Estimating dependent t-test for paired samples when only means and SDs are available

Question

I'm attempting to extract t-values from a series of published papers for a meta-analysis. The specific measure of interest is the difference in performance (proportion correct on a memory test) between a Treatment Group and Control Group.

Some papers report the results of a t-test between the Treatment and Control groups--exactly what I need. Others however, do not.

Instead, those papers only provide Means, SDs, and n. When the experiment is between subjects, that is sufficient for me to calculate the t-statistic.

However, when the experiment involves a within-subjects design--although I have Means, SDs, and n for the Treatment and Control conditions, I do not have the SD of the difference scores for each subject (Treatment minus Control).

Is there a way that I can estimate what the results of a t-test would be in that case?

Further detail: in some of these within-subjects studies, each subject went through a Treatment, Control, and an Alternate Treatment condition. An ANOVA reporting a main effect of condition (3 levels) is reported.

Thanks in advance for any assistance!

Answer 1

Is there a way that I can estimate what the results of a t-test would be in that case?

If you know the covariance (or the correlation) between (Treatment,Control) pairs, then yes, because if we let $T_i$ and $C_i$ be the treatment and control values for the $i$-th subject, then

$\text{Var}(T_i-C_i)= \text{Var}(T_i)+\text{Var}(C_i)-2\ \text{Cov}(T_i,C_i)$

Since $\text{Var}(T_i)$ and $\text{Var}(C_i)$ are known that leaves only the covariance being needed (and if you know the correlation, the covariance can be computed from the correlation and the two known variances).

If you have some reasonable estimate of the correlation (or bounds on it), you can produce a similarly reasonable estimate of the denominator of the t-statistic (or perhaps bounds on it).