metafor (version 1.9-8)

influence.rma.uni: Outlier and Influential Case Diagnostics for 'rma.uni' Objects

Description

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.uni".

Usage

"influence"(model, digits, ...)
"print"(x, digits, ...)
"cooks.distance"(model, ...)
"dfbetas"(model, ...)
"hatvalues"(model, type="diagonal", ...)

Arguments

model

an object of class "rma.uni".

x

an object of class "infl.rma.uni" (for print).

digits

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

type

character string indicating whether to return only the diagonal of the hat matrix ("diagonal") or the entire hat matrix ("matrix").

...

other arguments.

Value

An object of class "infl.rma.uni". The object is a list containing the following components:

inf

a data frame with columns equal to the externally standardized residuals, DFFITS values, Cook's distances, covariance ratios, leave-one-out \tau² estimates, leave-one-out (residual) heterogeneity test statistics, hat values, and weights.

dfb

a data frame with columns equal to the DFBETAS values.

is.infl

a logical vector indicating whether a particular study is considered to be ‘influential’.

...

some additional elements/values.

The results are printed with print.infl.rma.uni and plotted with plot.infl.rma.uni.

Details

The influence function calculates the following leave-one-out diagnostics for each study:

• externally standardized residual,
• DFFITS value,
• Cook's distance,
• covariance ratio,
• the leave-one-out amount of (residual) heterogeneity,
• the leave-one-out test statistic for the test of (residual) heterogeneity,
• DFBETAS value(s).

The diagonal elements of the hat matrix and the weights (in %) given to the observed effects or outcomes during the model fitting are also provided (except for their scaling, the hat values and weights are the same for models without moderators, but will differ when moderators are included).

For details on externally standardized residuals, see rstudent.rma.uni.

The DFFITS value essentially indicates how many standard deviations the predicted (average) effect for the $i$th study changes after excluding the $i$th study from the model fitting.

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

The covariance ratio is defined as the determinant of the variance-covariance matrix of the parameter estimates based on the dataset with the $i$th study removed divided by the determinant of the variance-covariance matrix of the parameter estimates based on the complete dataset. A value below 1 therefore indicates that removal of the $i$th study yields more precise estimates of the model coefficients.

The leave-one-out amount of (residual) heterogeneity is the estimated value of \tau² based on the dataset with the $i$th study removed. Note that this is always equal to 0 for fixed-effects models.

Similarly, the leave-one-out test statistic for the test of (residual) heterogeneity is the value of the test statistic of the test for (residual) heterogeneity calculated based on the dataset with the $i$th study removed.

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

A study may be considered to be ‘influential’ if at least one of the following is true:

• The absolute DFFITS value is larger than $3\sqrt(p/(k-p))$, where $p$ is the number of model coefficients and $k$ the number of studies.
• The lower tail area of a chi-square distribution with $p$ degrees of freedom cut off by the Cook's distance is larger than 50%.
• The hat value is larger than $3(p/k)$.
• Any DFBETAS value is larger than $1$.

Studies which are considered influential with respect to any of these measures are marked with an asterisk. Note that the chosen cut-offs are (somewhat) arbitrary. Substantively informed judgment should always be used when examining the influence of each study on the results.

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.

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

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.

See Also

plot.infl.rma.uni, rstudent.rma.uni, weights.rma.uni

Examples

### load BCG vaccine data
data(dat.bcg)

### meta-analysis of the log relative risks using a mixed-effects model
### with two moderators (absolute latitude and publication year)
res <- rma(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg,
           mods = ~ ablat + year, data=dat.bcg)
influence(res)
plot(influence(res))

cooks.distance(res)
dfbetas(res)
hatvalues(res)