# Transforming regression coefficients back to original values from square-root -transformed data

I've ran linear regression with square-root transformed dependent variable. Due to negative skew of the dependent variable, the formula for data transformation prior analysis was as follows:

NewVar = SQRT(147 - OldVar).

Now I have the transformed regression data. I wish to know what kind of equation I need to do in order to transform my regression coefficients back to original values (Standard error, unstandardized and standardized coefficients).

• The coefficients and their associated results apply on the scale on which they are estimated. The only transformation that makes sense is to transform predictions of the response back to the original scale by squaring and subtracting. That said, transforming just because you have a skewed response is a dubious thing to do, and a transformation that looks very ad hoc is dubious to a further extent. – Nick Cox Feb 15 '17 at 18:53
• If you can tell us (much) more about your original data, you may well get a (much) more positive suggestion. How many predictors do you have? Can you show the data? Can you show a scatter plot if just one predictor? – Nick Cox Feb 15 '17 at 18:55
• Thank you for your response! I transformed the data to make it better fit for the assumptions of linear regression (residual normality, homoscedasticity and linearity). I'm performing the analysis with and without covariates. All the covariates are continuous. My independent variable is binomial. Unfortunately I can't show the data. – Frans Suokas Feb 15 '17 at 19:07
• You are mentioning first the least important assumption of regression (residual normality) and mentioning last the most important assumption (linearity). At an absolute minimum you could tell us the smallest and largest values of your response. Then we can plot $\sqrt{147 - \text{ response}}$ over that range and suggest better alternatives. – Nick Cox Feb 15 '17 at 19:11
• Alright! The maximum value is 146 and the lowest is 40. – Frans Suokas Feb 15 '17 at 19:17

The coefficients and their associated results apply on the scale on which they are estimated. The only transformation that makes sense is to transform predictions of the response back to the original scale by squaring and subtracting. That said, transforming just because you have a skewed response is a dubious thing to do, and a transformation that looks very ad hoc is dubious to a further extent.

Comments reveal that the range of the response is from 40 to 146. Although the function won't seem challenging to those who remember their high school mathematics, it is always a good idea to plot the function to see exactly what it does:

The transformation is problematic in various ways, in increasing worry order:

1. It reverses the scale from high to low. This in itself is not a major issue, and the main consequence is to note that coefficients of predictors accordingly have reversed sign. But in general it is best avoided unless essential on other grounds.

2. It is approximately linear over most of its range, so does little to change anything over that range.

3. But conversely it necessarily behaves differently over the range from about 130 to 146. Does that correspond to any physical (biological, economic, whatever) knowledge of how the response will behave over that range?

4. It is not easily transferable to other similar datasets. Unless it is known that values above 147 are always impossible, it is impossible to compare easily results using this transformation with those for other data for which the upper values might differ.

So, contradictory though it may seem, I doubt that this transform is strong enough to change how your data behaves very much except that it is likely to work on the highest values in ways that may not make substantive sense.

Also, ad hoc transformations are a bad idea. Almost always, we should want to analyse data in ways that would always make perfect sense for other similar bodies of data.

Further comment would depend on learning more. The precise consequences in terms of increased or decreased skewness or linearity will depend on the exact distribution between the extremes and on the values of the predictors.

• Thank you for this clarification. I ended up in a different procedure, since data transformation to other than predictions is flawed. – Frans Suokas Feb 16 '17 at 20:30
• @Frans Suokas Good that you solved your problem, but your using a "different procedure" is not in itself informative to people who read this thread in the future and wonder what to do. It's open to you for example to post an answer yourself explaining what you did. – Nick Cox Feb 16 '17 at 20:47