ti is an estimate of effect size and θi

is the true effect size in the ith study

Ti = θi +ei ei is the error with which Ti estimates θi

var(Ti)= דθ2 +vi


random versus fixed effect models

Neither fixed nor random effect analysis can be considered ideal.

Random effect models has been criticized on grounds that unrealistic distributional assumption have to be made.

Random effect models are consistent with the specific aims of generalization.


Peto’s advocates

He suggested a critical value .01 instead of usual .05 to decide whether a treatment effect is statistically significant for a fixed effect model.

This more conservative approach has the effect of reducing the differences between fixed and random effect models.


Meta-regression

If more than two groups of studies have been formed and the characteristic used for grouping is ordered, greater power to identify sources of heterogeneity may be obtained by regressing study results on the characteristic .

With meta-regression, it is not necessary or even desirable to groups the studies.

The individual study results can be entered directly in the analysis.


Meta-Regresion

1- meta-Regression model( extension of fixed effect model)

2- Mixed model( extension of random effect model)


Fixed-effects regression

Θi=B0+B1xi1+...+Bpxip

It’s the covariate predictor variables that are responsible for the variation not a random effect; the variation is predictable, not random.


Mixed model

Θi=B0+B1xi1+...+Bpxip+ui

This model assumes that part of the variability in true effects is unexplainable by the model.


Between studies variation

You can and should allow for real differences between studies–heterogeneity–in the magnitude of the effect.

– The τ2 statistic quantifies % of variation due to real differences.


Fixed effects model and heterogeneity

In fixed-effects meta-analysis, you do so by testing for heterogeneity using the Q statistic.

If p<0.10, you exclude "outlier" studies and re-test, until p>0.10.

When p>0.10, you declare the effect homogeneous.

But the approach is unrealistic, limited, and suffers from all the problems of statistical significance.


In random-effect meta-analysis, you assume there are real differences between all studies in the magnitude of the effect.

The "random effect" is the standard deviation representing the variation in the true magnitude from study to study.

You need more studies than for traditional meta-analysis.

The analysis is not available in a spreadsheet.

Random effects model and heterogeneity


Fixed effects models: within-study variability

– "Did the treatment produce benefit on average

in the studies at hand?"

Random effects models: between-study and

within-study variability

– "Will the treatment produce benefit ‘on

average’?"

Concept of analysis in random versus fixed effect models


Limitations

It's focused on mean effects and differences between studies. But what really matters is effects on individuals.

(Aggression bias)

A meta-analysis reflects only what's published or searchable.


Aggregation bias

Relation between group rates or and means may not resemble the relation between individual values of exposure and outcome.

This phenomenon is known as aggregation bias or ecologic bias.


Ecological fallacy

BP

Education


Meta-analysis of neoadjuvant chemotherapy for cervical cancer

Word of Mouth14%

Trial Registers 14%

Hand Searching14%

Medline/Cancerlit

58%


Type of reporting

Published in full47%

Published as abstract24%

Unpublished24%

Ongoing5%


Selection bias in Meta analysis

English language bias Database biasPublication biasBias in reporting of data Citation biasMultiple publication biasSample size


Publication bias

The results of a meta-analysis may be biased if the included studies are a biased sample of studies in general.

The classic form of this problem is publication bias, a tendency of journals to accept preferentially papers reporting an association over papers reporting no association


Publication bias

If such a bias is operating, a meta-analysis based on only published reports will yield results biased away from the null.

Because small studies tend to display more publication bias, some authors attempt to avoid or minimize the problem by excluding studies below a certain size.


Some meta-analysts present the effect magnitude of all the studies as a funnel plot, to address the issue of publication bias.

A plot of 1/(standard error) vs effect magnitude has an inverted funnel shape.

Asymmetry in the plot can indicate non-significant studies that weren’t published.


Funnel plot

effectmagnitude

0

1/SE

“funnel” ofunbiasedstudies


Funnel plot


1- Linear Regression Approach (Egger’s method)

SND=a + b. precisionSND=OR/SE

The intercept “a” provides a measure of asymmetry- the larger its deviation from zero the more pronounced the asymmetry.

Measures of Funnel Plot Asymmetry


2- A rank correlation test

This method is based on association between the size of effect estimates and their variance. If publication bias is present, a positive correlation between effect size and variance emerges because the variance of the estimates from smaller studies will also be large.

Measures of Funnel Plot Asymmetry


Funnel plot


Key Messages

Funnel plot asymmetry was found in 38% of meta-analyses published in leading general medicine journals and in 13% of reviews from the Cochrane Database of Systematic Reviews.

Critical examination of systematic reviews for publication and related biases should be considered a routine procedure.


Sources of Funnel Plot asymmetry

Selection BiasTrue HeterogeneitySize of effect differs according to study size:

– Intensity of interventions– Difference on underlying risk– Data irregularities– Poor methodological design of small studies– Inadequate analyses– Fraud– Artefactual– Choice of effect measure– Chance


Sample size as source of bias

Consider a hypothetical literature summary stating, “of 17 studies to date, 5 have found a positive association,11 have found no association, and 1 has found a negative association; thus, the preponderance of evidence favors no association”.

Mere lack of power might cause most or all of the study results to be reported as null.


Quality score

Some meta-analysts score the quality of a study.– Examples (scored yes=1, no=0):

• Published in a peer-reviewed journal?• Experienced researchers?• Research funded by impartial agency? • Study performed by impartial researchers?• Subjects selected randomly from a population?• Subjects assigned randomly to treatments?• High proportion of subjects entered and/or finished the

study?• Subjects blind to treatment?• Data gatherers blind to treatment?• Analysis performed blind?


Quality score

Use the score to exclude some studies, and/or…

Include as a covariate in the meta-analysis, but…

Some statisticians advise caution when using quality.


Quality scoring

A very common practice is to weight studies on a quality score usually based on some subjective assignment .

For example, 10 quality points for a cohort design, 8 points for a nested case control design, and 4 points for a population based case control design.


Quality scoring

Quality scoring submerges important information by combining disparate study features into a single score.

It also introduces an unnecessary and somewhat arbitrary subjective element in to the analysis.


Quality scores as weighing factors

study weight=1/var.

Quality adjusted weight= quality score /var.


The judgment that the studies should or should not be combined should be stated and justified explicitly.

There is some of a tendency to make this judgment on the basis of the quantitative results, but it’s critical to make a qualitative judgment.

Quality scores


What is an IPD Meta-analysis?

Involves the central collection, checking and analysis of updated individual patient data

Include all properly randomised trials, published and unpublished

Include all patients in an intention-to-treat analysis


IPD Meta-analysis

Individual patient data used

Analysis stratified by trial

IPD does not mean that all patients are combined into a single mega trial


IPD Analyses

Collect raw data from related studies, whether or not the studies collaborated at the design stage, exposures measures and other covariates that can be applied uniformly across the studies combined.

The major advantage of a IPD over an MA is the use of individual-based rather than group-based data.


sensitivity analysis

In sensitivity analysis, the sensitivity of inference to variations in or violations of certain assumptions is investigated.

For example, the sensitivity of inference to the assumption about the bias produced by failure to control for smoking can be checked by repeating the meta-analysis using other plausible values of the bias.



If such reanalysis produces little change in an inference, one can be more confident that the inference is insensitive to assumptions about confounding by smoking.

In influence analysis, the extent to which inferences depend on a particular study or group of studies is examined; this can be accomplished by varying the weight of that study or group.



Thus , in looking at the influence of a study, one could repeat the meta-analysis without the study, or perhaps with half its usual weight .

If change in weight of a study produces little change in an inference, inclusion of the study can not produce a serious problem, even if unquantified biases exist in the study


Sensitivity and influence analysis

On the other hand, if an inference hinges on a single study or group of studies, one should refrain from making that inference


conclusion

Most meta-analysis will require from each study both a point estimate of effect and an estimate of its standard error .

A point estimate accompanied only by a P value will generally not provide for accurate computation of a standard error estimate, and should not be considered sufficient for reporting purposes.


Like large epidemiologic studies, meta-analysis run the risk of appearing to give results more precise and conclusive that warranted.

The lager number of subjects contributing to a meta-analysis will often lead to very narrow confidence intervals for the effect estimate.

Over conclusion

Documents

Meta-analysis The EBM workshop A.A.Haghdoost, MD; PhD of Epidemiology [email protected]