Estimated marginal means versus model-derived interaction effects

Question

I'm getting confused with the results of experimental data with two groups and three measurements. Imagine a mixed-model in lme4, where I am interested in the effect of the interaction group*measurement. My model is a random-intercept model comparable to model1 <- lme(outcome ~ group + measurement + group:measurement + baseline_outcome + (1|ID).

My research question is: Is there any interaction between group and measurement time?

Now my model shows that there indeed is an interaction between group and measurement1, but not for group and measurement2 and group and measurement3.

If I use the emmeans package (emmeans(model1, pairwise ~ Group|measurement, infer=TRUE)), I get the exactly same results for the coefficients as I do with the multiplicative interaction of my initial lmer model. However, now there is no evidence for any group differences at measurement1 (same differences, different SEs, different p-values).

Regarding my research question, which term should I interpret? The multiplicative interaction or rather the results of the estimated marginal means? If the latter, what for do I need the multiplicative interaction coefficients anyway?

EDIT: Here are the formulas and models. Thanks for pointing this out.

lme4 model model1 <- lme(outcome ~ group + measurement + baseline_outcome + group:measurement + (1|ID)

Here are the relevant outputs (not for measurement main effect and baseline-outcome):

group1: Estimate = -3.33, SE = 0.61, df = 428, t-value = -1.99, p-value = 0.043

measurement 2 * group1: Estimate = 0.88, SE = 0.59, df = 140.24, t-value = 1.09, p-value = 0.30

measurement 3 * group1: Estimate = 0.86, SE = 0.58, df = 141.36, t-value = 1.06, p-value 0.33

So the effect of measurement 1:group1 = -3.33, for measurement2:group1 = -3.33+0.88 = -2.45, for measurement3:group1 = -3.33 + 0.86 = -2.47

emmeans model: emmeans (model1, pairwise ~ group|measurement, infer=TRUE, adjust = "none")

contrast	estimate	SE	df	t ratio	p value
group0-group1 (measurement 1)	-3.33	0.61	441	1.93	0.054
group0-group1 (measurement 2)	-2.45	0.64	488	0.94	0.37
group0-group1 (measurement 3)	-2.47	0.64	483	0.97	0.35

Answer 1

Interaction terms in linear models assess if group effects differ across measurements. A significant interaction suggests the group's influence on outcomes varies at different times.

emmeans calculates adjusted average group means for each measurement, considering other factors in the model. This adjustment can lead to different significance levels compared to the interaction term due to refined standard errors and p-values which are adjusted for multiple testing. This would not appear to be the reason in yhour case since you have used adjust=none.

One possibility is that the emmeans function computes standard errors and confidence intervals based on the marginal distribution of the EMMs, which takes into account the uncertainty in the estimation of the fixed effects, as well as the random effects structure of the model. The lmer model, however, tests the interaction terms based on the conditional distribution given the random effects.

For research questions on interactions, consider the multiplicative interaction to determine if an effect exists between group and measurement time. The multiplicative interaction coefficients indicate whether there is something to look at, while the EMMs tell you what the effect looks like.

If there is a significant interaction, further investigation can be achieved with EMMs to see where those differences lie. But if the interaction is not significant, this often suggests that the main effects of group and measurement time may be sufficient to explain the outcome variable. In practice, researchers often report both the interaction test results and the EMMs to provide a complete picture of the effects.