Between-person level interaction

Question

I have a database called dat1. Each participant (id) had multiple measurements. One of the variables measures type of the stress called stress_type (acute=0, chronic =1) and another variable is the stressful level of the stressor (continuous variable 0-3).

I wanted to examine the effect of the stressor type, stressful level and interaction of type*level of stress on depression. I disentangled stress_type and stress level variables to get the within and between person mean, and calculated the within and between interaction as well.

mean_stress_type_rec = between person mean of stress type

mean_stress_level = between person mean of stress level

cent_stress_type_rec = within person mean of stress type

cent_stress_level = within person mean of stress level

I ran the following model:

res_intvsnonint_levels_int_nopleas <- lme(dep ~  mean_stress_type_rec * mean_stress_level +
 cent_stress_type_rec * cent_stress_level, 
 random = ~ cent_stress_type_rec + cent_stress_level _rec + obs_cent1 | id, data=dat1,
 na.action=na.omit,
 control=lmeControl(maxIter=1000, msMaxIter=1000, niterEM=1000, sing.tol=1e-20))

I found a very marginal interaction at the between level:

                    Coef.   SE         95% CI      t-value    p
Fixed effects                   
Intercept           29.28   4.42    20.61 – 37.96   6.62    <0.001
Time               -0.07    0.03    -0.13 – -0.01   -2.43   0.015
Type - between      4.46    6.47    -8.36 – 17.27   0.69    0.492
Type - within       0.41    0.59    -0.74 – 1.56    0.7     0.484
Level - between     2.14    2.6     -3.01 – 7.29    0.82    0.413
Level - within      0.41    0.35    -0.27 – 1.10    1.18    0.237
Type*Level - between-8.21   4.13    -16.40 – -0.03  -1.99   0.048
Type*Level - within  0.6    0.79    -0.96 – 2.16    0.76    0.448

I don't know what then I should do to further examine the between-level interaction. I'd appreciate your help.

Answer 1

In principle*, you can use emmeans emtrends for this. You need to choose a couple of "testing points" for your other continuous predictor. This is easiest if you standardize your predictors and choose for instance points -1, 0, 1 for predictor 1. This way you will be comparing predictor 2 slopes at predictor 1's levels of mean-1 SD, mean, and mean +1 SD, which I find is often reasonable way to test continuous-continuous interaction. So, first standardize your predictors, then run the model again with standardized predictors, then:

library(emmeans)
emtrends(modelname, pairwise ~ levelBetween, var = "levelWithin", at=list(levelBetween=c(-1,0,1)) 

You don't have to standardize though if you can find reasonable testing points from the predictor's raw values.

*However, I wonder about your stress type variable - can it really be understood as continuous? I understand you need to somehow get the mean value from it but I wonder what the resulting variable actually represents because it was originally categorical (?). But in principle you can use the above with a cont x cont interaction.