Agenda

[We're not going to get through it all]

• Review Homework 2

• Some equation practice

• Introduce Bayes theorem

  • Go through an example with estimating a mean

• Discuss Bayes in the context of regression modeling

The data

From an example in the Mplus manual. I made up the column names.

mplus_d <- read_csv(here::here("data", "mplus920.csv"))
mplus_dCopy Code

## # A tibble: 7,500 x 6
##        score  baseline sch_treatment  dist_ses schid distid
##        <dbl>     <dbl>         <dbl>     <dbl> <dbl>  <dbl>
##  1  5.559216  1.383101             1 -0.642262     1      1
##  2 -0.107394 -0.789654             1 -0.642262     1      1
##  3  0.049476 -0.760867             1 -0.642262     1      1
##  4 -2.387703 -0.798527             1 -0.642262     1      1
##  5  1.180393 -0.411377             1 -0.642262     1      1
##  6  3.959005 -0.987154             1 -0.642262     2      1
##  7 -0.895792 -1.966773             1 -0.642262     2      1
##  8  2.879087  0.42117              1 -0.642262     2      1
##  9  5.611088  1.67047              1 -0.642262     2      1
## 10  2.828119  0.001154             1 -0.642262     2      1
## # … with 7,490 more rowsCopy Code

Model 1

Fit the following model

scorei∼N(αj[i],σ2)αj∼N(μαj,σ2αj), for distid j = 1,…,J$$\begin{array}{cc}{\mathrm{score}}_i& \sim N({\alpha }_{j\left[i\right]},{\sigma }^2)\\ {\alpha }_j& \sim N({\mu }_{{\alpha }_j},{\sigma }_{{\alpha }_j}^2)\text{, for distid j = 1,}\dots \text{,J}\end{array}$$

lmer(score ~ 1 + (1|distid), data = mplus_d)Copy Code

Model 2

Fit the following model

scorei∼N(αj[i],k[i]+β1(baseline),σ2)αj∼N(μαj,σ2αj), for schid j = 1,…,Jαk∼N(μαk,σ2αk), for distid k = 1,…,K$$\begin{array}{cc}{\mathrm{score}}_i& \sim N({\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_1(\mathrm{baseline}),{\sigma }^2)\\ {\alpha }_j& \sim N({\mu }_{{\alpha }_j},{\sigma }_{{\alpha }_j}^2)\text{, for schid j = 1,}\dots \text{,J}\\ {\alpha }_k& \sim N({\mu }_{{\alpha }_k},{\sigma }_{{\alpha }_k}^2)\text{, for distid k = 1,}\dots \text{,K}\end{array}$$

lmer(score ~ baseline + (1|schid) + (1|distid),
     data = mplus_d)Copy Code

Model 3

Fit the following model

scorei∼N(αj[i],k[i]+β1j[i](baseline),σ2)(αjβ1j)∼N((μαjμβ1j),(σ2αjραjβ1jρβ1jαjσ2β1j)), for schid j = 1,…,Jαk∼N(γα0+γα1(dist_ses)+γα2(baseline×dist_ses),σ2αk), for distid k = 1,…,K$$\begin{array}{cc}{\mathrm{score}}_i& \sim N({\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_{1j\left[i\right]}(\mathrm{baseline}),{\sigma }^2)\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{1j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\mu }_{{\alpha }_j}\\ & {\mu }_{{\beta }_{1j}}\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_j}^2& {\rho }_{{\alpha }_j{\beta }_{1j}}\\ {\rho }_{{\beta }_{1j}{\alpha }_j}& {\sigma }_{{\beta }_{1j}}^2\end{array}\right))\text{, for schid j = 1,}\dots \text{,J}\\ {\alpha }_k& \sim N({\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }(dist\_ses)+{\gamma }_2^{\alpha }(\mathrm{baseline}\times dist\_ses),{\sigma }_{{\alpha }_k}^2)\text{, for distid k = 1,}\dots \text{,K}\end{array}$$

lmer(score ~ baseline * dist_ses + 
       (baseline|schid) + (1|distid),
     data = mplus_d)Copy Code

Model 4

Fit the following model

scorei∼N(αj[i],k[i]+β1j[i](baseline),σ2)(αjβ1j)∼N((γα0+γα1(sch_treatment)μβ1j),(σ2αjραjβ1jρβ1jαjσ2β1j)), for schid j = 1,…,Jαk∼N(γα0+γα1(dist_ses),σ2αk), for distid k = 1,…,K$$\begin{array}{cc}{\mathrm{score}}_i& \sim N({\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_{1j\left[i\right]}(\mathrm{baseline}),{\sigma }^2)\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{1j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left(sch\_treatment\right)\\ & {\mu }_{{\beta }_{1j}}\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_j}^2& {\rho }_{{\alpha }_j{\beta }_{1j}}\\ {\rho }_{{\beta }_{1j}{\alpha }_j}& {\sigma }_{{\beta }_{1j}}^2\end{array}\right))\text{, for schid j = 1,}\dots \text{,J}\\ {\alpha }_k& \sim N({\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }(dist\_ses),{\sigma }_{{\alpha }_k}^2)\text{, for distid k = 1,}\dots \text{,K}\end{array}$$

lmer(score ~ baseline + sch_treatment + dist_ses + 
       (baseline|schid) + (1|distid),
     data = mplus_d)Copy Code

Model 5

Fit the following model

scorei∼N(αj[i],k[i]+β1j[i](baseline),σ2)(αjβ1j)∼N((γα0+γα1k[i](sch_treatment)γβ10+γβ11(sch_treatment)),(σ2αjραjβ1jρβ1jαjσ2β1j)), for schid j = 1,…,J(αkγ1k)∼N((γα0+γα1(dist_ses)μγ1k),(σ2αkραkγ1kργ1kαkσ2γ1k)), for distid k = 1,…,K$$\begin{array}{cc}{\mathrm{score}}_i& \sim N({\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_{1j\left[i\right]}(\mathrm{baseline}),{\sigma }^2)\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{1j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_{1k\left[i\right]}^{\alpha }\left(sch\_treatment\right)\\ & {\gamma }_0^{{\beta }_1}+{\gamma }_1^{{\beta }_1}\left(sch\_treatment\right)\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_j}^2& {\rho }_{{\alpha }_j{\beta }_{1j}}\\ {\rho }_{{\beta }_{1j}{\alpha }_j}& {\sigma }_{{\beta }_{1j}}^2\end{array}\right))\text{, for schid j = 1,}\dots \text{,J}\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_k\\ & {\gamma }_{1k}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left(dist\_ses\right)\\ & {\mu }_{{\gamma }_{1k}}\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_k}^2& {\rho }_{{\alpha }_k{\gamma }_{1k}}\\ {\rho }_{{\gamma }_{1k}{\alpha }_k}& {\sigma }_{{\gamma }_{1k}}^2\end{array}\right))\text{, for distid k = 1,}\dots \text{,K}\end{array}$$

lmer(score ~ baseline * sch_treatment + dist_ses + 
       (baseline|schid) + (sch_treatment|distid),
     data = mplus_d)Copy Code

Model 6

Fit the following model

scorei∼N(αj[i],k[i]+β1j[i],k[i](baseline),σ2)(αjβ1j)∼N((γα0+γα1k[i](sch_treatment)γβ11k[i0+γβ11k[i](sch_treatment)),(σ2αjραjβ1jρβ1jαjσ2β1j)), for schid j = 1,…,J(αkβ1kγ1kγβ11k)∼N((γα0+γα1(dist_ses)μβ1kμγ1kμγβ11k),(σ2αk0000σ2β1k0000σ2γ1k0000σ2γβ11k)), for distid k = 1,…,K$$\begin{array}{cc}{\mathrm{score}}_i& \sim N({\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_{1j\left[i\right],k\left[i\right]}(\mathrm{baseline}),{\sigma }^2)\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{1j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_{1k\left[i\right]}^{\alpha }\left(sch\_treatment\right)\\ & {\gamma }_{1k[i0}^{{\beta }_1}+{\gamma }_{1k\left[i\right]}^{{\beta }_1}\left(sch\_treatment\right)\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_j}^2& {\rho }_{{\alpha }_j{\beta }_{1j}}\\ {\rho }_{{\beta }_{1j}{\alpha }_j}& {\sigma }_{{\beta }_{1j}}^2\end{array}\right))\text{, for schid j = 1,}\dots \text{,J}\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_k\\ & {\beta }_{1k}\\ & {\gamma }_{1k}\\ & {\gamma }_{1k}^{{\beta }_1}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left(dist\_ses\right)\\ & {\mu }_{{\beta }_{1k}}\\ & {\mu }_{{\gamma }_{1k}}\\ & {\mu }_{{\gamma }_{1k}^{{\beta }_1}}\end{array}\end{array}\right),\left(\begin{array}{cccc}{\sigma }_{{\alpha }_k}^2& 0& 0& 0\\ 0& {\sigma }_{{\beta }_{1k}}^2& 0& 0\\ 0& 0& {\sigma }_{{\gamma }_{1k}}^2& 0\\ 0& 0& 0& {\sigma }_{{\gamma }_{1k}^{{\beta }_1}}^2\end{array}\right))\text{, for distid k = 1,}\dots \text{,K}\end{array}$$