eval 6970: meta-analysis fixed-effect and random- effects models dr. chris l. s. coryn spring 2011
Post on 19-Dec-2015
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Agenda
β’ Fixed-effect models for meta-analysisβ’ Random-effects models for meta-
analysisβ’ Review questionsβ’ In-class activity
Fixed-Effect Models
β’ Under the fixed-effect model it is assumed that all studies share a common (true) effect size
β’ All factors that could influence the effect size are the same across all studies (the true effect is the same, thus fixed)
β’ The true (unknown) effect size is
Fixed-Effect Models
β’ Given that all studies share the same true effect, the observed effect size varies from study to study only because of random (sampling) error
β’ Although error is random, the sampling distribution of the errors can be estimated
Fixed-Effect Models
β’ To obtain an estimate of the population effect (to minimize variance) a weighted mean is computed using the inverse of each studyβs variance as the studyβs weight in computing the mean effect
Fixed-Effect Models
β’ The weighted mean () is computed as
π=βπ=1
π
π ππ π
βπ=1
π
π π
Fixed-Effect Models
β’ Where the weight assigned to each study is
β’ Where is the within-study variance for study
π π=1π π π
Fixed-Effect Models
β’ With
β’ And
β’ And
ππ=1
βπ=1
π
π π
ππΈπ=βππ
πΏπΏπ=πβ1.96ΓππΈπ
ππΏπ=π+1.96ΓππΈπ
Random-Effects Models
β’ Does not assume that the true effect is identical across studies
β’ Because study characteristics vary (e.g., participant characteristics, treatment intensity, outcome measurement) there may be different effect sizes underlying different studies
Random-Effects Models
β’ If the true effect size for a study is then the observed effect will be less than or greater than due to sampling error
β’ The distance between the summary mean and the observed effect consists of true variation in effect sizes () and sampling error ()
Random-Effects Models
π π=π+π π+ππRandom-effects model: True effect and observed effect
Random-Effects Models
β’ The distance from to each depends on the standard deviation of the true effects across studies, which is represented as and for its variance
β’ Each studyβs variance is a function of both the within-study variance and and is the sum of these two values
Random-Effects Models
β’ is estimated as
β’ Where
π 2=πβπππΆ
π=βπ=1
π
π ππ π2β
(βπ=1
π
π ππ π)2
βπ=1
π
π π
Random-Effects Models
β’ With
β’ Where is the number of studies
β’ And
ππ=πβ1
πΆ=βπ πββπ π
2
βπ π
Random-Effects Model
β’ Under the random-effects model the weight assigned to each study is
β’ Where is the within-study variance for study plus the between study variance
π πβ=
1
π π π
β
Random-Effects Models
β’ Where
β’ With the weighted mean computed as
π π π
β =π π π+π 2
πβ=βπ=1
π
π πβπ π
βπ=1
π
π πβ
Random-Effects Models
β’ With
β’ And
β’ And
ππ β=1
βπ=1
π
π πβ
ππΈπ β=βπ πβ
πΏπΏπβ=πββ1.96ΓππΈπ β
ππΏπβ=πβ+1.96ΓππΈπ β
Review Questions
1. When is it appropriate to use a fixed-effect model?
2. When is it appropriate to use a random-effects model?
3. How do the study weights differ for fixed-effect and random-effects models?
Todayβs In-Class Activity
β’ Individually, or in your working groups, download βData Sets 1-6 XLSXβ from the course Websiteβ Calculate the fixed-effect and random-effects
model weighted means for Data Sets 1, 2, 3, 4, 5, and 6
β Calculate the 95% confidence intervals (i.e., LL and UL) for the weighted means from Data Sets 1, 2, 3, 4, 5, and 6
β Use βMeta-Analysis with Means, Binary Data, and Correlations XLSXβ to assist with the calculations