Test for interaction effect when contrasting specific timepoints in mixed-model I'm having a mixed model (lmer in R) with 5 repeated measurements (time is categorical), and 6 groups, and several covariates.
I'm mainly interested in seeing whether there is a significant difference in 'trajectory' between groups comparing timepoint 3 and 5. Can I somehow test for the significance of the interaction specifically contrasting these two timepoints?
When I use an LRT (anova) to test for the interaction variable, of course I test for the overall significance. From the model summary, I only get the interaction between two contrasts (e.g. group 1-2 at timepoint 3-5).
When I use the emmeans package and contrasts, i.e.
PCSmeans <-emmeans(lmerFitPCS, ~ time*group6, adjust = "holm")
pairs(PCSmeans, simple = "each", adjust='Holm')

I only get the simple contrasts for time or group (e.g. comparing group 1 and 2 at timepoint 3, or comparing group 2 at timepoint 3 and 5). but I would like to get the overall significance of the differences in groups between timepoint 3 and 5.
If I use only a subset of the dataset and use the same model, with only data from timepoints 3 and 5, I can use the LRT test to test the significance I want. However, my model diagnostics (residual plots etc) when I run the model with the subset don't look very good, while the overall model (including all 5 timepoints) does seem to meet the assumptions.
Edit, to make question clear based on @EdM 's comment:
I'm looking for a test whether any of the groups differ between those two time points.
What can I do here?
 A: If for some reason you can't get the joint_tests() function in emmeans to do what you want, you can do this with a Wald test on multiple parameters. That's based on the variance-covariance matrix of the coefficient estimates (obtained via the vcov() function on the model object) and a matrix describing the combination of Q hypotheses and P parameters that you want to examine together.
In this case you would be doing that on a set of 6 differences, the difference between time_5 and time_3 for each group (for Q = 6 rows of the hypothesis matrix described in the above link). The null hypothesis would be that all of those 6 within-group time_5 - time_3 differences equal 0.
That gives "an overall test testing whether there are any differences between groups across timepoint 3 vs 5." The matrix representing your set of hypotheses has 6 rows (1 hypothesis for each group) and 30+ columns (the number of coefficients for a model with interactions between two categorical predictors, one with 5 levels and one with 6, plus extra coefficients for covariates).
Each row of that matrix then contains a representation of the linear combination of parameter values that you wish to test. For example, consider group $j$ at time 5. If the model used standard treatment coding for the categorical predictors, then the outcome estimate $y_{jt_5}$ is:*
$$y_{jt_5} = \beta_0 + \beta_j + \beta_{t_5} + \beta_{jt_5},$$
where $\beta_0$ is the intercept (for reference group and reference time), $\beta_j$ is the difference for group $j$ at the reference time, $\beta_{t_5}$ is the difference for the reference group at time $t_5$, and $\beta_{jt_5}$ is the extra difference represented by the interaction coefficient.
For group $j$ at time 3 you have:
$$y_{jt_3} = \beta_0 + \beta_j + \beta_{t_3} + \beta_{jt_3},$$
so the difference you seek to evaluate is:
$$y_{jt_5}- y_{jt_3}= (\beta_{t_5}-\beta_{t_3}) + (\beta_{jt_5}-\beta_{jt_3}).$$
So in each of the 6 rows there would be a +1 in the column for $\beta_{t_5}$ and a -1 in the column for $\beta_{t_3}$. For group $j$ (except for the reference group) there would be an additional +1 in the column for $\beta_{jt_5}$ and a -1 for $\beta_{jt_3}$. All other entries in the Q x P hypothesis matrix would be 0.
The wald.test() function in the R aod package might be adapted to this task.

*This formula assumes that other covariate values are 0. Provided that there are no interactions of covariates with group or time, then any extra contributions to outcome from non-zero covariate values would cancel when you take the difference between time_5 and time_3.
