The function calculates confidence intervals for the model coefficients and/or for variance/correlation parameters in the model (the latter only for rma.uni and rma.mv objects).

# S3 method for rma.uni
confint(object, parm, level, fixed=FALSE, random=TRUE,
        digits, transf, targs, verbose=FALSE, control, ...)

# S3 method for rma.mh
confint(object, parm, level, digits, transf, targs, ...)

# S3 method for rma.peto
confint(object, parm, level, digits, transf, targs, ...)

# S3 method for rma.glmm
confint(object, parm, level, digits, transf, targs, ...)

# S3 method for rma.mv
confint(object, parm, level, fixed=FALSE, sigma2, tau2, rho, gamma2, phi,
        digits, transf, targs, verbose=FALSE, control, ...)

Arguments

object

an object of class "rma.uni", "rma.mh", "rma.peto", or "rma.mv". The method is not yet implemented for objects of class "rma.glmm".

parm

this argument is here for compatibility with the generic function confint, but is (currently) ignored.

fixed

logical indicating whether confidence intervals for the model coefficients should be returned (the default is FALSE).

random

logical indicating whether a confidence interval for the amount of (residual) heterogeneity should be returned (the default is TRUE) (only relevant for "rma.uni" objects).

sigma2

integer specifying for which \(\sigma^2\) value a confidence interval should be obtained (only relevant for "rma.mv" objects).

tau2

integer specifying for which \(\tau^2\) value a confidence interval should be obtained (only relevant for "rma.mv" objects).

rho

integer specifying for which \(\rho\) value the confidence interval should be obtained (only relevant for "rma.mv" objects).

gamma2

integer specifying for which \(\gamma^2\) value a confidence interval should be obtained (only relevant for "rma.mv" objects).

phi

integer specifying for which \(\phi\) value a confidence interval should be obtained (only relevant for "rma.mv" objects).

level

numerical value between 0 and 100 specifying the confidence interval level (if unspecified, the default is to take the value from the object).

digits

integer specifying the number of decimal places to which the results should be rounded (if unspecified, the default is to take the value from the object).

transf

optional argument specifying the name of a function that should be used to transform the model coefficients and interval bounds (e.g., transf=exp; see also transf). If unspecified, no transformation is used.

targs

optional arguments needed by the function specified under transf.

verbose

logical indicating whether output should be generated on the progress of the iterative algorithms used to obtain the confidence intervals (the default is FALSE). See ‘Note’.

control

list of control values for the iterative algorithms. If unspecified, default values are defined inside the function. See ‘Note’.

...

other arguments.

Details

The confidence intervals for the model coefficients are simply the usual Wald-type intervals which are also shown when printing the fitted object.

For objects of class "rma.uni" obtained with the rma.uni function, the confidence interval for the amount of (residual) heterogeneity (i.e., \(\tau^2\)) is obtained iteratively either via the Q-profile method or via the generalized Q-statistic method (Hartung and Knapp, 2005; Viechtbauer, 2007; Jackson, 2013; Jackson et al., 2014). The latter is automatically used when the model was fitted with method="GENQ", the former is used in all other cases. Either method provides an exact confidence interval for \(\tau^2\) in random- and mixed-effects models. The square root of the interval bounds is also returned for easier interpretation. Confidence intervals for \(I^2\) and \(H^2\) are also provided (Higgins & Thompson, 2002). Since \(I^2\) and \(H^2\) are just monotonic transformations of \(\tau^2\) (for details, see print.rma.uni), the confidence intervals for \(I^2\) and \(H^2\) are also exact.

Usually, the estimate of \(\tau^2\) from the random/mixed-effects model will fall within the confidence interval provided by the Q-profile method. However, this is not guaranteed. Depending on the method used to estimate \(\tau^2\) and the width of the confidence interval, it can happen that the confidence interval does not actually contain the estimate (trying to explain this to reviewers can be tricky). However, using the empirical Bayes or Paule-Mandel estimator of \(\tau^2\) when fitting the model (i.e., using method="EB" or method="PM") guarantees that the estimate of \(\tau^2\) falls within the confidence interval. When method="GENQ" was used to fit the model, the corresponding CI obtained via the generalized Q-statistic method is also guaranteed to contain the estimate \(\tau^2\).

For objects of class "rma.mv" obtained with the rma.mv function, profile likelihood confidence intervals are obtained by default for all (non-fixed) variance and correlation components of the model. Alternatively, one can use the sigma2, tau2, rho, gamma2, or phi arguments to specify for which variance/correlation parameter a confidence interval should be obtained. Only one of these arguments can be used at a time. A single integer is used to specify the number of the parameter. The function then provides a profile likelihood confidence interval for the specified parameter. It is a good idea to examine the corresponding profile likelihood plot (via the profile.rma.mv function) to make sure that the bounds obtained are sensible.

The methods used to find a confidence interval for a variance/correlation component are iterative and require the 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 1000. These values can be adjusted with control=list(tol=value, maxiter=value), but the defaults should be adequate for most purposes. If verbose=TRUE, output is generated on the progress of the iterative algorithm. This is especially useful when model fitting is slow, in which case finding the confidence interval bounds can also take considerable amounts of time.

More problematic when using the uniroot function is setting appropriate end points of the interval to be searched for the confidence interval bounds. The function tries to set some sensible defaults for the end points, but it may happen that the function is only able to determine that a bound is below/above a certain limit (this is indicated in the output accordingly with < or > signs). It can also happen that the model cannot be fitted or does not converge especially at the extremes of the interval to be searched. This will result in missing (NA) bounds and corresponding warnings. It may then be necessary to adjust the end points manually (see ‘Note’).

Finally, it is also possible that the lower and upper confidence interval bounds for a variance component both fall below zero. Since both bounds then fall outside of the parameter space, the confidence interval then consists of the null/empty set. Alternatively, one could interpret this as a CI with bounds \([0,0]\) or as indicating 'highly (or overly) homogeneous' data.

Value

An object of class "confint.rma". The object is a list with either one or two elements (named fixed and random) with the following elements:

estimate

estimate of the model coefficient or variance component.

ci.lb

lower bound of the confidence interval.

ci.ub

upper bound of the confidence interval.

When obtaining CIs for all variance and correlation components of "rma.mv" models, the object is a list of class "list.confint.rma", where each element is a "confint.rma" object as described above. The results are formatted and printed with the print.confint.rma and print.list.confint.rma functions.

Note

For objects of class "rma.uni" with random=TRUE, the function provides a confidence interval for \(\tau^2\). The upper end point of the interval to be searched for the confidence interval bounds is set to 100 (which should be large enough for most cases). The upper end point can be adjusted with control=list(tau2.max=value). One can also adjust the lower end point of the interval to be searched with control=list(tau2.min=value) (the default is to take that value from the object, which is 0 by default). You should only play around with this value if you know what you are doing.

For objects of class "rma.mv", the function provides confidence intervals for the variance/correlation parameters in the model. For variance components, the lower end point of the interval to be searched is set to 0 and the upper end point to the larger of 10 and 100 times the value of the component. For correlations, the function tries to set the lower end point to a sensible default depending on the type of variance structure chosen, while the upper end point is set to 1. One can adjust the lower and/or upper end points with control=list(vc.min=value, vc.max=value). Also, the function tries to adjust the lower/upper end points when the model does not converge at these extremes (the end points are then moved closer to the estimated value of the component). The total number of tries for setting/adjusting the end points is determined via control=list(eptries=<value>), with the default being 10 tries.

The Q-profile and generalized Q-statistic methods are both exact under the assumptions of the random- and mixed-effects models (i.e., normally distributed observed and true outcomes and known sampling variances). In practice, these assumptions are usually only approximately true, turning confidence intervals for \(\tau^2\) also into approximations. Profile likelihood confidence intervals are not exact by construction and rely on the asymptotic behavior of the likelihood ratio statistic, so they may be inaccurate in small samples.

References

Hartung, J., & Knapp, G. (2005). On confidence intervals for the among-group variance in the one-way random effects model with unequal error variances. Journal of Statistical Planning and Inference, 127, 157--177.

Higgins, J. P. T., & Thompson, S. G. (2002). Quantifying heterogeneity in a meta-analysis. Statistics in Medicine, 21, 1539--1558.

Jackson, D. (2013). Confidence intervals for the between-study variance in random effects meta-analysis using generalised Cochran heterogeneity statistics. Research Synthesis Methods, 4, 220--229.

Jackson, D., Turner, R., Rhodes, K., & Viechtbauer, W. (2014). Methods for calculating confidence and credible intervals for the residual between-study variance in random effects meta-regression models. BMC Medical Research Methodology, 14, 103.

Mantel, N., & Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute, 22, 719--748.

Viechtbauer, W. (2007). Confidence intervals for the amount of heterogeneity in meta-analysis. Statistics in Medicine, 26, 37--52.

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/.

Yusuf, S., Peto, R., Lewis, J., Collins, R., & Sleight, P. (1985). Beta blockade during and after myocardial infarction: An overview of the randomized trials. Progress in Cardiovascular Disease, 27, 335--371.

See also

Examples

### meta-analysis of the log risk ratios using a random-effects model res <- rma(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg) ### confidence interval for the total amount of heterogeneity confint(res)
#> #> estimate ci.lb ci.ub #> tau^2 0.3132 0.1197 1.1115 #> tau 0.5597 0.3460 1.0543 #> I^2(%) 92.2214 81.9177 97.6781 #> H^2 12.8558 5.5303 43.0680 #>
### mixed-effects model with absolute latitude in the model res <- rma(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg, mods = ~ ablat, data=dat.bcg) ### confidence interval for the residual amount of heterogeneity confint(res)
#> #> estimate ci.lb ci.ub #> tau^2 0.0764 0.0167 0.7849 #> tau 0.2763 0.1292 0.8859 #> I^2(%) 68.3931 32.1080 95.6976 #> H^2 3.1639 1.4729 23.2428 #>
### multilevel random-effects model res <- rma.mv(yi, vi, random = ~ 1 | district/school, data=dat.konstantopoulos2011) ### profile plots and confidence intervals for the variance components if (FALSE) { par(mfrow=c(2,1)) profile(res, sigma2=1) abline(h=logLik(res) - qchisq(.95, df=1)/2, lty="dotted") sav <- confint(res, sigma2=1) sav abline(v=sav$random[1,2:3], lty="dotted") profile(res, sigma2=2) abline(h=logLik(res) - qchisq(.95, df=1)/2, lty="dotted") sav <- confint(res, sigma2=2) sav abline(v=sav$random[1,2:3], lty="dotted")} ### multivariate parameterization of the model res <- rma.mv(yi, vi, random = ~ factor(school) | district, data=dat.konstantopoulos2011) ### profile plots and confidence intervals for the variance component and correlation if (FALSE) { par(mfrow=c(2,1)) profile(res, tau2=1) abline(h=logLik(res) - qchisq(.95, df=1)/2, lty="dotted") sav <- confint(res, tau2=1) sav abline(v=sav$random[1,2:3], lty="dotted") profile(res, rho=1) abline(h=logLik(res) - qchisq(.95, df=1)/2, lty="dotted") sav <- confint(res, rho=1) sav abline(v=sav$random[1,2:3], lty="dotted")}