I have a between-group independent variable with two level (A and B) and a dependent variable Y that I transformed in order to normalize the distribution of the residual. I used a Box Cox transformation with the following formula : ((Y^3,169833)-1)/3,169833
With this transformation I observed the following results (Y' refers to the transformed variable)
I used this formula to back transform my data:(Y'*3,169833+1)^(1/3,169833) and I obtained the means and CI in red in the figure.
My question is how can I get the difference between my two means and the 95% CI for that difference? I tried to calculate the difference between the trasnformed means, to get its 95% CI and then to transform back. However I obtained an improbable result. I guess that maybe this is because I transformed Y and not the difference between my to groups. Is there a formula to compute the CI of a difference between means from the two CI of the two means?
Thank you very much in advance
Update after Alexy's answer
I have a question about the fact that "inferences about f(x) are not the same thing as inferences about x". I agree with that. However I thought that it was for that reason that we have to back transform the transformed data in order to interpret the result more easily. For example, Bland & Altman (1996)* have suggested that we can back-transformed the confidence limits of a log transform data using the anti-log in order interpret the results in the original unit. Do you agree with this possibility? Or does the fact that "inferences about f(x) are not the same thing as inferences about x" preclude this possibility? I you agree with Bland & Altman (1996), do you think that we can approximate the sd for x by following the following steps?
Does it make a little sense or do I am totally wrong in doing that?
*Of course I have forgotten that Bland & Altman (1996) have underlined the impossibility to back transform the standard deviation.
So I tried to compute the sd for back-transformed Y from the back-transformed CI and the critical t value and the square root of the sample size (see the results above). I wanted to find a method in order to check the validity of this method. My assumption was that if the method is valid, the inference made with transformed data would have to lead to the same conclusion as the inference made with the back-transformed data. If so the t-test with the two sets of data should lead to the same results and a standardized version of the effect size estimate (point and CI) should also be similar with the two sets of data. So I performed a t-test for independent samples with the transformed data and another t-test for independent samples with the back transformed data (using the sd computed from the above-mentioned formula). I also computed the Cohen's d and their 95% CI from the two sets of data. The following table presents the results of my analyses.
From that results it seems that the inference made with the back-transformed data and the approximate sd is consistent with those made with the transformed data. Of course the inference made with the back-transformed data is very very slightly liberal compared with the inference made with the transformed data. A more central issue with the formula is that as Alexy said, "inferences about f(x) are not the same thing as inferences about x". However, while keeping this fact in mind, it is easier to interpret the back-transformed data than the transformed data and since the MOE is similar for the two sets of data why we could not use the back-transformed data?
I aknowledge that the obtained back-transformed mean difference is not equivalent to the arithmetic mean difference. So I think that if we decide to use this method to compute the back-transformed mean difference and its CI, we have to remind that fact to the reader.
What do you think about my solution to the problem? Do you see other limits of my solution? Do you think that I underestimate the limits that I mentioned previously?
Thanks again for your time and for your answers.
Bland, J. M., & Altman, D. G. (1996). Statistics notes: Transformations, means, and confidence intervals. BMJ, 312(7032), 1079–1079. doi:10.1136/bmj.312.7038.1079.