# Using effect size estimates and p-values to determine effect size variance and standard error

I am in the process of conducting a meta-analysis. I am having difficulty replicating effect sizes that are in previous meta-analyses on the same topic. That is, for a given study I am not getting the same effect size estimate.

In these past meta-analyses, effect sizes and p-values are provided. Is it possible to use these estimates to obtain the variance of the effect size. For example, can I use the p-value and effect size to obtain the 95 % confidence interval, through which I can obtain the standard error.

As an aside, I have reached out to the authors of the previous meta-analyses and they have not responded.

• What kind of effect size measure are you referring to? Raw or standardized mean differences? Log odds/risk ratios or risk differences? Something else? Also, just to clarify: You have a bunch of these estimates based on individual studies, and corresponding p-values, and you want to compute the variances (standard errors) thereof? – Wolfgang Apr 10 '16 at 10:41
• Hello. I have standardized mean differences, both Cohen's d and Hedges G. Indeed, I have many effect size estimates with corresponding p-values and would like to calculate the variance for each effect. – D_Williams Apr 10 '16 at 15:32
• In addition to effect size estimates and p-values, I also have the sample sizes. – D_Williams Apr 10 '16 at 15:58
• Sample sizes of each group or just total (both groups combined)? – Wolfgang Apr 10 '16 at 16:46
• In one of the meta-analyses, the total is provided. In a different meta-analyses there are sample sizes for each group. From the original studies, however, I can obtain the sample sizes for each group. – D_Williams Apr 10 '16 at 18:10

## 1 Answer

If you know the standardized mean difference ($d$) for a study and the sample sizes of the two groups ($n_1$ and $n_2$), then you can compute (or to be precise: estimate) the sampling variance with: $$v = \frac{1}{n_1} + \frac{1}{n_2} + \frac{d^2}{2(n_1 + n_2)}.$$ If you only know the total sample size of a study ($N$), you could assume $n_1 = n_2 = N/2$. The square-root of the sampling variance then gives you the standard error of $d$.

A bias-correction is often applied to $d$ values. Roughly, the bias-corrected standardized mean difference (often called Hedges' g) is given by: $$g = \left(1 - \frac{3}{4m-1} \right)d,$$ where $m = n_1 + n_2 - 2$ (this may explain the discrepancy you found when you computed the standardized mean difference yourself versus what was reported in the paper). Then you can plug $g$ into the equation for the sampling variance.