Description

Function to fit the meta-analytic fixed- and random/mixed-effects models with or without moderators via linear (mixed-effects) models. See the documentation of the metafor-package for more details on these models.

Usage

Arguments

Details

Specifying the Data

The function can be used in conjunction with any of the usual effect size or outcome measures used in meta-analyses (e.g., log risk ratios, log odds ratios, risk differences, mean differences, standardized mean differences, raw correlation coefficients, correlation coefficients transformed with Fisher's r-to-z transformation, and so on). Simply specify the observed outcomes via the yi argument and the corresponding sampling variances via the vi argument. Instead of specifying vi, one can specify the standard errors (the square root of the sampling variances) via the sei argument. The escalc function can be used to compute a wide variety of effect size and outcome measures (and the corresponding sampling variances) based on summary statistics.

Alternatively, the function can automatically calculate the values of a chosen effect size or outcome measure (and the corresponding sampling variances) when supplied with the necessary data. The escalc function describes which effect size or outcome measures are currently implemented and what data/arguments should then be specified/used. The measure argument should then be set to the desired effect size or outcome measure.

Specifying the Model

The function can be used to fit fixed- and random/mixed-effects models, as well as meta-regression models including moderators (the difference between the various models is described in detail in the introductory metafor-package help file).

Assuming the observed outcomes and corresponding sampling variances are supplied via yi and vi, a fixed-effects model can be fitted with rma(yi, vi, method="FE"). Weighted estimation (with inverse-variance weights) is used by default. User-defined weights can be supplied via the weights argument. Unweighted estimation can be used by setting weighted=FALSE (same as setting the weights equal to a constant).

A random-effects model can be fitted with the same code but setting method to one of the various estimators for the amount of heterogeneity:

• method="DL" = DerSimonian-Laird estimator

• method="HE" = Hedges estimator

• method="HS" = Hunter-Schmidt estimator

• method="SJ" = Sidik-Jonkman estimator

• method="ML" = maximum-likelihood estimator

• method="REML" = restricted maximum-likelihood estimator

• method="EB" = empirical Bayes estimator

• method="PM" = Paule-Mandel estimator

• method="GENQ" = generalized Q-statistic estimator

For a description of the various estimators, see DerSimonian and Kacker (2007), Raudenbush (2009), Viechtbauer (2005), and Viechtbauer et al. (2015). Note that the Hedges estimator is also called the ‘variance component estimator’ or ‘Cochran estimator’, the Sidik-Jonkman estimator is also called the ‘model error variance estimator’, and the empirical Bayes estimator is actually identical to the Paule-Mandel estimator (Paule & Mandel, 1982). Finally, the generalized Q-statistic estimator is a general method-of-moments estimator (DerSimonian & Kacker, 2007) requiring the specification of weights (the HE and DL estimators are just special cases with equal and inverse variance weights, respectively).

One or more moderators can be included in these models via the mods argument. A single moderator can be given as a (row or column) vector of length k specifying the values of the moderator. Multiple moderators are specified by giving an appropriate model matrix (i.e., X) with k rows and as many columns as there are moderator variables (e.g., mods = cbind(mod1, mod2, mod3), where mod1, mod2, and mod3 correspond to the names of the variables for three moderator variables). The intercept is added to the model matrix by default unless intercept=FALSE.

Alternatively, one can use the standard formula syntax to specify the model. In this case, the mods argument should be set equal to a one-sided formula of the form mods = ~ model (e.g., mods = ~ mod1 + mod2 + mod3). Interactions, polynomial terms, and factors can be easily added to the model in this manner. When specifying a model formula via the mods argument, the intercept argument is ignored. Instead, the inclusion/exclusion of the intercept is controlled by the specified formula (e.g., mods = ~ mod1 + mod2 + mod3 - 1 would lead to the removal of the intercept).

A fixed-effects with moderators model is then fitted by setting method="FE", while a mixed-effects model is fitted by specifying one of the estimators for the amount of (residual) heterogeneity given earlier.

When the observed outcomes and corresponding sampling variances are supplied via the yi and vi (or sei) arguments, one can also specify moderators via the yi argument (e.g., rma(yi ~ mod1 + mod2 + mod3, vi)). In that case, the mods argument is ignored and the inclusion/exclusion of the intercept again is controlled by the specified formula.

Omnibus Test of Parameters

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.

Categorical Moderators

Categorical moderator variables can be included in the model via the mods argument in the same way that appropriately (dummy) coded categorical independent variables can be included in linear models. One can either do the dummy coding manually or use a model formula together with the factor function to let R handle the coding automatically. An example to illustrate these different approaches is provided below.

Knapp & Hartung Adjustment

By default, the test statistics of the individual coefficients in the model (and the corresponding confidence intervals) are based on a standard normal distribution, while the omnibus test is based on a chi-square distribution with m degrees of freedom (m being the number of coefficients tested). The Knapp and Hartung (2003) method (test="knha") is an adjustment to the standard errors of the estimated coefficients, which helps to account for the uncertainty in the estimate of the amount of (residual) heterogeneity and leads to different reference distributions. Tests of individual coefficients and confidence intervals are then based on the t-distribution with k-p degrees of freedom, while the omnibus test statistic then uses an F-distribution with m and k-p degrees of freedom (p being the total number of model coefficients including the intercept if it is present). The Knapp and Hartung (2003) adjustment is only meant to be used in the context of random- or mixed-effects models.

Test for (Residual) Heterogeneity

A test for (residual) heterogeneity is automatically carried out by the function. Without moderators in the model, this is simply Cochran's Q-test (Cochran, 1954), which tests whether the variability in the observed effect sizes or outcomes is larger than would be expected based on sampling variability alone. A significant test suggests that the true effects or outcomes are heterogeneous. When moderators are included in the model, this is the Q_E-test for residual heterogeneity, which tests whether the variability in the observed effect sizes or outcomes not accounted for by the moderators included in the model is larger than would be expected based on sampling variability alone.

Value

An object of class c("rma.uni","rma"). The object is a list containing the following components:

Methods

The results of the fitted model are formatted and printed with the print.rma.uni function. If fit statistics should also be given, use summary.rma (or use the fitstats.rma function to extract them). Full versus reduced model comparisons in terms of fit statistics and likelihoods can be obtained with anova.rma. Wald-type tests for sets of model coefficients or linear combinations thereof can be obtained with the same function. Permutation tests for the model coefficient(s) can be obtained with permutest.rma.uni. Tests and confidence intervals based on (cluster) robust methods can be obtained with robust.rma.uni.

Predicted/fitted values can be obtained with predict.rma and fitted.rma. For best linear unbiased predictions, see blup.rma.uni and ranef.rma.uni.

The residuals.rma, rstandard.rma.uni, and rstudent.rma.uni functions extract raw and standardized residuals. Additional case diagnostics (e.g., to determine influential studies) can be obtained with the influence.rma.uni function. For models without moderators, leave-one-out diagnostics can also be obtained with leave1out.rma.uni. For models with moderators, variance inflation factors can be obtained with vif.rma.

A confidence interval for the amount of (residual) heterogeneity in the random/mixed-effects model can be obtained with confint.rma.uni.

Forest, funnel, radial, L'Abbé, and Baujat plots can be obtained with forest.rma, funnel.rma, radial.rma, labbe.rma, and baujat (radial and L'Abbé plots only for models without moderators). The qqnorm.rma.uni function provides normal QQ plots of the standardized residuals. One can also just call plot.rma.uni on the fitted model object to obtain various plots at once. For random/mixed-effects models, the profile.rma.uni function can be used to obtain a plot of the (restricted) log-likelihood as a function of τ².

Tests for funnel plot asymmetry (which may be indicative of publication bias) can be obtained with ranktest.rma and regtest.rma. For models without moderators, the trimfill.rma.uni method can be used to carry out a trim and fill analysis and hc.rma.uni provides a random-effects model analysis that is more robust to publication bias (based on the method by Henmi & Copas, 2010).

For models without moderators, a cumulative meta-analysis (i.e., adding one observation at a time) can be obtained with cumul.rma.uni.

Other extractor functions include coef.rma, vcov.rma, logLik.rma, deviance.rma, AIC.rma, BIC.rma, hatvalues.rma.uni, and weights.rma.uni.

Note

While the HS, HE, DL, SJ, and GENQ estimators of τ² are based on closed-form solutions, the ML, REML, and EB estimators must be obtained numerically. For this, the function makes use of the Fisher scoring algorithm, which is robust to poor starting values and usually converges quickly (Harville, 1977; Jennrich & Sampson, 1976). By default, the starting value is set equal to the value of the Hedges (HE) estimator and the algorithm terminates when the change in the estimated value of τ² is smaller than 10⁻⁵ from one iteration to the next. The maximum number of iterations is 100 by default (which should be sufficient in most cases). Information on the progress of the algorithm can be obtained by setting verbose=TRUE. One can also set verbose to an integer (verbose=2 yields even more information and verbose=3 also sets option(warn=1) temporarily).

A different starting value, threshold, and maximum number of iterations can be specified via the control argument by setting control=list(tau2.init=value, threshold=value, maxiter=value). The step length of the Fisher scoring algorithm can also be manually adjusted by a desired factor with control=list(stepadj=value) (values below 1 will reduce the step length). If using verbose=TRUE shows the estimate jumping around erratically (or cycling through a few values), decreasing the step length (and increasing the maximum number of iterations) can often help with convergence (e.g., control=list(stepadj=0.5, maxiter=1000)).

The PM estimator also involves an iterative algorithm, which makes use of the uniroot function. By default, the desired accuracy (tol) is set equal to .Machine$double.eps^0.25 and the maximum number of iterations (maxiter) to 100 (as above). The upper bound of the interval searched (tau2.max) is set to 100 (which should be large enough for most cases). These values can be adjusted with control=list(tol=value, maxiter=value, tau2.max=value).

All of the heterogeneity estimators except SJ can in principle yield negative estimates for the amount of (residual) heterogeneity. However, negative estimates of τ² are outside of the parameter space. For the HS, HE, DL, and GENQ estimators, negative estimates are therefore truncated to zero. For the ML, REML, and EB estimators, the Fisher scoring algorithm makes use of step halving (Jennrich & Sampson, 1976) to guarantee a non-negative estimate. Finally, for the PM estimator, the lower bound of the interval searched is set by default to zero. For those brave enough to step into risky territory, there is the option to set the lower bound for all these estimators equal to some other value besides zero (even a negative one) with control=list(tau2.min=value), but the lowest value permitted is -min(vi) (to ensure that the marginal variances are always non-negative).

The Hunter-Schmidt estimator for the amount of heterogeneity is defined in Hunter and Schmidt (1990) only in the context of the random-effects model when analyzing correlation coefficients. A general version of this estimator for random- and mixed-effects models not specific to any particular outcome measure is described in Viechtbauer (2005) and Viechtbauer et al. (2015) and is implemented here.

The Sidik-Jonkman estimator starts with a crude estimate of τ², which is then updated as described in Sidik and Jonkman (2005b, 2007). If, instead of the crude estimate, one wants to use a better a priori estimate, one can do so by passing this value via control=list(tau2.init=value).

Outcomes with non-positive sampling variances are problematic. If a sampling variance is equal to zero, then its weight will be 1/0 for fixed-effects models when using weighted estimation. Switching to unweighted estimation is a possible solution then. For random/mixed-effects model, some estimators of τ² are undefined when there is at least one sampling variance equal to zero. Other estimators may work, but it may still be necessary to switch to unweighted model fitting, especially when the estimate of τ² turns out to be zero.

When including moderators in the model, it is possible that the model matrix is not of full rank (i.e., there is a linear relationship between the moderator variables included in the model). The function automatically tries to reduce the model matrix to full rank by removing redundant predictors, but if this fails the model cannot be fitted and an error will be issued. Deleting (redundant) moderator variables from the model as needed should solve this problem.

Finally, some general words of caution about the assumptions underlying the models:

• The sampling variances (i.e., the vi values) are treated as if they are known constants. This (usually) implies that the distributions of the test statistics and corresponding confidence intervals are only exact and have nominal coverage when the within-study sample sizes are large (i.e., when the error in the sampling variance estimates is small). Certain outcome measures (e.g., the arcsine square root transformed risk difference and Fisher's r-to-z transformed correlation coefficient) are based on variance stabilizing transformations that also help to make the assumption of known sampling variances much more reasonable.

• When fitting a mixed/random-effects model, τ² is estimated and then treated as a known constant thereafter. This ignores the uncertainty in the estimate of τ². As a consequence, the standard errors of the parameter estimates tend to be too small, yielding test statistics that are too large and confidence intervals that are not wide enough. The Knapp and Hartung (2003) adjustment can be used to counter this problem, yielding test statistics and confidence intervals whose properties are closer to nominal.

• Most effect size measures are not exactly normally distributed as assumed under the various models. However, the normal approximation usually becomes more accurate for most effect size or outcome measures as the within-study sample sizes increase. Therefore, sufficiently large within-study sample sizes are (usually) needed to be certain that the tests and confidence intervals have nominal levels/coverage. Again, certain outcome measures (e.g., Fisher's r-to-z transformed correlation coefficient) may be preferable from this perspective as well.

Author(s)

Wolfgang Viechtbauer wvb@metafor-project.org http://www.metafor-project.org/

References

Berkey, C. S., Hoaglin, D. C., Mosteller, F., & Colditz, G. A. (1995). A random-effects regression model for meta-analysis. Statistics in Medicine, 14, 395–411.

Cochran, W. G. (1954). The combination of estimates from different experiments. Biometrics, 10, 101–129.

DerSimonian, R., & Laird, N. (1986). Meta-analysis in clinical trials. Controlled Clinical Trials, 7, 177–188.

DerSimonian, R., & Kacker, R. (2007). Random-effects model for meta-analysis of clinical trials: An update. Contemporary Clinical Trials, 28, 105–114.

Harville, D. A. (1977). Maximum likelihood approaches to variance component estimation and to related problems. Journal of the American Statistical Association, 72, 320–338.

Hedges, L. V. (1983). A random effects model for effect sizes. Psychological Bulletin, 93, 388–395.

Hedges, L. V., & Olkin, I. (1985). Statistical methods for meta-analysis. San Diego, CA: Academic Press.

Henmi, M., & Copas, J. B. (2010). Confidence intervals for random effects meta-analysis and robustness to publication bias. Statistics in Medicine, 29, 2969–2983.

Hunter, J. E., & Schmidt, F. L. (2004). Methods of meta-analysis: Correcting error and bias in research findings (2nd ed.). Thousand Oaks, CA: Sage.

Jennrich, R. I., & Sampson, P. F. (1976). Newton-Raphson and related algorithms for maximum likelihood variance component estimation. Technometrics, 18, 11–17.

Knapp, G., & Hartung, J. (2003). Improved tests for a random effects meta-regression with a single covariate. Statistics in Medicine, 22, 2693–2710.

Morris, C. N. (1983). Parametric empirical Bayes inference: Theory and applications (with discussion). Journal of the American Statistical Association, 78, 47–65.

Paule, R. C., & Mandel, J. (1982). Consensus values and weighting factors. Journal of Research of the National Bureau of Standards, 87, 377–385.

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.

Sidik, K., & Jonkman, J. N. (2005a). A note on variance estimation in random effects meta-regression. Journal of Biopharmaceutical Statistics, 15, 823–838.

Sidik, K., & Jonkman, J. N. (2005b). Simple heterogeneity variance estimation for meta-analysis. Journal of the Royal Statistical Society, Series C, 54, 367–384.

Sidik, K., & Jonkman, J. N. (2007). A comparison of heterogeneity variance estimators in combining results of studies. Statistics in Medicine, 26, 1964–1981.

Viechtbauer, W. (2005). Bias and efficiency of meta-analytic variance estimators in the random-effects model. Journal of Educational and Behavioral Statistics, 30, 261–293.

Viechtbauer, W. (2010). Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36(3), 1–48. https://www.jstatsoft.org/v036/i03.

Viechtbauer, W., López-López, J. A., Sánchez-Meca, J., & Marín-Martínez, F. (2015). A comparison of procedures to test for moderators in mixed-effects meta-regression models. Psychological Methods, 20, 360–374.

See Also

rma.mh, rma.peto, rma.glmm, and rma.mv for other model fitting functions.

Examples