When is performing back-transformation of inferences on transformaed variables Ok, and when is it not Ok?

by Alexis   Last Updated October 09, 2019 21:19 PM

Caveat: This question may be a tad rambly, and I welcome comments with specific directions for me to improve it.

During a too brief exchange with the worthy @NickCox I got to thinking about transformation/back transformation and inference.

It seems to me pretty apparent that frequentist inference—confidence intervals, hypothesis tests—on a transformed variable $$f(x)$$, is not inference about the untransformed variable $$x$$, even when back-transforming inferential quantities, because, generally, $$\sigma^{2}(f(x)) \ne f(\sigma^{2}(x))$$, unless $$f(x) = x$$, and both CIs and hypothesis tests rely upon an estimate of the variance. To quote from my answer here:

Basing CIs on transformed variables + back-transformation produces intervals without the nominal coverage probabilities, so back-transformed confidence about an estimate based on $$f(x)$$ is not confidence on an estimate based on $$x$$.

Likewise, inferences about untransformed variables based on hypothesis tests on transformed variables means that any of the following can be true, for example, when making inferences about $$x$$ based on some grouping variable $$y$$:

1. $$x$$ differs significantly across $$y$$, but $$f(x)$$ does not differ significantly across $$y$$.

2. $$x$$ differs significantly across $$y$$, and $$f(x)$$ differs significantly across $$y$$.

3. $$x$$ does not differ significantly across $$y$$, and $$f(x)$$ does not differ significantly across $$y$$.

4. $$x$$ does not differ significantly across $$y$$, but $$f(x)$$ differs significantly across $$y$$.

It is also very easy to imagine examples setting this point down sharply. For example, if $$y_{i} = x_{i}$$ has Pearson's $$r=1.0$$ for $$y$$ and $$x$$, but Pearson 's $$r=0.0$$ for $$y$$ and $$x^{2}$$ if the range of $$x$$ is symmetric about 0.

On the other hand, tricks like Oehlert's Delta method can provide a 'back-transformation' that approximates the correct variance of $$x$$ as an alternative to simply calculating it directly, or calculating it as $$f^{-1}(\sigma^{2}(x))$$.

Good Nick Cox however, points out that to "estimate on a link scale and report on the original scale is central to generalized linear models," and that (if I understood correctly) inference on the geometric mean entails such back-transformation in the form $$exp\left(\frac{\sum \log (x)}{n}\right)$$.

When is it Ok to base inferences about $$\boldsymbol{x}$$ on back-transformations of estimates and inferences on $$\boldsymbol{f(x)}$$, and when is it not?

Second caveat: I am not calling Nick Cox out to defend any position with this question, and am genuinely interested in understanding when performing inference on $$\boldsymbol{f(x)}$$ but drawing conclusions about $$\boldsymbol{x}$$ based on back-transformation makes sense and does not make sense.

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