Moderator analyses: Categorical models and Meta-regression

The Campbell Collaboration www.campbellcollaboration.org

Moderator analyses: Categorical models and Meta-regression

Terri Pigott, C2 Methods Editor & co-ChairProfessor, Loyola University Chicago

[email protected]

Campbell Collaboration Colloquium – May 2012 www.campbellcollaboration.org

Moderator analyses in meta-analysis

• We often want to test our hypotheses about whether variation among studies in effect size is associated with differences in study methods or participants

• We have these ideas a priori, incorporating these characteristics of studies into our coding forms

• Two major forms of moderator analyses in meta-analysis: categorical models analogous to ANOVA, and meta-regression


Assumptions for this session• We will focus on random effects models as these are the

most common in Campbell reviews• I will assume that we have computed the random effects

variance component (as you did if you were in my session yesterday - though you may feel like this right now)

• We will use two software packages:– RevMan – available here:

http://ims.cochrane.org/revman/download– Comprehensive Meta-analysis – available for free download

and limited trial here: http://www.meta-analysis.com/pages/demo.html


Categorical moderators

• When the moderator variable is categorical, we can estimate models analogous to ANOVA

• Typically, we are interested in comparing the group mean effect sizes for 2 or more groups

• For example, we will look at a meta-analysis where we compare the mean effect size for studies published in three different sources: journals, dissertations, and unpublished studies


Categorical moderator models

• With a one-way random effects ANOVA model, recall that we will compute– A mean effect size and standard error for each group, and then

test whether these means are significantly different from one another

– The mean effect size and standard error require an estimate of the variance component

– QUESTION: Will we assume that each group has the same variance component? Or, will we assume that each group has its own variance component?


What are our assumptions if we decide to use separate estimates within subgroups?• We believe that the variation among studies is different

between groups.• For example, if we are testing out an intervention and we

have studies that use either a low-income and a high-income group of students, we might believe that there will be more variation in effectiveness among studies that have mostly low-income participants

• Another example: the effectiveness of an intervention for juvenile delinquents will vary more for the group that had a prior arrest than for those that do not have a prior arrest


What are our assumptions when we use a pooled estimate?• We believe that the variation among effect sizes are the

same no matter the group.• For an intervention review, we may assume that the variation

among studies does not differ within the groups of interest• Caveat: We might have to use a pooled estimate if we have

small sample sizes within subgroups. We need at least 5 cases (in general) to be able to estimate a separate variance component for each subgroup


Flowchart from Borenstein


Steps for a random effects ANOVA

• Make a decision about the use of a pooled or a separate estimate of the variance component

• Compute the group mean effect sizes, and their standard errors

• Compare the group mean effect sizes to see if they are statistically different from one another


Eagly, Johannesen-Schmidt & van Engen (2003)

• This synthesis examines the standardized mean difference estimated in primary studies for the difference between men and women in their use of transformational leadership.

• Transformational leadership involves “establishing onself as a role model by gaining the trust and confidence of followers” (Eagly et al. 2003, p. 570).

• The sample data is a subset of the studies in the full meta-analysis, a set of 24 studies that compare men and women in their use of transformational leadership

• Positive effect sizes indicate males score higher than females


To follow along:

• Open RevMan• Open a review from a file• Open the file named:

Gender_differences_for_transformational_leadership.rm5• Go to Data and analyses on the left-hand menu• Double-click on 1.1 Transformational leadership• NOTE: RevMan uses the assumption that each group has a

different variance component


Summary of results – separate variance estimates for each group

Group k Mean 95% CI τ2 p

Journals 13 -0.05 [-0.24, 0.12] 0.09 <0.001

Dissertations 7 -0.47 [-0.69,-0.26] 0.02 0.22

Unpublished 4 -0.16 [-0.30,-0.03] 0.00 0.87

TOTAL 24 -0.16 [-0.29, -0.03] 0.08 <0.001

•Journals have a significant variance component, and the mean is not different from zero•Dissertations and unpublished studies both have a non-significant variance component, but both find that women score higher on transformational leadership


Summary of results – separate variance estimates for each group (continued)

Group k Mean 95% CI τ2 p

Journals 13 -0.05 [-0.24, 0.12] 0.09 <0.001

Dissertations 7 -0.47 [-0.69,-0.26] 0.02 0.22

Unpublished 4 -0.16 [-0.30,-0.03] 0.00 0.87

TOTAL 24 -0.16 [-0.29, -0.03] 0.08 <0.001

•The test of the variance component as different from zero is exactly the fixed effects test of homogeneity. •To get this test, we compute the test of homogeneity within each group of studies.


Test of between group differences

• To test between group differences in a random effects model, we test whether the variance component for the variation among the random effects means is equal to zero

• There are several ways to obtain this value• We will use a test of homogeneity of the three means – we

will treat the three group means as a meta-analysis


Test of between-group differences

• We will compute a test of homogeneity using our three means as if this is a meta-analysis

• We will use the means and their estimated variances to compute the sums we need to compute the homogeneity test

• These computations are all done “behind the scenes” by RevMan


Computation of Q between groups

Source Mean Var Wt Wt*Mean Wt*Mean2

Journals -0.05 0.008 122.53 -6.13 0.031

Dissertations -0.47 0.012 86.46 -40.64 19.10

Unpublished -0.16 0.005 211.32 -33.82 5.41

SUM 420.31 -80.59 24.54

( 80.59* 80.59)24.54

420.319.09

Q

Compare 9.09 to a chi-square with df=3-1=2. p-value is 0.011


What happens if we use the same variance component for all groups?

• We will need to try this in Comprehensive Meta-analysis• Open your trial version of Comprehensive Meta-analysis• Check that you will run the trial• Open the file called: leaderage.cma• Data is here:

https://my.vanderbilt.edu/emilytannersmith/training-materials/


Run analyses


Group bypub source

Study name Statistics for each study Hedges's g and 95% CI

Hedges's Standard Lower Upper g error Variance limit limit Z-Value p-Value

1.00 B23 -0.090 0.066 0.004 -0.220 0.040 -1.355 0.1761.00 CA -0.170 0.083 0.007 -0.333 -0.007 -2.045 0.0411.00 CH -0.220 0.118 0.014 -0.451 0.011 -1.868 0.0621.00 CW1 0.610 0.092 0.008 0.430 0.790 6.656 0.0001.00 CW2 0.200 0.150 0.023 -0.095 0.495 1.330 0.1831.00 DF -0.250 0.147 0.022 -0.539 0.039 -1.696 0.0901.00 JB -0.130 0.117 0.014 -0.360 0.100 -1.109 0.2671.00 JL -0.290 0.105 0.011 -0.495 -0.085 -2.768 0.0061.00 KO 0.310 0.078 0.006 0.157 0.463 3.982 0.0001.00 KUU 0.090 0.214 0.046 -0.330 0.510 0.420 0.6751.00 LJ -0.350 0.068 0.005 -0.482 -0.218 -5.185 0.0001.00 RH -0.360 0.087 0.008 -0.531 -0.189 -4.116 0.0001.00 SM 0.000 0.283 0.080 -0.556 0.556 0.000 1.0001.00 -0.055 0.084 0.007 -0.219 0.108 -0.662 0.5083.00 BO -0.620 0.262 0.069 -1.134 -0.106 -2.365 0.0183.00 CU -0.170 0.185 0.034 -0.533 0.193 -0.917 0.3593.00 DA -0.150 0.281 0.079 -0.701 0.401 -0.534 0.5933.00 HI -0.360 0.356 0.127 -1.059 0.339 -1.010 0.3123.00 MCG -0.720 0.242 0.059 -1.194 -0.246 -2.975 0.0033.00 RO -0.440 0.225 0.050 -0.880 0.000 -1.960 0.0503.00 WI -0.870 0.239 0.057 -1.339 -0.401 -3.639 0.0003.00 -0.477 0.141 0.020 -0.752 -0.201 -3.389 0.0014.00 BJ -0.100 0.148 0.022 -0.390 0.190 -0.675 0.5004.00 GM -0.100 0.252 0.064 -0.594 0.394 -0.397 0.6924.00 MA -0.080 0.199 0.040 -0.471 0.311 -0.401 0.6884.00 SP -0.210 0.090 0.008 -0.385 -0.035 -2.346 0.0194.00 -0.130 0.162 0.026 -0.447 0.187 -0.805 0.421Overall -0.159 0.066 0.004 -0.288 -0.031 -2.429 0.015

-1.00 -0.50 0.00 0.50 1.00

Favours A Favours B

Meta Analysis

Meta Analysis


Notes about the CMA forest plot

• Like RevMan, the confidence intervals around each study are the fixed effects confidence intervals (they use the within-study fixed effects variance)

• The group means are the random effects means computed using random effects weights. Their confidence intervals are also use random effects.

• In this example, we are using the same variance component for all groups


Summary from CMA


Notes about CMA results

• We assumed that the variance component was 0.08 for all studies

• Compared to our separate variance estimates, this value is smaller than the separate variance estimate for journals, but larger than the separate estimate for dissertations and unpublished articles


Reporting results from a random effects categorical analysis

Group RE mean RE CI τ3• The assumption made about the random effects variance: separate estimate for each group, or the same estimate for all groups.

• Rationale for the choice of variance component

• The random effects mean and CI• The value of the variance

components (or variance component)

• The test of the between-group differences, and its significance

for between-group means

df and p-value

Q


What is meta-regression?

• Meta-regression is a statistical technique used in a meta –analysis to examine how characteristics of studies are related to variation in effect sizes across studies

• Meta-regression is analogous to regression analysis but using effect sizes as our outcomes, and information extracted from studies as moderators/predictors

• NOTE: We can conduct a meta-regression in any statistical program. Here we will use CMA. BUT, note that using other standard programs may necessitate some adjustments to the results since they don’t produce exactly what we want.


Meta-regression used to examine heterogeneity

• When we have a heterogeneous set of effect sizes, we can use statistical techniques to examine the association among characteristics of the study and variation among effect sizes

• We have a plan for these analyses a priori – based on our understanding of the literature, and a logic model or framework

• Meta-regression used when we have more than one predictor or moderator (either continuous or categorical)


Form of the meta-regression model

0 1 1 ...i i p ipES B B x B x

1 2

0 1

is our generic effect size for study i

, ,... are the values for the predictor variables for study i

, ,..., are the unknown regression coefficients

i

i i ip

p

ES

x x x

B B B


Recall that the variance of the effect size

• Depends on the sample size for all types of effects we have talked about

• Thus, the precision of each study’s effect size depends on sample size

• This is different from our typical application of regression where we assume every person has the same “weight”

• Thus, we need to use weighted least squares regression to account for the fact that the precision of each effect size depends on sample size


Random effects meta-regression

• As in the categorical analysis discussion, we will need an estimate of the random effects variance for our studies that will be used as our weights in the regression

• There are many ways to compute the variance component in a random effects meta-regression

• For now, let’s assume a single variance component for all studies.


Test for the fit of the meta-regression model• As in a standard regression model, we can use the regression

ANOVA table for diagnostics about the fit of a meta-regression• Recall that in a standard regression analysis, we would get the

following regression ANOVA table:

ANOVAb,c

556.483 2 278.241 6.927 .004a

923.856 23 40.168

1480.338 25

Regression

Residual

Total

Model1

Sum ofSquares df Mean Square F Sig.

Predictors: (Constant), grade, percmina.

Dependent Variable: ztransb.

Weighted Least Squares Regression - Weighted by wtc.


Test of Model Fit in Meta-regression

• In meta-regression, we use the ANOVA table to get two different Q statistics:

• QM – model sum of squares, compare to chi-square distribution with p – 1 df (p is number of predictors in the model)

• QR – residual sum of squares, compare to chi-square distribution with k - p – 1 df (k is the number of studies)

• See Lipsey & Wilson. 2001. Practical Meta-analysis. Thousand Oaks, CA: Sage. pp. 122-124


QM , the model sum of squares

• Qmodel is the test of whether at least one of the regression coefficients (not including the intercept) is different from zero

• We compare QM to a chi-square distribution with p – 1 degrees of freedom with p = # of predictors in model

• If QM is significant, then at least one of the regression coefficients is different from zero


QR , the error or residual sum of squares

• QR is the test of whether there is more residual variation than we would expect IF the model “fits” the data

• We compare QR to a chi-square distribution with k - p – 1 degrees of freedom with k = # of studies/effect sizes, and p = # of predictors in model

• If QR is significant, then we have more error or residual variation to explain, or that is not accounted for by the variables we have in the model


Testing significance of individual regression coefficients in meta-regression• In a standard regression analysis, we find the t-tests on the printout to

see which regression coefficients are significantly different from zero• Those significant regression coefficients indicate that these predictors

are associated with the outcome• We will use CMA which gives us the z-tests for the regression

coefficients• NOTE: When doing meta-regression in a standard program like

SPSS, we have to make some adjustments since these programs do not compute the weighted regression in the way we need for meta-analysis


To conduct a meta-regression in CMA: run an analysis to get to the table below


Under Analyses, choose meta-regression


On the next page, choose the continuous outcome, averageage


By default, the analysis will be fixed effects. Choose method of moments under Computational options


Regression of averageage on Hedges's g

averageage

Hed

ges's

g

24.90 28.62 32.34 36.06 39.78 43.50 47.22 50.94 54.66 58.38 62.10

0.80

0.62

0.44

0.26

0.08

-0.10

-0.28

-0.46

-0.64

-0.82

-1.00

Plot of points and regression line


Results


Example for meta-regression


Objective of the review


Interventions


Example from Wilson & Lipsey


What to report in a random effects meta-regression?• The software and/or method used to compute the results• The method used to compute the random effects variance

component• The goodness of fit tests: Qmodel , and QResidual

• The regression coefficients and their test of significance


Final notes

• Software may be a problem in meta-regression as only CMA computes meta-regression. RevMan does not have the capacity for meta-regression

• CMA only allows one predictor in the meta-regression• To conduct the analyses as seen in the Wilson & Lipsey

example, you need to use other general statistical programs like SPSS, or STATA

• There are R programs available to conduct meta-analyses as well

The Campbell Collaboration www.campbellcollaboration.org

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0130 Oslo, Norway

E-mail: [email protected]

http://www.campbellcollaboration.org

Moderator analyses: Categorical models and Meta-regression

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