For two (nested) models of class
"rma.mv", the function provides a full versus reduced model comparison in terms of model fit statistics and a likelihood ratio test. When a single model is specified, a Wald-type test of one or more model coefficients or linear combinations thereof is carried out.
# S3 method for rma anova(object, object2, btt, L, digits, ...)
an object of class
an (optional) object of class
optional vector of indices to specify which coefficients should be included in the Wald-type test. Can also be a string to grep for. See ‘Details’.
optional numeric vector or matrix to specify one or more linear combinations of the coefficients in the model that should be tested. See ‘Details’.
integer to specify the number of decimal places to which the printed results should be rounded (if unspecified, the default is to take the value from the object).
When a single model is specified, the function provides Wald-type tests of one or more model coefficients or linear combinations thereof.
In particular, for a fixed- or random-effects model (i.e., a model without moderators), this is just the test of the single coefficient of the model. For models including moderators, an omnibus test of all the model coefficients is conducted that excludes the intercept (the first coefficient) if it is included in the model. If no intercept is included in the model, then the omnibus test includes all of the coefficients in the model including the first.
Alternatively, one can manually specify the indices of the coefficients to test via the
btt argument. For example, with
btt=c(3,4), only the third and fourth coefficient from the model would be included in the test (if an intercept is included in the model, then it corresponds to the first coefficient in the model). Instead of specifying the coefficient numbers, one can specify a string for
btt. In that case,
grep will be used to search for all coefficient names that match the string. Using the
btt argument, one can for example select all coefficients corresponding to a particular factor to test if the factor as a whole is significant. See ‘Examples’.
Instead, one can use the
L argument to specify a linear combination of the coefficients in the model that should be tested (i.e., whether the linear combination is significantly different from zero). If a matrix of linear combinations is specified, each row defines a particular linear combination to be tested. If the matrix is of full rank, an omnibus Wald-type test of all linear combinations is also provided. Linear combinations can also be obtained with the
predict.rma function, which will provide corresponding confidence intervals.
When specifying two models for comparison, the function provides a likelihood ratio test comparing the two models. The two models must be based on the same set of data, must be of the same class, and should be nested for the likelihood ratio test to make sense. Note that likelihood ratio tests are not meaningful when using REML estimation and the two models have different fixed effects.
An object of class
"anova.rma". When a single model is specified (without any further arguments or together with the
btt argument), the object is a list containing the following components:
test statistic of the Wald-type test of the model coefficients.
indices of the coefficients tested by the Wald-type test.
number of outcomes included in the analysis.
number of coefficients in the model (including the intercept).
number of coefficients included in the Wald-type test.
some additional elements/values.
test statistic of the omnibus Wald-type test of all linear combinations.
description of the linear combinations tested.
values of the linear combinations.
standard errors of the linear combinations.
test statistics of the linear combinations.
log-likelihood, deviance, AIC, BIC, and AICc for the full model.
log-likelihood, deviance, AIC, BIC, and AICc for the reduced model.
number of parameters in the full model.
number of parameters in the reduced model.
likelihood ratio test statistic.
test statistic of the test for (residual) heterogeneity from the full model.
test statistic of the test for (residual) heterogeneity from the reduced model.
estimated \(\tau^2\) value from the full model.
estimated \(\tau^2\) value from the reduced model.
amount (in percent) of the heterogeneity in the reduced model that is accounted for in the full model (
NA for fixed-effects models or for
"rma.mv" objects). This can be regarded as a pseudo \(R^2\) statistic (Raudenbush, 2009). Note that the value may not be very accurate unless \(k\) is large (Lopez-Lopez et al., 2014).
some additional elements/values.
In principle, one can consider likelihood ratio tests for the amount of (residual) heterogeneity in random- and mixed-effects models. The full model should then be fitted with either
method="REML" and the reduced model with
method="FE". The p-value for that test is based on a chi-square distribution with 1 degree of freedom, but actually needs to be adjusted for the fact that the parameter (i.e., \(\tau^2\)) falls on the boundary of the parameter space under the null hypothesis. Moreover, the Q-test usually keeps better control of the Type I error rate and therefore should be preferred (see Viechtbauer, 2007, for more details).
Hardy, R. J., & Thompson, S. G. (1996). A likelihood approach to meta-analysis with random effects. Statistics in Medicine, 15(6), 619--629. https://doi.org/10.1002/(sici)1097-0258(19960330)15:6%3C619::aid-sim188%3E3.0.co;2-a
Huizenga, H. M., Visser, I., & Dolan, C. V. (2011). Testing overall and moderator effects in random effects meta-regression. British Journal of Mathematical and Statistical Psychology, 64(1), 1--19. https://doi.org/10.1348/000711010X522687
López-López, J. A., Marín-Martínez, F., Sánchez-Meca, J., Van den Noortgate, W., & Viechtbauer, W. (2014). Estimation of the predictive power of the model in mixed-effects meta-regression: A simulation study. British Journal of Mathematical and Statistical Psychology, 67(1), 30--48. https://doi.org/10.1111/bmsp.12002
Raudenbush, S. W. (2009). Analyzing effect sizes: Random effects models. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis (2nd ed., pp. 295--315). New York: Russell Sage Foundation.
Viechtbauer, W. (2007). Hypothesis tests for population heterogeneity in meta-analysis. British Journal of Mathematical and Statistical Psychology, 60(1), 29--60. https://doi.org/10.1348/000711005X64042
Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1--48. https://doi.org/10.18637/jss.v036.i03
### calculate log risk ratios and corresponding sampling variances dat <- escalc(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg) ### random-effects model res1 <- rma(yi, vi, data=dat, method="ML") ### mixed-effects model with two moderators (absolute latitude and publication year) res2 <- rma(yi, vi, mods = ~ ablat + year, data=dat, method="ML") ### Wald-type test of the two moderators anova(res2)#> #> Test of Moderators (coefficients 2:3): #> QM(df = 2) = 33.9192, p-val < .0001 #>#> #> Hypotheses: #> 1: ablat = 0 #> 2: year = 0 #> #> Results: #> estimate se zval pval #> 1: -0.0308 0.0063 -4.9057 <.0001 #> 2: -0.0032 0.0092 -0.3471 0.7285 #> #> Omnibus Test of Hypotheses: #> QM(df = 2) = 33.9192, p-val < .0001 #>### corresponding likelihood ratio test anova(res1, res2)#> #> df AIC BIC AICc logLik LRT pval QE tau^2 #> Full 4 23.2922 25.5520 28.2922 -7.6461 28.3251 0.0269 #> Reduced 2 29.3302 30.4601 30.5302 -12.6651 10.0379 0.0066 152.2330 0.2800 #> R^2 #> Full #> Reduced 90.3948% #>### test of a linear combination anova(res2, L=c(1,35,1970))#> #> Hypothesis: #> 1: intrcpt + 35*ablat + 1970*year = 0 #> #> Results: #> estimate se zval pval #> 1: -0.7533 0.0837 -9.0039 <.0001 #> #> Test of Hypothesis: #> QM(df = 1) = 81.0700, p-val < .0001 #>### use predict() to obtain the same linear combination (with its CI) predict(res2, newmods=c(35,1970))#> #> pred se ci.lb ci.ub pi.lb pi.ub #> -0.7533 0.0837 -0.9173 -0.5893 -1.1142 -0.3925 #>### mixed-effects model with three moderators res3 <- rma(yi, vi, mods = ~ ablat + year + alloc, data=dat, method="ML") res3#> #> Mixed-Effects Model (k = 13; tau^2 estimator: ML) #> #> tau^2 (estimated amount of residual heterogeneity): 0.0329 (SE = 0.0272) #> tau (square root of estimated tau^2 value): 0.1814 #> I^2 (residual heterogeneity / unaccounted variability): 33.24% #> H^2 (unaccounted variability / sampling variability): 1.50 #> R^2 (amount of heterogeneity accounted for): 88.24% #> #> Test for Residual Heterogeneity: #> QE(df = 8) = 26.2030, p-val = 0.0010 #> #> Test of Moderators (coefficients 2:5): #> QM(df = 4) = 31.2609, p-val < .0001 #> #> Model Results: #> #> estimate se zval pval ci.lb ci.ub #> intrcpt -10.0824 23.9973 -0.4201 0.6744 -57.1162 36.9514 #> ablat -0.0277 0.0071 -3.9298 <.0001 -0.0416 -0.0139 *** #> year 0.0053 0.0122 0.4371 0.6620 -0.0185 0.0292 #> allocrandom -0.2826 0.2540 -1.1127 0.2658 -0.7804 0.2152 #> allocsystematic -0.1104 0.2551 -0.4330 0.6650 -0.6104 0.3895 #> #> --- #> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 #>### test the 'alloc' factor anova(res3, btt=4:5)#> #> Test of Moderators (coefficients 4:5): #> QM(df = 2) = 1.4674, p-val = 0.4801 #>### instead of specifying the coefficient numbers, grep for "alloc" anova(res3, btt="alloc")#> #> Test of Moderators (coefficients 4:5): #> QM(df = 2) = 1.4674, p-val = 0.4801 #>