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metafor (version 1.5-0)

rma.uni: Meta-Analysis via the Linear (Mixed-Effects) Model

Description

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

Usage

rma.uni(yi, vi, sei, ai, bi, ci, di, n1i, n2i, 
        x1i, x2i, t1i, t2i, m1i, m2i, sd1i, sd2i, 
        xi, mi, ri, ni, ti, mods, measure="GEN", 
        intercept=TRUE, data, slab, subset, 
        add=1/2, to="only0", vtype="LS", 
        method="REML", weighted=TRUE, knha=FALSE,
        level=95, digits=4, btt, tau2, control)
rma(yi, vi, sei, ai, bi, ci, di, n1i, n2i, 
        x1i, x2i, t1i, t2i, m1i, m2i, sd1i, sd2i,
        xi, mi, ri, ni, ti, mods, measure="GEN", 
        intercept=TRUE, data, slab, subset, 
        add=1/2, to="only0", vtype="LS", 
        method="REML", weighted=TRUE, knha=FALSE,
        level=95, digits=4, btt, tau2, control)

Arguments

yi
vector of length $k$ with the observed effect sizes or outcomes. See Details.
vi
vector of length $k$ with the corresponding sampling variances. See Details.
sei
vector of length $k$ with the corresponding standard errors. See Details.
ai
see below and the documentation of the escalc function for more details.
bi
see below and the documentation of the escalc function for more details.
ci
see below and the documentation of the escalc function for more details.
di
see below and the documentation of the escalc function for more details.
n1i
see below and the documentation of the escalc function for more details.
n2i
see below and the documentation of the escalc function for more details.
x1i
see below and the documentation of the escalc function for more details.
x2i
see below and the documentation of the escalc function for more details.
t1i
see below and the documentation of the escalc function for more details.
t2i
see below and the documentation of the escalc function for more details.
m1i
see below and the documentation of the escalc function for more details.
m2i
see below and the documentation of the escalc function for more details.
sd1i
see below and the documentation of the escalc function for more details.
sd2i
see below and the documentation of the escalc function for more details.
xi
see below and the documentation of the escalc function for more details.
mi
see below and the documentation of the escalc function for more details.
ri
see below and the documentation of the escalc function for more details.
ni
see below and the documentation of the escalc function for more details.
ti
see below and the documentation of the escalc function for more details.
mods
an optional argument to include one or more moderators in the model. A single moderator can be given as a vector of length $k$ specifying the values of the moderator. Multiple moderators are specified by giving a matrix with $k$ rows and $p'$ columns. Alt
measure
a character string indicating the type of data supplied to the function. When measure="GEN" (default), the observed effect sizes or outcomes and corresponding sampling variances (or standard errors) should be supplied to the function via the
intercept
logical, indicating whether an intercept term should be added to the model (default is TRUE).
data
an optional data frame containing the data supplied to the function.
slab
an optional vector with unique labels for the $k$ studies.
subset
an optional vector indicating the subset of studies that should be used for the analysis. This can be a logical vector of length $k$ or a numeric vector indicating the indices of the observations to include.
add
see the documentation of the escalc function.
to
see the documentation of the escalc function.
vtype
see the documentation of the escalc function.
method
a character string specifying whether a fixed- or a random/mixed-effects model should be fitted. A fixed-effects model (with or without moderators) is fitted when using method="FE". Random/mixed-effects models are fitted by setting meth
weighted
logical indicating whether weighted (default) or unweighted least squares should be used to fit the model.
knha
logical specifying whether the method by Knapp and Hartung (2003) should be used for adjusting test statistics and confidence intervals (default is FALSE). See Details.
level
a numerical value between 0 and 100 specifying the confidence interval level (default is 95).
digits
an integer specifying the number of decimal places to which the printed results should be rounded (default is 4).
btt
an optional vector of indices specifying which coefficients to include in the omnibus test of moderators. See Details.
tau2
an optional numerical value to specify the amount of (residual) heterogeneity in a random- or mixed-effects model (instead of estimating it). Useful for sensitivity analyses (e.g., for plotting results as a function of $\tau^2$). When unspecified, the val
control
optional list of control values for the iterative estimation algorithms. Defaults to an empty list, which means that default values are defined inside the function. See Note.

Value

  • An object of class c("rma.uni","rma"). The object is a list containing the following components:
  • bestimated coefficients of the model.
  • sestandard errors of the coefficients.
  • zvaltest statistics of the coefficients.
  • pvalp-values for the test statistics.
  • ci.lblower bound of the confidence intervals for the coefficients.
  • ci.ubupper bound of the confidence intervals for the coefficients.
  • vbvariance-covariance matrix of the estimated coefficients.
  • tau2estimated amount of (residual) heterogeneity. Always 0 when method="FE".
  • se.tau2estimated standard error of the estimated amount of (residual) heterogeneity when using ML or REML estimation (NA otherwise).
  • knumber of outcomes included in the model fitting (equal to length(yi) unless subset was used or if there are missing data).
  • pnumber of coefficients in the model (including the intercept).
  • mnumber of coefficients included in the omnibus test of coefficients.
  • QEtest statistic for the test of (residual) heterogeneity.
  • QEpp-value for the test of (residual) heterogeneity.
  • QMtest statistic for the omnibus test of coefficients.
  • QMpp-value for the omnibus test of coefficients.
  • I2value of $I^2$ (only for the random-effects model; NA otherwise).
  • H2value of $H^2$ (only for the random-effects model; NA otherwise).
  • int.onlylogical that indicates whether the model only includes an intercept.
  • yi, vi, Xthe vector of outcomes, the corresponding sampling variances, and the design matrix of the model.
  • fit.statsa list with the log likelihood, deviance, AIC, and BIC values under the unrestricted and restricted likelihood.
  • ...some additional elements/values.
  • The results of the fitted model are neatly formated and printed with the print.rma.uni function. If you also want the fit statistics, 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.uni. Permutation tests for the model coefficient(s) can be obtained with permutest.rma.uni. Predicted/fitted values can be obtained with predict.rma and fitted.rma. For best linear unbiased predictions, see blup.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. 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, and L'abbe plots (the latter two only for models without moderators) can be obtained with forest.rma, funnel.rma, radial.rma, and labbe.rma. 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. Tests for publication bias (or more accurately, for funnel plot asymmetry) 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. For models without moderators, a cumulative meta-analysis (i.e., adding one obervation at a time) can be obtained with cumul.rma.uni. Other assessor functions include coef.rma, vcov.rma, logLik.rma, deviance.rma, AIC.rma, BIC.rma, hatvalues.rma.uni, and weights.rma.uni.

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 odds ratios, log relative risks, 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 (or specify the standard errors, the square root of the sampling variances, via the sei argument). When specifying the data in this way, one must set measure="GEN" (which is the default). 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 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 Assuming the observed outcomes and corresponding sampling variances are supplied via yi and vi, the fixed-effects model is fitted with rma(yi, vi, method="FE"). The random-effects model is fitted with the same code but setting method to one of the various estimators for the amount of heterogeneity:
  • method="HS"= Hunter-Schmidt estimator
  • method="HE"= Hedges estimator
  • method="DL"= DerSimonian-Laird estimator
  • method="SJ"= Sidik-Jonkman estimator
  • method="ML"= maximum-likelihood estimator
  • method="REML"= restricted maximum-likelihood estimator
  • method="EB"= empirical Bayes estimator.
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 design matrix with $k$ rows and $p'$ columns (e.g., using mods = cbind(mod1, mod2, mod3), where mod1, mod2, and mod3 correspond to the names of the variables for the three moderator variables). The intercept is included in the model by default unless intercept=FALSE. Alternatively, one can use the model formula syntax to specify the model. In this case, the mods argument should be set equal to a one-sided formula of the form ~ model (e.g., ~ 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 term is controlled by the specified formula (e.g., ~ mod1 + mod2 + mod3 - 1 would exclude the intercept term). In this case, the model formula determines whether an intercept is included in the model or not. 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. 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, use btt=c(3,4) to only include the third and fourth coefficient from the model in the test (if an intercept is included in the model, then it corresponds to the first coefficient in the model). Categorical Moderators Categorical moderator variables can be included in the model in the same way that appropriately (dummy) coded categorical independent variables can be included in linear models. You can either do the dummy coding manually or use a model formula together with the factor function to let Rhandle 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 the 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 (knha=TRUE) 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. 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.

References

DerSimonian, R. & Laird, N. (1986). Meta-analysis in clinical trials. Controlled Clinical Trials, 7, 177--188. 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. Hunter, J. E. & Schmidt, F. L. (1990). Methods of meta-analysis: Correcting error and bias in research findings. Newbury Park, 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. Raudenbush, S. W. (2009). Analyzing effect sizes: Random effects models. In H. C. 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. 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. http://www.jstatsoft.org/v36/i03/.

See Also

rma.mh, rma.peto

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, append=TRUE)

### random-effects model
rma(yi, vi, data=dat, method="REML")

### mixed-effects model with two moderators (absolute latitude and publication year)
rma(yi, vi, mods=cbind(ablat, year), data=dat, method="REML")

### using a model formula to specify the same model
rma(yi, vi, mods=~ablat+year, data=dat, method="REML")

### supplying the cell frequencies directly to the function
rma(ai=tpos, bi=tneg, ci=cpos, di=cneg, mods=cbind(ablat, year), 
    data=dat, measure="RR", method="REML")

### manual dummy coding of the allocation factor
alloc.random     <- ifelse(dat$alloc == "random",     1, 0)
alloc.alternate  <- ifelse(dat$alloc == "alternate",  1, 0)
alloc.systematic <- ifelse(dat$alloc == "systematic", 1, 0)

### test the allocation factor (in the presence of the other moderators)
### note: "alternate" is the reference level of the allocation factor
### note: the intercept is the first coefficient, so btt=c(2,3)
rma(yi, vi, mods=cbind(alloc.random, alloc.systematic, year, ablat), 
    data=dat, method="REML", btt=c(2,3))

### using a model formula to specify the same model
rma(yi, vi, mods=~factor(alloc)+year+ablat, data=dat, method="REML", btt=c(2,3))

### subgrouping versus using a single model with a factor (subgrouping provides 
### an estimate of tau^2 within each subgroup, but the number of studies in each 
### subgroup get quite small; the model with the allocation factor provides a
### single estimate of tau^2 based on a larger number of studies, but assumes
### that tau^2 is the same within each subgroup)
res.a <- rma(yi, vi, data=dat, subset=(alloc=="alternate"))
res.r <- rma(yi, vi, data=dat, subset=(alloc=="random"))
res.s <- rma(yi, vi, data=dat, subset=(alloc=="systematic"))
res.a
res.r
res.s
res <- rma(yi, vi, mods=~factor(alloc)-1, data=dat)
res

### demonstrating that Q_E + Q_M = Q_total for fixed-effects models
### note: this does not work for random/mixed-effects models, since Q_E and
### Q_total are calculated under the assumption that tau^2 = 0, while the
### calculation of Q_M incorporates the estimate of tau^2
res <- rma(yi, vi, data=dat, method="FE")
res
res <- rma(yi, vi, mods=cbind(ablat, year), data=dat, method="FE")
res
res$QE + res$QM

### decomposition of Q_E into subgroup Q-values
res <- rma(yi, vi, mods=~factor(alloc), data=dat)
res

res.a <- rma(yi, vi, data=dat, subset=(alloc=="alternate"))
res.r <- rma(yi, vi, data=dat, subset=(alloc=="random"))
res.s <- rma(yi, vi, data=dat, subset=(alloc=="systematic"))

res.a$QE ### Q-value within subgroup
res.r$QE ### Q-value within subgroup
res.s$QE ### Q-value within subgroup

res$QE
res.a$QE + res.r$QE + res.s$QE

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