# Back Transformation of Predicted Y from Log 10 Transformed Model Data

I am using a General Linear Model for Analysis in Minitab.

I have two questions.

I had two responses variables which I transformed using Log10 as there was evidence from the residual plots of some issue with normality and with order and the transformation seemed to improve that quite well. (I got some advice here on that before, thanks very much)

From all of that I then ran my analysis for my Log10 transformed responses to get some details on significance and crossed effects etc of the various factors. I also used the regression equation to predict Y values. Now these predicted Y values can of course be easily compared to the transformed Y values. However to this part I have questions.

1. What are the issues with me conducting a back transformation (antilog in Minitab) of the predicted Y values, i.e., the predicted means for the purpose of comparing them to the original untransformed means as a way of commenting on the value of the model for the various relevant factors?
2. Is this legitimate as I have read somewhere that there are issues with comparing the original arithmetic means with what I believe are so called geometric means, is there a straightforward work around?.
3. Is there a book or paper someone can refer to me on this?.
4. What can I do about back transforming the Std Errors if I want to report those?
• Four questions is not two questions. Four may be too many.
– Carl
Commented Apr 11 at 17:51
• Search here for Duan smearing transformation. It's for base $e$ logs, but easy enough to modify. I would avoid this and use a het-robust Poisson. Commented Apr 11 at 18:11

When you perform standard linear regression, you are estimating the mean value of the outcome as a function of predictor values. Thus, when you do a log transformation on the outcome values before a linear regression, you are estimating the mean value of the log of the outcome as a function of predictor values. As you note in point 2, that's the geometric mean when back-transformed to the original scale. For point 3, the Wikipedia page explains how the mean of a set of log-transformed values is related to the geometric mean.

You also can work in the original scale in this type of situation with a generalized linear model with a log link function. That can be preferable, as it directly models the mean values in the original scale. See this page for an explanation of the difference between modeling log outcomes in linear regression and using a log link in a generalized linear model.

With regard to point 1, there is no simple way to estimate the mean in the original scale from the geometric mean that you get by doing linear regression on log-transformed outcomes. See this page for the issues involved. You have to apply your understanding of the subject matter to decide whether to work in and report in the original scale or the log scale.

Sometimes the log scale is better. For example, pH values are base-10 logarithms of hydrogen-ion activity. In quantitative polymerase chain reaction analyses, the "Cq" values are related to base-2 logarithms of the nucleic acid being analyzed, and they often are reported on that logarithmic scale instead of back-transforming to some absolute scale.

For point 4, you don't back-transform standard errors themselves. If you want error estimates in the original scale, you might get 95% confidence limits in the log scale and then back-transform those confidence limits. It's good to get comfortable with that process, as it's often done in many types of regressions involving logarithms, for example for confidence limits of hazard ratios in Cox survival regressions that directly model the log hazard. Remember, however, that those back-transformed confidence limits will not be symmetric about the point estimate and, in your case, they will be confidence limits for the geometric mean in the original scale.

• Thanks EdM. So exploring the extent of the possibility to keep this simple. Can I generate an acceptable Geometrical mean for the original untransformed data set, say multiplying each of the original responses in a set of responses and taking the appropriate root (based on number of them) for any given factor combination. Then as the general linear model I have created will predict a Y response which is a geometric mean would it be reasonable to compare this predicted geometric mean to the estimated geometric mean for the original dataset?. Thanks
– GTK
Commented Apr 12 at 9:40
• @GTK if you want to compare observed against predicted values it's simpler just to work in the log scale. Then you get case-by-case comparisons. If all of your predictors are factors then you could do what you suggest, but with a large number of cases you might end up against numerical limits in computing the product. I find it simpler, and probably more numerically stable, to compute the geometric mean as the equivalent "exponential of the arithmetic mean of logarithms" instead, as the Wikipedia link in the question explains.
– EdM
Commented Apr 12 at 13:44
• Thank EdM. I looked at that Wikipedia page and I was kind of put off by all the notations. I’m not so hot on those and as there was no worked example I was not so confident I would nail that correctly. Is there a text book explainer you might be familiar with. Sorry for troubling you. I also found that if I compare predictions generated by the model using the transformed Log10 data vs mean of Log10data that the values are exactly the same. For this comparison would I get mean of observed first and then get log10 of that mean to compare to predicted. Thanks
– GTK
Commented Apr 12 at 16:51
• @GTK this web page shows how to calculate a geometric mean either by taking the root of the product or by the logarithmic method. That page doesn't seem to have any practice problems, but you can practice and check your work on any of many web pages about geometric means.
– EdM
Commented Apr 13 at 18:00

Questions 1 and 2. It is not clear exactly what was log-transformed. x and y or only y? If x and y then a linear result in the transformed coordinate system is a power function in the untransformed coordinates. Power functions would not rely on mean values. You should check if you need offsets or not, that is, a power function would go through the origin [0,0] so check to see if that is plausible and if it isn't then subtract a constant(s) from the x and/or y original data to make it so.

Question 3. I don't know, that's not how I learn. I learn by doing and checking algorithms and literature when I have done my first attempted solution.

Question 4. The standard errors become asymmetric when back transformed. Compute the principal value plus the standard error and then the principal value minus the standard error in transformed coordinates, back transform both and express as transformed principal value (e.g., geometric mean) +some value and -some other value for deviations.

• Thanks Very Much Carl.
– GTK
Commented Apr 12 at 9:40
• OK, (+1) on your question. You need 15 points before you can upvote answers, which you do by left clicking on the up pointing triangle inside the circle. I have given you 10 points by upvoting your question, that gives you 21 points so you can now upvote. You can also accept answers. To do that you left click the check mark next to an answer. That gives the answer you choose (only one) fifteen points. That is how we say "thank you" on this site.
– Carl
Commented Apr 13 at 8:52