# Interpreting a three-way interaction in a multilevel growth model

Background

I have the following multilevel (mixed effects) growth model:

$$y_{it} =\beta_{0} + \beta_{1}Time_{it} + \beta_{2}Time^2_{it} + \beta_{3}GHP_{i} + \beta_4Baseline_{i} + \beta_5(Time_{it}*GHP_{i}) + \beta_6(Time^2_{it}*GHP_{i}) + u_{0i} + u_{1i}Time_{it} + u_{2i}Time^2_{it} + e_{it}$$

where $y$ reflects depression scores, $i$ reflects subject, $t$ reflects linear time (i.e. $t_1 = 1$, $t_2 = 2$ ... $t_8 = 8$), $Time^2$ reflects quadratic time (i.e. $t_1 = 1$, $t_2 = 4$ ... $t_8 = 64$), $\beta_{1i}Time_{it}$ and $\beta_{2i}Time^2_{it}$ reflect the the linear and quadratic effects of time, respectively, $\beta_3(GHP_{i})$ is the fixed-effect of general health problems, $\beta_4Baseline_{i}$ is the fixed-effect of baseline depression scores, $\beta_5(Time_{it}*GHP_{i})$ and $\beta_6(Time^2_{it}*GHP_{i})$ reflect the cross-level interactions between general health problems and linear time, and general health problems and quadratic time, respectively, and $u_{0i}$, $u_{1i}Time_{it}$, and $u_{2i}Time^2_{it}$ are random intercept, random slope for linear time, and random slope for quadratic time, respectively.

Question

I am looking for some guidance on how to interpret the cross-level interactions, particularly the $\beta_6(Time^2_{it}*GHP_{i})$ interaction (which, I know, is technically a three-level interaction, i.e. time x time x general health problems).

Assisting Information

First, it is important to know what the direction of the fixed effects of linear time, quadratic time, and general health problems were on the outcome (all eefects were significant at $a$ = .05).

a) $\beta_1Time_{it}$ = -2.74, hence depression scores declined linearly over time.

b) $\beta_2Time^2_{it}$ = .27, hence depression scores show a (weak) convex curve, or the rate at which they decline decreases over time. This looks something like: c) $\beta_3GHQ_{i}$ = -1.35, hence higher general health problem scores were associated with lower baseline depression values.

Now that we've set the scene, I'll introduce the cross-level interaction effects:

d) $\beta_5(Time_{it}*GHQ_i)$ = 1, so the negative linear slope of time on depression scores was more positive, or flatter, at higher levels of general health problems.

e) $\beta_6(Time^2_{it}*GHQ_i)$ = -.18. This is what I am having trouble interpreting (if I have not already misinterpreted the prior effects).

My take is that the convex slope gets more negative, and hence more concave, with more general health problems, because the convex slope is positive.

This might look like the solid red line in the image below (the dotted red lines are meant to signify the relationship, i.e. the stronger the three-level interaction, the more negative and hence concave the convex slope becomes ): But I would appreciate your thoughts.

Bonus Question

What would $\beta_6$ look like if $\beta_3GHQ_{i}$ = 1.8, $\beta_5(Time_{it}*GHQ_i)$ = -.6, and $\beta_6(Time^2_{it}*GHQ_i)$ = .2? (Note, I mean this in the most general sense, i.e. without looking too much into the strength of the coefficients, just their direction). 