# Linear mixed model - change in results of fixed effects log transformed

I hope you can help me with this question. I will try to explain my data:

I have a repeated measure design (with 16 measures at 'visit' 1 and 'visit' 2 for two groups: treatment and placebo). I am using linear mixed model before doing post-hoc testing(multiple comparisons). I have following variables:

Exhaled hydrogen             (response variable)
drink [treatment]            (factor, fixed effect)
Time                         (factor, fixed effect)
visit                        (factor, fixed effect)
Exhaled hydrogen at baseline (covariate, fixed effect)


The data are not normally distributed and therefore I did a log transfer.

My model looks like this:

hydrogen ~ drink*time+visit-1 + hydrogen_baseline, random=~1|participant, method="ML", data=breath, na.action = na.exclude)

My problem is that after the log transformation, the factor 'visit/treatment' is not significant anymore. Since I want to compare the effects after treatment, I am not sure how to deal with this problem.

• In your post, please clarify the difference between "Time" and "Visit". By "visit/treatment" factor , are you actually referring to the drink*time interaction? – AOGSTA Nov 19 '17 at 17:17

I don't think you should expect the log-transform to preserve significance results. This is because after you transform the data, you're testing a different null hypothesis.

Consider the un-transformed model with single binary treatment indicator $trt$, a Gaussian error $\epsilon_{ij} \sim N(0, \phi)$, and a subject-level random effect $\alpha_i \sim N(0, \psi)$. A mixed model assumes $\alpha \perp \epsilon$.

$Y_{ij} = \beta_0 + \beta_1trt_{i} + \alpha_i + \epsilon_{ij}$

In this model $Y_{ij}$ is Gaussian for both treatment groups - for treated, the mean is $\beta_0 + \beta_1$ and for untreated, the mean is $\beta_0$. The difference is $\beta_1$. Roughly speaking, a p-value on $\beta_1$ tests whether the difference in the two group means is different from 0.

Now if you think $Y_{ij}$ is not Gaussian, but that it is Gaussian after a log transformation, then you are saying that $Y$ actually has a log-normal distribution. But the mean of a lognormal distribution for the treated group is not $\beta_0 + \beta_1$. It is $exp(\beta_0 + \beta_1 + \frac{\phi + \psi}{2})$. For the untreated group, the mean is $exp(\beta_0 + \frac{\phi + \psi}{2})$. Check the wikipedia page for the log-normal distribution.

Testing

$H_0: (\beta_1 + \beta_0) - \beta_0 = 0$

is not the same as testing

$H_0: exp(\beta_0 + \beta_1 + \frac{\phi + \psi}{2}) - exp(\beta_0 + \frac{\phi + \psi}{2}) = 0$

This is the main result of the following paper: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4120293/