anova.rma: Likelihood Ratio and Wald-Type Tests for 'rma' Objects

Description

For two (nested) models of class "rma.uni" or "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.

Usage

# S3 method for rma
anova(object, object2, btt, X, att, Z, rhs, adjust, digits, refit=FALSE, ...)

Value

An object of class "anova.rma". When a single model is specified (without any further arguments or together with the btt or att argument), the object is a list containing the following components:

QM: test statistic of the Wald-type test of the model coefficients.
QMdf: corresponding degrees of freedom.
QMp: corresponding p-value.
btt: indices of the coefficients tested by the Wald-type test.
k: number of outcomes included in the analysis.
p: number of coefficients in the model (including the intercept).
m: number of coefficients included in the Wald-type test.
...: some additional elements/values.

When btt or att was a list, then the object is a list of class "list.anova.rma", where each element is an "anova.rma" object as described above.

When argument X is used, the object is a list containing the following components:

QM: test statistic of the omnibus Wald-type test of all linear combinations.
QMdf: corresponding degrees of freedom.
QMp: corresponding p-value.
hyp: description of the linear combinations tested.
Xb: values of the linear combinations.
se: standard errors of the linear combinations.
zval: test statistics of the linear combinations.
pval: corresponding p-values.

When two models are specified, the object is a list containing the following components:

fit.stats.f: log-likelihood, deviance, AIC, BIC, and AICc for the full model.
fit.stats.r: log-likelihood, deviance, AIC, BIC, and AICc for the reduced model.
parms.f: number of parameters in the full model.
parms.r: number of parameters in the reduced model.
LRT: likelihood ratio test statistic.
pval: corresponding p-value.
QE.f: test statistic of the test for (residual) heterogeneity from the full model.
QE.r: test statistic of the test for (residual) heterogeneity from the reduced model.
tau2.f: estimated ^2 value from the full model. NA for "rma.mv" objects.
tau2.r: estimated ^2 value from the reduced model. NA for "rma.mv" objects.
R2: amount (in percent) of the heterogeneity in the reduced model that is accounted for in the full model (NA 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.

The results are formatted and printed with the print function. To format the results as a data frame, one can use the as.data.frame function.

Arguments

object: an object of class "rma.uni" or "rma.mv".
object2: an (optional) object of class "rma.uni" or "rma.mv". Only relevant when conducting a model comparison and likelihood ratio test. See ‘Details’.
btt: optional vector of indices (or list thereof) to specify which coefficients should be included in the Wald-type test. Can also be a string to grep for. See ‘Details’.
X: optional numeric vector or matrix to specify one or more linear combinations of the coefficients in the model that should be tested. See ‘Details’.
att: optional vector of indices (or list thereof) to specify which scale coefficients should be included in the Wald-type test. Can also be a string to grep for. See ‘Details’. Only relevant for location-scale models (see rma.uni).
Z: optional numeric vector or matrix to specify one or more linear combinations of the scale coefficients in the model that should be tested. See ‘Details’. Only relevant for location-scale models (see rma.uni).
rhs: optional scalar or vector of values for the right-hand side of the null hypothesis when testing a set of coefficients (via btt or att) or linear combinations thereof (via X or Z). If unspecified, this defaults to a vector of zeros of the appropriate length. See ‘Details’.
adjust: optional argument to specify (as a character string) a method for adjusting the p-values of Wald-type tests for multiple testing. See p.adjust for possible options. Can be abbreviated. Can also be a logical and if TRUE, then a Bonferroni correction is used.
digits: optional 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.
refit: logical to specify whether models fitted with REML estimation and differing in their fixed effects should be refitted with ML estimation when conducting a likelihood ratio test (the default is FALSE).
...: other arguments.

Author

Wolfgang Viechtbauer (wvb@metafor-project.org, https://www.metafor-project.org).

Details

The function can be used in three different ways:

When a single model is specified (via argument object), the function provides a Wald-type test of one or more model coefficients, that is, H_0:\; _j btt = 0,H_0: _j ∈ btt = 0, where _j btt_j ∈ btt is the set of coefficients to be tested (by default whether the set of coefficients is significantly different from zero, but one can specify a different value under the null hypothesis via argument rhs).

In particular, for equal- or random-effects models (i.e., models without moderators), this is just the test of the single coefficient of the model (i.e., H_0:\; = 0H_0: = 0 or H_0:\; = 0H_0: = 0). For models including moderators, an omnibus test of all 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 coefficients in the model including the first.

Alternatively, one can manually specify the indices of the coefficients to test via the btt (‘betas to test’) argument. For example, with btt=c(3,4), only the third and fourth coefficients from the model are 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 (and hence, one can use regular expressions to fine-tune the search for matching strings). 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. One can also specify a list of indices/strings, in which case tests of all list elements will be conducted. See ‘Examples’.

For location-scale models fitted with the rma.uni function, one can use the att argument in an analogous manner to specify the indices of the scale coefficients to test (i.e., H_0:\; _j att = 0H_0: _j ∈ att = 0, where _j att_j ∈ att is the set of coefficients to be tested).
When a single model is specified (via argument object), one can use the X argument^1 to specify a linear combination of the coefficients in the model that should be tested using a Wald-type test, that is, H_0:\; x = 0,H_0: x = 0, where xx is a (row) vector of the same length as there are coefficients in the model (by default whether the linear combination is significantly different from zero, but one can specify a different value under the null hypothesis via argument rhs). One can also specify a matrix of linear combinations via the X argument to test H_0:\; X = 0,H_0: X = 0, where each row of XX defines a particular linear combination to be tested (if rhs is used, then it should either be a scalar or of the same length as the number of combinations 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 function, which provides corresponding confidence intervals. See also the pairmat function for constructing a matrix of pairwise contrasts for testing the levels of a categorical moderator against each other.

For location-scale models fitted with the rma.uni function, one can use the Z argument in an analogous manner to specify one or multiple linear combinations of the scale coefficients in the model that should be tested (i.e., H_0:\; Z = 0H_0: Z = 0).
When specifying two models for comparison (via arguments object and object2), the function provides a likelihood ratio test (LRT) 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 LRT to make sense. Also, LRTs are not meaningful when using REML estimation and the two models differ in terms of their fixed effects (setting refit=TRUE automatically refits the two models using ML estimation). Also, the theory underlying LRTs is only really applicable when comparing models that were fitted with ML/REML estimation, so if some other estimation method was used to fit the two models, the results should be treated with caution.

---------

^1 This argument used to be called L, but was renamed to X (but using L in place of X still works).

References

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

Viechtbauer, W., & López-López, J. A. (2022). Location-scale models for meta-analysis. Research Synthesis Methods. 13(6), 697--715. https://doi.org/10.1002/jrsm.1562

Examples

Run this code

### calculate log risk ratios and corresponding sampling variances
dat <- escalc(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg)

### fit random-effects model
res1 <- rma(yi, vi, data=dat, method="ML")
res1

### fit mixed-effects model with two moderators (absolute latitude and publication year)
res2 <- rma(yi, vi, mods = ~ ablat + year, data=dat, method="ML")
res2

### Wald-type test of the two moderators
anova(res2)

### alternative way of specifying the same test
anova(res2, X=rbind(c(0,1,0), c(0,0,1)))

### corresponding likelihood ratio test
anova(res1, res2)

### Wald-type test of a linear combination
anova(res2, X=c(1,35,1970))

### use predict() to obtain the same linear combination (with its CI)
predict(res2, newmods=c(35,1970))

### Wald-type tests of several linear combinations
anova(res2, X=cbind(1,seq(0,60,by=10),1970))

### adjust for multiple testing with the Bonferroni method
anova(res2, X=cbind(1,seq(0,60,by=10),1970), adjust="bonf")

### mixed-effects model with three moderators
res3 <- rma(yi, vi, mods = ~ ablat + year + alloc, data=dat, method="ML")
res3

### Wald-type test of the 'alloc' factor
anova(res3, btt=4:5)

### instead of specifying the coefficient numbers, grep for "alloc"
anova(res3, btt="alloc")

### specify a list for the 'btt' argument
anova(res3, btt=list(2,3,4:5))

### adjust for multiple testing with the Bonferroni method
anova(res3, btt=list(2,3,4:5), adjust="bonf")

############################################################################

### an example of doing LRTs of variance components in more complex models
dat <- dat.konstantopoulos2011
res <- rma.mv(yi, vi, random = ~ 1 | district/school, data=dat)

### likelihood ratio test of the district-level variance component
res0 <- rma.mv(yi, vi, random = ~ 1 | district/school, data=dat, sigma2=c(0,NA))
anova(res, res0)

### likelihood ratio test of the school-level variance component
res0 <- rma.mv(yi, vi, random = ~ 1 | district/school, data=dat, sigma2=c(NA,0))
anova(res, res0)

### likelihood ratio test of both variance components simultaneously
res0 <- rma.mv(yi, vi, data=dat)
anova(res, res0)

############################################################################

### an example illustrating a workflow involving cluster-robust inference
dat <- dat.assink2016

### assume that the effect sizes within studies are correlated with rho=0.6
V <- vcalc(vi, cluster=study, obs=esid, data=dat, rho=0.6)

### fit multilevel model using this approximate V matrix
res <- rma.mv(yi, V, random = ~ 1 | study/esid, data=dat)
res

### likelihood ratio tests of the two variance components
res0 <- rma.mv(yi, V, random = ~ 1 | study/esid, data=dat, sigma2=c(0,NA))
anova(res, res0)
res0 <- rma.mv(yi, V, random = ~ 1 | study/esid, data=dat, sigma2=c(NA,0))
anova(res, res0)

### use cluster-robust methods for inferences about the fixed effects
sav <- robust(res, cluster=study, clubSandwich=TRUE)
sav

### examine if 'deltype' is a potential moderator
res <- rma.mv(yi, V, mods = ~ deltype, random = ~ 1 | study/esid, data=dat)
sav <- robust(res, cluster=study, clubSandwich=TRUE)
sav

### note: the (denominator) dfs for the omnibus F-test are very low, so the results
### of this test may not be trustworthy; consider using cluster wild bootstrapping
if (FALSE) {
library(wildmeta)
Wald_test_cwb(res, constraints=constrain_zero(2:3), R=1000, seed=1234)
}

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