A basic introduction to fixedeffect and randomeffects ...

1 research ArticleReceived 1 December 2009,Revised 19 August 2010,Accepted 25 August 2010 Published online in Wiley Online Library( ) DOI: basic introduction to fixed-effect andrandom-effects models for meta-analysisMichael Borensteina , Larry V. Hedgesb, Julian Higginscand Hannah R. RothsteindThere are two popular statistical models for meta-analysis, the fixed-effect model and the random-effects model. Thefact that these two models employ similar sets of formulas to compute statistics, and sometimes yield similar estimatesfor the various parameters, may lead people to believe that the models are interchangeable.

2 In fact, though, the modelsrepresent fundamentally different assumptions about the data. The selection of the appropriate model is important toensure that the various statistics are estimated correctly. Additionally, and more fundamentally, the model serves toplace the analysis in context. It provides a framework for the goals of the analysis as well as for the interpretation ofthe this paper we explain the key assumptions of each model, and then outline the differences between the conclude with a discussion of factors to consider when choosing between the two models.

3 Copyright 2010 JohnWiley & Sons, :meta-analysis; fixed-effect; random-effects; statistical models; research synthesis; systematic reviewsIntroductionThere are two popular statistical models for meta-analysis, the fixed-effect model and the random-effects model. Under thefixed-effect model we assume that there is one true effect size that underlies all the studies in the analysis, and that all differencesin observed effects are due to sampling error. While we follow the practice of calling this a fixed-effect model, a more descriptiveterm would be a common-effect model.

4 In either case, we use the singular (effect) since there is only one true contrast, under the random-effects model we allow the true effect sizes to differ it is possible that all studies share acommon effect size, but it is also possible that the effect size varies from study to study. For example, the effect size might behigher (or lower) in studies where the participants are older, or more educated, or healthier than in other studies, or when a moreintensive variant of an intervention is used.

5 Because studies will differ in the mixes of participants and in the implementationsof interventions, among other reasons, there may be different effect sizes underlying different studies. If it were possible toperform an infinite number of studies (based on the inclusion criteria for the review), the true effect sizes for these studieswould be distributed about some mean. In a random-effects meta-analysis model, the effect sizes in the studies that actuallywere performed are assumed to represent a random sample from a particular distribution of these effect sizes (hence the termrandom effects).

6 Here, we use the plural (effects) since there is an array of true selection of the model is critically important. In addition to affecting the computations, the model helps to define thegoals of the analysis and the interpretation of the statistics. In this paper we explain the similarities and differences between themodels and discuss how to select an appropriate model for a given exampleFor illustrative purposes, we use fictional scenarios in which the goal is to estimate the mean score on a science aptitude example is a bit unusual, in that the effect size is a simple mean, whereas most meta-analyses employ an effect size thataBiostat, Inc.

7 , Englewood, NJ, of Statistics, Northwestern University, Evanston, IL, Biostatistics Unit, Cambridge, Department, Baruch College City University of New York, NY, Correspondence to: Michael Borenstein, Biostat, Inc., Englewood, NJ, E-mail: 2010 John Wiley & Sons, Syn. ,1 97--11197M. BORENSTEINET 1. Example of a fixed-effect 2. Example of a random-effects the impact of an intervention or the strength of a relationship. However, the same procedures apply in all cases, andthe selected example will allow for a simpler presentation of the relevant exampleThe defining feature of the fixed-effect model is that all studies in the analysis share a common effect size.

8 Suppose that wewant to estimate the mean aptitude score for freshmen at a specific college. Suppose further that the true mean at this collegeis 100, with a standard deviation of 20 points and a variance of 400. We generate a list of 1600 freshmen (selected at random)from that college. The testing facility cannot accommodate all these students at one sitting, and so we divide the names intofive groups, each of which is considered a separate study. In studies A, B, D, and E, the sample size is 200, whereas in studyC the sample size is 800.

9 If we assume that the assignment to one group or another has no impact on the score, then all fivestudies share a common (true) effect size, and the fixed-effect model analysis based on a fixed-effect model is shown in Figure 1. The effect size and confidence interval for each study appearon a separate row. The summary effect and its confidence interval are displayed at the exampleThe defining feature of the random-effects model is that there isa distributionof true effect sizes, and our goal is to estimate themean of this distribution.

10 Suppose that we want to estimate the mean score for freshmen atanycollege in California. Supposefurther that the true mean across all of these colleges is 100 and that within any college the scores are distributed with a standarddeviation of 20 points and variance of 400. First, we select five colleges at random. Then, we sample students at random fromeach of these colleges, using a sample size of 200 for colleges A, B, D, and E, and of 800 for college C. We will be using themean of these colleges to estimate the mean at all colleges.