The function computes best linear unbiased predictions (BLUPs) of the study-specific true effect sizes or outcomes by combining the fitted values based on the fixed effects and the estimated contributions of the random effects for objects of class "rma.uni". Corresponding standard errors and prediction interval bounds are also provided.

blup(x, ...)

# S3 method for rma.uni
blup(x, level, digits, transf, targs, ...)

## Arguments

x

an object of class "rma.uni".

level

numeric value between 0 and 100 to specify the prediction interval level. If unspecified, the default is to take the value from the object.

digits

optional integer to specify 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.

transf

optional argument to specify a function to transform the predicted values 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.

...

other arguments.

## Value

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

pred

predicted values.

se

corresponding standard errors.

pi.lb

lower bound of the prediction intervals.

pi.ub

upper bound of the prediction intervals.

...

The object is formatted and printed with the print function. To format the results as a data frame, one can use the as.data.frame function.

## Note

For best linear unbiased predictions of only the random effects, see ranef.

For predicted/fitted values that are based only on the fixed effects of the model, see fitted and predict.

For conditional residuals (the deviations of the observed effect sizes or outcomes from the BLUPs), see rstandard.rma.uni with type="conditional".

Equal-effects models do not contain random study effects. The BLUPs for these models will therefore be equal to the fitted values, that is, those obtained with fitted and predict.

When using the transf argument, the transformation is applied to the predicted values and the corresponding interval bounds. The standard errors are then set equal to NA and are omitted from the printed output.

By default, a standard normal distribution is used to construct the prediction intervals. When the model was fitted with test="t" or test="knha", then a t-distribution with $$k-p$$ degrees of freedom is used.

To be precise, it should be noted that the function actually computes empirical BLUPs (eBLUPs), since the predicted values are a function of the estimated value of $$\tau^2$$.

## Author

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

## References

Kackar, R. N., & Harville, D. A. (1981). Unbiasedness of two-stage estimation and prediction procedures for mixed linear models. Communications in Statistics, Theory and Methods, 10(13), 1249–1261. https://doi.org/10.1080/03610928108828108

Raudenbush, S. W., & Bryk, A. S. (1985). Empirical Bayes meta-analysis. Journal of Educational Statistics, 10(2), 75–98. https://doi.org/10.3102/10769986010002075

Robinson, G. K. (1991). That BLUP is a good thing: The estimation of random effects. Statistical Science, 6(1), 15–32. https://doi.org/10.1214/ss/1177011926

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 for the function to fit models for which BLUPs can be extracted.

predict.rma and fitted.rma for functions to compute the predicted/fitted values based only on the fixed effects and ranef.rma.uni for a function to compute the BLUPs based only on the random effects.

## Examples

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

### meta-analysis of the log risk ratios using a random-effects model
res <- rma(yi, vi, data=dat)

### BLUPs of the true risk ratios for each study
blup(res, transf=exp)
#>
#>      pred  pi.lb  pi.ub
#> 1  0.4492 0.2012 1.0032
#> 2  0.2860 0.1431 0.5716
#> 3  0.3727 0.1590 0.8740
#> 4  0.2471 0.1887 0.3236
#> 5  0.7502 0.4958 1.1352
#> 6  0.4563 0.3883 0.5362
#> 7  0.2882 0.1400 0.5936
#> 8  1.0029 0.8871 1.1338
#> 9  0.6024 0.3911 0.9279
#> 10 0.2873 0.1775 0.4651
#> 11 0.7021 0.5665 0.8702
#> 12 0.7522 0.3064 1.8469
#> 13 0.8635 0.5359 1.3915
#>

### illustrate shrinkage of BLUPs towards the (estimated) population average
res <- rma(yi, vi, data=dat)
blups <- blup(res)$pred plot(NA, NA, xlim=c(.8,2.4), ylim=c(-2,0.5), pch=19, xaxt="n", bty="n", xlab="", ylab="Log Risk Ratio") segments(rep(1,13), dat$yi, rep(2,13), blups, col="darkgray")
points(rep(1,13), dat$yi, pch=19) points(rep(2,13), blups, pch=19) axis(side=1, at=c(1,2), labels=c("Observed\nValues", "BLUPs"), lwd=0) segments(0, res$beta, 2.15, res$beta, lty="dotted") text(2.3, res$beta, substitute(hat(mu)==muhat, list(muhat=round(res\$beta[[1]], 2))), cex=1)