# How do I report and interpret the output from linear mixed models with interaction terms?

So, we are doing a linear mixed effects model for analyzing some results of our study. In short, we have performed two different meal tests (i.e., two groups), and measured the response in various biomarkers at baseline as well as 1, 2, 3, and 4 hours after the meal.

I had a meeting with a statistician who explained that we should use linear mixed models for this and as such, using the nlme package in R the syntax looks like this:

model <-lme(biomarker~ as.factor(group)*visit, random = ~1|ID, data=data, method="ML")
summary(model)


Linear mixed-effects model fit by maximum likelihood
Data: data
AIC      BIC    logLik
137.593 149.0651 -62.79649

Random effects:
Formula: ~1 | ID
(Intercept)  Residual
StdDev:    1.462879 0.6039689

Fixed effects: biomarker ~ as.factor(group) * visit
Value Std.Error DF   t-value p-value
(Intercept)              7.869766 0.7157143 38 10.995681  0.0000
as.factor(group)3        1.295118 1.0121729  8  1.279542  0.2366
visit                   -0.096024 0.0679003 38 -1.414191  0.1654
as.factor(group)3:visit -0.358905 0.0960255 38 -3.737606  0.0006
Correlation:
(Intr) as.()3 visit
as.factor(group)3       -0.707
visit                   -0.247  0.174
as.factor(group)3:visit  0.174 -0.247 -0.707

Standardized Within-Group Residuals:
Min           Q1          Med           Q3          Max
-3.107751422 -0.303320567  0.004573801  0.377750437  1.967646127

Number of Observations: 50
Number of Groups: 10


To summarize:

• Exposure = one of two meal tests (group in the syntax)
• Outcome = Biomarker
• Time variable = Visit (5 in total for each participant, continuous)

My questions are:

1. Am I correct in interpreting this that there is an interaction between group and visit?

2. I am unclear as to how I should interpret the estimates here. Am I correct in saying that at time = 0, then the group difference (3 vs. 2) is 1.29? And further that this effect depends on the visit? What about the other timepoints?

3. Is it sufficient to report this model or should we also include a model without the interaction term that is just including group and visit?

• You need to repeat this without as.factor(group)3 in the model. It's contribution is not significant to the model: $p=0.2366$ is not significant. You may have to eliminate visit as well, but first check what the model shows without as.factor(group)3. – Carl Mar 8 '17 at 2:54
• @Carl can you elaborate on why would one need to refit the model without non-significant predictors? That is not what I would do. – juod Mar 8 '17 at 9:10
• No, you should not remove as.factor(group)3. It is part of an interaction. And you shouldn't remove variables just based on significance, either. – Peter Flom Mar 8 '17 at 11:40
• @Carl The possible overdetermination (we can't really tell if it's overdetrmined without more information) is a separate issue, but an important one. But if you have a large interaction, then you need both main effects. I am not sure why one variable is using 38 df. – Peter Flom Mar 8 '17 at 22:16
• The linear mixed effects model is not suitable if the objective is what I perceived. – Subhash C. Davar Mar 10 '17 at 13:51

1) Am I correct in interpreting this that there is an interaction between group and visit?

Yes.

2) I am unclear as to how I should interpret the estimates here. Am I correct in saying that at time = 0, then the group difference (3 vs. 2) is 1.29? And further that this effect depends on the visit? What about the other timepoints?

Yes, that is the group difference at time = 0 and yes, it depends on time. You can calculate the estimate at any combination of the variables by using the formula:

$7.87 + 1.30*(I(\text{group} = 3)) - 0.10*\text{visit} - 0.36*(I(\text{group} = 3))*\text{visit}$

3) Is it sufficient to report this model or should we also include a model without the interaction term that is just including group and visit?

That depends on what you are interested in, but when there is an interaction, the model with only main effects can be misleading.

• Thank you for your thorough response. Regarding your last point, I regularly see studies reporting the p-values and/or coefficients for the entire model with the interaction term. However, does a model like this really test for the differences between groups and does it make sense to report those estimates/p-values in a model with an interaction term? Finally, a means around this would maybe be to do post-hoc tests for the outcomes where the interaction is significant. – Tho Mar 9 '17 at 9:12
• There are differences between groups that differ over time. – Peter Flom Mar 9 '17 at 11:47
• After some reading, I think I understand. This has been the most relevant answer so far, and I'll mark it as such. Thank you very much, Peter. – Tho Mar 10 '17 at 8:46