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

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EBM workshopMeta analysis 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).

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EBM workshopMeta 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 hypothesiscontinued

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EBM workshopMeta 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

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EBM workshopMeta 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)

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EBM workshopMeta 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.

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EBM workshopMeta analysis Main magnitude of effectsDescriptive Mean PrevalenceAnalyticalAdditive Mean difference Standardized mean difference Risk, rate or hazard difference Correlation coefficientMultiplicative Odds ratio, Risk, Rate or Hazard Ratio

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EBM workshopMeta 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.

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

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EBM workshopMeta 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.

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EBM workshopMeta analysis Forest plot

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

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EBM workshopMeta 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

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EBM workshopMeta 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 X i = θ + e i e i is N (0,δ 2 ) X i = Observed study effect θ = Fixed effect common to all studies

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EBM workshopMeta analysis General Fixed effect model Ť=∑ w i T i / ∑ w i The weights that minimize the variance of Ť are inversely proportional to the conditional variance in each study W i =1/v i Var(Ť)=1/ ∑ w i

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EBM workshopMeta analysis Mantel- Haenszel method Each study is considered a strata. Ť=∑a i di / n i / ∑b i c i /n i

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EBM workshopMeta 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.

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EBM workshopMeta 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

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EBM workshopMeta 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)

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EBM workshopMeta 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= X 2 homo =∑ i=1 n w i (m i -M) 2 df=n-1 w i =weight M=meta analytic estimate of effect m i =effect measure of each study

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EBM workshopMeta 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

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EBM workshopMeta 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)

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EBM workshopMeta 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

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EBM workshopMeta 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.

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EBM workshopMeta analysis Random effect X i = θ i + e i e i is N (0,δ 2 ) X i = Observed study effect θ i = Random effect specific to each study θ i =U+d i U=Grand mean (common effect) d i is N (0, ד 2 ) – Random term

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EBM workshopMeta analysis Weighted least squares for Random Effect Ŵ=∑w i /k S 2 w =1/k-1(∑w i 2 -k Ŵ 2 ) U=(k-1)(Ŵ-S 2 w /kŴ) ד 2 =0 if Q

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EBM workshopMeta analysis Weighted least squares for Random Effect (WLS) Ť.RND =∑ w i * T i / ∑ w i * Var(Ť.RND )=1/ ∑ w i * Where T i is an estimate of effect size and θ i is the true effect size in the ith study T i = θ i +e i e i is the error with which T i estimates θ i var(T i )= ד θ 2 +v i

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EBM workshopMeta 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.

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EBM workshopMeta 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.

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EBM workshopMeta 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.

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

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EBM workshopMeta analysis Fixed-effects regression Θ i =B 0 +B 1 x i B p x ip It’s the covariate predictor variables that are responsible for the variation not a random effect; the variation is predictable, not random.

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EBM workshopMeta analysis Mixed model Θ i =B 0 +B 1 x i B p x ip +u i This model assumes that part of the variability in true effects is unexplainable by the model.

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EBM workshopMeta analysis Between studies variation You can and should allow for real differences between studies– heterogeneity–in the magnitude of the effect. –The τ 2 statistic quantifies % of variation due to real differences.

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EBM workshopMeta analysis Fixed effects model and heterogeneity In fixed-effects meta-analysis, you do so by testing for heterogeneity using the Q statistic. If p When p>0.10, you declare the effect homogeneous. But the approach is unrealistic, limited, and suffers from all the problems of statistical significance.

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EBM workshopMeta analysis 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

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EBM workshopMeta analysis 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

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EBM workshopMeta 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.

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EBM workshopMeta 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.

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EBM workshopMeta analysis Ecological fallacy BP Education

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

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

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EBM workshopMeta 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

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EBM workshopMeta 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

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EBM workshopMeta 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.

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EBM workshopMeta 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.

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EBM workshopMeta analysis Funnel plot effect magnitude 0 1/SE “funnel” of unbiased studies

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EBM workshopMeta analysis Funnel plot

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EBM workshopMeta analysis 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. Measures of Funnel Plot Asymmetry

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EBM workshopMeta analysis 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

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EBM workshopMeta analysis Funnel plot

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EBM workshopMeta 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.

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EBM workshopMeta 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

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EBM workshopMeta 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.

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EBM workshopMeta 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?

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EBM workshopMeta 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.

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EBM workshopMeta 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.

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EBM workshopMeta 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.

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

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EBM workshopMeta analysis 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

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EBM workshopMeta 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

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EBM workshopMeta 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

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EBM workshopMeta 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.

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EBM workshopMeta 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.

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EBM workshopMeta 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.

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EBM workshopMeta 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

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EBM workshopMeta 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

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EBM workshopMeta 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.

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EBM workshopMeta analysis 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

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