# Inverse Hyperbolic Sine Transformation (IHS) for dependent variable - How to back transform predictions?

I am doing IHS transformation for the dependent variable (count data, mostly 0 and small counts) while training a non-parametric tree-based machine learning model. I've seen posts saying it will induce bias if the predictions are back transformed using

For example, this post.

Instead of transforming using the above formula, the author proposed to multiply the formula by exp(s^2 /2) to adjust for biases.

I want to back transform my predictions to the original scale and calculate the RMSE and then compare the performance of models with and without transformation. Is it a feasible comparison? In addition, the back transformed predictions will then be used further as an input to another model. There are also several questions which I cannot answer.

Firstly, I do understand that back transforming without bias adjustment will bring some problems. The same thing applies to log transformation as well. The only thing I don't understand is why kagglers only use exp(pred) to transform the predictions to the normal scale? Will that not induce bias?

Secondly, I am not sure if I have understood the above post correctly. My understanding is that s is the standard deviation of a list of predictions on the IHS scale before back transformation. Please correct me if I am wrong.

Thirdly, does this formula also apply to prediction results from a non-linear regression model?

• You should explicitly credit the author you're quoting (as well as linking as you do now). Commented Aug 21, 2019 at 23:24
• It's interesting that the analysis ignores the effect of the bias in the estimate of the mean due to estimating $\sigma^2$ by $s^2$. If the corresponding situation in the lognormal is anything to go by (as we might expect because of their close connection), that's going to be a much more substantial effect. Commented Aug 21, 2019 at 23:30

## 1 Answer

Your proposed comparison seems like a solid one.

My guess on the first question is that they repress all memories of Jensen's inequality from traumatic statistics classes. More seriously, even advanced people do all kinds of silly things with statistics until they learn better.

$$s^2$$ is the sum of the squared residuals on the transformed scale, divided by (n - k): $$\frac{\sum_i^N \left( \hat \sinh^{-1}(y_i)- \sinh^{-1}(y_i) \right)^2}{n-k}.$$

This formula only applies to linear models.

Also note that this retransformation should improve the mean prediction, but does not ensure that predictions for individual cases are particularly good. You might also try some sort of GLM (like a het-robust poison) that avoids all these transformations.