## 4.4 Correlations

Performing a meta-analysis of correlations is not too different from the methods we described before. Commonly, the generic inverse-variance pooling method is also used to combine correlations from different studies into one pooled correlation estimate. When pooling correlations, it is advised to perform Fisher’s $$z$$-transformation to obtain accurate weights for each study. Luckily, we do not have to do this transformation ourselves. There is an additional function for meta-analyses of correlations included in the meta package, the metacor function, which does most of the calculations for us.

The parameters of the metacor function are mostly identical to the metagen and metacont function we described before (see Chapter 4.1 and 4.2), as is the statistics behind it. To use the function, we again need a dataset with the study label (Author), the correlation $$r$$ (cor) reported for each study, and the sample size (n) for each study. Let us have a look at my cordata dataset, which follows this structure:

str(cordata)
## 'data.frame':    8 obs. of  3 variables:
##  $Author: Factor w/ 8 levels "Altman2011","Borenstein2013",..: 7 5 1 8 2 3 4 6 ##$ cor   : num  0.456 0.234 0.235 0.297 0.374 0.561 0.102 0.377
##  $n : num 150 67 45 80 230 110 78 45 We can use this dataset to calculate a meta-analysis using the metacor function. Like in the chapters before, we will use a random-effects model with the Sidik-Jonkman estimator for the between-study heterogeneity $$\tau^2$$. In addition, we set sm to "ZCOR" to use the $$z$$-transformed correlations for the meta-analysis (as is advised). The rest of the syntax remains identical. I will call the results of the meta-analysis m.cor. m.cor <- metacor(cor, n, data = cordata, studlab = cordata$Author,
sm = "ZCOR",
method.tau = "SJ")

Let us have a look at the output, which has the same structure as the one of the metagen, metabin and metainc functions.

##                   COR            95%-CI %W(fixed) %W(random)
## Mantel1999     0.4560 [ 0.3191; 0.5743]      18.8       15.3
## Haenszel1987   0.2340 [-0.0066; 0.4490]       8.2       11.3
## Altman2011     0.2350 [-0.0629; 0.4944]       5.4        9.1
## Olkin1999      0.2970 [ 0.0827; 0.4851]       9.9       12.3
## Borenstein2013 0.3740 [ 0.2571; 0.4801]      29.1       16.8
## Egger2001      0.5610 [ 0.4176; 0.6771]      13.7       13.9
## Glass2007      0.1020 [-0.1233; 0.3173]       9.6       12.1
## Ioannidis2010  0.3770 [ 0.0939; 0.6037]       5.4        9.1
##
## Number of studies combined: k = 8
##
##                         COR           95%-CI     z  p-value
## Fixed effect model   0.3693 [0.3072; 0.4282] 10.83 < 0.0001
## Random effects model 0.3491 [0.2378; 0.4515]  5.85 < 0.0001
##
## Quantifying heterogeneity:
## tau^2 = 0.0186; H = 1.56 [1.06; 2.31]; I^2 = 59.0% [10.6%; 81.2%]
##
## Test of heterogeneity:
##      Q d.f. p-value
##  17.09    7  0.0168
##
## Details on meta-analytical method:
## - Inverse variance method
## - Sidik-Jonkman estimator for tau^2
## - Fisher's z transformation of correlations

As can be seen from the output, the pooled correlation in this dataset is $$r=0.35$$ (for the random-effects model), which is significant ($$p<0.0001$$). The $$I^2$$-heterogeneity in this analysis is substantial (58%), supporting the use of a random-effects model.