# Compute Becker's (1988) standardized mean change based only on difference score and t-statistic (meta-analysis)

In my meta-analysis, I'm combining independent-sample designs with single-group pretest-posttest designs. To ensure interpretability across designs, I would like to quantify the effects from pretest-posttest designs as standardized mean changes using raw score standardization.

When pre and post means and SDs are available, the effect size can be easily computed via metafor::escalc(measure = "SMCR"). However, at least one relevant paper in my database reports only the change from pretest to posttest, $$\Delta$$ = 0.15, as well as the results of the paired-samples t-test, t(49) = 2.84, p = .007.

The metafor documentation explains how to compute SMRCs on the basis of change scores, but to do so one would still need the SD of the pretest. While one can easily back-calculate the SD of the change score based on the t-statistic, I'm not aware the same can be done for the pretest SD. Of course I could also compute the repeated-measures effect size drm (Morris & DeShon, 2002, Eq. 28), and then transform it to a raw-score standardized d (Morris & DeShon, 2002, Eq. 11).

$$d_{rm} = \frac{t}{\sqrt{n}} = \frac{2.84}{\sqrt{50}} = 0.402$$

$$d = d_{rm}\sqrt{2(1-\rho)} = 0.402\sqrt{2(1-0.7)} = 0.311$$

However, this computation differs from that in metafor::escalc(measure = "SMCR"), which I understand follows the approach proposed by Becker (1988). As Morris and DeShon (2002) point out, "the two approaches will not produce exactly the same value" (p. 111), especially when variances are not equal.

My question is thus whether I can compute the SMCR, as per Becker (1988), on the basis of just the change score and the t statistic -- either in metafor or by hand. Thanks for any pointers!

References:

• Becker, B. J. (1988). Synthesizing standardized mean-change measures. British Journal of Mathematical and Statistical Psychology, 41, 257–278.
• Morris, S. B., & DeShon, R. P. (2002). Combining effect size estimates in meta-analysis with repeated measures and independent-groups designs. Psychological Methods, 7, 105-125.