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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.

Best regards, Selma

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

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Answer to the following:

In your post, please clarify the difference between "Time" and "Visit". By "visit/treatment" factor , are you actually referring to the drinktime interaction? – AOGSTA*

Time is from 0 minutes to 360 minutes (measurement taken almost every half hour), while visit is the time the participants visit the facility to take the measurements ( the first visit is baseline measure (before they get any treatment) and the second visit is after 6 weeks treatment - this baseline measure shall not be confused by baseline measurement of hydrogen, which is the first measurement taken upon fasting).

So no I am not refering to the drink*time interaction, drink is the treatment and time is (0 - 360 minutes [as factors]) and visit is 'before / after' treatment. The response is dependent on the time.

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  • $\begingroup$ Thanks for the clarification. Could you please update your original question with this remark instead of posting an "answer"? Then delete this answer. If my answer above is sufficient, mark the question as answered so it doesn't keep showing up in the "unanswered" section. Thanks! $\endgroup$ – AOGSTA Nov 19 '17 at 19:54
  • $\begingroup$ I am still new on this page, so I'm not sure how it works. I understand your first answer, thank you for that. However, I am not sure how to deal with my results and if there is another way. Since 'visit' is not significant, it means that there was no difference in the response after 6 weeks intervention on the exhaled hydrogen. I am still not sure how to deal with the remaining results when doing post hoc since 'visit' is not significant anymore, since I initially had to compare differences before and after treatment $\endgroup$ – Selma R Nov 19 '17 at 20:12
  • $\begingroup$ I would not even take the log transform. Linear mixed models are robust to departures from the normal assumption in that they still yield unbiased point estimates (and asymptotically consistent standard errors) even if the errors or the random effects aren't actually normally distributed. See Gelman & Hill 2006 book on multilevel models. You might have some efficiency loss in small samples (idk what your N is), but the cost of using a log transform is higher (unclear hypothesis testing, difficult interpretation) in my opinion. $\endgroup$ – AOGSTA Nov 19 '17 at 20:36

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