# Interpretation of linear mixed model with log(x+1)-transformed response variable

Before running a linear mixed model I transformed my response variable with log(x+1) to get closer to a normal distribution of residuals. Doing so I get these results (for a simplified example):

           Estimate      Upper CI Limits    Lower CI Limits   p-value
level1     0.6518415     0.8720254          0.4316577
level2     0.8431060     1.0625152          0.6236968         0.071
level3     0.5089360     0.7258301          0.2920420         0.170
level4     0.3987420     0.6166745          0.1808096         0.017


Am I right that p-values can be interpreted without back-transformation?

Can I back-transform estimates and CI-Limits by exp(estimate)-1 or exp(limit)-1 which results in the following?

           Estimate      Upper CI Limits    Lower CI Limits   p-value
level1     0.9190715     1.3917500          0.5398079
level2     1.3235730     1.8936400          0.8658128         0.071
level3     0.6635203     1.0664460          0.3391593         0.170
level4     0.4899492     0.8527564          0.1981870         0.017

• You can consider the p-values of your log-linear model to assess variable importance without back transforming. However, if you want to make predictions or infer predicted values by interpreting the model coefficient, you must most definitely back-transform your model. You don't need to "touch" your estimated coefficients and associated confidence intervals at all, you just use them on the back-transformed model (which won't be linear anymore). – Digio Jul 24 '17 at 8:51
• Thanks for your comment! How would the back-transformation with log(x+1) work? – HerthaBSC Jul 24 '17 at 8:59
• For prediction, you may use this model where you simply solve for y. For coefficient interpretation, as explained here, you will interpret the exp(.) of your coefficients as such: exp(β0) = effect on the mean of y, when x = 0, and exp(β1) = the average change of Y with every unit increase in x. NB: If your independent variables are categorical, you will interpret them as dummy variables where instead of unit increase you have a change in factor level. – Digio Jul 24 '17 at 9:15
• Instead of transforming the data to fit the model, why not use a model that doesn't make that assumption? I suggest using robust regression or quantile regression. – Peter Flom - Reinstate Monica Jul 24 '17 at 11:56
• You can't do that. Digio meant you could transform your prediction from the log scale back to the raw scale. But you can't just convert the coefficients like you do. – SmallChess Jul 24 '17 at 13:41

Short answer: Back-transformed coeffient estimator is biased, and not a good estimator. Back-transformed confidence interval is valid, but sub-optimal.

Longer answer: Since you have not included any residual plots in your post, it is unclear whether your transformed model actually fits the data well. (This is not something you can tell from the coefficient estimates table.) The OLS coefficient estimators in the Gaussian linear regression model are MLEs, so if you take the corresponding back-transformed estimators these will be MLEs of the corresponding back-transformed parameters (by the invariance properties of MLEs). However, you should bear in mind that back-transformed estimators using a non-linear transform will be biased, which is why many statisticians counsel against their use (see e.g., this related question). In the case of an exponential back-transformation you will get an estimator that is positively biased, meaning that it will tend to overestimate the true back-transformed parameter on average. The back-transformed coefficient estimator as not a particularly good point estimator, for this reason.

You can also back-transform the limits of the confidence intervals, and these remain valid interval estimators, with the same confidence level as in the original model. However, by using a non-linear back-transformation you will end up with a confidence interval that is a little wider than it needs to be (since the equal-tail interval used for the linear model is no longer the shortest interval with that confidence level). It is possible to get a shorter interval at the same confidence level in the space of the back-transformed parameter, but this requires a bit more work, and it requires more knowledge of the underlying properties of the pivotal quantity used to form the interval estimator.

You shouldn't do that. While it's fine to transform the prediction, your coefficients stay in your non-linear transformed scale.

If you want to make a model in the original scale, you might select another model that doesn't assume normality.

Back-transformation of regression coefficients

• Thanks! I’m relatively new to new to linear models. So I’m not completely familiar with terminology yet: Aren’t the „estimate“-values I gave above predictions? They are calculated from these values: (Intercept): 0.6518 level2: 0.1913 level3: -0.1429 level4: -0.2531 – HerthaBSC Jul 24 '17 at 14:28
• They are model parameters, not predictions. You predict something only when you give some predictor values. – SmallChess Jul 24 '17 at 14:49
• Ok, so I’d like to plot the effect of the different fixed effect levels on my response variable. Am I right that I should do this without any back-transformation meaning plotting log(response) +1 against factor levels? – HerthaBSC Jul 24 '17 at 15:00