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Note: while I asked if is ok or not to add a constant to the log response ratio here, with this question I focus specifically on the formula of the variance of the log response ratio when adding a constant.

@Wolfgang gave a very clear and useful answer to this similar question. My question builds on his answer: In his example, given the observed means $\bar{x}_1$ and $\bar{x}_2$, if one adds the same constant to the two means, say $\bar{x}_1+k$ and $\bar{x}_2+k$ , then is the variance of the log response ratio this?

$$ \mbox{Var}(\mbox{lnRR}) = \frac{s_1^2}{n_1 (\bar{x}_1+k)^2} + \frac{s_2^2}{n_2 (\bar{x}_2+k)^2} $$

I am also aware that adding a constant is not an encouraged fix to zero values: Koricheva, Handbook of Meta-analysis:

A consequence of using log-transformed ratios, however, is that the ratio must be positive […] In this case, the response ratio is an inappropriate measure and Hedges’ d is instead recommended. One should also not use a ratio as an effect size measure when either the numerator or denominator would be equal to zero; the transform is undefined and trying to adjust the values by adding a tiny fraction to the numerator and denominator usually results in abnormally large estimates of effect size.

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