# Combining correlation coefficient and t-statistics in meta-analysis

I have some studies that report a correlation coefficient between a predictor variable and and outcome of interest. Other studies have dichotomised the predictor variable and so instead present a t-statistic. Would it be justified to combine these two types of study in a single meta-analysis?

If so could I then just convert the r statistic to a t-statistic $$r=t/(t^2 +dof)^{0.5}$$? And then convert all t-statistics to Hedge's g and combine?

No, this would not be appropriate, since $$r$$ will be a point-biserial correlation, which does not provide an estimate of the correlation between the underlying continuous variables. For this, you need the biserial correlation coefficient. See:

Jacobs, P., & Viechtbauer, W. (2017). Estimation of the biserial correlation and its sampling variance for use in meta-analysis. Research Synthesis Methods, 8(2), 161-180. https://doi.org/10.1002/jrsm.1218

To demonstrate, say that the true correlation between X and Y is 0.6. Let's generate a really large sample based on a bivariate normal distribution with this correlation:

library(MASS)
n <- 100000
XY <- mvrnorm(n, mu=c(0,0), Sigma=matrix(c(1,0.6,0.6,1), nrow=2))
cor(XY[,1], XY[,2])


Not surprisingly, the observed correlation is essentially equal to 0.6. Now let's dichotomize one of the two variables and conduct a t-test and use the given equation, which yields the point-biserial correlation between the dichotomized variable and the continuous one:

res <- t.test(XY[,1] ~ XY[,2] > 0, var.equal=TRUE)
rpb <- -res$$statistic / sqrt(res$$statistic^2 + (n-2))
rpb
cor(XY[,1], XY[,2] > 0)


But if our goal is to estimate the underlying correlation between the two continuous variables, then we need the biserial correlation coefficient, which we can get with:

p <- mean(XY[,2] > 0)
sqrt(p*(1-p)) / dnorm(qnorm(p)) * rpb


This is again essentially equal to 0.6. Of course, in smaller samples, the (biserial) correlation coefficients will vary more around the true correlation, but bias should be negligible. Note that computation of the sampling variance of the biserial correlation coefficient is a bit more tricky than for a regular correlation coefficient (see again the paper above). However, all of this is implemented in the metafor package in case you are using R. See:

https://wviechtb.github.io/metafor/reference/escalc.html#-d-measures-for-mixed-variable-types