Meta-analysis and systematic reviews (AS12)

EPM304 Advanced Statistical Methods in Epidemiology

Course: PG Diploma/ MSc Epidemiology

This document contains a copy of the study material located within the computer assisted learning (CAL) session. If you have any questions regarding this document or your course, please contact DLsupport via DLsupport@lshtm.ac.uk. Important note: this document does not replace the CAL material found on your module CDROM. When studying this session, please ensure you work through the CDROM material first. This document can then be used for revision purposes to refer back to specific sessions. These study materials have been prepared by the London School of Hygiene & Tropical Medicine as part of the PG Diploma/MSc Epidemiology distance learning course. This material is not licensed either for resale or further copying.

London School of Hygiene & Tropical Medicine September 2013 v1.0

Section 1: Meta-analysis and systematic reviews Aim

To learn how to review and summarise information from many studies. Objectives By the end of this session, you should be able to:

Describe how to combine estimates from different studies, a method known as meta-analysis

Carry out two statistical approaches used in meta-analysis and know when to use them

Describe how to carry out a systematic review of the medical literature Understand the sources of bias in meta-analysis and know how to deal with

them This session should take you between 1.5 and 2 hours to complete. Section 2: Planning your study The aim of this session is to learn how to combine estimates from different studies and systematically review the literature. Meta-analysis is a large area, and so in this session you are given a brief introduction and overview of meta-analysis. The contents of this session are listed opposite. What is meta-analysis? Why is it used? Two main statistical approaches Why are there differences in estimates between studies? Issues in collecting data and various types of bias Final thoughts and conclusions 2.1: Planning your study To work through this session you should know about estimates of effect (e.g. odds ratio, rate ratio, risk ratio) which can be obtained from a regression model or classical methods of analysis. The statistical methods used in meta-analysis are similar to those of the Mantel-Haenzsel method. You may wish to review the sessions below.

Framework for Regression Models AS01 Cohort Studies SM02 To illustrate methods in this session the example below is used.

Randomised controlled trial of diuretics in preganancy

Section 3: What is meta-analysis? It is now common for important clinical questions in medical research to be addressed in several studies. This can be confusing for a medical practitioner. Which studies' results should be followed? How can the information from the mass of data published be summarised? Meta-analysis is a quantitative tool that can be used to summarise information from many studies. An informal literature review can be too subjective and misleading, whereas meta-analysis can assist with an overall conclusion. This combines results from different studies to give an overall summary estimate (and confidence interval). Popularity of Meta-Analysis

3.1: What is meta-analysis? The concept of meta-analysis is fairly easy to understand. The purpose of meta-analysis is to summarise information from different studies. Statistical techniques are used to combine results from studies that address the same clinical or epidemiological question. There are two extreme views on this and some people may view meta-analysis as making unjustified generalisations 1: An objective quantitative approach to combining evidence from separate (but similar) studies. 2: Statistical tricks that make unjustified assumptions in producing oversimplified generalisations out of a complex of disparate studies. Note: Meta-analysis is common within the area of clinical trials, and is also widely used within the general field of epidemiology. 3.2: What is meta-analysis? The rationale behind a meta-analysis is to address 'problems' with the original studies. Any one study is often: Interaction: Tabs: Too small: Studies that are too small fail to give clear-cut results. Would you rely on the results of a small study? What would you expect of the confidence interval around the sample estimate? Interaction: Button: clouds picture (pop up box appears): The confidence interval for the sample estimate of a small study would be wide, as a result it may be difficult to determine the true effect. Interaction: Tabs: Not generalisable: Often studies use a very select group of individuals, this makes it difficult to generalise study results to other types of individuals. 3.3: What is meta-analysis? What do you think the main purposes of a meta-analysis are? Click on one of the boxes below.

Interaction: Hotspot: Combine information from all studies to give an overall summary Correct Response(pop up box appears and card appears on RHS): Yes, meta-analysis attempts to combine information from studies addressing the same research question to give an overall summary. There are a number of reasons why meta-analysis is used to combine information from relevant studies:

1. Provide a data display and objective review

2. Give a summary interpretation

3. Test an overall hypothesis

4. Estimate an average exposure effect

5. Assess whether the data are compatible with the effect of the exposure being the same in all studies

Yes, using meta-analysis we can assess whether the data are compatible with the effect of interest being the same in all studies. 3.4: What is meta-analysis? So why is a meta-analysis useful? It is basically because the increased total size of the combined analysis increases the chances of detecting a moderate but clinically and/or epidemiologically important effect Often the aim of a meta-analysis is more general than that of a single study.

Combine information from all studies to give an overall summary

Increase the sample size and simulate a very large study that would otherwise be impossible to do

Assess whether the effect of interest is consistent for all studies

Interaction: Button: Example (card appears on right handside): The aim of a smaller study could be to study the effect of beta-blocker in patients of a certain type after a heart attack. What do you think the aim of a meta-analysis that includes this study may be? Interaction: Button: clouds picture: (pop up box appears): The aim of the meta-analysis could be to evaluate beta-blockers in general after a heart attack. 3.5: What is meta-analysis? Before you look at the statistical methodology, first consider the following example in which a number of trials assessed the effect of diureticsduring pregnancy. Interaction: Hyperlink: diuretics (pop up box appears): Diuretic An agent that promotes urination. Diuretics in pregnancy for pre-eclampsia An early meta-analysis looked at randomised controlled trials of diuretics in pregnancy. Note: Diuretics help to reduce blood pressure. Pre-eclampsia, a rapid increase in blood pressure or proteinuria, is a complication in pregnancy. On the next page you can see the results of this meta-analysis. Interaction: Hyperlink: Pre-eclampsia (pop up box appears): Pre-eclampsia Blood poisoning in late pregnancy, characterised by hypertension, oedema (excess fluid in the subcutaneous tissues) and proteinuria (excess protein in the urine). 3.6: What is meta-analysis? The 9 trials shown in the table below investigated whether diuretics helped in reducing the risk of pre-eclampsia. First author Preeclampsia

/total in treated patients

Preeclampsia/total in control patients

Odds ratio (95% CI)

Weseley 14 / 131 14 / 136 1.04 (0.48 2.28)

Flowers 21 / 385 17 / 134 0.40 (0.20 0.78)

Menzies 14 / 57 24 / 48 0.33 (0.14 0.74)

Fallis 6 / 38 18 / 40 0.23 (0.08 0.67)

Cuadros 12 / 1011 35 / 760 0.25 (0.13 0.48)

Landesman 138 / 1370 175 / 1336 0.74 (0.59 0.94)

Kraus 15 / 506 20 / 524 0.77 (0.39 1.52)

Tervila 6 / 108 2 / 103 2.97 (0.59 15.1)

Campbell 65 / 153 40 / 102 1.15 (0.69 1.91)

Total 291 / 3759 345 / 3183 0.69 (0.59 0.81)

The 2nd column shows the number of events in treated patients. The 3rd column show the number of events in untreated (control) patients. 3.7: What is meta-analysis? The table shows the odds ratio estimate for each study together with corresponding confidence intervals. What is the odds ratio estimating the effect of? Interaction: Button: clouds picture (pop up box appears and text appears on top RHS): The odds ratio estimates the effect of diuretic on reducing pre-eclampsia in pregnancy. Do you think it is reasonable to combine the odds ratio estimates to give a summary estimate? Interaction: Button: clouds picture (pop up box appears): Most ORs are considerably less than 1, indicating a protective effect of diuretics. For 3 studies the ORs are greater than 1 - but their confidence intervals are quite wide and overlap 1, so these studies do not provide evidence that diuretics are harmful. Overall, there is quite wide variation in the value of the ORs, suggesting that there may be real differences among the studies for the effect of diuretics 3.8: What is meta-analysis? There are two main approaches to meta-analysis:

1. Fixed effects method 2. Random effects method The latter helps to deal with heterogeneity between studies. You will now learn both methods, when and how to apply them. Section 4: Fixed-effects method This is the simplest method for calculating a summary estimate. However, this method makes the assumption that each study is measuring the same true effect, e.g. odds ratio, rate ratio. The question you should address to check if this assumption is valid is: Q. Are the differences observed between studies only due to sampling variation? You can think of this as similar to stratum-specific estimates with no interaction. An overall estimate assumes the effect is the same in each strata. The fixed-effects method:

Assumes the true effect is the same in all studies Gives a weight to each individual study estimate Calculates a summary estimate by calculating a weighted average of the

individual study estimates. 4.1: Fixed-effects method So, we obtain a weighted average of the study estimates. The tabs opposite show how this is obtained for a summary odds ratio Note: The same formats can be used for any outcome measure. Interaction: Tabs: Step 1: The weight for each study is the inverse of the variance of the study, this gives more weight to studies that are more precise. Choosing the weights in this way minimises the variability of the summary log odds ratio.

wi =

1

vi

where vi is the variance of the estimated log odds ratio in study i. Interaction: Tabs: Step 2: Using the weights and the estimated log odds ratios in each study i , the summary odds ratio is calculated as:

F =

u

Wii i-1

u

Wi i-1

Where F (for 'fixed') denotes the assumption that the effect of diuretic is the same in each study. Interaction: Tabs: Step 3: To calculate confidence intervals and do hypothesis tests on the summary estimate you need the variance of the summary estimate. This is calculated as

1

u

Wi i-1

4.2: Fixed-effects method Most statistical packages will perform a fixed-effects meta-analysis. Below you can see the results for the studies that assessed the effect of pre-eclampsia during pregnancy. Fixed effects meta-analysis (exponential form)

Metho

d

Pooled Estima

te

95% confidence interval

Asymptotic No. of studies

Lower Upper z value P-value

Fixed 0.672 0.564 0.800 4.455 < 0.001 9

So the summary estimate shows a reduction in pre-eclampsia of 33% and a 95% confidence interval of 20% to 44%. This is a statistically significant reduction (P < 0.001). 4.3: Fixed-effects method The standard way of presenting results of a meta-analysis graphically is shown opposite. This is known as a 'forest plot'. Click 'show' to see a forest plot for the pre-eclampsia meta-analysis. Interaction: Button: Show (graph pops up in new window):

Interaction: Tabs: 1: For each study, the box area is proportional to the weight for the study, this is to stop attention being drawn to extreme estimates. What do you notice about the confidence interval for the studies with larger boxes? Interaction: Button: clouds picture (pop up box appears): The confidence interval for studies with a larger box are narrower, i.e. more precise. Interaction: Tabs: 2:

The diamond and broken vertical line represent the overall summary estimate. The confidence interval is given by the width of the diamond. Interaction: Tabs: 3: The unbroken vertical line is the null value, i.e. odds ratio of 1 = no effect. Interaction: Tabs: 4: Notice that the x-axis for the odds ratio is on a log scale. This is to obtain symmetry in the plot. 4.4: Fixed-effects method Look again at the forest plot. Notice that the large trial (Landesman) dominates. Notice also that this trial has the smallest confidence interval. The combined estimate of the odds ratio is closest to this trial. Do you think you can assume homogeneity across these trials, i.e. are the results compatible with the true effect being the same in all studies? Interaction: Button: clouds picture (pop up box appears): Look at the degree of overlap in the CIs of the studies with the combined estimate (indicated by the dotted line). One study (Cuadros) shows no overlap, and two others (Fallis and Campbell) show only borderline overlap with the combined estimate. Also, the variation in ORs among studies is quite wide. These findings suggest that there may be real differences among the studies for the effect of diuretics Section 5: Tests for heterogeneity The fixed-effects method is based on the assumption that the true effect does not differ between studies. This assumption should be checked, and if there is a difference then a random effects method should be used to obtain a summary estimate. You will find out more about this on the next page. Interaction: Button: Show (pop up box appears):

5.1: Tests for heterogeneity The test for heterogeneity is based on the distance between the individual study estimates and the summary estimate from the fixed-effects method. Q = u

Wi(i - F) i-1

This is large if the average distance between the individual study effects and the summary effect is large. This statistic is referred to the distribution, if statistically significant then there is evidence against the null hypothesis of a common effect for all studies. if not statistically significant, there is no evidence for a heterogenous effect across trials. ie we would conclude that there is a common effect (homogeneity) across trials.

You do not have to do this calculation by hand statistics packages that perform meta-analysis will include a test for heterogeneity Section 6: Random effects method If the effects shown in each study differ, then a random effects method should be used to obtain a summary estimate. The interpretation of the summary estimate is that it is a mean effect about which it is assumed that the true study effects vary In a random effects model: the true effects are allowed to differ, i.e. it allows for interaction the true effects vary randomly about the population average the between-study variance is estimated the variance is used to modify the study weights 6.1: Random effects method The formula for the random-effects summary estimate is similar to that for the fixed-effects summary estimate. The difference is the weighting. The weights include a between study variance. This is complex and is not discussed here, you only need to know that this variance is taken into account and that is how the random effects method differs from the fixed effects method.

R =

u

W*ii i-1

u

W*i i-1

R indicates random effects estimate. 6.2: Random effects method Most statistical packages will perform a random-effects meta-analysis. Below you can see the results for the trials that assessed the effect of diuretics during pregnancy. Random effects meta-analysis Metho

d Pooled estima

te

95% confidence interval

Asymptotic No. of studies

Lower Upper z value P-value

Random 0.596 0.400 0.889 2.537 0.011 9 What does the summary estimate show?

Interaction: Hotspot: While the effect of diuretics varies between studies, on average their effect is to reduce the odds of pre-eclampsia by around 40%: Correct Response (pop up box appears): Yes, we estimate that on average diuretics reduce the odds of pre-eclampsia by around 40% (the estimate of the OR is 0.6). We estimate this overall OR using a random-effects meta-analysis, because we have evidence that the true effect of diuretics on pre-eclampsia varied among the studies Interaction: Hotspot: The true effect of diuretics is to reduce pre-eclampsia by 40%: Incorrect Response (pop up box appears): No, you cannot say what the true effect is. The summary estimate can only estimate the true effect. You can be 95% confident that the true effect will lie within the confidence intervals. ORR = 0.60 (0.40 to 0.89). Interaction: Hotspot: There is evidence that the effect of diuretics reduces pre-eclampsia: Correct Response (pop up box appears): Yes, that's correct, the sample estimate shows evidence that the effect of diuretics reduces pre-eclampsia. ORR = 0.60 (0.40 to 0.89), P = 0.01. 6.3: Random effects method Use the 'swap' button below to swap between the fixed-effects and random-effects results.

While the effect of diuretics varies between studies, on average their effect is to reduce the odds of pre-eclampsia by around 40%

There is evidence that the effect of diuretics reduces pre-eclampsia

The true effect of diuretics is to reduce pre-eclampsia by 40%

How does the estimate from the random-effects method compare to that from the fixed-effects method? Interaction: Button: clouds picture (pop up box appears and text appears on top RHS) The random-effects summary estimate is somewhat smaller than the fixed-effects estimate, but the confidence interval is much wider. This means that it overlaps more with the individual trial estimates. Fixed-effects summary estimate ORF = 0.67 (0.56 to 0.80) Random-effects summary estimate ORR = 0.60 (0.40 to 0.89) Interaction: Button: Swap (table at bottom LHS changes to the following): Fixed effects meta-analysis (exponential form) Method

Pooled estimate

95% confidence interval

Asymptotic No. of studies

Lower Upper z value P-value

Fixed 0.672 0.564 0.800 4.455 < 0.001 9 6.4: Random effects method The results below now include the test for heterogeneity. What can you conclude from this test? Interaction: Button: clouds picture (pop up box appears): There is strong evidence (P = 0.001) that the effect of diuretics differs between studies. The random effects method should be used for this meta-analysis. Metho

d Pooled estima

te

95% confidence interval

Asymptotic No.of studies

Lower Upper z value P-value

Random 0.596 0.400 0.889 2.537 0.011 9 Test for heterogeneity: Q = 27.265 on 8 degrees of freedom (P

= 0.001)

6.5: Random effects method It is instructive to compare the weights used in the fixed- and random-effects meta-analyses. Consider the table below. What is different about the random effect weights, and how do you think this affects the summary estimate? Interaction: Button: clouds picture (pop up box appears and text appears on top RHS): The weights are much more similar for the random-effects estimate than the fixed-effect estimate, because of the inclusion of the estimated between-study variance. This means that smaller studies will be given greater weight in random-effects meta-analysis. The wider confidence intervals for the random-effects estimate reflect the greater uncertainty because it is assumed that, in addition to sampling variation, the true effect varies between studies. Random-effects meta-analysis will, in general, be more conservative than fixed-effects meta-analysis: confidence intervals will be wider and P-values will be larger. Therefore, if you are unsure which to use then the random effects method is safer.

Study Fixed weights Random weights

Estimate (95% CI)

Weseley 6.27 2.57 1.04 (0.48,2.28)

Flowers 8.49 2.88 0.40 (0.20,0.78)

Menzies 5.62 2.45 0.33 (0.14,0.74)

Fallis 3.35 1.89 0.23 (0.08,0.67)

Cuadros 8.75 2.91 0.25 (0.13,0.48)

Landesman 68.34 4.09 0.74 (0.59,0.94)

Kraus 8.29 2.85 0.77 (0.39,1.52)

Tervila 1.46 1.09 2.97 (0.59,15.10)

Campbell 14.73 3.36 1.14 (0.69,1.91)

6.6: Random effects method Characteristics of the fixed effects and random effects method are summarised below. Interaction: Tabs: Fixed effects:

assumes the true effect is the same in each study does not allow for heterogeneity produces a narrow confidence interval Interaction: Tabs: Random effects: assumes the true effect differs between studies the true effects vary randomly about the population 'average' the between-study variance needs to be estimated weights for each trial incorporate the between-study variance 6.7: Random effects method You have seen that the two methods produce different results; in fact, the interpretation of the fixed and random estimates is very different. In a fixed-effects meta-analysis, it is assumed that the only reason for variation in the individual estimates is due to sampling error. In a random-effects meta-analysis, the summary estimate is a mean effect about which it is assumed that the true study effects vary. There is disagreement over whether it is appropriate to use random-effects models to combine study estimates in the presence of heterogeneity. This is because a single overall estimate may be unhelpful. For example, the treatment/exposure may have an effect in some populations but not in others. This is the case for BCG vaccination - there is strong evidence that it protects against tuberculosis in the UK, but no evidence that it does so in India or Malawi. So here, calculating a single overall estimate of the efficacy of BCG would be "combining apples and pears" - combining results from a setting where the intervention works with results from settings in which it doesn't. A more useful statement would be, for example, that BCG works in some settings (e.g. high latitude countries) but not in others (e.g. tropical countries with high levels of exposure to environmental mycobacteria). It is clear, therefore, that investigation into the sources of heterogeneity may yield important insights. The sources of heterogeneity are biological or methodological differences between the studies. For example, study latitude in the example of BCG vaccination is a source of heterogeneity. Section 7: Systematic reviews You have seen the methods that can be used to perform a meta-analysis. As always, the results of a statistical analysis are only as good as the data from which they are derived. One of the key issues in this area of analysis is obtaining the data. Interaction: Button: More (card appears on right handside as a timed pop up):

In recent decades there has been a massive increase in the number of research studies carried out. Over 2 million articles are published in the biomedical literature each year. A summary of the research evidence for any particular area is a difficult task. 7.1: Systematic reviews A conventional literature review has the potential to be too subjective. It is based on the information available to the author and is presented from the viewpoint of the author. To avoid this occurring in 1995 guidelines on how to systematically review the literature were developed. A systematic review of the literature is a 'systematic assembly, critical appraisal and synthesis of all relevant studies on a particular topic'. Guidelines for systematic review of the literature are given below. Interaction: Tabs: 1: A systematic review must address a specific health care question. The question determines which studies are relevant and how their data should be combined. Interaction: Tabs: 2: The methodology of a study must serve the biology, users and providers of health care. The systematic review study team should therefore include expertise in the content area and methodology. Interaction: Tabs: 3:

In order to collate all the necessary information, collaboration with the investigators of the primary studies is necessary. Interaction: Tabs: 4: Systematic reviews are retrospective, and can be affected by the same bias that affects other retrospective studies. A good systematic review should therefore be based on good review methodology. Interaction: Tabs: 5: For many reasons, review methods can vary. The review methods employed should be described for all systematic reviews. Interaction: tabs: 6: Some randomised trials, case-control studies and cohort studies may be unsatisfactory. This does not mean they should simply be excluded from a review. They should be critically appraised, empirically studied, and the analysis improved. Overviews of observational studies require a great deal of methodological development. 7.2: Systematic reviews The inclusion of studies in a systematic review should be based on: The methods of a study, not the results Acceptable standards should be defined before study reviews are carried out. For example, whether individuals were blind to the treatment under study, and the quality of the follow-up, need to be considered The quality of a trial associated with treatment effects The treatment benefits can be greatly exaggerated in uncontrolled trials where, for example, randomisation was inadequate. 7.3: Systematic reviews Of the millions of studies carried out, only a small percentage is published. It is well known that there is strong publication bias in favour of studies with the more 'positive' or 'interesting' results. The people behind this bias are listed on the tabs below. Interaction: Tabs: Investigators: If an investigator obtains the result they expected at the start of their research they are more likely to submit their findings for publication than if they feel the results are 'negative'. Whatever the study results, a good quality study should be reported.

Interaction: Tabs: Editor: If the editor of a journal is enthusiastic about the findings of a study, or if the study area has a currently high profile, then it is more likely to be sent for review. Interaction: Tabs: Referee: A referee is more likely to favour studies with a 'positive' result, which appear more interesting than studies with a so-called 'negative' result.

7.4: Systematic reviews This problem has been made worse by the following issues. Click each one for details: Small studies The 'magical' P = 0.05 cut-off Interaction: Hyperlink: Small studies (card appears on right handside): Studies have been too small Small studies can show a significant effect by chance. They are then more likely to be published than a non-significant study. This is known as publication bias We saw earlier that a random-effects meta-analysis gives studies more equal weights than a fixed-effects analysis does. If a random-effects summary estimate differs from the fixed-effects estimate, this is a sign that the average estimate from the smaller studies differs from the average of the large ones. Given that small studies are more subject to publication bias than large ones, this is clearly a disadvantage of random-effects analysis.

Interaction: Hyperlink: publication bias (pop up box appears): Publication bias Studies which produce negative or non-significant effect estimates are often considered less important and less appealing to scientific journals. As a consequence one problem that arises in drawing conclusions from a number of published studies is that the studies may not be truly representative, they are more likely to be the ones which showed strong positive effect estimates. Interaction: Hyperlink: The 'magical' P = 0.05 cut-off (card appears on right handside): Too much emphasis on the 'magical' cut-off value, P < 0.05 The emphasis should be on the confidence intervals that reflect the precision of the study estimate. 7.5: Systematic reviews A solution to overcome the problem of publication bias has been to establish registers of all studies carried out in a particular subject area. It has also been proposed that journals consider studies for publication "blind" of the actual results. It is clear that the active discouragement of studies that do not have power to detect an important effect would reduce the problem. Publication bias is not so great a problem for larger studies, for which there tends to be general agreement that the results are of interest, whatever they are. For more information on publication bias, see the Dickersin and Berlin paper in your Reader. 7.6: Systematic reviews We have seen that: to perform a systematic review is a substantial undertaking, medical practice needs to be based on the results of systematic reviews, rather than (non-systematic) literature reviews. The Cochrane Collaboration, which started in 1993, is an attempt to address these issues. It aims to produce systematic, periodically updated reviews of randomised controlled trials. These are available in electronic form (via CD-ROM), which means that reviews can be updated as new evidence becomes available or mistakes have been identified. Already, some hundreds of systematic reviews are available as part of the Cochrane Collaboration, and at least 150,000 studies are indexed in the database of randomised trials. The Cochrane Collaboration have also carried out systematic reviews and meta analyses of observational studies

Section 8: Summary The main points of this session will appear below as you click through the step card opposite. Click on any of the list entries below to go back to that page. Why use meta-analysis? Individual studies are often too small to obtain a precise estimate of effect. The objective of meta-analysis is to combine evidence from several studies. Common use in research Systematic reviews and meta-analysis (the quantitative analysis of such reviews) are now accepted as an important part of medical research. While the analytical methods are relatively simple, there is still controversy over appropriate methods of analysis. Increasing importance Systematic reviews are substantial undertakings, and those conducting such reviews need to be aware of the potential biases that may affect their conclusions. However, the explosion in medical research information and the availability of reviews on-line mean that summaries of research findings are likely to be of increasing importance to the practice of medicine. Fixed-effects versus random-effects If it is correct to assume that the only reason for variation in the study estimates is due to sampling error, then you can apply the fixed-effects method in a meta-analysis. If the study effects vary randomly about the population average, the random-effects method should be applied. The variation in study effects can be assessed using a test for heterogeneity. Helping the decision process Meta-analysis cannot tell a researcher what to do, but it can help with the decision-making process.

2.1: Planning your study3.1: What is meta-analysis?3.2: What is meta-analysis?3.3: What is meta-analysis?3.4: What is meta-analysis?3.5: What is meta-analysis?3.6: What is meta-analysis?3.7: What is meta-analysis?3.8: What is meta-analysis?4.1: Fixed-effects method4.2: Fixed-effects method4.3: Fixed-effects method4.4: Fixed-effects method5.1: Tests for heterogeneity6.1: Random effects method6.2: Random effects method6.3: Random effects method6.4: Random effects method6.5: Random effects method6.6: Random effects method6.7: Random effects method7.1: Systematic reviews7.2: Systematic reviews7.3: Systematic reviews7.4: Systematic reviews7.5: Systematic reviews7.6: Systematic reviews