Agenda

• Equation practice

• Fitting multilevel logistic regression models with brms

  • Applied walkthrough 1: Twitter data

  • Applied walkthrough 2: Lung cancer data

• Missing values

Data

Read in the nurses.csv data. Note - each row represents data for one nurse.

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

Data

nursesCopy Code

## # A tibble: 1,000 x 11
##    hospital  ward wardid nurse   age gender experien stress wardtype     hospsize expcon    
##       <dbl> <dbl>  <dbl> <dbl> <dbl> <chr>     <dbl>  <dbl> <chr>        <chr>    <chr>     
##  1        1     1     11     1    36 Male         11      7 general care large    experiment
##  2        1     1     11     2    45 Male         20      7 general care large    experiment
##  3        1     1     11     3    32 Male          7      7 general care large    experiment
##  4        1     1     11     4    57 Female       25      6 general care large    experiment
##  5        1     1     11     5    46 Female       22      6 general care large    experiment
##  6        1     1     11     6    60 Female       22      6 general care large    experiment
##  7        1     1     11     7    23 Female       13      6 general care large    experiment
##  8        1     1     11     8    32 Female       13      7 general care large    experiment
##  9        1     1     11     9    60 Male         17      7 general care large    experiment
## 10        1     2     12    10    45 Male         21      6 special care large    experiment
## # … with 990 more rowsCopy Code

Model 1

Fit the following model

$$\begin{array}{cc}{\mathrm{stress}}_i& \sim N({\alpha }_{j\left[i\right]}+{\beta }_{1j\left[i\right]}(\mathrm{experien}),{\sigma }^2)\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{1j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left({\mathrm{wardtype}}_{specialcare}\right)\\ & {\gamma }_0^{{\beta }_1}+{\gamma }_1^{{\beta }_1}\left({\mathrm{wardtype}}_{specialcare}\right)\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_j}^2& {\rho }_{{\alpha }_j{\beta }_{1j}}\\ {\rho }_{{\beta }_{1j}{\alpha }_j}& {\sigma }_{{\beta }_{1j}}^2\end{array}\right))\text{, for wardid j = 1,}\dots \text{,J}\end{array}$$

lmer(stress ~ experien * wardtype + (experien|wardid),
     data = nurses)
# or
lmer(stress ~ experien + wardtype + experien:wardtype + 
       (experien|wardid),
     data = nurses)Copy Code

Model 2

Fit the following model

$$\begin{array}{cc}{\mathrm{stress}}_i& \sim N({\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_{1j\left[i\right],k\left[i\right]}(\mathrm{experien}),{\sigma }^2)\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{1j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left({\mathrm{wardtype}}_{specialcare}\right)\\ & {\gamma }_0^{{\beta }_1}+{\gamma }_1^{{\beta }_1}\left({\mathrm{wardtype}}_{specialcare}\right)\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_j}^2& {\rho }_{{\alpha }_j{\beta }_{1j}}\\ {\rho }_{{\beta }_{1j}{\alpha }_j}& {\sigma }_{{\beta }_{1j}}^2\end{array}\right))\text{, for wardid j = 1,}\dots \text{,J}\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_k\\ & {\beta }_{1k}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left({\mathrm{hospsize}}_{\mathrm{medium}}\right)+{\gamma }_2^{\alpha }\left({\mathrm{hospsize}}_{\mathrm{small}}\right)\\ & {\mu }_{{\beta }_{1k}}\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_k}^2& {\rho }_{{\alpha }_k{\beta }_{1k}}\\ {\rho }_{{\beta }_{1k}{\alpha }_k}& {\sigma }_{{\beta }_{1k}}^2\end{array}\right))\text{, for hospital k = 1,}\dots \text{,K}\end{array}$$

lmer(stress ~ experien * wardtype + hospsize +
       (experien|wardid) + (experien|hospital),
     data = nurses)Copy Code

Model 3

Fit the following model

$$\begin{array}{cc}{\mathrm{stress}}_i& \sim N({\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_{1j\left[i\right],k\left[i\right]}(\mathrm{experien})+{\beta }_2(\mathrm{age}),{\sigma }^2)\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{1j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left({\mathrm{expcon}}_{\mathrm{experiment}}\right)\\ & {\mu }_{{\beta }_{1j}}\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_j}^2& 0\\ 0& {\sigma }_{{\beta }_{1j}}^2\end{array}\right))\text{, for wardid j = 1,}\dots \text{,J}\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_k\\ & {\beta }_{1k}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\mu }_{{\alpha }_k}\\ & {\mu }_{{\beta }_{1k}}\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_k}^2& 0\\ 0& {\sigma }_{{\beta }_{1k}}^2\end{array}\right))\text{, for hospital k = 1,}\dots \text{,K}\end{array}$$

lmer(stress ~ experien + age + expcon +
       (experien||wardid) + (experien||hospital),
     data = nurses)
# or
lmer(stress ~ experien + age + expcon +
       (1|wardid) + (0 + experien|wardid) + 
       (1|hospital) + (0 + experien|hospital),
     data = nurses)Copy Code

Model 4

Fit the following model

$$\begin{array}{cc}{\mathrm{stress}}_i& \sim N({\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_{1j\left[i\right],k\left[i\right]}(\mathrm{experien})+{\beta }_2(\mathrm{age}),{\sigma }^2)\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{1j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left({\mathrm{expcon}}_{\mathrm{experiment}}\right)+{\gamma }_2^{\alpha }\left({\mathrm{wardtype}}_{specialcare}\right)\\ & {\mu }_{{\beta }_{1j}}\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_j}^2& {\rho }_{{\alpha }_j{\beta }_{1j}}\\ {\rho }_{{\beta }_{1j}{\alpha }_j}& {\sigma }_{{\beta }_{1j}}^2\end{array}\right))\text{, for wardid j = 1,}\dots \text{,J}\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_k\\ & {\beta }_{1k}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left({\mathrm{hospsize}}_{\mathrm{medium}}\right)+{\gamma }_2^{\alpha }\left({\mathrm{hospsize}}_{\mathrm{small}}\right)\\ & {\gamma }_0^{{\beta }_1}+{\gamma }_1^{{\beta }_1}\left({\mathrm{hospsize}}_{\mathrm{medium}}\right)+{\gamma }_2^{{\beta }_1}\left({\mathrm{hospsize}}_{\mathrm{small}}\right)\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_k}^2& {\rho }_{{\alpha }_k{\beta }_{1k}}\\ {\rho }_{{\beta }_{1k}{\alpha }_k}& {\sigma }_{{\beta }_{1k}}^2\end{array}\right))\text{, for hospital k = 1,}\dots \text{,K}\end{array}$$

lmer(stress ~ experien * hospsize + age + expcon + wardtype + 
       (experien|wardid) + (experien|hospital),
     data = nurses)Copy Code

Model 5

Fit the following model

$$\begin{array}{cc}{\mathrm{expcon}}_i& \sim \mathrm{Binomial}(n=1,{\mathrm{prob}}_{\mathrm{expcon}=\mathrm{experiment}}=\hat{P})\\ \log \left[\frac{\hat{P}}{1-\hat{P}}\right]& ={\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_{1j\left[i\right]}\left(\mathrm{age}\right)\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{1j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\mu }_{{\alpha }_j}\\ & {\mu }_{{\beta }_{1j}}\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_j}^2& {\rho }_{{\alpha }_j{\beta }_{1j}}\\ {\rho }_{{\beta }_{1j}{\alpha }_j}& {\sigma }_{{\beta }_{1j}}^2\end{array}\right))\text{, for wardid j = 1,}\dots \text{,J}\\ {\alpha }_k& \sim N({\mu }_{{\alpha }_k},{\sigma }_{{\alpha }_k}^2)\text{, for hospital k = 1,}\dots \text{,K}\end{array}$$

nurses <- nurses %>% 
  mutate(expcon = factor(expcon))
glmer(expcon ~ age + 
        (age|wardid) + (1|hospital),
      data = nurses,
      family = binomial(link = "logit"),
      data = nurses)Copy Code

Model 6

Fit the following model

$$\begin{array}{cc}{\mathrm{expcon}}_i& \sim \mathrm{Binomial}(n=1,{\mathrm{prob}}_{\mathrm{expcon}=\mathrm{experiment}}=\hat{P})\\ \log \left[\frac{\hat{P}}{1-\hat{P}}\right]& ={\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_{1j\left[i\right]}\left(\mathrm{age}\right)\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{1j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left({\mathrm{wardtype}}_{specialcare}\right)\\ & {\mu }_{{\beta }_{1j}}\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_j}^2& {\rho }_{{\alpha }_j{\beta }_{1j}}\\ {\rho }_{{\beta }_{1j}{\alpha }_j}& {\sigma }_{{\beta }_{1j}}^2\end{array}\right))\text{, for wardid j = 1,}\dots \text{,J}\\ {\alpha }_k& \sim N({\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }({\mathrm{hospsize}}_{\mathrm{medium}})+{\gamma }_2^{\alpha }({\mathrm{hospsize}}_{\mathrm{small}}),{\sigma }_{{\alpha }_k}^2)\text{, for hospital k = 1,}\dots \text{,K}\end{array}$$

glmer(expcon ~ hospsize + age + wardtype + 
        (age|wardid) + (1|hospital),
      data = nurses,
      family = binomial(link = "logit"))Copy Code

Model 7

Fit the following model

$$\begin{array}{cc}{\mathrm{expcon}}_i& \sim \mathrm{Binomial}(n=1,{\mathrm{prob}}_{\mathrm{expcon}=\mathrm{experiment}}=\hat{P})\\ \log \left[\frac{\hat{P}}{1-\hat{P}}\right]& ={\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_{1j\left[i\right],k\left[i\right]}\left(\mathrm{age}\right)+{\beta }_2\left({\mathrm{gender}}_{\mathrm{Male}}\right)+{\beta }_3(\mathrm{age}\times {\mathrm{gender}}_{\mathrm{Male}})\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{1j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left({\mathrm{wardtype}}_{specialcare}\right)\\ & {\mu }_{{\beta }_{1j}}\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_j}^2& {\rho }_{{\alpha }_j{\beta }_{1j}}\\ {\rho }_{{\beta }_{1j}{\alpha }_j}& {\sigma }_{{\beta }_{1j}}^2\end{array}\right))\text{, for wardid j = 1,}\dots \text{,J}\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_k\\ & {\beta }_{1k}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left({\mathrm{hospsize}}_{\mathrm{medium}}\right)+{\gamma }_2^{\alpha }\left({\mathrm{hospsize}}_{\mathrm{small}}\right)\\ & {\mu }_{{\beta }_{1k}}\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_k}^2& {\rho }_{{\alpha }_k{\beta }_{1k}}\\ {\rho }_{{\beta }_{1k}{\alpha }_k}& {\sigma }_{{\beta }_{1k}}^2\end{array}\right))\text{, for hospital k = 1,}\dots \text{,K}\end{array}$$

glmer(expcon ~ age * gender + wardtype + hospsize +
        (age|wardid) + (age|hospital),
      data = nurses,
      family = binomial(link = "logit"))Copy Code

Model 8

Fit the following model

$$\begin{array}{cc}{\mathrm{expcon}}_i& \sim \mathrm{Binomial}(n=1,{\mathrm{prob}}_{\mathrm{expcon}=\mathrm{experiment}}=\hat{P})\\ \log \left[\frac{\hat{P}}{1-\hat{P}}\right]& ={\alpha }_{j\left[i\right],k\left[i\right]}+{\beta }_1\left(\mathrm{age}\right)+{\beta }_2\left({\mathrm{gender}}_{\mathrm{Male}}\right)+{\beta }_{3j\left[i\right],k\left[i\right]}\left(\mathrm{experien}\right)\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_j\\ & {\beta }_{3j}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left({\mathrm{wardtype}}_{specialcare}\right)\\ & {\gamma }_0^{{\beta }_3}+{\gamma }_1^{{\beta }_3}\left({\mathrm{wardtype}}_{specialcare}\right)\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_j}^2& {\rho }_{{\alpha }_j{\beta }_{3j}}\\ {\rho }_{{\beta }_{3j}{\alpha }_j}& {\sigma }_{{\beta }_{3j}}^2\end{array}\right))\text{, for wardid j = 1,}\dots \text{,J}\\ \left(\begin{array}{c}\begin{array}{cc}& {\alpha }_k\\ & {\beta }_{3k}\end{array}\end{array}\right)& \sim N(\left(\begin{array}{c}\begin{array}{cc}& {\gamma }_0^{\alpha }+{\gamma }_1^{\alpha }\left({\mathrm{hospsize}}_{\mathrm{medium}}\right)+{\gamma }_2^{\alpha }\left({\mathrm{hospsize}}_{\mathrm{small}}\right)\\ & {\gamma }_0^{{\beta }_3}+{\gamma }_1^{{\beta }_3}\left({\mathrm{hospsize}}_{\mathrm{medium}}\right)+{\gamma }_2^{{\beta }_3}\left({\mathrm{hospsize}}_{\mathrm{small}}\right)\end{array}\end{array}\right),\left(\begin{array}{cc}{\sigma }_{{\alpha }_k}^2& {\rho }_{{\alpha }_k{\beta }_{3k}}\\ {\rho }_{{\beta }_{3k}{\alpha }_k}& {\sigma }_{{\beta }_{3k}}^2\end{array}\right))\text{, for hospital k = 1,}\dots \text{,K}\end{array}$$