influence.rma.uni: Case Diagnostics for rma.uni Objects

Description

The function calculates various case diagnostics that indicate the influence of deleting one case at a time on the model fit and the fitted/residual values for objects of class "rma.uni".

Usage

## S3 method for class 'rma.uni':
influence(model, digits=model$digits, ...)

Arguments

model

an object of class "rma.uni".

digits

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

...

other arguments.

Value

An object of class "rma.uni.infl". The object is a list containing the following components:
infA data frame with columns equal to the externally standardized residuals, DFFITS values, Cook's distances, covariance ratios, leave-one-out $\tau^2$ estimates, leave-one-out (residual) heterogeneity test statistics, hat values, and weights.
dfbA data frame with columns equal to the DFBETAS values.
...some additional elements/values.
The results are printed with print.rma.uni.infl and plotted with plot.rma.uni.infl.

Details

The following leave-one-out diagnostics are calculated for each case:

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, and
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. For details on externally standardized residuals, see rstudent.rma.uni. For the meta-analytic models underlying the rma.uni function, the DFFITS value for the $i^{th}$ case is here defined as the difference between the predicted value based on the fitted model using the complete dataset and the predicted value based on the fitted model using the dataset with the $i^{th}$ case removed, divided by the standard error of the predicted value based on the fitted model using the dataset with the $i^{th}$ case removed. Cook's distance is here defined as $(\hat{\beta}-\hat{\beta}_{(i)})' \hat{\Sigma}^{-1} (\hat{\beta}-\hat{\beta}_{(i)})$, where $\hat{\beta}$ is the vector of parameter estimates based on the complete dataset, $\hat{\beta}_{(i)}$ is the vector of parameter estimates based on the dataset with the $i^{th}$ case removed, and $\hat{\Sigma}$ is the variance-covariance matrix of the parameter estimates based on the full dataset. The covariance ratio is defined as the determinant of the variance-covariance matrix of the parameter estimates based on the dataset with the $i^{th}$ case removed divided by the determinant of the variance-covariance matrix of the parameter estimates based on the complete dataset. The leave-one-out amount of (residual) heterogeneity is the estimated value of $\tau^2$ based on the dataset with the $i^{th}$ case removed. 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 $i^{th}$ case removed. Finally, the DFBETAS values for the $i^{th}$ case are defined as the difference between the parameter estimates based on the complete dataset and the parameter estimates based on the dataset with the $i^{th}$ case removed, divided by the standard error of the parameter estimates based on the dataset with the $i^{th}$ case removed.

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.

Examples

Run this code

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

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

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