Calculating Pre/Post Correlation from the Standard Deviation of Change Scores

Welcome to the first blog post of Meta-Analysis Magic! In this post we will go over a couple of ways to directly calculate pre/post correlations from alternative statistics such as standard deviations of change scores. Pre/post correlations are necessary to obtain various types of effect sizes and standard errors such as the repeated measures variant of Cohen’s \(d\), also referred to as Cohen’s \(d_{rm}\).

Step 1: Obtain Pre-test, Post-test, and Change score standard deviations

In order to calculate the pre/post correlation (\(r\)), we need an estimate of the standard deviation (SD) of pre-test scores (\(S_{pre}\)), the SD of post-test scores (\(S_{post}\)), and the SD of change scores (\(S_{change}\), where \(x_{change}=x_{post}-x_{pre}\)). If the post-test SD is unavailable, but the pre-test SD is available, you can approximate the post-test SD by, first, taking average ratio of the pre-test SD and post-test SD from \(k\) studies in the current meta-analysis,

\[ \bar{S}_{ratio}=\frac{1}{k}\sum_{i=1}^k \frac{S_{post,i}}{S_{pre,i}} \] Then we can make an approximation of the post-test SD by multiplying the pre-test SD by the average SD ratio,

\[ S_{post}\approx \bar{S}_{ratio}\times S_{pre} \]

A rougher approximation would be to simply set the pre-test SD and post-test SD to be equal.

# Define standard deviations
S_pre <- 9
S_post <- 11
S_change <- 8

Step 2: Calculate the Pre/Post Correlation

The correlation between pre-test and post-test scores (\(r\)) can be calculated by re-arranging the equation for change score SD:

\[ S_{change} = \sqrt{S^2_{pre} + S^2_{post} - 2rS_{pre}S_{post}} \] Then we can solve for \(r\),

\[ r = \frac{S^2_{pre} + S^2_{post} - S^2_{change}}{2S_{pre}S_{post}} \] Note that this is a direct conversion and not merely an approximation.

# Calculate pre/post correlation
r <- (S_pre^2 + S_post^2 - S_change^2) / (2*S_pre*S_post)

# Print results
print(paste0('r = ',round(r,3)))

[1] "r = 0.697"

Applying it to a simulated dataset

We can simulate correlated pre/post scores from a bivariate Gaussian with known parameters. It can be seen that the correlation calculated from the formulas above is perfectly precise.

# install.packages('MASS')
library(MASS)

# Define parameters
S_pre <- 9
S_post <- 11
r_true <- .70

# Simulate correlated pre/post scores from bivariate gaussian
data <- mvrnorm(n=200,
               mu=c(0,0),
               Sigma = data.frame(x=c(S_pre^2,r_true*S_pre*S_post),
                                  y=c(r_true*S_pre*S_post,S_post^2)),
               empirical = TRUE)

# Obtain simulated scores
x_pre <- data[,1] # Pre-test scores
x_post <- data[,2] # Post-test scores
x_change <- x_post - x_pre # Calculate change scores

# Calculate standard deviations
S_pre <- sd(x_pre)
S_post <- sd(x_post)
S_change <- sd(x_change)

# Calculate pre/post correlation
r <- (S_pre^2 + S_post^2 - S_change^2) / (2*S_pre*S_post)

print(paste0('r = ',r))

[1] "r = 0.7"