# Meta-analysis when response variable is already a ratio

I want to use meta-analysis on a number of variables, including percentages (like percent nitrogen) and ratios (like ratio of carbon to nitrogen). I'm estimating mean effect sizes using meta-analysis (fixed effects) of log response ratios following Hedges (1999):

$$L = ln(R) = ln(\bar{X}_{E})- ln(\bar{X}_{C})$$

Variance for L is estimated as:

$$v = \frac{(SD_{E})^2}{n_{E}\bar{X}_{E}^2}+\frac{(SD_{C})^2}{n_{C}\bar{X}_{C}^2}$$

Where $$\bar{X}_{E}, SD_{E}, n_{E}$$ are the mean, standard deviation, and sample size of the experimental group, and $$\bar{X}_{C}, SD_{C}, n_{C}$$ are the mean, standard deviation, and sample size of the control group. After L and v are calculated for multiple studies, then they are used to estimate an overall weighted mean (details in Hedges 1999).

My question is whether I can use the above approach with response variables that are percentages or ratios. The above approach assumes that the experimental and control means come from normal distributions, but ratios and percentages are bounded. Is it advisable to transform the percentage or ratio response variables before calculating the means?

• Strictly speaking the variables must be bounded as you can only take logs of a positive number. Taking logs of a ratio seem eminently sensible as it makes it symmetric about zero. For proportions I would have thought there were better options if they have the form $r$ out of $n$. Can you elaborate a bit on what they are like? – mdewey Feb 23 '17 at 14:17