# Meta-analysis - effect size from multiple severity groups

I am currently performing a meta analysis to evaluate whether a measure (let's call it Meas) is correlated to the severity of a disease (Sev). The severity of the disease is reported as a score from 0 to 10. Most studies would report the pearson coefficient between Meas and Sev.

However, some others compute means and sds of Meas for different severity groups. For these latter, I want to compute an effect size to convert to Cohen's d and then report it as a Person coefficient (following chapter 7 from Introduction to meta-analysis by Borenstein).

Thus, I have the following question: Let's say a study has 3 severity groups:

$$\begin{array}{lllll} & Sev␣mean & Meas␣mean & Meas␣SD & nb␣in␣group \\ group1 & 3 & 1 & 0.1 & 20 \\ group2 & 5.5 & 2 & 0.2 & 30 \\ group3 & 7 & 4 & 0.1 & 30 \end{array}$$

How can I compute an effect size to combine the evolution on Meas accross the 3 groups?

I think of 2 things: - I compute a Pearson correlation coefficient between the means of the severity groups () and the means of the measures of the severity groups:

$$Pearson (\begin{bmatrix} mean␣Sev␣group1 & mean␣Meas␣group 1\\mean␣Sev␣group2 & mean␣Meas␣group 2\\mean␣Sev␣group3 & mean␣Meas␣group3\end{bmatrix})$$ - I compute the standardised mean difference between group2 and group1 and then between group3 and group2. Then I take the mean of the two as the effect size.

Does one of them make sense? If not, any other idea?

Thank you very much!

• It isn't clear to me how the three groups in your example are constructed. Based on two cutoff values for the severity variable? In any case, you probably want to compute a polyserial correlation coefficient here. – Wolfgang Sep 9 '18 at 12:43
• Thank you for your answer Wolfgang. Indeed, severity groups are defined based on cut-off values (but these values vary depending on the studies: for instance, some would take 0-4, 5-7, 8-10 while other 0-5, 6-10 etc.... I guess they try to homogenise the number of patients in each group. Indeed, I had not thought of polyserial correlation coefficient. How should I transform it to Cohen's d later one? Shoudl I apply the same transformation as for Pearson's moment product? $$d=\frac{2r}{\sqrt(1-r^2)}$$ and $$Vd=\frac{4Vr}{(1-r^2)^3}$$ Thanks! – Epicure Sep 9 '18 at 17:52
• In fact I won't go through Cohen's d so please forget my last question. However, can I use the same Fisher Z transform as for Pearson's coefficient (arctan(r)/variance)? And if so, would it keep its normalizing properties? Thank you! – Epicure Sep 9 '18 at 18:06
• The variance of a polyserial correlation coefficient is different than the variance of a Pearson product-moment correlation coefficient. So, if you apply Fisher's r-to-z transformation to a polyserial correlation coefficient, this won't be a variance stabilizing transformation and the variance is not $1/(n-3)$. – Wolfgang Sep 9 '18 at 19:22
• Dear Wolfgang, Thank you very much. Indeed, the polyserial function in R. I will look more into this to understand how it is computed. Thanks again! (and for all the other questions you answered and also helped me). – Epicure Sep 10 '18 at 12:29