Hi,
I'm measuring a specific area of phalanges in different primate species. As species and individuals vary in size (and I'm not interested in these size differences), I divide this area of interest by the total area of the bone, obtaining a measurement that can be compared among individuals. This ratio is not normally distributed and is asymmetric (has a lot of zeros), so I transformed the data using this function:
log (x+1)
However, I have some doubts about all of this:
(i) I have read that using ratios for statistical purposes is complicated and not allways adviceable, although I don't see any issue in using them for the problem I'm dealing with. Does anyone see any problem?
(ii) I add the constant in the fuction (1) because I have zeros in the data (meaning the area of interest is not present in that bone). This changed the distribution to normallity in most cases. But, as there are various functions to transform data, I wonder if my transfomation is appropiate or are there others more appropiate for my problem.
(iii) Finally, I wonder about how to back-transform the results (specifically, mean, mean differences and confidence intervals from t-test) so I can interpret them easily. Which is the opposite matemathical function for log (x+1)?
Lots of questions :) I'll appreciate your responses
The transormation you use is common and sensible. An alternative approach and perhaps more elegant is the use of "offset" to normalize for the exposure (in your case bone area of the subject). Check this:
https://en.wikipedia.org/wiki/Poisson_regression#"Exposure"_and_offset
and the section "Exposure and offset".
To back-tranform a point-estimate (s, say) computed on y=ln(1+x) you apply the inverse function which is exp(s)-1. Similarly you can compute the standard error by using the Delta Method:
https://en.wikipedia.org/wiki/Delta_method
Note that the Delta Method is an approximation where the approximation improves with the sample size.
Given e^x is the inverse of ln(x), e^x + 1 is the inverse of ln(x+1).
To test in R:
x
Also, you said your problem is related to lots of zeroes. There are lots of approaches designed for this (e.g., zero inflated models, hurdle models, the Tobit model, each for different situations, see Figure 4 of Article Statistical techniques for juror and jury research
).
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