Function to profile the (restricted) log-likelihood for objects of class "rma.uni" and "rma.mv".

# S3 method for rma.uni
profile(fitted, xlim, ylim, steps=20,
lltol=1e-04, progbar=TRUE, parallel="no", ncpus=1, cl=NULL,
plot=TRUE, pch=19, cline=FALSE, ...)

# S3 method for rma.mv
profile(fitted, sigma2, tau2, rho, gamma2, phi,
xlim, ylim, steps=20, lltol=1e-04, progbar=TRUE,
parallel="no", ncpus=1, cl=NULL,
plot=TRUE, pch=19, cline=FALSE, ...)

# S3 method for profile.rma
print(x, ...)
# S3 method for profile.rma
plot(x, xlim, ylim, pch=19,
xlab, ylab, main, cline=FALSE, ...)

Arguments

fitted an object of class "rma.uni" or "rma.mv". an object of class "profile.rma" (for plot and print). integer to specify for which $$\sigma^2$$ value the likelihood should be profiled (only relevant for "rma.mv" objects). integer to specify for which $$\tau^2$$ value the likelihood should be profiled (only relevant for "rma.mv" objects). integer to specify for which $$\rho$$ value the likelihood should be profiled (only relevant for "rma.mv" objects). integer to specify for which $$\gamma^2$$ value the likelihood should be profiled (only relevant for "rma.mv" objects). integer to specify for which $$\phi$$ value the likelihood should be profiled (only relevant for "rma.mv" objects). optional vector to specify the lower and upper limit of the parameter over which the profiling should be done. If unspecified, the function tries to set these limits automatically. optional vector to specify the y-axis limits when plotting the profiled likelihood. If unspecified, the function tries to set these limits automatically. number of points between xlim[1] and xlim[2] (inclusive) for which the likelihood should be evaluated (the default is 20). numerical tolerance used when comparing values of the profiled log-likelihood with the log-likelihood of the fitted model (the default is 1e-06). logical to specify whether a progress bar should be shown (the default is TRUE). character string to specify whether parallel processing should be used (the default is "no"). For parallel processing, set to either "snow" or "multicore". See ‘Details’. integer to specify the number of processes to use in the parallel processing. optional snow cluster to use if parallel="snow". If not supplied, a cluster on the local machine is created for the duration of the call. logical to specify whether the profile plot should be drawn after profiling is finished (the default is TRUE). plotting symbol to use. By default, a filled circle is used. See points for other options. logical to specify whether a horizontal reference line should be added to the plot that indicates the log-likelihood value corresponding to the 95% profile confidence interval (the default is FALSE). title for the x-axis. If unspecified, the function tries to set an appropriate axis title. title for the y-axis. If unspecified, the function tries to set an appropriate axis title. title for the plot. If unspecified, the function tries to set an appropriate title. other arguments.

Details

The function fixes a particular variance component or correlation parameter of the model and then computes the maximized (restricted) log-likelihood over the remaining parameters of the model. By doing this for a range of values for the parameter that was fixed, a profile of the (restricted) log-likelihood is constructed.

For objects of class "rma.uni" obtained with the rma.uni function, the function profiles over parameter $$\tau^2$$. If the model was fitted with method="ML" or method="REML", the profiled (restricted) log-likelihood should be maximized at the ML/REML estimate of $$\tau^2$$ (the function checks whether any of the profiled log-likelihood values is larger than the log-likelihood of the fitted model, using a numerical tolerance of lltol; if so, a warning is issued as this might indicate that the optimizer did not identify the actual ML/REML estimate).

For objects of class "rma.mv" obtained with the rma.mv function, profiling is done by default over all (non-fixed) variance and correlation components of the model. Alternatively, one can use the sigma2, tau2, rho, gamma2, or phi arguments to specify over which parameter the profiling should be done. Only one of these arguments can be used at a time. A single integer is used to specify the number of the parameter.

A profile plot should show a single peak at the corresponding ML/REML estimate (as described above, the function checks for this). If the profiled likelihood has multiple peaks, this indicates that the likelihood surface is not unimodal, which implies that the ML/REML estimates may correspond to a local optimum. If the profile is flat (over the entire parameter space or large portions of it), then this suggests that at least some of the parameters of the model are not identifiable (and the parameter estimates obtained are to some extent arbitrary). Some further discussion of parameter identifiability (structurally and practically) and the use of profile likelihoods to check for this can be found in Raue et al. (2009).

Profiling requires repeatedly refitting the same model, which can be slow when $$k$$ is large and/or the model is complex (the latter especially applies to "rma.mv" objects). On machines with multiple cores, one can usually speed things up by delegating the model fitting to separate worker processes, that is, by setting parallel="snow" or parallel="multicore" and ncpus to some value larger than 1. Parallel processing makes use of the parallel package, using the makePSOCKcluster and parLapply functions when parallel="snow" or using mclapply when parallel="multicore" (the latter only works on Unix/Linux-alikes). With parallel::detectCores(), one can check on the number of available cores on the local machine.

Value

An object of class "profile.rma". The object is a list (or list of lists) containing the following components:

sigma2

values of $$\sigma^2$$ over which the likelihood was profiled (only when profiling was actually done over $$\sigma^2$$).

tau2

values of $$\tau^2$$ over which the likelihood was profiled (only when profiling was actually done over $$\tau^2$$).

rho

values of $$\rho$$ over which the likelihood was profiled (only when profiling was actually done over $$\rho$$).

gamma2

values of $$\gamma^2$$ over which the likelihood was profiled (only when profiling was actually done over $$\gamma^2$$).

phi

values of $$\phi$$ over which the likelihood was profiled (only when profiling was actually done over $$\phi$$).

ll

(restricted) log-likelihood at the corresponding parameter value.

beta

a matrix with the estimated model coefficients at the corresponding parameter value.

ci.lb

a matrix with the lower confidence interval bounds for the model coefficients at the corresponding parameter value.

ci.ub

a matrix with the upper confidence interval bounds for the model coefficients at the corresponding parameter value.

...

Note that the list is returned invisibly.

Author

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

References

Raue, A., Kreutz, C., Maiwald, T., Bachmann, J., Schilling, M., Klingmuller, U., & Timmer, J. (2009). Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood. Bioinformatics, 25(15), 1923--1929. https://doi.org/10.1093/bioinformatics/btp358

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

rma.uni, rma.mv

Examples

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

### random-effects model using rma.uni()
res <- rma(yi, vi, data=dat)

### profile over tau^2
profile(res, progbar=FALSE)

### change data into long format
dat.long <- to.long(measure="OR", ai=tpos, bi=tneg, ci=cpos, di=cneg, data=dat.bcg)

### set levels of group variable ("exp" = experimental/vaccinated; "con" = control/non-vaccinated)
levels(dat.long$group) <- c("exp", "con") ### set "con" to reference level dat.long$group <- relevel(dat.long\$group, ref="con")

### calculate log odds and corresponding sampling variances
dat.long <- escalc(measure="PLO", xi=out1, mi=out2, data=dat.long)

### bivariate random-effects model using rma.mv()
res <- rma.mv(yi, vi, mods = ~ group, random = ~ group | study, struct="UN", data=dat.long)
res
#>
#> Multivariate Meta-Analysis Model (k = 26; method: REML)
#>
#> Variance Components:
#>
#> outer factor: study (nlvls = 13)
#> inner factor: group (nlvls = 2)
#>
#>             estim    sqrt  k.lvl  fixed  level
#> tau^2.1    2.6173  1.6178     13     no    con
#> tau^2.2    1.5486  1.2444     13     no    exp
#>
#>      rho.con  rho.exp    con  exp
#> con        1   0.9450      -   no
#> exp   0.9450        1     13    -
#>
#> Test for Residual Heterogeneity:
#> QE(df = 24) = 5270.3863, p-val < .0001
#>
#> Test of Moderators (coefficient 2):
#> QM(df = 1) = 15.5470, p-val < .0001
#>
#> Model Results:
#>
#>           estimate      se     zval    pval    ci.lb    ci.ub
#> intrcpt    -4.0960  0.4529  -9.0432  <.0001  -4.9837  -3.2082  ***
#> groupexp   -0.7414  0.1880  -3.9430  <.0001  -1.1099  -0.3729  ***
#>
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
### profile over tau^2_1, tau^2_2, and rho
### note: for rho, adjust region over which profiling is done ('zoom in' on area around estimate)
# \dontrun{
par(mfrow=c(3,1))
profile(res, tau2=1)
profile(res, tau2=2)
profile(res, rho=1, xlim=c(.90, .98))# }