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

"anova"(object, object2, btt, L, digits, ...)

Arguments

object

an object of class "rma.uni" or "rma.mv".

object2

an (optional) object of class "rma.uni" or "rma.mv". See ‘Details’.

btt

optional vector of indices specifying which coefficients to include in the Wald-type test. See ‘Details’.

optional numeric vector or matrix specifying linear combinations of the coefficients in the model that should be tested to be zero. See ‘Details’.

digits

integer specifying 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).

...

other arguments.

Value

QM: test statistic for the Wald-type test of model coefficients.
QMp: p-value for the Wald-type test of model coefficients.
btt: indices of the coefficients tested by the Wald-type test.
k: number of outcomes included in the model fitting.
p: number of coefficients in the model (including the intercept).
m: number of coefficients included in the Wald-type test of coefficients.
...: some additional elements/values.
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.
p.f: number of parameters in the full model.
p.r: number of parameters in the reduced model.
LRT: likelihood ratio test statistic.
pval: p-value for the likelihood ratio test.
QE.f: test statistic for the test of (residual) heterogeneity from the full model.
QE.r: test statistic for the test of (residual) heterogeneity from the reduced model.
tau2.f: estimated \tau² value from the full model. NA for "rma.mv" objects.
tau2.r: estimated \tau² value from the reduced model. NA for "rma.mv" objects.
R2: amount of (residual) heterogeneity in the reduced model that is accounted for in the full model (in percent). NA for fixed-effects models, if the amount of heterogeneity in the reduced model is equal to zero, or for "rma.mv" objects. This can be regarded as a pseudo R² 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.

Details

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, one can also specify one or multiple linear combinations of the coefficients in the model that should be tested to be zero via the L argument. If given as a matrix, its rows define the linear combinations to be tested. If the matrix is of full rank, an omnibus Wald-type test of all linear combinations is also provided.

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.

References

Hardy, R. J., & Thompson, S. G. (1996). A likelihood approach to meta-analysis with random effects. Statistics in Medicine, 15, 619--629.

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--19.

Lopez-Lopez, J. A., Marin-Martinez, F., Sanchez-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, 30--48.

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, 29--60.

Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1--48. http://www.jstatsoft.org/v36/i03/.

Examples

Run this code

### load BCG vaccine data
data(dat.bcg)

### calculate log relative risks 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)

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

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

### test of a linear combination
anova(res2, L=c(1,35,1970))

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