# A.A.Haghdoost, MD; PhD of Epidemiology

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A.A.Haghdoost, MD; PhD of Epidemiology
Meta-analysis The EBM workshop A.A.Haghdoost, MD; PhD of Epidemiology

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). EBM workshop Meta analysis

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 continued EBM workshop Meta analysis

Function of Meta-Analysis(2)
continuation 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 EBM workshop Meta analysis

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) EBM workshop Meta analysis

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. EBM workshop Meta analysis

Main magnitude of effects
Descriptive Mean Prevalence Analytical Additive Mean difference Standardized mean difference Risk, rate or hazard difference Correlation coefficient Multiplicative Odds ratio, Risk, Rate or Hazard Ratio EBM workshop Meta analysis

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. EBM workshop Meta analysis

Statistical concepts(2)
The impact of fish oil consumption on Cardio-vascular diseases EBM workshop Meta analysis

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. EBM workshop Meta analysis

Forest plot EBM workshop Meta analysis

Statistical concepts(3)
There are two basic approach to Quantitative meta –analysis: Weighted-sum Fixed effect model Random effect model Meta-regression model EBM workshop Meta analysis

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 EBM workshop Meta analysis

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 EBM workshop Meta analysis

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 EBM workshop Meta analysis

Mantel- Haenszel method
Each study is considered a strata. Ť=∑ai di / ni / ∑bi ci /ni EBM workshop Meta analysis

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. EBM workshop Meta analysis

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 EBM workshop Meta analysis

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) EBM workshop Meta analysis

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=1nwi (mi -M)2 df=n-1 wi =weight M=meta analytic estimate of effect mi =effect measure of each study EBM workshop Meta analysis

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 analysis Exclusion of study Choose another scale Random effect model Meta-regression EBM workshop Meta analysis

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) EBM workshop Meta analysis

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 EBM workshop Meta analysis

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. EBM workshop Meta analysis

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, ד2 ) – Random term EBM workshop Meta analysis

Weighted least squares for Random Effect
Ŵ=∑wi/k S2w=1/k-1(∑wi2-k Ŵ2) U=(k-1)(Ŵ-S2w/kŴ) ד2=0 if Q<k-1 ד2=(Q-(k-1))/U if Q>k-1 wi* = 1/var.+ ד var.=within study variances EBM workshop Meta analysis

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 EBM workshop Meta analysis

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. EBM workshop Meta analysis

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. EBM workshop Meta analysis

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. EBM workshop Meta analysis

Meta-Regresion 1- meta-Regression model( extension of fixed effect model) 2- Mixed model( extension of random effect model) EBM workshop Meta analysis

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. EBM workshop Meta analysis

Mixed model Θi=B0+B1xi1+...+Bpxip+ui
This model assumes that part of the variability in true effects is unexplainable by the model. EBM workshop Meta analysis

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. EBM workshop Meta analysis

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. EBM workshop Meta analysis

Random effects model and heterogeneity
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. EBM workshop Meta analysis

Concept of analysis in random versus fixed effect models
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’?" EBM workshop Meta analysis

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. EBM workshop Meta analysis

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. EBM workshop Meta analysis

Ecological fallacy BP Education EBM workshop Meta analysis

Meta-analysis of neoadjuvant chemotherapy for cervical cancer
Word of Mouth 14% Trial Registers Hand Searching Medline/Cancerlit 58% EBM workshop Meta analysis

Type of reporting Published in full 47% Published as abstract 24%
Unpublished Ongoing 5% EBM workshop Meta analysis

Selection bias in Meta analysis
English language bias Database bias Publication bias Bias in reporting of data Citation bias Multiple publication bias Sample size EBM workshop Meta analysis

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 EBM workshop Meta analysis

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. EBM workshop Meta analysis

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. EBM workshop Meta analysis

“funnel” of unbiased studies
Funnel plot effect magnitude 1/SE “funnel” of unbiased studies EBM workshop Meta analysis

Funnel plot EBM workshop Meta analysis

Measures of Funnel Plot Asymmetry
1- Linear Regression Approach (Egger’s method) SND=a + b. precision SND=OR/SE The intercept “a” provides a measure of asymmetry- the larger its deviation from zero the more pronounced the asymmetry. EBM workshop Meta analysis

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. EBM workshop Meta analysis

Funnel plot EBM workshop Meta analysis

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. EBM workshop Meta analysis

Sources of Funnel Plot asymmetry
Selection Bias True Heterogeneity Size 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 EBM workshop Meta analysis

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. EBM workshop Meta analysis

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? EBM workshop Meta analysis

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. EBM workshop Meta analysis

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. EBM workshop Meta analysis

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. EBM workshop Meta analysis

Quality scores as weighing factors
study weight=1/var. Quality adjusted weight= quality score /var. EBM workshop Meta analysis

Quality scores 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. EBM workshop Meta analysis

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 EBM workshop Meta 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 EBM workshop Meta analysis

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. EBM workshop Meta analysis

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. EBM workshop Meta analysis

sensitivity analysis 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. EBM workshop Meta analysis

sensitivity analysis 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 EBM workshop Meta analysis

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 EBM workshop Meta analysis

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. EBM workshop Meta analysis

Over conclusion 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. EBM workshop Meta analysis

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