Calculating Pre/Post Correlation from a Paired T-Test

In this post we will go over how to convert a t-statistic from a paired t-test into a pre/post correlation. This is useful when change score standard deviations are unknown. 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 the necessary statistics

In order to calculate the pre/post correlation (\(r\)), we need the standard deviation (SD) of pre-test scores (\(SD_{pre}\)), the SD of post-test scores (\(SD_{post}\)), the mean change (\(M_{change}\)), the paired t-statistic (\(t\)), the sample size (\(N\)). In this blog post, we are assuming that the change score standard deviation (\(SD_{change}\)) is unavailable to us. If the pre-test SD is available, but the post-test SD is unavailable, 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 (\(\overline{SD}_{ratio}\); see the blog post on 9/8/2023), then we can approximate the post-test SD by multiplying the pre-test SD by the average SD ratio,

\[ SD_{post}\approx \bar{SD}_{ratio}\times SD_{pre} \]

A rougher approximation would be to simply set the pre-test SD and post-test SD to be equal. If the study reports an F-statistic from a one-way repeated measures ANOVA, the F-statistic is equal to the square of the t-statistic,

\[ t = \sqrt{F} \]

Ensure that you apply the correct sign (negative or positive) to the t-statistic, since the F-statistic is always positive.

# Obtain values
M_change <- 3
SD_post <- 11
SD_pre <- 9
t <- 2.50
N <- 50

Step 2: Calculate the Pre/Post Correlation

Lets start by figuring out how to find the change score SD. The paired t-statistic is defined as the mean change score divided by the standard error of change scores, such that, \[ t = \frac{M_{change}}{SE_{change}} \] Since we need the change score SD, we can use the definition of the standard error of the mean to put the t-statistic in terms of \(SD_{change}\): \[ SE_{change}=\frac{SD_{change}}{\sqrt{N}} \] and therefore \(t\) can be expressed as,

\[ t=\frac{M_{change}}{\left(\frac{SD_{change}}{\sqrt{N}}\right)} \]

then we just need to solve for \(SD_{change}\):

\[ SD_{change}=\frac{M_{change}\times\sqrt{N}}{t} \]

Okay so now let us recall the definition of change score SDs from the blog post on 9/8/2023. In that blog we discussed how to obtain the pre/post correlation from the change score SD, now that we have converted \(t\) to \(SD_{change}\), we can solve for the correlation in a similar way. First things first, the change score SD can be defined as, \[ SD_{change} = \sqrt{SD^2_{pre} + SD^2_{post} - 2\times r\times SD_{pre}SD_{post}} \]

We can re-arrange this to isolate the pre/post correlation (\(r\)),

\[ r = \frac{SD^2_{pre} + SD^2_{post} - SD^2_{change}}{2 \times SD_{pre}\times SD_{post}} \]

In our case, the study did not report the change score SD, therefore we can replace it with our derived \(SD_{change}\) from a paired t-test:

\[ r = \frac{SD^2_{pre} + SD^2_{post} - \left(\frac{M_{change}\times\sqrt{N}}{t}\right) ^2}{2 \times SD_{pre}\times SD_{post}} \] Lets neaten this formulation up a tad:

\[ r = \frac{t^2\left(SD^2_{pre} + SD^2_{post}\right) - N\times M^2_{change} }{2 \times t^2 \times SD_{pre}\times SD_{post} } \]

Isn’t that just a beautiful thing?? So there you have it! the full equation for the pre/post correlation from a paired t-test! Note that this is a direct conversion and not merely an approximation.

# Calculate pre/post correlation
r <- (t^2*(SD_pre^2 + SD_post^2) - N * M_change^2) / (2*t^2*SD_pre*SD_post)

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

[1] "r = 0.657"

Applying it to a simulated dataset

We can simulate correlated pre/post scores from a bivariate Gaussian with known parameters. The calculated correlation is exactly correct!

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

# Define parameters
SD_pre <- 9
SD_post <- 11
r_true <- .70
M_pre <- 20
M_post <- 25
N <- 100

# Simulate correlated pre/post scores from bivariate gaussian
data <- mvrnorm(n=N,
               mu=c(M_pre,M_post),
               Sigma = data.frame(x=c(SD_pre^2,r_true*SD_pre*SD_post),
                                  y=c(r_true*SD_pre*SD_post,SD_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, t-stats, and mean change
SD_pre <- sd(x_pre)
SD_post <- sd(x_post)
t <- mean(x_change) / (sd(x_change)/sqrt(N))
M_change <- mean(x_change)

# Calculate pre/post correlation
r <- (t^2*(SD_pre^2 + SD_post^2) - N * M_change^2) / (2*t^2*SD_pre*SD_post)

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

[1] "r = 0.7"