# Stouffer’s meta-analysis with weight and direction effect in R

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I often have to perform meta-analysis of past experiments for which the only info I have is the fold change and the p-value (of any measure you may imagine: species richness, gene expression, depth of coverage, plates of carbonara eaten in 5 minutes, everything).

The hardest thing to find out for me was how to take the direction of the changes into account. E.g. if monday I eat 10 carbonara more than my brother (p=0.01), on tuesday 10 more (p=0.01), on wednesday 5 more (p=0.1), on thursday 10 less (p=0.01), on friday ten less (p=0.01) and on saturday 5 less (p=0.1), a standard Stouffer meta-analysis would return a highly significant p-value, completely disregarding the fact that the significant changes were half in my favor, and half in favor of my brother.

How can I take into account the information on the direction?

I had no idea on how to proceed, and I wasn’t able to find any reference (I am not an expert of meta-analysis), but then I was so lucky to stumble upon the manual page of the software package METAINTER for performing meta-analysis.

The authors described there a method to perform Stouffer meta-analysis accounting for the direction of the effects. Their explanation was so clear that it was easy for me to write a simple function in R: I paste it below.

signed.Stouffer.meta <- function(p, w, sign) { # p is a vector of p-values if (missing(w)) { w <- rep(1, length(p))/length(p) } else { if (length(w) != length(p)) stop("Length of p and w must equal!") } if(length(p)==1) Zi1) { Zi<-qnorm(p/2,lower.tail=FALSE) Zi[sign<0]<-(-Zi[sign<0]) } Z <- sum(w*Zi)/sqrt(sum(w^2)) p.val <- 2*(1-pnorm(abs(Z))) return(c(Z = Z, p.value = p.val)) }

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