Agenda

• More practice with equation/models

• Thinking flexibly about time

  • Coefficient interpretation by time coding

• A few methods for handling non-linearity

Note

The last bullet is not necessarily specific to growth models

The data

Please load the following data

library(tidyverse)
d <- read_csv(here::here("data", "longitudinal-sim.csv"))Copy Code

Model 1

Please translate the following model into lme4::lmer() code

g5_springi∼N(αj[i],k[i]+β1(g4_spring)+β2(g3_spring),σ2)αj∼N(μαj,σ2αj), for scid j = 1,…,Jαk∼N(μαk,σ2αk), for distid k = 1,…,K$$\begin{array}{cc}{g5\_spring}_i& \sim N({\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_1(g4\_spring)+{\beta }_2(g3\_spring),{\sigma }^2)\\ {\alpha }_j& \sim N({\mu }_{{\alpha }_j},{\sigma }_{{\alpha }_j}^2)\text{, for scid j = 1,}\dots \text{,J}\\ {\alpha }_k& \sim N({\mu }_{{\alpha }_k},{\sigma }_{{\alpha }_k}^2)\text{, for distid k = 1,}\dots \text{,K}\end{array}$$

Three equivalent specifications

m1a <- lmer(g5_spring ~ g4_spring + g3_spring + 
              (1|scid) + (1|distid),
            data = d)
m1b <- lmer(g5_spring ~ g4_spring + g3_spring + (1|distid/scid),
            data = d)
m1c <- lmer(g5_spring ~ g4_spring + g3_spring + 
              (1|distid) + (1|distid:scid),
            data = d)Copy Code

Compute starting group means

d <- d %>% 
  group_by(scid) %>% 
  mutate(sch_mean_start = mean(g3_fall)) %>% 
  group_by(distid) %>% 
  mutate(dist_mean_start = mean(g3_fall))Copy Code

Model 2

Don't worry if you run into convergence warnings

g5_springi∼N(αj[i],k[i]+β1(g4_spring)+β2j[i](g3_spring),σ2)(αjβ2j)∼N((γα0+γα1(sch_mean_start)μβ2j),(σ2αjραjβ2jρβ2jαjσ2β2j)), for scid j = 1,…,Jαk∼N(μαk,σ2αk), for distid k = 1,…,K$$\begin{array}{cc}{g5\_spring}_i& \sim N({\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_1(g4\_spring)+{\beta }_{2j\left[i\right]}(g3\_spring),{\sigma }^2)\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{2j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left(sch\_mean\_start\right)\\ & {\mu }_{{\beta }_{2j}}\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_j}^2& {\rho }_{{\alpha }_j{\beta }_{2j}}\\ {\rho }_{{\beta }_{2j}{\alpha }_j}& {\sigma }_{{\beta }_{2j}}^2\end{array}\right))\text{, for scid j = 1,}\dots \text{,J}\\ {\alpha }_k& \sim N({\mu }_{{\alpha }_k},{\sigma }_{{\alpha }_k}^2)\text{, for distid k = 1,}\dots \text{,K}\end{array}$$

lmer(g5_spring ~ g4_spring + g3_spring + sch_mean_start + 
       (g3_spring|scid) + (1|distid),
     data = d)Copy Code

Model 3

g5_springi∼N(αj[i],k[i]+β1j[i],k[i](g4_spring)+β2j[i],k[i](g3_spring),σ2)(αjβ1jβ2j)∼N((γα0+γα1(sch_mean_start)μβ1jμβ2j),(σ2αjραjβ1jραjβ2jρβ1jαjσ2β1jρβ1jβ2jρβ2jαjρβ2jβ1jσ2β2j)), for scid j = 1,…,J(αkβ1kβ2k)∼N((μαkμβ1kμβ2k),(σ2αkραkβ1kραkβ2kρβ1kαkσ2β1kρβ1kβ2kρβ2kαkρβ2kβ1kσ2β2k)), for distid k = 1,…,K$$\begin{array}{cc}{g5\_spring}_i& \sim N({\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_{1j\left[i\right],k\left[i\right]}(g4\_spring)+{\beta }_{2j\left[i\right],k\left[i\right]}(g3\_spring),{\sigma }^2)\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{1j}\\ & {\beta }_{2j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left(sch\_mean\_start\right)\\ & {\mu }_{{\beta }_{1j}}\\ & {\mu }_{{\beta }_{2j}}\end{array}\end{array}\right),\left(\begin{array}{ccc}{\sigma }_{{\alpha }_j}^2& {\rho }_{{\alpha }_j{\beta }_{1j}}& {\rho }_{{\alpha }_j{\beta }_{2j}}\\ {\rho }_{{\beta }_{1j}{\alpha }_j}& {\sigma }_{{\beta }_{1j}}^2& {\rho }_{{\beta }_{1j}{\beta }_{2j}}\\ {\rho }_{{\beta }_{2j}{\alpha }_j}& {\rho }_{{\beta }_{2j}{\beta }_{1j}}& {\sigma }_{{\beta }_{2j}}^2\end{array}\right))\text{, for scid j = 1,}\dots \text{,J}\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_k\\ & {\beta }_{1k}\\ & {\beta }_{2k}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\mu }_{{\alpha }_k}\\ & {\mu }_{{\beta }_{1k}}\\ & {\mu }_{{\beta }_{2k}}\end{array}\end{array}\right),\left(\begin{array}{ccc}{\sigma }_{{\alpha }_k}^2& {\rho }_{{\alpha }_k{\beta }_{1k}}& {\rho }_{{\alpha }_k{\beta }_{2k}}\\ {\rho }_{{\beta }_{1k}{\alpha }_k}& {\sigma }_{{\beta }_{1k}}^2& {\rho }_{{\beta }_{1k}{\beta }_{2k}}\\ {\rho }_{{\beta }_{2k}{\alpha }_k}& {\rho }_{{\beta }_{2k}{\beta }_{1k}}& {\sigma }_{{\beta }_{2k}}^2\end{array}\right))\text{, for distid k = 1,}\dots \text{,K}\end{array}$$

lmer(g5_spring ~ g4_spring + g3_spring + sch_mean_start + 
       (g4_spring + g3_spring|scid) + 
       (g4_spring + g3_spring|distid),
     data = d)Copy Code

Model 4

g5_springi∼N(αj[i],k[i]+β1j[i],k[i](g4_spring)+β2j[i],k[i](g3_spring),σ2)(αjβ1jβ2j)∼N((γα0+γα1(sch_mean_start)μβ1jγβ20+γβ21(sch_mean_start)),(σ2αjραjβ1jραjβ2jρβ1jαjσ2β1jρβ1jβ2jρβ2jαjρβ2jβ1jσ2β2j)), for scid j = 1,…,J(αkβ1kβ2k)∼N((μαkμβ1kμβ2k),(σ2αkραkβ1kραkβ2kρβ1kαkσ2β1kρβ1kβ2kρβ2kαkρβ2kβ1kσ2β2k)), for distid k = 1,…,K$$\begin{array}{cc}{g5\_spring}_i& \sim N({\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_{1j\left[i\right],k\left[i\right]}(g4\_spring)+{\beta }_{2j\left[i\right],k\left[i\right]}(g3\_spring),{\sigma }^2)\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{1j}\\ & {\beta }_{2j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left(sch\_mean\_start\right)\\ & {\mu }_{{\beta }_{1j}}\\ & {\gamma }_0^{{\beta }_2}+{\gamma }_1^{{\beta }_2}\left(sch\_mean\_start\right)\end{array}\end{array}\right),\left(\begin{array}{ccc}{\sigma }_{{\alpha }_j}^2& {\rho }_{{\alpha }_j{\beta }_{1j}}& {\rho }_{{\alpha }_j{\beta }_{2j}}\\ {\rho }_{{\beta }_{1j}{\alpha }_j}& {\sigma }_{{\beta }_{1j}}^2& {\rho }_{{\beta }_{1j}{\beta }_{2j}}\\ {\rho }_{{\beta }_{2j}{\alpha }_j}& {\rho }_{{\beta }_{2j}{\beta }_{1j}}& {\sigma }_{{\beta }_{2j}}^2\end{array}\right))\text{, for scid j = 1,}\dots \text{,J}\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_k\\ & {\beta }_{1k}\\ & {\beta }_{2k}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\mu }_{{\alpha }_k}\\ & {\mu }_{{\beta }_{1k}}\\ & {\mu }_{{\beta }_{2k}}\end{array}\end{array}\right),\left(\begin{array}{ccc}{\sigma }_{{\alpha }_k}^2& {\rho }_{{\alpha }_k{\beta }_{1k}}& {\rho }_{{\alpha }_k{\beta }_{2k}}\\ {\rho }_{{\beta }_{1k}{\alpha }_k}& {\sigma }_{{\beta }_{1k}}^2& {\rho }_{{\beta }_{1k}{\beta }_{2k}}\\ {\rho }_{{\beta }_{2k}{\alpha }_k}& {\rho }_{{\beta }_{2k}{\beta }_{1k}}& {\sigma }_{{\beta }_{2k}}^2\end{array}\right))\text{, for distid k = 1,}\dots \text{,K}\end{array}$$