Introduction

Comparisons between two- and three-level models with Cooper et al.’s (2003) dataset

As an illustration, I first conduct the tradition (two-level) meta-analysis using the meta() function. Then I conduct a three-level meta-analysis using the meta3() function. We may compare the similarities and differences between these two sets of results.

Inspecting the data

Before running the analyses, we need to load the metaSEM library. The datasets are stored in the library. It is always a good idea to inspect the data before the analyses. We may display the first few cases of the dataset by using the head() command.

#### Cooper et al. (2003)
library("metaSEM")

head(Cooper03)
  District Study     y     v Year
1       11     1 -0.18 0.118 1976
2       11     2 -0.22 0.118 1976
3       11     3  0.23 0.144 1976
4       11     4 -0.30 0.144 1976
5       12     5  0.13 0.014 1989
6       12     6 -0.26 0.014 1989

Two-level meta-analysis

Similar to other R packages, we may use summary() to extract the results after running the analyses. I first conduct a random-effects meta-analysis and then a fixed- and mixed-effects meta-analyses.

  • Random-effects model The Q statistic on testing the homogeneity of effect sizes was \(578.86, df = 55, p < .001\). The estimated heterogeneity \(\tau^2\) (labeled Tau2_1_1 in the output) and \(I^2\) were 0.0866 and 0.9459, respectively. This indicates that the between-study effect explains about 95% of the total variation. The average population effect (labeled Intercept1 in the output; and its 95% Wald CI) was 0.1280 (0.0428, 0.2132).
#### Two-level meta-analysis

## Random-effects model  
summary( meta(y=y, v=v, data=Cooper03) )

Call:
meta(y = y, v = v, data = Cooper03)

95% confidence intervals: z statistic approximation
Coefficients:
           Estimate Std.Error   lbound   ubound z value  Pr(>|z|)    
Intercept1 0.128003  0.043472 0.042799 0.213207  2.9445  0.003235 ** 
Tau2_1_1   0.086537  0.019485 0.048346 0.124728  4.4411 8.949e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 578.864
Degrees of freedom of the Q statistic: 55
P value of the Q statistic: 0

Heterogeneity indices (based on the estimated Tau2):
                             Estimate
Intercept1: I2 (Q statistic)   0.9459

Number of studies (or clusters): 56
Number of observed statistics: 56
Number of estimated parameters: 2
Degrees of freedom: 54
-2 log likelihood: 33.2919 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • Fixed-effects model A fixed-effects meta-analysis can be conducted by fixing the heterogeneity of the random effects at 0 with the RE.constraints argument (random-effects constraints). The estimated common effect (and its 95% Wald CI) was 0.0464 (0.0284, 0.0644).
## Fixed-effects model
summary( meta(y=y, v=v, data=Cooper03, RE.constraints=0) )

Call:
meta(y = y, v = v, data = Cooper03, RE.constraints = 0)

95% confidence intervals: z statistic approximation
Coefficients:
            Estimate Std.Error    lbound    ubound z value Pr(>|z|)    
Intercept1 0.0464072 0.0091897 0.0283957 0.0644186  5.0499 4.42e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 578.864
Degrees of freedom of the Q statistic: 55
P value of the Q statistic: 0

Heterogeneity indices (based on the estimated Tau2):
                             Estimate
Intercept1: I2 (Q statistic)        0

Number of studies (or clusters): 56
Number of observed statistics: 56
Number of estimated parameters: 1
Degrees of freedom: 55
-2 log likelihood: 434.2075 
OpenMx status1: 1 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • Mixed-effects model Year was used as a covariate. It is easier to interpret the intercept by centering the Year with scale(Year, scale=FALSE). The scale=FALSE argument means that it is centered, but not standardized. The estimated regression coefficient (labeled Slope1_1 in the output; and its 95% Wald CI) was 0.0051 (-0.0033, 0.0136) which is not significant at \(\alpha=.05\). The \(R^2\) is 0.0164.
## Mixed-effects model
summary( meta(y=y, v=v, x=scale(Year, scale=FALSE), data=Cooper03) )

Call:
meta(y = y, v = v, x = scale(Year, scale = FALSE), data = Cooper03)

95% confidence intervals: z statistic approximation
Coefficients:
             Estimate  Std.Error     lbound     ubound z value  Pr(>|z|)
Intercept1  0.1259125  0.0432028  0.0412367  0.2105884  2.9145  0.003563
Slope1_1    0.0051307  0.0043248 -0.0033457  0.0136071  1.1864  0.235483
Tau2_1_1    0.0851153  0.0190462  0.0477856  0.1224451  4.4689 7.862e-06
              
Intercept1 ** 
Slope1_1      
Tau2_1_1   ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 578.864
Degrees of freedom of the Q statistic: 55
P value of the Q statistic: 0

Explained variances (R2):
                           y1
Tau2 (no predictor)    0.0865
Tau2 (with predictors) 0.0851
R2                     0.0164

Number of studies (or clusters): 56
Number of observed statistics: 56
Number of estimated parameters: 3
Degrees of freedom: 53
-2 log likelihood: 31.88635 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)

Three-level meta-analysis

  • Random-effects model The Q statistic on testing the homogeneity of effect sizes was the same as that under the two-level meta-analysis. The estimated heterogeneity at level 2 \(\tau^2_{(2)}\) (labeled Tau2_2 in the output) and at level 3 \(\tau^2_{(3)}\) (labeled Tau2_3 in the output) were 0.0329 and 0.0577, respectively. The level 2 \(I^2_{(2)}\) (labeled I2_2 in the output) and the level 3 \(I^2_{(3)}\) (labeled I2_3 in the output) were 0.3440 and 0.6043, respectively. Schools (level 2) and districts (level 3) explain about 34% and 60% of the total variation, respectively. The average population effect (and its 95% Wald CI) was 0.1845 (0.0266, 0.3423).
#### Three-level meta-analysis
## Random-effects model
summary( meta3(y=y, v=v, cluster=District, data=Cooper03) )

Call:
meta3(y = y, v = v, cluster = District, data = Cooper03)

95% confidence intervals: z statistic approximation
Coefficients:
            Estimate  Std.Error     lbound     ubound z value Pr(>|z|)   
Intercept  0.1844554  0.0805411  0.0265977  0.3423130  2.2902 0.022010 * 
Tau2_2     0.0328648  0.0111397  0.0110314  0.0546982  2.9502 0.003175 **
Tau2_3     0.0577384  0.0307423 -0.0025154  0.1179921  1.8781 0.060362 . 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 578.864
Degrees of freedom of the Q statistic: 55
P value of the Q statistic: 0

Heterogeneity indices (based on the estimated Tau2):
                              Estimate
I2_2 (Typical v: Q statistic)   0.3440
I2_3 (Typical v: Q statistic)   0.6043

Number of studies (or clusters): 11
Number of observed statistics: 56
Number of estimated parameters: 3
Degrees of freedom: 53
-2 log likelihood: 16.78987 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • Mixed-effects model Year was used as a covariate. The estimated regression coefficient (labeled Slope_1 in the output; and its 95% Wald CI) was 0.0051 (-0.0116, 0.0218) which is not significant at \(\alpha=.05\). The estimated level 2 \(R^2_{(2)}\) and level 3 \(R^2_{(3)}\) were 0.0000 and 0.0221, respectively.
## Mixed-effects model
summary( meta3(y=y, v=v, cluster=District, x=scale(Year, scale=FALSE), 
               data=Cooper03) )

Call:
meta3(y = y, v = v, cluster = District, x = scale(Year, scale = FALSE), 
    data = Cooper03)

95% confidence intervals: z statistic approximation
Coefficients:
            Estimate  Std.Error     lbound     ubound z value Pr(>|z|)   
Intercept  0.1780268  0.0805219  0.0202067  0.3358469  2.2109 0.027042 * 
Slope_1    0.0050737  0.0085266 -0.0116382  0.0217856  0.5950 0.551814   
Tau2_2     0.0329390  0.0111620  0.0110618  0.0548162  2.9510 0.003168 **
Tau2_3     0.0564628  0.0300330 -0.0024007  0.1153264  1.8800 0.060104 . 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 578.864
Degrees of freedom of the Q statistic: 55
P value of the Q statistic: 0

Explained variances (R2):
                        Level 2 Level 3
Tau2 (no predictor)    0.032865  0.0577
Tau2 (with predictors) 0.032939  0.0565
R2                     0.000000  0.0221

Number of studies (or clusters): 11
Number of observed statistics: 56
Number of estimated parameters: 4
Degrees of freedom: 52
-2 log likelihood: 16.43629 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • Model comparisons Many research hypotheses involve model comparisons among nested models. anova(), a generic function to comparing nested models, may be used to conduct a likelihood ratio test which is also known as a chi-square difference test.

  • Testing \(H_0: \tau^2_{(3)} = 0\)
    • Based on the data structure, it is clear that a 3-level meta-analysis is preferred to a traditional 2-level meta-analysis. It is still of interest to test whether the 3-level model is statistically better than the 2-level model by testing \(H_0: \tau^2_{(3)}=0\). Since the models with \(\tau^2_{(3)}\) being freely estimated and with \(\tau^2_{(3)}=0\) are nested, we may compare them by the use of a likelihood ratio test.
    • It should be noted, however, that \(H_0: \tau^2_{(3)}=0\) is tested on the boundary. The likelihood ratio test is not distributed as a chi-square variate with 1 df. A simple strategy to correct this bias is to reject the null hypothesis when the observed p value is larger than .10 for \(\alpha=.05\).
    • The likelihood-ratio test was 16.5020 (df =1), p < .001. This clearly demonstrates that the three-level model is statistically better than the two-level model.
## Model comparisons
  
model2 <- meta(y=y, v=v, data=Cooper03, model.name="2 level model", silent=TRUE) 
#### An equivalent model by fixing tau2 at level 3=0 in meta3()
## model2 <- meta3(y=y, v=v, cluster=District, data=Cooper03, 
##                 model.name="2 level model", RE3.constraints=0) 
model3 <- meta3(y=y, v=v, cluster=District, data=Cooper03, 
                model.name="3 level model", silent=TRUE) 
anova(model3, model2)
           base    comparison ep minus2LL df       AIC   diffLL diffdf
1 3 level model          <NA>  3 16.78987 53 -89.21013       NA     NA
2 3 level model 2 level model  2 33.29190 54 -74.70810 16.50203      1
             p
1           NA
2 4.859793e-05
  • Testing \(H_0: \tau^2_{(2)} = \tau^2_{(3)}\)
    • From the results of the 3-level random-effects meta-analysis, it appears the level 3 heterogeneity is much larger than that at level 2.
    • We may test the null hypothesis \(H_0: \tau^2_{(2)} = \tau^2_{(3)}\) by the use of a likelihood-ratio test.
    • We may impose an equality constraint on \(\tau^2_{(2)} = \tau^2_{(3)}\) by using the same label in meta3(). For example, Eq_tau2 is used as the label in RE2.constraints and RE3.constraints meaning that both the level 2 and level 3 random effects heterogeneity variances are constrained equally. The value of 0.1 was used as the starting value in the constraints.
    • The likelihood-ratio test was 0.6871 (df = 1), p = 0.4072. This indicates that there is not enough evidence to reject \(H_0: \tau^2_2=\tau^2_3\).
## Testing \tau^2_2 = \tau^2_3
modelEqTau2 <- meta3(y=y, v=v, cluster=District, data=Cooper03, 
                     model.name="Equal tau2 at both levels",
                     RE2.constraints="0.1*Eq_tau2", RE3.constraints="0.1*Eq_tau2") 
anova(model3, modelEqTau2)
           base                comparison ep minus2LL df       AIC
1 3 level model                      <NA>  3 16.78987 53 -89.21013
2 3 level model Equal tau2 at both levels  2 17.47697 54 -90.52303
     diffLL diffdf         p
1        NA     NA        NA
2 0.6870959      1 0.4071539
  • Likelihood-based confidence interval
    • A Wald CI is constructed by \(\hat{\theta} \pm 1.96 SE\) where \(\hat{\theta}\) and \(SE\) are the parameter estimate and its estimated standard error.
    • A LBCI can be constructed by the use of the likelihood ratio statistic (e.g., Cheung, 2009; Neal & Miller, 1997).
    • It is well known that the performance of Wald CI on variance components is very poor. For example, the 95% Wald CI on \(\hat{\tau}^2_{(3)}\) in the three-level random-effects meta-analysis was (-0.0025, 0.1180). The lower bound falls outside 0.
    • A LBCI on the heterogeneity variance is preferred. Since \(I^2_{(2)}\) and \(I^2_{(3)}\) are functions of \(\tau^2_{(2)}\) and \(\tau^2_{(3)}\), LBCI on these indices may also be requested and used to indicate the precision of these indices.
    • LBCI may be requested by specifying LB in the intervals.type argument.
    • The 95% LBCI on \(\hat{\tau}^2_{(3)}\) is (0.0198, 0.1763) that stay inside the meaningful boundaries. Regarding the \(I^2\), the 95% LBCIs on \(I^2_{(2)}\) and \(I^2_{(3)}\) were (0.1274, 0.6573) and (0.2794, 0.8454), respectively.
## LBCI for random-effects model
summary( meta3(y=y, v=v, cluster=District, data=Cooper03, 
               I2=c("I2q", "ICC"), intervals.type="LB") ) 

Call:
meta3(y = y, v = v, cluster = District, data = Cooper03, intervals.type = "LB", 
    I2 = c("I2q", "ICC"))

95% confidence intervals: Likelihood-based statistic
Coefficients:
          Estimate Std.Error   lbound   ubound z value Pr(>|z|)
Intercept 0.184455        NA 0.011668 0.358535      NA       NA
Tau2_2    0.032865        NA 0.016330 0.063126      NA       NA
Tau2_3    0.057738        NA 0.019805 0.176328      NA       NA

Q statistic on the homogeneity of effect sizes: 578.864
Degrees of freedom of the Q statistic: 55
P value of the Q statistic: 0

Heterogeneity indices (I2) and their 95% likelihood-based CIs:
                               lbound Estimate ubound
I2_2 (Typical v: Q statistic) 0.12736  0.34396 0.6573
ICC_2 (tau^2/(tau^2+tau^3))   0.13102  0.36273 0.7015
I2_3 (Typical v: Q statistic) 0.27937  0.60429 0.8454
ICC_3 (tau^3/(tau^2+tau^3))   0.29847  0.63727 0.8690

Number of studies (or clusters): 11
Number of observed statistics: 56
Number of estimated parameters: 3
Degrees of freedom: 53
-2 log likelihood: 16.78987 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
## LBCI for mixed-effects model
summary( meta3(y=y, v=v, cluster=District, x=scale(Year, scale=FALSE), 
               data=Cooper03, intervals.type="LB") ) 

Call:
meta3(y = y, v = v, cluster = District, x = scale(Year, scale = FALSE), 
    data = Cooper03, intervals.type = "LB")

95% confidence intervals: Likelihood-based statistic
Coefficients:
            Estimate Std.Error     lbound     ubound z value Pr(>|z|)
Intercept  0.1780268        NA  0.0052929  0.3515622      NA       NA
Slope_1    0.0050737        NA -0.0128356  0.0237979      NA       NA
Tau2_2     0.0329390        NA  0.0163732  0.0632779      NA       NA
Tau2_3     0.0564628        NA  0.0192355  0.1720022      NA       NA

Q statistic on the homogeneity of effect sizes: 578.864
Degrees of freedom of the Q statistic: 55
P value of the Q statistic: 0

Explained variances (R2):
                        Level 2 Level 3
Tau2 (no predictor)    0.032865  0.0577
Tau2 (with predictors) 0.032939  0.0565
R2                     0.000000  0.0221

Number of studies (or clusters): 11
Number of observed statistics: 56
Number of estimated parameters: 4
Degrees of freedom: 52
-2 log likelihood: 16.43629 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • Restricted maximum likelihood estimation
    • REML may also be used in three-level meta-analysis. The parameter estimates for \(\tau^2_{(2)}\) and \(\tau^2_{(3)}\) were 0.0327 and 0.0651, respectively.
## REML
summary( reml1 <- reml3(y=y, v=v, cluster=District, data=Cooper03) )

Call:
reml3(y = y, v = v, cluster = District, data = Cooper03)

95% confidence intervals: z statistic approximation
Coefficients:
         Estimate  Std.Error     lbound     ubound z value Pr(>|z|)   
Tau2_2  0.0327365  0.0110922  0.0109963  0.0544768  2.9513 0.003164 **
Tau2_3  0.0650619  0.0355102 -0.0045368  0.1346607  1.8322 0.066921 . 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Number of studies (or clusters): 56
Number of observed statistics: 55
Number of estimated parameters: 2
Degrees of freedom: 53
-2 log likelihood: -81.14044 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • We may impose an equality constraint on \(\tau^2_{(2)}\) and \(\tau^2_{(3)}\) and test whether this constraint is statistically significant. The estimated value for \(\tau^2_{(2)}=\tau^2_{(3)}\) was 0.0404. When this model is compared against the unconstrained model, the test statistic was 1.0033 (df = 1), p = .3165, which is not significant.
summary( reml0 <- reml3(y=y, v=v, cluster=District, data=Cooper03,
                        RE.equal=TRUE, model.name="Equal Tau2") )

Call:
reml3(y = y, v = v, cluster = District, data = Cooper03, RE.equal = TRUE, 
    model.name = "Equal Tau2")

95% confidence intervals: z statistic approximation
Coefficients:
     Estimate Std.Error   lbound   ubound z value  Pr(>|z|)    
Tau2 0.040418  0.010290 0.020249 0.060587  3.9277 8.576e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Number of studies (or clusters): 56
Number of observed statistics: 55
Number of estimated parameters: 1
Degrees of freedom: 54
-2 log likelihood: -80.1371 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
anova(reml1, reml0)
                          base comparison ep  minus2LL df AIC   diffLL
1 Variance component with REML       <NA>  2 -81.14044 -2  NA       NA
2 Variance component with REML Equal Tau2  1 -80.13710 -1  NA 1.003336
  diffdf         p
1     NA        NA
2      1 0.3165046
  • We may also estimate the residual heterogeneity after controlling for the covariate. The estimated residual heterogeneity for \(\tau^2_{(2)}\) and \(\tau^2_{(3)}\) were 0.0327 and 0.0723, respectively.
summary( reml3(y=y, v=v, cluster=District, x=scale(Year, scale=FALSE),
               data=Cooper03) )

Call:
reml3(y = y, v = v, cluster = District, x = scale(Year, scale = FALSE), 
    data = Cooper03)

95% confidence intervals: z statistic approximation
Coefficients:
         Estimate  Std.Error     lbound     ubound z value Pr(>|z|)   
Tau2_2  0.0326502  0.0110529  0.0109870  0.0543134  2.9540 0.003137 **
Tau2_3  0.0722656  0.0405349 -0.0071813  0.1517125  1.7828 0.074619 . 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Number of studies (or clusters): 56
Number of observed statistics: 54
Number of estimated parameters: 2
Degrees of freedom: 52
-2 log likelihood: -72.09405 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)

More complex 3-level meta-analyses with Bornmann et al.’s (2007) dataset

This section replicates the findings in Table 3 of Marsh et al. (2009). Several additional analyses on model comparisons were conducted. Missing data were artificially introduced to illustrate how missing data might be handled with FIML.

Inspecting the data

The effect size and its sampling variance are logOR (log of the odds ratio) and v, respectively. Cluster is the variable representing the cluster effect, whereas the potential covariates are Year (year of publication), Type (Grants vs. Fellowship), Discipline (Physical sciences, Life sciences/biology, Social sciences/humanities and Multidisciplinary) and Country (United States, Canada, Australia, United Kingdom and Europe).

#### Bornmann et al. (2007)
head(Bornmann07)
  Id                       Study Cluster    logOR          v Year
1  1 Ackers (2000a; Marie Curie)       1 -0.40108 0.01391692 1996
2  2 Ackers (2000b; Marie Curie)       1 -0.05727 0.03428793 1996
3  3 Ackers (2000c; Marie Curie)       1 -0.29852 0.03391122 1996
4  4 Ackers (2000d; Marie Curie)       1  0.36094 0.03404025 1996
5  5 Ackers (2000e; Marie Curie)       1 -0.33336 0.01282103 1996
6  6 Ackers (2000f; Marie Curie)       1 -0.07173 0.01361189 1996
        Type                 Discipline Country
1 Fellowship          Physical sciences  Europe
2 Fellowship          Physical sciences  Europe
3 Fellowship          Physical sciences  Europe
4 Fellowship          Physical sciences  Europe
5 Fellowship Social sciences/humanities  Europe
6 Fellowship          Physical sciences  Europe

Model 0: Intercept

The Q statistic was 221.2809 (df = 65), p < .001. The estimated average effect (and its 95% Wald CI) was -0.1008 (-0.1794, -0.0221). The \(\hat{\tau}^2_{(2)}\) and \(\hat{\tau}^3_{(3)}\) were 0.0038 and 0.0141, respectively. The \(I^2_{(2)}\) and \(I^2_{(3)}\) were 0.1568 and 0.5839, respectively.

## Model 0: Intercept  
summary( Model0 <- meta3(y=logOR, v=v, cluster=Cluster, data=Bornmann07, 
                         model.name="3 level model") )

Call:
meta3(y = logOR, v = v, cluster = Cluster, data = Bornmann07, 
    model.name = "3 level model")

95% confidence intervals: z statistic approximation
Coefficients:
            Estimate  Std.Error     lbound     ubound z value Pr(>|z|)  
Intercept -0.1007784  0.0401327 -0.1794370 -0.0221198 -2.5111  0.01203 *
Tau2_2     0.0037965  0.0027210 -0.0015367  0.0091297  1.3952  0.16295  
Tau2_3     0.0141352  0.0091445 -0.0037877  0.0320580  1.5458  0.12216  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 221.2809
Degrees of freedom of the Q statistic: 65
P value of the Q statistic: 0

Heterogeneity indices (based on the estimated Tau2):
                              Estimate
I2_2 (Typical v: Q statistic)   0.1568
I2_3 (Typical v: Q statistic)   0.5839

Number of studies (or clusters): 21
Number of observed statistics: 66
Number of estimated parameters: 3
Degrees of freedom: 63
-2 log likelihood: 25.80256 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • Testing \(H_0: \tau^2_3 = 0\) We may test whether the three-level model is necessary by testing \(H_0: \tau^2_{(3)} = 0\). The likelihood ratio statistic was 10.2202 (df = 1), p < .01. Thus, the three-level model is statistically better than the two-level model.
## Testing tau^2_3 = 0
Model0a <- meta3(logOR, v, cluster=Cluster, data=Bornmann07, 
                 RE3.constraints=0, model.name="2 level model")
anova(Model0, Model0a)
           base    comparison ep minus2LL df        AIC   diffLL diffdf
1 3 level model          <NA>  3 25.80256 63 -100.19744       NA     NA
2 3 level model 2 level model  2 36.02279 64  -91.97721 10.22024      1
            p
1          NA
2 0.001389081
  • Testing \(H_0: \tau^2_2 = \tau^2_3\) The likelihood-ratio statistic in testing \(H_0: \tau^2_{(2)} = \tau^2_{(3)}\) was 1.3591 (df = 1), p = 0.2437. Thus, there is no evidence to reject the null hypothesis.
## Testing tau^2_2 = tau^2_3
Model0b <- meta3(logOR, v, cluster=Cluster, data=Bornmann07, 
                 RE2.constraints="0.1*Eq_tau2", RE3.constraints="0.1*Eq_tau2", 
                 model.name="tau2_2 equals tau2_3")
anova(Model0, Model0b)
           base           comparison ep minus2LL df       AIC   diffLL
1 3 level model                 <NA>  3 25.80256 63 -100.1974       NA
2 3 level model tau2_2 equals tau2_3  2 27.16166 64 -100.8383 1.359103
  diffdf        p
1     NA       NA
2      1 0.243693

Model 1: Type as a covariate

  • Conventionally, one level (e.g., Grants) is used as the reference group. The estimated intercept (labeled Intercept in the output) represents the estimated effect size for Grants and the regression coefficient (labeled Slope_1 in the output) is the difference between Fellowship and Grants.
    • The estimated slope on Type (and its 95% Wald CI) was -0.1956 (-0.3018, -0.0894) which is statistically significant at \(\alpha=.05\). This is the difference between Fellowship and Grants. The \(R^2_{(2)}\) and \(R^2_{(3)}\) were 0.0693 and 0.7943, respectively.
## Model 1: Type as a covariate  
## Convert characters into a dummy variable
## Type2=0 (Grants); Type2=1 (Fellowship)    
Type2 <- ifelse(Bornmann07$Type=="Fellowship", yes=1, no=0)
summary( Model1 <- meta3(logOR, v, x=Type2, cluster=Cluster, data=Bornmann07)) 

Call:
meta3(y = logOR, v = v, cluster = Cluster, x = Type2, data = Bornmann07)

95% confidence intervals: z statistic approximation
Coefficients:
            Estimate  Std.Error     lbound     ubound z value  Pr(>|z|)
Intercept -0.0066071  0.0371125 -0.0793462  0.0661320 -0.1780 0.8587001
Slope_1   -0.1955869  0.0541649 -0.3017483 -0.0894256 -3.6110 0.0003051
Tau2_2     0.0035335  0.0024306 -0.0012303  0.0082974  1.4538 0.1460058
Tau2_3     0.0029079  0.0031183 -0.0032039  0.0090197  0.9325 0.3510704
             
Intercept    
Slope_1   ***
Tau2_2       
Tau2_3       
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 221.2809
Degrees of freedom of the Q statistic: 65
P value of the Q statistic: 0

Explained variances (R2):
                         Level 2 Level 3
Tau2 (no predictor)    0.0037965  0.0141
Tau2 (with predictors) 0.0035335  0.0029
R2                     0.0692595  0.7943

Number of studies (or clusters): 21
Number of observed statistics: 66
Number of estimated parameters: 4
Degrees of freedom: 62
-2 log likelihood: 17.62569 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • Alternative model: Grants and Fellowship as indicator variables
    • If we want to estimate the effects for both Grants and Fellowship, we may create two indicator variables to represent them. Since we cannot estimate the intercept and these coefficients at the same time, we need to fix the intercept at 0 by specifying the intercept.constraints=0 argument in meta3(). We are now able to include both Grants and Fellowship in the analysis. The estimated effects (and their 95% CIs) for Grants and Fellowship were -0.0066 (-0.0793, 0.0661) and -0.2022 (-0.2805, -0.1239), respectively.
## Alternative model: Grants and Fellowship as indicators  
## Indicator variables
Grants <- ifelse(Bornmann07$Type=="Grants", yes=1, no=0)
Fellowship <- ifelse(Bornmann07$Type=="Fellowship", yes=1, no=0)
  
summary(Model1b <- meta3(logOR, v, x=cbind(Grants, Fellowship), 
                         cluster=Cluster, data=Bornmann07,
                         intercept.constraints=0, model.name="Model 1"))

Call:
meta3(y = logOR, v = v, cluster = Cluster, x = cbind(Grants, 
    Fellowship), data = Bornmann07, intercept.constraints = 0, 
    model.name = "Model 1")

95% confidence intervals: z statistic approximation
Coefficients:
           Estimate   Std.Error      lbound      ubound z value  Pr(>|z|)
Slope_1  0.10000000          NA          NA          NA      NA        NA
Slope_2 -0.20209280  0.03953671 -0.27958332 -0.12460227 -5.1115 3.196e-07
Tau2_2   0.00357518  0.00229434 -0.00092164  0.00807201  1.5583    0.1192
Tau2_3   0.00271391  0.00037933  0.00197044  0.00345737  7.1546 8.396e-13
           
Slope_1    
Slope_2 ***
Tau2_2     
Tau2_3  ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 221.2809
Degrees of freedom of the Q statistic: 65
P value of the Q statistic: 0

Explained variances (R2):
                         Level 2 Level 3
Tau2 (no predictor)    0.0037965  0.0141
Tau2 (with predictors) 0.0035752  0.0027
R2                     0.0582930  0.8080

Number of studies (or clusters): 21
Number of observed statistics: 66
Number of estimated parameters: 4
Degrees of freedom: 62
-2 log likelihood: 17.65814 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)

Model 2: Year and Year^2 as covariates

  • When there are several covariates, we may combine them with the cbind() command. For example, cbind(Year, Year^2) includes both Year and its squared as covariates. In the output, Slope_1 and Slope_2 refer to the regression coefficients for Year and Year^2, respectively. To increase the numerical stability, the covariates are usually centered before creating the quadratic terms. Since Marsh et al. (2009) standardized Year in their analysis, I follow this practice here.
  • The estimated regression coefficients (and their 95% CIs) for =Year= and =Year^2= were -0.0010 (-0.0473, 0.0454) and -0.0118 (-0.0247, 0.0012), respectively. The \(R^2_{(2)}\) and \(R^2_{(3)}\) were 0.2430 and 0.0000, respectively.
## Model 2: Year and Year^2 as covariates
summary( Model2 <- meta3(logOR, v, x=cbind(scale(Year), scale(Year)^2), 
                         cluster=Cluster, data=Bornmann07,
                         model.name="Model 2") ) 

Call:
meta3(y = logOR, v = v, cluster = Cluster, x = cbind(scale(Year), 
    scale(Year)^2), data = Bornmann07, model.name = "Model 2")

95% confidence intervals: z statistic approximation
Coefficients:
             Estimate   Std.Error      lbound      ubound z value Pr(>|z|)
Intercept -0.08627312  0.04125581 -0.16713302 -0.00541321 -2.0912  0.03651
Slope_1   -0.00095287  0.02365224 -0.04731040  0.04540466 -0.0403  0.96786
Slope_2   -0.01176840  0.00659995 -0.02470407  0.00116727 -1.7831  0.07457
Tau2_2     0.00287389  0.00206817 -0.00117965  0.00692744  1.3896  0.16466
Tau2_3     0.01479446  0.00926095 -0.00335666  0.03294558  1.5975  0.11015
           
Intercept *
Slope_1    
Slope_2   .
Tau2_2     
Tau2_3     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 221.2809
Degrees of freedom of the Q statistic: 65
P value of the Q statistic: 0

Explained variances (R2):
                         Level 2 Level 3
Tau2 (no predictor)    0.0037965  0.0141
Tau2 (with predictors) 0.0028739  0.0148
R2                     0.2430133  0.0000

Number of studies (or clusters): 21
Number of observed statistics: 66
Number of estimated parameters: 5
Degrees of freedom: 61
-2 log likelihood: 22.3836 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • Testing \(H_0: \beta_{Year} = \beta_{Year^2}=0\) The test statistic was 3.4190 (df = 2), p = 0.1810. Thus, there is no evidence supporting that =Year= has a quadratic effect on the effect size.
## Testing beta_{Year} = beta_{Year^2}=0
anova(Model2, Model0)
     base    comparison ep minus2LL df       AIC   diffLL diffdf         p
1 Model 2          <NA>  5 22.38360 61  -99.6164       NA     NA        NA
2 Model 2 3 level model  3 25.80256 63 -100.1974 3.418955      2 0.1809603

Model 3: Discipline as a covariate

  • There are four categories in Discipline. multidisciplinary is used as the reference group in the analysis.
  • The estimated regression coefficients (and their 95% Wald CIs) for DisciplinePhy, DisciplineLife and DisciplineSoc were -0.0091 (-0.2041, 0.1859), -0.1262 (-0.2804, 0.0280) and -0.2370 (-0.4746, 0.0007), respectively. The \(R^2_2\) and \(R^2_3\) were 0.0000 and 0.4975, respectively.
## Model 3: Discipline as a covariate
## Create dummy variables using multidisciplinary as the reference group
DisciplinePhy <- ifelse(Bornmann07$Discipline=="Physical sciences", yes=1, no=0)
DisciplineLife <- ifelse(Bornmann07$Discipline=="Life sciences/biology", yes=1, no=0)
DisciplineSoc <- ifelse(Bornmann07$Discipline=="Social sciences/humanities", yes=1, no=0)
summary( Model3 <- meta3(logOR, v, x=cbind(DisciplinePhy, DisciplineLife, DisciplineSoc), 
                         cluster=Cluster, data=Bornmann07,
                         model.name="Model 3") )

Call:
meta3(y = logOR, v = v, cluster = Cluster, x = cbind(DisciplinePhy, 
    DisciplineLife, DisciplineSoc), data = Bornmann07, model.name = "Model 3")

95% confidence intervals: z statistic approximation
Coefficients:
            Estimate  Std.Error     lbound     ubound z value Pr(>|z|)  
Intercept -0.0147478  0.0638994 -0.1399885  0.1104928 -0.2308  0.81747  
Slope_1   -0.0091306  0.0994920 -0.2041313  0.1858701 -0.0918  0.92688  
Slope_2   -0.1261796  0.0786627 -0.2803557  0.0279966 -1.6041  0.10870  
Slope_3   -0.2369570  0.1212309 -0.4745652  0.0006513 -1.9546  0.05063 .
Tau2_2     0.0039094  0.0028395 -0.0016559  0.0094747  1.3768  0.16857  
Tau2_3     0.0071034  0.0064321 -0.0055033  0.0197101  1.1044  0.26944  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 221.2809
Degrees of freedom of the Q statistic: 65
P value of the Q statistic: 0

Explained variances (R2):
                         Level 2 Level 3
Tau2 (no predictor)    0.0037965  0.0141
Tau2 (with predictors) 0.0039094  0.0071
R2                     0.0000000  0.4975

Number of studies (or clusters): 21
Number of observed statistics: 66
Number of estimated parameters: 6
Degrees of freedom: 60
-2 log likelihood: 20.07571 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • Testing whether Discipline is significant
    • The test statistic was 5.7268 (df = 3), p = 0.1257 which is not significant. Therefore, there is no evidence supporting that =Discipline= explains the variation of the effect sizes.
## Testing whether Discipline is significant
anova(Model3, Model0)
     base    comparison ep minus2LL df        AIC   diffLL diffdf
1 Model 3          <NA>  6 20.07571 60  -99.92429       NA     NA
2 Model 3 3 level model  3 25.80256 63 -100.19744 5.726842      3
          p
1        NA
2 0.1256832

Model 4: Country as a covariate

  • There are five categories in Country. United States is used as the reference group in the analysis.
  • The estimated regression coefficients (and their 95% Wald CIs) for CountryAus, CountryCan, CountryEur, and CountryUK are -0.0240 (-0.2405, 0.1924), -0.1341 (-0.3674, 0.0993), -0.2211 (-0.3660, -0.0762) and 0.0537 (-0.1413, 0.2487), respectively. The \(R^2_2\) and \(R^2_3\) were 0.1209 and 0.6606, respectively.
## Model 4: Country as a covariate
## Create dummy variables using the United States as the reference group
CountryAus <- ifelse(Bornmann07$Country=="Australia", yes=1, no=0)
CountryCan <- ifelse(Bornmann07$Country=="Canada", yes=1, no=0)
CountryEur <- ifelse(Bornmann07$Country=="Europe", yes=1, no=0)
CountryUK <- ifelse(Bornmann07$Country=="United Kingdom", yes=1, no=0)
  
summary( Model4 <- meta3(logOR, v, x=cbind(CountryAus, CountryCan, CountryEur, 
                         CountryUK), cluster=Cluster, data=Bornmann07,
                         model.name="Model 4") )  

Call:
meta3(y = logOR, v = v, cluster = Cluster, x = cbind(CountryAus, 
    CountryCan, CountryEur, CountryUK), data = Bornmann07, model.name = "Model 4")

95% confidence intervals: z statistic approximation
Coefficients:
            Estimate  Std.Error     lbound     ubound z value Pr(>|z|)   
Intercept  0.0025681  0.0597768 -0.1145922  0.1197284  0.0430 0.965732   
Slope_1   -0.0240109  0.1104328 -0.2404552  0.1924333 -0.2174 0.827876   
Slope_2   -0.1340800  0.1190667 -0.3674465  0.0992865 -1.1261 0.260127   
Slope_3   -0.2210801  0.0739174 -0.3659556 -0.0762046 -2.9909 0.002782 **
Slope_4    0.0537251  0.0994803 -0.1412527  0.2487030  0.5401 0.589157   
Tau2_2     0.0033376  0.0023492 -0.0012667  0.0079420  1.4208 0.155383   
Tau2_3     0.0047979  0.0044818 -0.0039862  0.0135820  1.0705 0.284379   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 221.2809
Degrees of freedom of the Q statistic: 65
P value of the Q statistic: 0

Explained variances (R2):
                         Level 2 Level 3
Tau2 (no predictor)    0.0037965  0.0141
Tau2 (with predictors) 0.0033376  0.0048
R2                     0.1208598  0.6606

Number of studies (or clusters): 21
Number of observed statistics: 66
Number of estimated parameters: 7
Degrees of freedom: 59
-2 log likelihood: 14.18259 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • Testing whether Discipline is significant
    • The test statistic was 11.6200 (df = 4), p = 0.0204 which is statistically significant.
  ## Testing whether Discipline is significant
  anova(Model4, Model0)
     base    comparison ep minus2LL df       AIC   diffLL diffdf
1 Model 4          <NA>  7 14.18259 59 -103.8174       NA     NA
2 Model 4 3 level model  3 25.80256 63 -100.1974 11.61996      4
           p
1         NA
2 0.02041284

Model 5: Type and Discipline as covariates

  • The \(R^2_{(2)}\) and \(R^2_{(3)}\) were 0.3925 and 1.0000, respectively. The \(\hat{\tau}^2_{(3)}\) was near 0 after controlling for the covariates.
## Model 5: Type and Discipline as covariates
summary( Model5 <- meta3(logOR, v, x=cbind(Type2, DisciplinePhy, DisciplineLife, 
                         DisciplineSoc), cluster=Cluster, data=Bornmann07,
                         model.name="Model 5") )

Call:
meta3(y = logOR, v = v, cluster = Cluster, x = cbind(Type2, DisciplinePhy, 
    DisciplineLife, DisciplineSoc), data = Bornmann07, model.name = "Model 5")

95% confidence intervals: z statistic approximation
Coefficients:
             Estimate   Std.Error      lbound      ubound z value
Intercept  6.7036e-02  1.8553e-02  3.0672e-02  1.0340e-01  3.6132
Slope_1   -1.9004e-01  4.0234e-02 -2.6890e-01 -1.1118e-01 -4.7233
Slope_2    1.9511e-02  6.5942e-02 -1.0973e-01  1.4875e-01  0.2959
Slope_3   -1.2779e-01  3.5914e-02 -1.9818e-01 -5.7400e-02 -3.5582
Slope_4   -2.3950e-01  9.4054e-02 -4.2384e-01 -5.5154e-02 -2.5464
Tau2_2     2.3062e-03  1.4270e-03 -4.9059e-04  5.1030e-03  1.6162
Tau2_3     1.0000e-10          NA          NA          NA      NA
           Pr(>|z|)    
Intercept 0.0003025 ***
Slope_1    2.32e-06 ***
Slope_2   0.7673210    
Slope_3   0.0003734 ***
Slope_4   0.0108849 *  
Tau2_2    0.1060586    
Tau2_3           NA    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 221.2809
Degrees of freedom of the Q statistic: 65
P value of the Q statistic: 0

Explained variances (R2):
                         Level 2 Level 3
Tau2 (no predictor)    0.0037965  0.0141
Tau2 (with predictors) 0.0023062  0.0000
R2                     0.3925434  1.0000

Number of studies (or clusters): 21
Number of observed statistics: 66
Number of estimated parameters: 7
Degrees of freedom: 59
-2 log likelihood: 4.66727 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • Testing whether Discipline is significant after controlling for Type
    • The test statistic was 12.9584 (df = 3), p = 0.0047 which is significant. Therefore, Discipline is still significant after controlling for Type.
## Testing whether Discipline is significant after controlling for Type
anova(Model5, Model1)
     base            comparison ep minus2LL df       AIC   diffLL diffdf
1 Model 5                  <NA>  7  4.66727 59 -113.3327       NA     NA
2 Model 5 Meta analysis with ML  4 17.62569 62 -106.3743 12.95842      3
            p
1          NA
2 0.004727388

Model 6: Type and Country as covariates

  • The \(R^2_{(2)}\) and \(R^2_{(3)}\) were 0.3948 and 1.0000, respectively. The \(\hat{\tau}^2_{(3)}\) was near 0 after controlling for the covariates.
## Model 6: Type and Country as covariates
summary( Model6 <- meta3(logOR, v, x=cbind(Type2, CountryAus, CountryCan, 
                         CountryEur, CountryUK), cluster=Cluster, data=Bornmann07,
                         model.name="Model 6") ) 

Call:
meta3(y = logOR, v = v, cluster = Cluster, x = cbind(Type2, CountryAus, 
    CountryCan, CountryEur, CountryUK), data = Bornmann07, model.name = "Model 6")

95% confidence intervals: z statistic approximation
Coefficients:
             Estimate   Std.Error      lbound      ubound z value
Intercept  6.7507e-02  1.8933e-02  3.0399e-02  1.0461e-01  3.5656
Slope_1   -1.5167e-01  4.1113e-02 -2.3225e-01 -7.1092e-02 -3.6892
Slope_2   -6.9580e-02  8.5164e-02 -2.3650e-01  9.7339e-02 -0.8170
Slope_3   -1.4231e-01  7.5204e-02 -2.8970e-01  5.0879e-03 -1.8923
Slope_4   -1.6116e-01  4.0203e-02 -2.3995e-01 -8.2361e-02 -4.0086
Slope_5    9.0419e-03  7.0074e-02 -1.2830e-01  1.4639e-01  0.1290
Tau2_2     2.2976e-03  1.4407e-03 -5.2618e-04  5.1213e-03  1.5947
Tau2_3     1.0000e-10          NA          NA          NA      NA
           Pr(>|z|)    
Intercept 0.0003631 ***
Slope_1   0.0002250 ***
Slope_2   0.4139266    
Slope_3   0.0584497 .  
Slope_4   6.108e-05 ***
Slope_5   0.8973315    
Tau2_2    0.1107693    
Tau2_3           NA    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 221.2809
Degrees of freedom of the Q statistic: 65
P value of the Q statistic: 0

Explained variances (R2):
                         Level 2 Level 3
Tau2 (no predictor)    0.0037965  0.0141
Tau2 (with predictors) 0.0022976  0.0000
R2                     0.3948192  1.0000

Number of studies (or clusters): 21
Number of observed statistics: 66
Number of estimated parameters: 8
Degrees of freedom: 58
-2 log likelihood: 5.076592 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • Testing whether Country is significant after controlling for Type
    • The test statistic was 12.5491 (df = 4), p = 0.0137. Thus, Country is significant after controlling for Type.
## Testing whether Country is significant after controlling for Type
anova(Model6, Model1)
     base            comparison ep  minus2LL df       AIC  diffLL diffdf
1 Model 6                  <NA>  8  5.076592 58 -110.9234      NA     NA
2 Model 6 Meta analysis with ML  4 17.625692 62 -106.3743 12.5491      4
           p
1         NA
2 0.01370262

Model 7: Discipline and Country as covariates

  • The \(R^2_{(2)}\) and \(R^2_{(3)}\) were 0.1397 and 0.7126, respectively.
## Model 7: Discipline and Country as covariates
summary( meta3(logOR, v, x=cbind(DisciplinePhy, DisciplineLife, DisciplineSoc,
                         CountryAus, CountryCan, CountryEur, CountryUK), 
                         cluster=Cluster, data=Bornmann07,
                         model.name="Model 7") )

Call:
meta3(y = logOR, v = v, cluster = Cluster, x = cbind(DisciplinePhy, 
    DisciplineLife, DisciplineSoc, CountryAus, CountryCan, CountryEur, 
    CountryUK), data = Bornmann07, model.name = "Model 7")

95% confidence intervals: z statistic approximation
Coefficients:
            Estimate  Std.Error     lbound     ubound z value Pr(>|z|)  
Intercept  0.0029305  0.0576743 -0.1101090  0.1159700  0.0508  0.95948  
Slope_1    0.1742169  0.1702553 -0.1594774  0.5079113  1.0233  0.30618  
Slope_2    0.0826806  0.1599802 -0.2308748  0.3962359  0.5168  0.60528  
Slope_3   -0.0462265  0.1715773 -0.3825118  0.2900588 -0.2694  0.78761  
Slope_4   -0.0486321  0.1306918 -0.3047834  0.2075192 -0.3721  0.70981  
Slope_5   -0.2169132  0.1915703 -0.5923841  0.1585576 -1.1323  0.25751  
Slope_6   -0.3036578  0.1670720 -0.6311128  0.0237973 -1.8175  0.06914 .
Slope_7   -0.0605272  0.1809419 -0.4151667  0.2941123 -0.3345  0.73799  
Tau2_2     0.0032661  0.0022784 -0.0011994  0.0077317  1.4335  0.15171  
Tau2_3     0.0040618  0.0038459 -0.0034759  0.0115996  1.0562  0.29090  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 221.2809
Degrees of freedom of the Q statistic: 65
P value of the Q statistic: 0

Explained variances (R2):
                         Level 2 Level 3
Tau2 (no predictor)    0.0037965  0.0141
Tau2 (with predictors) 0.0032661  0.0041
R2                     0.1396973  0.7126

Number of studies (or clusters): 21
Number of observed statistics: 66
Number of estimated parameters: 10
Degrees of freedom: 56
-2 log likelihood: 10.31105 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)

Model 8: Type, Discipline and Country as covariates

  • The \(R^2_{(2)}\) and \(R^2_{(3)}\) were 0.4466 and 1.0000, respectively. The \(\hat{\tau}^2_{(3)}\) was near 0 after controlling for the covariates.
## Model 8: Type, Discipline and Country as covariates
Model8 <- meta3(logOR, v, x=cbind(Type2, DisciplinePhy, DisciplineLife, DisciplineSoc,
                           CountryAus, CountryCan, CountryEur, CountryUK), 
                           cluster=Cluster, data=Bornmann07,
                           model.name="Model 8") 
## There was an estimation error. The model was rerun again.
summary(rerun(Model8))
[1] "0.0535236181791552,-0.173644299023977,0.225829442961052,0.109119799000635,-0.00998438234060444,-0.0965364646916492,-0.23908250881336,-0.328450166224618,-0.125433589747216,0.00229312698904285,7.77101331739686e-11"

Call:
meta3(y = logOR, v = v, cluster = Cluster, x = cbind(Type2, DisciplinePhy, 
    DisciplineLife, DisciplineSoc, CountryAus, CountryCan, CountryEur, 
    CountryUK), data = Bornmann07, model.name = "Model 8")

95% confidence intervals: z statistic approximation
Coefficients:
             Estimate   Std.Error      lbound      ubound z value
Intercept  6.8563e-02  1.8630e-02  3.2049e-02  1.0508e-01  3.6802
Slope_1   -1.6885e-01  4.1545e-02 -2.5028e-01 -8.7425e-02 -4.0643
Slope_2    2.5329e-01  1.5814e-01 -5.6670e-02  5.6324e-01  1.6016
Slope_3    1.2689e-01  1.4774e-01 -1.6268e-01  4.1646e-01  0.8589
Slope_4   -8.3549e-03  1.5796e-01 -3.1795e-01  3.0124e-01 -0.0529
Slope_5   -1.1530e-01  1.1147e-01 -3.3377e-01  1.0317e-01 -1.0344
Slope_6   -2.6412e-01  1.6402e-01 -5.8559e-01  5.7343e-02 -1.6103
Slope_7   -2.9029e-01  1.4859e-01 -5.8152e-01  9.5188e-04 -1.9536
Slope_8   -1.5975e-01  1.6285e-01 -4.7893e-01  1.5943e-01 -0.9810
Tau2_2     2.1010e-03  1.2925e-03 -4.3226e-04  4.6342e-03  1.6255
Tau2_3     9.9973e-11          NA          NA          NA      NA
           Pr(>|z|)    
Intercept  0.000233 ***
Slope_1   4.818e-05 ***
Slope_2    0.109239    
Slope_3    0.390410    
Slope_4    0.957818    
Slope_5    0.300948    
Slope_6    0.107323    
Slope_7    0.050754 .  
Slope_8    0.326609    
Tau2_2     0.104051    
Tau2_3           NA    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 221.2809
Degrees of freedom of the Q statistic: 65
P value of the Q statistic: 0

Explained variances (R2):
                         Level 2 Level 3
Tau2 (no predictor)    0.0037965  0.0141
Tau2 (with predictors) 0.0021010  0.0000
R2                     0.4466073  1.0000

Number of studies (or clusters): 21
Number of observed statistics: 66
Number of estimated parameters: 11
Degrees of freedom: 55
-2 log likelihood: -1.645211 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)

Handling missing covariates with FIML

When there are missing data in the covariates, data with missing values are excluded before the analysis in meta3(). The missing covariates can be handled by the use of FIML in meta3X(). We illustrate two examples on how to analyze data with missing covariates with missing completely at random (MCAR) and missing at random (MAR) data.

MCAR

  • About 25% of the level-2 covariate Type was introduced by the MCAR mechanism.
#### Handling missing covariates with FIML
  
## MCAR
## Set seed for replication
set.seed(1000000)
  
## Copy Bornmann07 to my.df
my.df <- Bornmann07
## "Fellowship": 1; "Grant": 0
my.df$Type_MCAR <- ifelse(Bornmann07$Type=="Fellowship", yes=1, no=0)

## Create 17 out of 66 missingness with MCAR
my.df$Type_MCAR[sample(1:66, 17)] <- NA
  
summary(meta3(y=logOR, v=v, cluster=Cluster, x=Type_MCAR, data=my.df))

Call:
meta3(y = logOR, v = v, cluster = Cluster, x = Type_MCAR, data = my.df)

95% confidence intervals: z statistic approximation
Coefficients:
             Estimate   Std.Error      lbound      ubound z value
Intercept -0.00484542  0.03934429 -0.08195881  0.07226797 -0.1232
Slope_1   -0.21090081  0.05346221 -0.31568482 -0.10611680 -3.9449
Tau2_2     0.00446788  0.00549282 -0.00629784  0.01523361  0.8134
Tau2_3     0.00092884  0.00336491 -0.00566625  0.00752394  0.2760
           Pr(>|z|)    
Intercept    0.9020    
Slope_1   7.985e-05 ***
Tau2_2       0.4160    
Tau2_3       0.7825    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 151.643
Degrees of freedom of the Q statistic: 48
P value of the Q statistic: 1.115552e-12

Explained variances (R2):
                         Level 2 Level 3
Tau2 (no predictor)    0.0042664  0.0145
Tau2 (with predictors) 0.0044679  0.0009
R2                     0.0000000  0.9361

Number of studies (or clusters): 20
Number of observed statistics: 49
Number of estimated parameters: 4
Degrees of freedom: 45
-2 log likelihood: 13.13954 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • There is no need to specify whether the covariates are level 2 or level 3 in meta3() because the covariates are treated as a design matrix. When meta3X() is used, users need to specify whether the covariates are at level 2 (x2) or level 3 (x3).
summary(meta3X(y=logOR, v=v, cluster=Cluster, x2=Type_MCAR, data=my.df))

Call:
meta3X(y = logOR, v = v, cluster = Cluster, x2 = Type_MCAR, data = my.df)

95% confidence intervals: z statistic approximation
Coefficients:
            Estimate  Std.Error     lbound     ubound z value Pr(>|z|)   
Intercept -0.0106318  0.0397685 -0.0885766  0.0673131 -0.2673 0.789207   
SlopeX2_1 -0.1753249  0.0582619 -0.2895162 -0.0611336 -3.0093 0.002619 **
Tau2_2     0.0030338  0.0026839 -0.0022266  0.0082941  1.1304 0.258324   
Tau2_3     0.0036839  0.0042817 -0.0047082  0.0120759  0.8604 0.389586   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Explained variances (R2):
                         Level 2 Level 3
Tau2 (no predictor)    0.0037965  0.0141
Tau2 (with predictors) 0.0030338  0.0037
R2                     0.2009070  0.7394

Number of studies (or clusters): 21
Number of observed statistics: 115
Number of estimated parameters: 7
Degrees of freedom: 108
-2 log likelihood: 49.76372 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)

MAR

  • For the case for missing covariates with MAR, the missingness in Type depends on the values of Year. Type is missing when Year is smaller than 1996.
## MAR
Type_MAR <- ifelse(Bornmann07$Type=="Fellowship", yes=1, no=0)
  
## Create 27 out of 66 missingness with MAR for cases Year<1996
index_MAR <- ifelse(Bornmann07$Year<1996, yes=TRUE, no=FALSE)
Type_MAR[index_MAR] <- NA
  
summary(meta3(logOR, v, x=Type_MAR, cluster=Cluster, data=Bornmann07)) 

Call:
meta3(y = logOR, v = v, cluster = Cluster, x = Type_MAR, data = Bornmann07)

95% confidence intervals: z statistic approximation
Coefficients:
             Estimate   Std.Error      lbound      ubound z value Pr(>|z|)
Intercept -0.01587052  0.03952546 -0.09333900  0.06159795 -0.4015 0.688032
Slope_1   -0.17573648  0.06328326 -0.29976939 -0.05170356 -2.7770 0.005487
Tau2_2     0.00259266  0.00171596 -0.00077056  0.00595588  1.5109 0.130811
Tau2_3     0.00278384  0.00267150 -0.00245221  0.00801989  1.0421 0.297388
            
Intercept   
Slope_1   **
Tau2_2      
Tau2_3      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q statistic on the homogeneity of effect sizes: 151.11
Degrees of freedom of the Q statistic: 38
P value of the Q statistic: 1.998401e-15

Explained variances (R2):
                         Level 2 Level 3
Tau2 (no predictor)    0.0029593  0.0097
Tau2 (with predictors) 0.0025927  0.0028
R2                     0.1238925  0.7121

Number of studies (or clusters): 12
Number of observed statistics: 39
Number of estimated parameters: 4
Degrees of freedom: 35
-2 log likelihood: -24.19956 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)
  • It is possible to include level 2 (av2) and level 3 (av3) auxiliary variables. Auxiliary variables are those that predict the missing values or are correlated with the variables that contain missing values. The inclusion of auxiliary variables can improve the efficiency of the estimation and the parameter estimates.
## Include auxiliary variable
summary(meta3X(y=logOR, v=v, cluster=Cluster, x2=Type_MAR, av2=Year, data=my.df))

Call:
meta3X(y = logOR, v = v, cluster = Cluster, x2 = Type_MAR, av2 = Year, 
    data = my.df)

95% confidence intervals: z statistic approximation
Coefficients:
            Estimate  Std.Error     lbound     ubound z value Pr(>|z|)   
Intercept -0.0264057  0.0572041 -0.1385237  0.0857122 -0.4616 0.644364   
SlopeX2_1 -0.2003999  0.0691082 -0.3358495 -0.0649503 -2.8998 0.003734 **
Tau2_2     0.0029970  0.0022371 -0.0013877  0.0073816  1.3397 0.180358   
Tau2_3     0.0030212  0.0032463 -0.0033414  0.0093838  0.9307 0.352029   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Explained variances (R2):
                         Level 2 Level 3
Tau2 (no predictor)    0.0049237  0.0088
Tau2 (with predictors) 0.0029970  0.0030
R2                     0.3913242  0.6571

Number of studies (or clusters): 21
Number of observed statistics: 171
Number of estimated parameters: 14
Degrees of freedom: 157
-2 log likelihood: 377.3479 
OpenMx status1: 0 ("0" or "1": The optimization is considered fine.
Other values indicate problems.)

Implementing three-level meta-analyses as structural equation models in OpenMx

This section illustrates how to formulate three-level meta-analyses as structural equation models using the OpenMx package. The steps outline how to create the model-implied mean vector and the model-implied covariance matrix to fit the three-level meta-analyses. y is the effect size (standardized mean difference on the modified school calendars) and v is its sampling variance. =District= is the variable for the cluster effect, whereas Year is the year of publication.

Preparing data

  • Data in a three-level meta-analysis are usually stored in the long format, e.g., Cooper03 in this example, whereas the SEM approach uses the wide format.
  • Suppose the maximum number of effect sizes in the level-2 unit is \(k\) (\(k=11\) in this example). Each cluster is represented by one row with \(k=11\) variables representing the outcome effect size, say y_1 to y_11 in this example. The incomplete data are represented by NA (missing value).
  • Similarly, \(k=11\) variables are required to represent the known sampling variances, say v_1 to v_11 in this example.
  • If the covariates are at level 2, \(k=11\) variables are also required to represent each of them. For example, Year is a level-2 covariate, Year_1 to Year_11 are required to represent it.
  • Several extra steps are required to handle missing values. Missing values (represented by NA in R) are not allowed in v_1 to v_11 as they are definition variables. The missing data are converted into some arbitrary values, say 1e10 in this example. The actual value does not matter because the missing values will be removed before the analysis. It is because missing values in y_1 to y_11 (and v_1 to v_11) will be filtered out automatically by the use of FIML.
#### Steps in Analyzing Three-level Meta-analysis in OpenMx

#### Preparing data
## Load the library
library("OpenMx")
  
## Get the dataset from the metaSEM library
data(Cooper03, package="metaSEM")
  
## Make a copy of the original data
my.long <- Cooper03

## Show the first few cases in my.long
head(my.long)
  District Study     y     v Year
1       11     1 -0.18 0.118 1976
2       11     2 -0.22 0.118 1976
3       11     3  0.23 0.144 1976
4       11     4 -0.30 0.144 1976
5       12     5  0.13 0.014 1989
6       12     6 -0.26 0.014 1989
## Center the Year to increase numerical stability
my.long$Year <- scale(my.long$Year, scale=FALSE)
  
## maximum no. of effect sizes in level-2
k <- 11
  
## Create a variable called "time" to store: 1, 2, 3, ... k
my.long$time <- c(unlist(sapply(split(my.long$y, my.long$District), 
                                function(x) 1:length(x))))

## Convert long format to wide format by "District"
my.wide <- reshape(my.long, timevar="time", idvar=c("District"), 
                   sep="_", direction="wide")

## NA in v is due to NA in y in wide format
## Replace NA with 1e10 in "v"
temp <- my.wide[, paste("v_", 1:k, sep="")]
temp[is.na(temp)] <- 1e10
my.wide[, paste("v_", 1:k, sep="")] <- temp
  
## Replace NA with 0 in "Year"
temp <- my.wide[, paste("Year_", 1:k, sep="")]
temp[is.na(temp)] <- 0
my.wide[, paste("Year_", 1:k, sep="")] <- temp

## Show the first few cases in my.wide
head(my.wide)
   District Study_1   y_1   v_1      Year_1 Study_2   y_2   v_2
1        11       1 -0.18 0.118 -13.5535714       2 -0.22 0.118
5        12       5  0.13 0.014  -0.5535714       6 -0.26 0.014
9        18       9  0.45 0.023   4.4464286      10  0.38 0.043
12       27      12  0.16 0.020 -13.5535714      13  0.65 0.004
16       56      16  0.08 0.019   7.4464286      17  0.04 0.007
20       58      20 -0.18 0.020 -13.5535714      21  0.00 0.018
        Year_2 Study_3  y_3   v_3      Year_3 Study_4   y_4      v_4
1  -13.5535714       3 0.23 0.144 -13.5535714       4 -0.30 1.44e-01
5   -0.5535714       7 0.19 0.015  -0.5535714       8  0.32 2.40e-02
9    4.4464286      11 0.29 0.012   4.4464286      NA    NA 1.00e+10
12 -13.5535714      14 0.36 0.004 -13.5535714      15  0.60 7.00e-03
16   7.4464286      18 0.19 0.005   7.4464286      19 -0.06 4.00e-03
20 -13.5535714      22 0.00 0.019 -13.5535714      23 -0.28 2.20e-02
        Year_4 Study_5   y_5   v_5    Year_5 Study_6  y_6     v_6
1  -13.5535714      NA    NA 1e+10   0.00000      NA   NA 1.0e+10
5   -0.5535714      NA    NA 1e+10   0.00000      NA   NA 1.0e+10
9    0.0000000      NA    NA 1e+10   0.00000      NA   NA 1.0e+10
12 -13.5535714      NA    NA 1e+10   0.00000      NA   NA 1.0e+10
16   7.4464286      NA    NA 1e+10   0.00000      NA   NA 1.0e+10
20 -13.5535714      24 -0.04 2e-02 -13.55357      25 -0.3 2.1e-02
      Year_6 Study_7  y_7   v_7    Year_7 Study_8 y_8   v_8    Year_8
1    0.00000      NA   NA 1e+10   0.00000      NA  NA 1e+10   0.00000
5    0.00000      NA   NA 1e+10   0.00000      NA  NA 1e+10   0.00000
9    0.00000      NA   NA 1e+10   0.00000      NA  NA 1e+10   0.00000
12   0.00000      NA   NA 1e+10   0.00000      NA  NA 1e+10   0.00000
16   0.00000      NA   NA 1e+10   0.00000      NA  NA 1e+10   0.00000
20 -13.55357      26 0.07 6e-03 -13.55357      27   0 7e-03 -13.55357
   Study_9  y_9   v_9    Year_9 Study_10  y_10  v_10   Year_10 Study_11
1       NA   NA 1e+10   0.00000       NA    NA 1e+10   0.00000       NA
5       NA   NA 1e+10   0.00000       NA    NA 1e+10   0.00000       NA
9       NA   NA 1e+10   0.00000       NA    NA 1e+10   0.00000       NA
12      NA   NA 1e+10   0.00000       NA    NA 1e+10   0.00000       NA
16      NA   NA 1e+10   0.00000       NA    NA 1e+10   0.00000       NA
20      28 0.05 7e-03 -13.55357       29 -0.08 7e-03 -13.55357       30
    y_11  v_11   Year_11
1     NA 1e+10   0.00000
5     NA 1e+10   0.00000
9     NA 1e+10   0.00000
12    NA 1e+10   0.00000
16    NA 1e+10   0.00000
20 -0.09 7e-03 -13.55357

Random-effects model

  • To implement a three-level meta-analysis as a structural equation model, we need to specify both the model-implied mean vector \(\mu(\theta)\), say =expMean=, and the model-implied covariance matrix \(\Sigma(\theta)\), say expCov.
  • When there is no covariate, the expected mean is a \(k \times 1\) vector with all elements of =beta0= (the intercept), i.e., \(\mu(\theta) = \left[ \begin{array}{c} 1 \\ \vdots \\ 1 \end{array} \right]\beta_0\). Since OpenMx expects a row vector rather than a column vector in the model-implied means, we need to transpose the expMean in the analysis.
  • Tau2 (\(T^2_{(2)}\)) and =Tau3= (\(T^2_{(3)}\)) are the level 2 and level 3 matrices of heterogeneity, respectively. =Tau2= is a diagonal matrix with elements of \(\tau^2_{(2)}\), whereas =Tau3= is a full matrix with elements of \(\tau^2_{(3)}\). =V= is a diagonal matrix of the known sampling variances \(v_{ij}\).
  • The model-implied covariance matrix is \(\Sigma(\theta) = T^2_{(3)} + T^2_{(2)} + V\).
  • All of these matrices are stored into a model called random.model.
#### Random-effects model  
## Intercept
Beta0 <- mxMatrix("Full", ncol=1, nrow=1, free=TRUE, labels="beta0", 
                  name="Beta0")
## 1 by k row vector of ones
Ones <- mxMatrix("Unit", nrow=k, ncol=1, name="Ones")
  
## Model implied mean vector 
## OpenMx expects a row vector rather than a column vector.
expMean <- mxAlgebra( t(Ones %*% Beta0), name="expMean")
  
## Tau2_2
Tau2 <- mxMatrix("Symm", ncol=1, nrow=1, values=0.01, free=TRUE, labels="tau2_2", 
                 name="Tau2")
Tau3 <- mxMatrix("Symm", ncol=1, nrow=1, values=0.01, free=TRUE, labels="tau2_3", 
                 name="Tau3")
  
## k by k identity matrix
Iden <- mxMatrix("Iden", nrow=k, ncol=k, name="Iden")
  
## Conditional sampling variances
## data.v_1, data.v_2, ... data.v_k represent values for definition variables
V <- mxMatrix("Diag", nrow=k, ncol=k, free=FALSE, 
              labels=paste("data.v", 1:k, sep="_"), name="V")
  
## Model implied covariance matrix
expCov <- mxAlgebra( Ones%*% Tau3 %*% t(Ones) + Iden %x% Tau2 + V, name="expCov")
  
## Model stores everthing together
random.model <- mxModel(model="Random effects model", 
                        mxData(observed=my.wide, type="raw"), 
                        Iden, Ones, Beta0, Tau2, Tau3, V, expMean, expCov,
                        mxExpectationNormal("expCov","expMean", 
                        dimnames=paste("y", 1:k, sep="_")),
                        mxFitFunctionML() )
  • We perform a random-effects three-level meta-analysis by running the model with the mxRun() command. The parameter estimates (and their _SE_s) for \(\beta_0\), \(\tau^2_{(2)}\) and \(\tau^2_{(3)}\) were 0.1845 (0.0805), 0.0329 (0.0111) and 0.0577 (0.0307), respectively.
summary( mxRun(random.model) )
Summary of Random effects model 
 
free parameters:
    name matrix row col   Estimate  Std.Error
1  beta0  Beta0   1   1 0.18445537 0.08054110
2 tau2_2   Tau2   1   1 0.03286479 0.01113968
3 tau2_3   Tau3   1   1 0.05773836 0.03074229

observed statistics:  56 
estimated parameters:  3 
degrees of freedom:  53 
-2 log likelihood:  16.78987 
number of observations:  11 
Information Criteria: 
      |  df Penalty  |  Parameters Penalty  |  Sample-Size Adjusted
AIC:      -89.21013               22.78987                       NA
BIC:     -110.29858               23.98356                 14.95056
Some of your fit indices are missing.
  To get them, fit saturated and independence models, and include them with
  summary(yourModel, SaturatedLikelihood=..., IndependenceLikelihood=...). 
timestamp: 2014-11-28 18:32:14 
Wall clock time (HH:MM:SS.hh): 00:00:00.08 
optimizer:  NPSOL 
OpenMx version number: 2.0.0.4004 
Need help?  See help(mxSummary) 

Mixed-effects model

  • We may extend a random-effects model to a mixed-effects model by including a covariate (Year in this example).
  • beta1 is the regression coefficient, whereas X stores the value of Year via definition variables.
  • The conditional model-implied mean vector is \(\mu(\theta|Year_{ij}) = \left[ \begin{array}{c} 1 \\ \vdots \\ 1 \end{array} \right]\beta_0 + \left[ \begin{array}{c} Year_{1j} \\ \vdots \\ Year_{kj} \end{array} \right]\beta_1\).
  • The conditional model-implied covariance matrix is the same as that in the random-effects model, i.e., \(\Sigma(\theta|Year_{ij}) = T^2_{(3)} + T^2_{(2)} + V\).
#### Mixed-effects model
  
## Design matrix via definition variable
X <- mxMatrix("Full", nrow=k, ncol=1, free=FALSE, 
              labels=paste("data.Year_", 1:k, sep=""), name="X")
  
## Regression coefficient
Beta1 <- mxMatrix("Full", nrow=1, ncol=1, free=TRUE, values=0,
                  labels="beta1", name="Beta1")
  
## Model implied mean vector
expMean <- mxAlgebra( t(Ones%*%Beta0 + X%*%Beta1), name="expMean")
  
mixed.model <- mxModel(model="Mixed effects model", 
                       mxData(observed=my.wide, type="raw"), 
                       Iden, Ones, Beta0, Beta1, Tau2, Tau3, V, expMean, expCov, 
                       X, mxExpectationNormal("expCov","expMean", 
                       dimnames=paste("y", 1:k, sep="_")),
                       mxFitFunctionML() )
  • The parameter estimates (and their _SE_s) for \(\beta_0\), \(\beta_1\), \(\tau^2_2\) and \(\tau^2_3\) were 0.1780 (0.0805), 0.0051 (0.0085), 0.0329 (0.0112) and 0.0565 (0.0300), respectively.
summary ( mxRun(mixed.model) )
Summary of Mixed effects model 
 
free parameters:
    name matrix row col    Estimate   Std.Error
1  beta0  Beta0   1   1 0.178026727 0.080521929
2  beta1  Beta1   1   1 0.005073715 0.008526627
3 tau2_2   Tau2   1   1 0.032939012 0.011162043
4 tau2_3   Tau3   1   1 0.056462847 0.030032975

observed statistics:  56 
estimated parameters:  4 
degrees of freedom:  52 
-2 log likelihood:  16.43629 
number of observations:  11 
Information Criteria: 
      |  df Penalty  |  Parameters Penalty  |  Sample-Size Adjusted
AIC:      -87.56371               24.43629                       NA
BIC:     -108.25427               26.02787                 13.98387
Some of your fit indices are missing.
  To get them, fit saturated and independence models, and include them with
  summary(yourModel, SaturatedLikelihood=..., IndependenceLikelihood=...). 
timestamp: 2014-11-28 18:32:14 
Wall clock time (HH:MM:SS.hh): 00:00:00.10 
optimizer:  NPSOL 
OpenMx version number: 2.0.0.4004 
Need help?  See help(mxSummary) 

References

Bornmann, L., Mutz, R., & Daniel, H.-D. (2007). Gender differences in grant peer review: A meta-analysis. Journal of Informetrics, 1(3), 226–238.

Cheung, M.W.-L. (2009). Constructing approximate confidence intervals for parameters with structural equation models. Structural Equation Modeling, 16(2), 267-294.

Cheung, M.W.-L. (2014a). Modeling dependent effect sizes with three-level meta-analyses: A structural equation modeling approach. Psychological Methods, 19, 211-229.

Cheung, M.W.-L. (2014b). metaSEM: Meta-analysis using structural equation modeling. Retrieved from http://courses.nus.edu.sg/course/psycwlm/Internet/metaSEM/

Cooper, H., Valentine, J.C., Charlton, K., & Melson, A. (2003). The effects of modified school calendars on student achievement and on school and community attitudes. Review of Educational Research, 73(1), 1 –52.

Konstantopoulos, S. (2011). Fixed effects and variance components estimation in three-level meta-analysis. Research Synthesis Methods, 2(1), 61–76.

Marsh, H.W., Bornmann, L., Mutz, R., Daniel, H.-D., & O’Mara, A. (2009). Gender effects in the peer reviews of grant proposals: A comprehensive meta-analysis comparing traditional and multilevel approaches. Review of Educational Research, 79(3), 1290–1326.

Neale, M.C., & Miller, M.B. (1997). The use of likelihood-based confidence intervals in genetic models. Behavior Genetics, 27(2), 113–120.