Hello Philipp,

many thanks for the prompt and detailed answer!

Unfortunately, my response var. shows negative values in some observations (in this case, meaning a raise of surface, after machine passed. What is reasonable due to some effects.)

I now checked the changed output: when response is (y+v), obviously the intercept increases by v, but all remaining estimates stay the same.

No here's the problem: If I wanted to transform my response (with negative values) by log() or sqrt(), it certainly doesn't work.

If sqrt(y+v) is used as response var., coefficients of the linear model change. Although it shows similar patterns as the one with sqrt(y).

I will definitely keep glm with Gamma link in mind, and will try to apply and check it for this purpose.

Cheers, Marian

Hello Marian,

Using an additive constant (e.g., X' = X + MIN(X) + 1) does not change: (a) the shape of a distribution; (b) the variance of a distribution; or (c) relationship of scores on the variable with those of any other variable (the latter was noted in your query). As well, the mean change equals whatever constant you apply.

However, non-linear transforms (e.g., log, square root) will change the shape of a distribution, measures of location and/or dispersion, and will change relationships with other variables. That is why your model coefficients can and do change after transformation. The question is, does the transformed set of scores better help you to address your research question? If so, use them. If not, think about another method.

The ideal method would be a complete permutation/exact test. However, relatively few statistical software packages offer these except for very simplistic models. One option that you might consider (and which is increasingly available in statistical software) is using bootstrapped estimates for your model (and model error); doing so reduces the requirement for specific distributional form for model residuals.

Good luck with your work.

Hello David,

thanks a lot for this answer, completely solved my issue. I really like the suggestion "If so, use them."

Kind regards, Marian

shifting the data variable is one possible solution since the data variance remains unchanged

Marian -

I don't know, but it did not sound like you were talking about a GLM application in the question. If not, why did you need to do log transformations? It sounds like maybe this was a linear regression and you took logs on both sides, trying to reduce heteroscedasticity. Is that right? Well, if you are looking at a regression, and a graphical residual analysis does not tell you that you need to take a log of anything, and interpretation would be better without logs, I would not suggest using any transformations if you don't need to do so.

In many cases for linear regression we have skewed data, and no transformations are needed.

(A poison regression for count data would be a different matter.)

Consider the first, second, and last examples in here: https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity

And examples in here:

https://www.researchgate.net/publication/319914742_Quasi-Cutoff_Sampling_and_the_Classical_Ratio_Estimator_-_Application_to_Establishment_Surveys_for_Official_Statistics_at_the_US_Energy_Information_Administration_-_Historical_Development, using prediction.

Were you concerned about 'normality'? That is perhaps a minor concern for estimated residuals, but not for the unconditionally distributed data. (In an ANOVA the data are conditional by category, so they are like the residuals - that's different.)

If you were always talking about GLM, then I didn't understand the objective/application. Sorry.

Cheers - Jim

You haven't said why you want to transform a variable or variables. Is this just because you've chosen a model that makes these assumptions? Have you considered a different model or different way to do the estimation?

Also, you haven't said things like your sample size. Also part 2, testing for distribution assumptions is problematic. I'll let others expand on these.

Hi James R Knaub and Daniel Wright ,

thank you for the responses. I used a lm with generalized least square and included weights (weights = varIdent(group), in nlme::gls, R) to account for heteroscedasticity.

But, the residuals of this model are not normally distributed.

I assume, that this is a major assumption, also for gls. That's why I wanted to transform the response variable in the first hand. Considering boxcox, different ways for a transformation are recommended, e.g. reciprocal values, log(x), which do not work when negative values are included, o. c.

Best wishes, Marian

I may not be following. Are you saying you are using GLS, and you want normality for the estimated residuals? That is not really very strictly important. And the unconditional data can be of any distributions.

I suggest you research "graphical residual analysis" and "cross-validation." Just randomly removing some y-data, and re-estimating regression coefficients, and then seeing how well you predict for those cases - maybe plotting y and predicted y for those cases - might tell you all you need to know about the suitability of your model. Checking this on a new sample could help.

Cheers.