The functions compute various outlier and influential case diagnostics (some of which indicate the influence of deleting one case/study at a time on the model fit and the fitted/residual values) for objects of class "rma.mv".

# S3 method for rma.mv
cooks.distance(model, progbar=FALSE, cluster,
reestimate=TRUE, parallel="no", ncpus=1, cl=NULL, ...)
# S3 method for rma.mv
dfbetas(model, progbar=FALSE, cluster,
reestimate=TRUE, parallel="no", ncpus=1, cl=NULL, ...)
# S3 method for rma.mv
hatvalues(model, type="diagonal", ...)

## Arguments

model an object of class "rma.mv". logical indicating whether a progress bar should be shown (the default is FALSE). Ignored when using parallel processing. optional vector specifying a clustering variable to use for computing the Cook's distances. If not specified, Cook's distances are computed for all individual observed outcomes. logical indicating whether variance/correlation components should be re-estimated after deletion of the $$i$$th study/cluster (the default is TRUE). character string indicating whether parallel processing should be used (the default is "no"). For parallel processing, set to either "snow" or "multicore". See ‘Details’. integer specifying 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. character string indicating whether to return only the diagonal of the hat matrix ("diagonal") or the entire hat matrix ("matrix"). other arguments.

## Details

Cook's distance for the $$i$$th study/cluster can be interpreted as the Mahalanobis distance between the entire set of predicted values once with the $$i$$th study/cluster included and once with the $$i$$th study/cluster excluded from the model fitting.

The DFBETAS value(s) essentially indicate(s) how many standard deviations the estimated coefficient(s) change(s) after excluding the $$i$$th study/cluster from the model fitting.

## Value

The cooks.distance function returns a vector. The dfbetas function returns a data frame. The hatvalues function returns either a vector with the diagonal elements of the hat matrix or the entire hat matrix.

## Note

Right now, leave-one-out diagnostics are calculated by refitting the model $$k$$ times (where $$k$$ is the number of studies/clusters). Depending on how large $$k$$ is, it may take a few moments to finish the calculations. For complex models fitted with rma.mv, this can become computationally expensive.

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.

Alternatively (or in addition to using parallel processing), one can also set reestimate=FALSE, in which case any variance/correlation components in the model are not re-estimated after deleting the $$i$$th study/cluster from the dataset. Doing so only yields an approximation to the Cook's distances and DFBETAS values that ignores the influence of the $$i$$th study/cluster on the variance/correlation components, but is considerably faster (and often yields similar results).

It may not be possible to fit the model after deletion of the $$i$$th study/cluster from the dataset. This will result in NA values for that study/cluster.

## References

Belsley, D. A., Kuh, E., & Welsch, R. E. (1980). Regression diagnostics. New York: Wiley.

Cook, R. D., & Weisberg, S. (1982). Residuals and influence in regression. London: Chapman and Hall.

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/.

Viechtbauer, W., & Cheung, M. W.-L. (2010). Outlier and influence diagnostics for meta-analysis. Research Synthesis Methods, 1, 112--125.

rstudent.rma.mv, weights.rma.mv

## Examples

### copy data from Konstantopoulos (2011) into 'dat'
dat <- dat.konstantopoulos2011

### multilevel random-effects model
res <- rma.mv(yi, vi, random = ~ 1 | district/school, data=dat)
print(res, digits=3)#>
#> Multivariate Meta-Analysis Model (k = 56; method: REML)
#>
#> Variance Components:
#>
#>            estim   sqrt  nlvls  fixed           factor
#> sigma^2.1  0.065  0.255     11     no         district
#> sigma^2.2  0.033  0.181     56     no  district/school
#>
#> Test for Heterogeneity:
#> Q(df = 55) = 578.864, p-val < .001
#>
#> Model Results:
#>
#> estimate     se   zval   pval  ci.lb  ci.ub
#>    0.185  0.085  2.185  0.029  0.019  0.350  *
#>
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
### Cook's distances for each observed outcome
x <- cooks.distance(res)
x#>            1            2            3            4            5            6
#> 1.965212e-03 2.989768e-03 2.638140e-03 3.873368e-03 1.954562e-04 2.049672e-02
#>            7            8            9           10           11           12
#> 1.172821e-03 3.913806e-03 3.646294e-03 3.850068e-04 4.111673e-04 5.636933e-03
#>           13           14           15           16           17           18
#> 8.697463e-03 6.052503e-04 4.759729e-03 1.527261e-05 9.905788e-05 2.095987e-03
#>           19           20           21           22           23           24
#> 2.978160e-03 1.173555e-04 7.600104e-05 7.315289e-05 4.966021e-04 2.024080e-05
#>           25           26           27           28           29           30
#> 6.448093e-04 3.866063e-04 1.220054e-04 2.886483e-04 7.676490e-08 1.391702e-06
#>           31           32           33           34           35           36
#> 6.193156e-02 2.406239e-02 6.513879e-02 2.075376e-05 8.763819e-07 3.485720e-05
#>           37           38           39           40           41           42
#> 1.141739e-04 6.503505e-06 5.671804e-05 8.326350e-05 1.048009e-04 3.320742e-03
#>           43           44           45           46           47           48
#> 6.412251e-03 3.831098e-05 3.937144e-03 1.780433e-03 1.374408e-03 2.479767e-02
#>           49           50           51           52           53           54
#> 2.281846e-02 2.803636e-04 3.453698e-03 1.101867e-02 1.405349e-04 1.257869e-02
#>           55           56
#> 1.606173e-03 4.733739e-03 plot(x, type="o", pch=19, xlab="Observed Outcome", ylab="Cook's Distance")
### Cook's distances for each district
x <- cooks.distance(res, cluster=dat\$district)
x#>           11           12           18           27           56           58
#> 0.0693784140 0.0161880606 0.0419849759 0.1192089452 0.0213566370 0.1014506985
#>           71           86           91          108          644
#> 0.5585174853 0.0656500671 0.0037438517 0.0548839759 0.0003441647 plot(x, type="o", pch=19, xlab="District", ylab="Cook's Distance", xaxt="n")axis(side=1, at=seq_along(x), labels=as.numeric(names(x)))
### hat values
hatvalues(res)#>           1           2           3           4           5           6
#> 0.018246414 0.018246414 0.015562154 0.015562154 0.024305990 0.024305990
#>           7           8           9          10          11          12
#> 0.023796820 0.020021978 0.028628711 0.021068630 0.035668057 0.018288352
#>          13          14          15          16          17          18
#> 0.026253550 0.026253550 0.024271478 0.018700077 0.024347299 0.025637683
#>          19          20          21          22          23          24
#> 0.026335563 0.008149594 0.008470846 0.008307115 0.007851819 0.008149594
#>          25          26          27          28          29          30
#> 0.007997936 0.011094988 0.010815774 0.010815774 0.010815774 0.010815774
#>          31          32          33          34          35          36
#> 0.027875751 0.030425177 0.031137102 0.012900198 0.012900198 0.012900198
#>          37          38          39          40          41          42
#> 0.012900198 0.012900198 0.012900198 0.012900198 0.012900198 0.016836925
#>          43          44          45          46          47          48
#> 0.016451963 0.016836925 0.017240335 0.015732539 0.015732539 0.018231717
#>          49          50          51          52          53          54
#> 0.018231717 0.018522325 0.018522325 0.018522325 0.017484652 0.018246600
#>          55          56
#> 0.020990821 0.020990821