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I have the following situation:

I have some dataset in the form of samples $y_i, x_{i,j}$. I'm doing a GAM regression after a power transform (Yeo–Johnson - similar to Box-Cox). So I first learn the $\lambda$ for the transformation with MLE, then transform the $y$ values, then learn the GAM, then make the predictions and then apply the inverse power transform. So far so good. But I have doubts that it's methodologically valid to do the inverse power transformation on the confidence intervals I obtain form my predictions. In fact, in my case doing that makes them apparently much wider than they should be. What's the correct way to do this?

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  • $\begingroup$ What parameter(s) do you need confidence intervals for? $\endgroup$
    – whuber ♦
    Commented Dec 16, 2021 at 20:03
  • $\begingroup$ I mean the confidence of the predictions. So that I can say e.g.: $P(y \in [\hat{y}-c,\hat{y}+c]) \ge 0.8$ $\endgroup$
    – user344577
    Commented Dec 17, 2021 at 10:42
  • $\begingroup$ Let $f$ be any monotonic increasing transformation, such as the one you have applied. Because the event $y\in[\hat y-c,\hat y+c]$ is the same as the event $f(y)\in[f(\hat y-c),f(\hat y+c)],$ you can transform confidence intervals after the fact. $\endgroup$
    – whuber ♦
    Commented Dec 17, 2021 at 15:03
  • $\begingroup$ That doesn't seem to produce correct results in my case. (I may be doing some other mistake perhaps?) I can see how that should work for a linear transform, but could you please provide a (link to) proof why monotonicity is enough? $\endgroup$
    – user344577
    Commented Dec 20, 2021 at 7:44
  • $\begingroup$ My previous comment included the proof: "is the same as" is a direct consequence of the definition of monotonic increasing. $\endgroup$
    – whuber ♦
    Commented Dec 20, 2021 at 14:21

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First, some background from Wikipedia, to quote:

a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution.[1][2] Equivalently, if Y has a normal distribution, then the exponential function of Y, X = exp(Y), has a log-normal distribution.

And further:

Since the log-transformed variable ${Y=\ln X }$ has a normal distribution, and quantiles are preserved under monotonic transformations,...

Actual my experience with Box-Cox is that for generated normal random deviates to which one raises to a power, the Box-Cox correctly identified a log-transformation.

However, while the starting normal random deviates nicely produced a confidence interval centered close to the true normal population mean, applying the transform Exp() produces a value that corresponds now to a value reflective of median of the lognormal population. The latter is in accord with the fact that the median for a lognormal equates to the ${Exp(\mu)}$ where ${\mu}$ relates to the mean of the underlying normal.

So, transformation entail perhaps more of an interpretation issue as to what the center of the interval actually pertains to with respect to your original data set, albeit, the percentage coverage band is likely accurate.

Here is a prior answer that you may find more informative.

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