---
title: "Correlated Systems of Statistical Equations with Multiple Variable Types"
author: "Allison C Fialkowski"
date: "`r Sys.Date()`"
output: 
  bookdown::html_document2: 
    fig_caption: yes
bibliography: Bibliography.bib
vignette: >
  %\VignetteIndexEntry{Correlated Systems of Statistical Equations with Multiple Variable Types}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

<style type="text/css">

h1.title {
  text-align: center;
}
h4.author { /* Header 4 - and the author and data headers use this too  */
  text-align: center;
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```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, warning=FALSE, message=FALSE, fig.align = 'center', fig.width = 6, fig.height = 4, cache = FALSE)
```

```{r, include=FALSE}
library("bookdown")
```

The following examples demonstrate the use of the `corrsys` and `corrsys2` functions within the **SimRepeat** package.  These functions generate correlated systems of `M` equations representing a **system of repeated measures** at `M` time points.  The equations may contain 1) ordinal ($r \geq 2$ categories), continuous (normal, non-normal, and mixture distributions), and/or count (regular and zero-inflated, Poisson and Negative Binomial) independent variables $X$; 2) continuous error terms $E$; 3) a discrete time variable $Time$; and 4) random effects $U$.  The random effects may be a random intercept, a random slope for time, or a random slope for any of the $X$ variables.  The **important assumptions** are:    

1) There are at least 2 equations with at least 1 independent variable total.      
2) The independent variables, random effect terms, and error terms are uncorrelated.    
3) Each equation has an error term.      
4) All error terms have continuous non-mixture distributions or all have continuous mixture distributions.      
5) All random effects are continuous.     
6) Growth is linear with respect to time.  

The outcomes $Y$ are generated using a hierarchical linear models (HLM) approach.  See [The Hierarchical Linear Models Approach for a System of Correlated Equations with Multiple Variable Types](HLM_Approach.html) vignette for a description of the HLM model.  The independent variables, error terms, and random effects are generated from multivariate normal variables with intermediate correlations calculated using either `SimCorrMix::intercorr` and **correlation method 1** for `corrsys` or `SimCorrMix::intercorr2` and **correlation method 2** for `corrsys2`.  See the **SimCorrMix** package for a description of the correlation methods and the techniques used to generate each variable type.  

The `corrsys` and `corrsys2` functions contain no parameter checks in order to decrease simulation time.  That should be done first using `checkpar`.  Summaries of the system can be obtained using `summary_sys`.  More information regarding function inputs can be found by consulting the function documentation.  Some code has been adapted from the **SimMultiCorrData** [@SMCD] and **SimCorrMix** [@SimCorrMix] packages.     

**Example 1** demonstrates the `corrsys` function to generate a system of 3 equations for 5 independent variables with no random effects.  **Example 2** uses the `corrsys2` function to demonstrate how to handle missing variables by removing the independent variables from $Y_2$ in Example 1.  **Example 3** generates a system of system of 4 equations with random effects.  

#### Notes on fitting linear mixed models in R: {-}
There are two main options for fitting mixed effects models in R.  The **nlme** package [@Nlme] permits linear and nonlinear mixed models.  Linear models are fit using **lme** (see @Laird); nonlinear models are fit using **nlme** (see @Lind).  Both allow nested random effects and the within-group errors can be correlated and/or have unequal variances.  The `correlation` argument permits several different correlation structures, including AR(1) and CS (see that package's documentation).  Setting `weights = varIdent(form =` $\sim$ `1 | time)` achieves unequal variances across time.        

The **lme4** package [@Lme4] permits linear (`lmer`), generalized linear (`glmer`), and nonlinear (`nlmer`) mixed models.  Since there are no specific function arguments, implementing different error term correlation structures is more difficult than in **nlme**.  Certain error structures can be achieved through manipulation of the data and models (see @Bates).  However, **lme4** implements crossed random effects more efficiently and contains facilities for likelihood profiling and parametric bootstrapping.  The **lmerTest** package [@LmerTest] provides $p$-values for beta coefficients.

## Example 1: System of 3 equations for 5 independent variables {-}

### Description of Variables {-}

1. **Ordinal variable:** $X_{ord(1)}$ has 3 categories (i.e., drug treatment) and is the same in each equation
2. **Continuous variables:** 

 a) $X_{cont(1)}$ is a time-varying covariate (subject-level term) with a Rice distribution with increasing variance and an AR(1, p = 0.5) correlation structure 
 
   i. $X_{cont(11)}$ has a Rice(1, 0.5) distribution which requires a sixth cumulant correction of 0.08
   ii. $X_{cont(21)}$ has a Rice(2, 2) distribution which requires a sixth cumulant correction of 0.12
   iii. $X_{cont(31)}$ has a Rice(4, 8) distribution which requires a sixth cumulant correction of 0.36   
   
 b) $X_{mix(1)}$ is a mixture of Normal(-5, 2) and Normal(3, 1) with mixing parameters 0.3 and 0.7; it is a time-varying covariate (subject-level term) and the components have an AR(1, p = 0.4) correlation structure across Y   
 
3. **Poisson variable:** $X_{pois(1)}$ is a zero-inflated Poisson variable with $\lambda = 15$, the probability of a structural zero set at $0.10$, and is the same in each equation
4. **Negative Binomial variable:** $X_{nb(1)}$ is a regular NB time-varying covariate (subject-level term) with an AR(1, p = 0.3) correlation structure and increasing mean and variance     

  a) $X_{nb(11)}$ has a size of 10 and mean of 3     
  b) $X_{nb(21)}$ has a size of 10 and mean of 4     
  c) $X_{nb(31)}$ has a size of 10 and mean of 5      
  
5. **Error terms** have Skewnormal distributions which become more skewed over time with an AR(1, p = 0.4) correlation structure:   

  a) $E_1$ has a Skewnormal(0, 1, 1) distribution which requires a sixth cumulant correction of 0.06    
  b) $E_2$ has a Skewnormal(0, 1, 5) distribution   
  c) $E_3$ has a Skewnormal(0, 1, 25) distribution which requires a sixth cumulant correction of 0.15

### Description of Variables {-}     

There is an interaction between $X_{ord(1)}$ and $X_{pois(1)}$ for each $Y$.  Since they are both group-level covariates, the interaction is also a group-level covariate that will interact with the subject-level covariates $X_{cont(1)}$, $X_{mix(1)}$, and $X_{nb(1)}$.  However, only $X_{ord(1)}$ and $X_{pois(1)}$ interact with time in this example (normally their interaction would also interact with time).  The ordering in the equations below reflects the ordering in the simulation process.      

\begin{equation}  
\begin{split}   
Y_1 &= \beta_0 + \beta_1 * X_{ord(1)} + \beta_2 * X_{cont(11)} + \beta_3 * X_{mix(11)} + \beta_4 * X_{pois(1)} + \beta_5 * X_{nb(11)} + \beta_{int} * X_{ord(1)} * X_{pois(1)} \\ 
&+ \beta_{subj1} * X_{ord(1)} * X_{cont(11)} + \beta_{subj2} * X_{pois(1)} * X_{cont(11)} + \beta_{subj3} * X_{ord(1)} * X_{pois(1)} * X_{cont(11)} \\
&+ \beta_{subj4} * X_{ord(1)} * X_{mix(11)} + \beta_{subj5} * X_{pois(1)} * X_{mix(11)} + \beta_{subj6} * X_{ord(1)} * X_{pois(1)} * X_{mix(11)} \\
&+ \beta_{subj7} * X_{ord(1)} * X_{nb(11)} + \beta_{subj8} * X_{pois(1)} * X_{nb(11)} + \beta_{subj9} * X_{ord(1)} * X_{pois(1)} * X_{nb(11)} \\
&+ \beta_{tint1} * X_{ord(1)} * Time_1 + \beta_{tint2} * X_{pois(1)} * Time_1 + \beta_{t} * Time_1 + E_1   
\end{split}   
(\#eq:System1)
\end{equation}      

\begin{equation}  
\begin{split}   
Y_2 &= \beta_0 + \beta_1 * X_{ord(1)} + \beta_2 * X_{cont(21)} + \beta_3 * X_{mix(21)} + \beta_4 * X_{pois(1)} + \beta_5 * X_{nb(21)} + \beta_{int} * X_{ord(1)} * X_{pois(1)} \\ 
&+ \beta_{subj1} * X_{ord(1)} * X_{cont(21)} + \beta_{subj2} * X_{pois(1)} * X_{cont(21)} + \beta_{subj3} * X_{ord(1)} * X_{pois(1)} * X_{cont(21)} \\
&+ \beta_{subj4} * X_{ord(1)} * X_{mix(21)} + \beta_{subj5} * X_{pois(1)} * X_{mix(21)} + \beta_{subj6} * X_{ord(1)} * X_{pois(1)} * X_{mix(21)} \\
&+ \beta_{subj7} * X_{ord(1)} * X_{nb(21)} + \beta_{subj8} * X_{pois(1)} * X_{nb(21)} + \beta_{subj9} * X_{ord(1)} * X_{pois(1)} * X_{nb(21)} \\
&+ \beta_{tint1} * X_{ord(1)} * Time_2 + \beta_{tint2} * X_{pois(1)} * Time_2 + \beta_{t} * Time_2 + E_2   
\end{split}   
(\#eq:System2)
\end{equation}      

\begin{equation}  
\begin{split}   
Y_3 &= \beta_0 + \beta_1 * X_{ord(1)} + \beta_2 * X_{cont(31)} + \beta_3 * X_{mix(31)} + \beta_4 * X_{pois(1)} + \beta_5 * X_{nb(31)} + \beta_{int} * X_{ord(1)} * X_{pois(1)} \\ 
&+ \beta_{subj1} * X_{ord(1)} * X_{cont(31)} + \beta_{subj2} * X_{pois(1)} * X_{cont(31)} + \beta_{subj3} * X_{ord(1)} * X_{pois(1)} * X_{cont(31)} \\
&+ \beta_{subj4} * X_{ord(1)} * X_{mix(31)} + \beta_{subj5} * X_{pois(1)} * X_{mix(31)} + \beta_{subj6} * X_{ord(1)} * X_{pois(1)} * X_{mix(31)} \\
&+ \beta_{subj7} * X_{ord(1)} * X_{nb(31)} + \beta_{subj8} * X_{pois(1)} * X_{nb(31)} + \beta_{subj9} * X_{ord(1)} * X_{pois(1)} * X_{nb(31)} \\
&+ \beta_{tint1} * X_{ord(1)} * Time_3 + \beta_{tint2} * X_{pois(1)} * Time_3 + \beta_{t} * Time_3 + E_3 
\end{split}   
(\#eq:System3)
\end{equation}      

```{r}
library("SimRepeat")
library("printr")
library("nlme")
library("reshape2")
options(scipen = 999)
```

### Step 1: Set up parameter inputs {-}

This is the most time-consuming part of the simulation process.  It is important to read the function documentation carefully to understand the formats for each parameter input.  Incorrect formatting will lead to errors.  Most of these can be prevented by using the `checkpar` function in **Step 2**.   

```{r}
seed <- 137
n <- 10000
M <- 3

# Ordinal variable
marginal <- lapply(seq_len(M), function(x) list(c(1/3, 2/3)))
support <- lapply(seq_len(M), function(x) list(c(0, 1, 2)))

# Non-mixture continuous variables
method <- "Polynomial"
Stcum1 <- calc_theory("Rice", c(1, 0.5))
Stcum2 <- calc_theory("Rice", c(2, 2))
Stcum3 <- calc_theory("Rice", c(4, 8))

# Error terms
error_type <- "non_mix"
Error1 <- calc_theory("Skewnormal", c(0, 1, 1))
Error2 <- calc_theory("Skewnormal", c(0, 1, 5))
Error3 <- calc_theory("Skewnormal", c(0, 1, 25))
corr.e <- matrix(c(1, 0.4, 0.4^2, 0.4, 1, 0.4, 0.4^2, 0.4, 1), M, M, 
  byrow = TRUE)

skews <- list(c(Stcum1[3], Error1[3]), c(Stcum2[3], Error2[3]), 
  c(Stcum3[3], Error3[3]))
skurts <- list(c(Stcum1[4], Error1[4]), c(Stcum2[4], Error2[4]), 
  c(Stcum3[4], Error3[4]))
fifths <- list(c(Stcum1[5], Error1[5]), c(Stcum2[5], Error2[5]), 
  c(Stcum3[5], Error3[5]))
sixths <- list(c(Stcum1[6], Error1[6]), c(Stcum2[6], Error2[6]), 
  c(Stcum3[6], Error3[6]))
Six <- list(list(0.08, 0.06), list(0.12, NULL), list(0.36, 0.15))

# Mixture continuous variable
mix_pis <- lapply(seq_len(M), function(x) list(c(0.3, 0.7)))
mix_mus <- lapply(seq_len(M), function(x) list(c(-5, 3)))
mix_sigmas <- lapply(seq_len(M), function(x) list(c(2, 1)))
mix_skews <- mix_skurts <- mix_fifths <- mix_sixths <- 
  lapply(seq_len(M), function(x) list(c(0, 0)))
mix_Six <- list()
Nstcum <- calc_mixmoments(mix_pis[[1]][[1]], mix_mus[[1]][[1]], 
  mix_sigmas[[1]][[1]], mix_skews[[1]][[1]], mix_skurts[[1]][[1]], 
  mix_fifths[[1]][[1]], mix_sixths[[1]][[1]])

means <- list(c(Stcum1[1], Nstcum[1], 0),
  c(Stcum2[1], Nstcum[1], 0),
  c(Stcum3[1], Nstcum[1], 0))
vars <- list(c(Stcum1[2]^2, Nstcum[2]^2, Error1[2]^2),
  c(Stcum2[2]^2, Nstcum[2]^2, Error2[2]^2),
  c(Stcum3[2]^2, Nstcum[2]^2, Error3[2]^2))

# Poisson variable
lam <- list(15, 15, 15)
p_zip <- 0.10

# Negative Binomial variables
size <- list(10, 10, 10)
mu <- list(3, 4, 5)
prob <- list()
p_zinb <- 0

# X_ord(1) and X_pois(1) are the same across Y
same.var <- c(1, 5)

# set up X correlation matrix
corr.x <- list()
corr.x[[1]] <- list(matrix(0.4, 6, 6), matrix(0.35, 6, 6), matrix(0.25, 6, 6))
diag(corr.x[[1]][[1]]) <- 1
# set correlations between components of X_mix(11) to 0
corr.x[[1]][[1]][3:4, 3:4] <- diag(2)
# set correlations between time-varying covariates of Y1 and Y2
corr.x[[1]][[2]][2, 2] <- 0.5
corr.x[[1]][[2]][3:4, 3:4] <- matrix(0.4, 2, 2)
corr.x[[1]][[2]][6, 6] <- 0.3
# set correlations between time-varying covariates of Y1 and Y3
corr.x[[1]][[3]][2, 2] <- 0.5^2
corr.x[[1]][[3]][3:4, 3:4] <- matrix(0.4^2, 2, 2)
corr.x[[1]][[3]][6, 6] <- 0.3^2
# set correlations for the same variables equal across outcomes
corr.x[[1]][[2]][, same.var] <- corr.x[[1]][[3]][, same.var] <-
  corr.x[[1]][[1]][, same.var]

corr.x[[2]] <- list(t(corr.x[[1]][[2]]), matrix(0.35, 6, 6), 
  matrix(0.25, 6, 6))
diag(corr.x[[2]][[2]]) <- 1
# set correlations between components of X_mix(21) to 0
corr.x[[2]][[2]][3:4, 3:4] <- diag(2)
# set correlations between time-varying covariates of Y2 and Y3
corr.x[[2]][[3]][2, 2] <- 0.5
corr.x[[2]][[3]][3:4, 3:4] <- matrix(0.4, 2, 2)
corr.x[[2]][[3]][6, 6] <- 0.3
# set correlations for the same variables equal across outcomes
corr.x[[2]][[2]][same.var, ] <- corr.x[[1]][[2]][same.var, ]
corr.x[[2]][[2]][, same.var] <- corr.x[[2]][[3]][, same.var] <- 
  t(corr.x[[1]][[2]][same.var, ])
corr.x[[2]][[3]][same.var, ] <- corr.x[[1]][[3]][same.var, ]

corr.x[[3]] <- list(t(corr.x[[1]][[3]]), t(corr.x[[2]][[3]]), 
  matrix(0.3, 6, 6))
diag(corr.x[[3]][[3]]) <- 1
# set correlations between components of X_mix(31) to 0
corr.x[[3]][[3]][3:4, 3:4] <- diag(2)
# set correlations for the same variables equal across outcomes
corr.x[[3]][[3]][same.var, ] <- corr.x[[1]][[3]][same.var, ]
corr.x[[3]][[3]][, same.var] <- t(corr.x[[3]][[3]][same.var, ])

Time <- 1:M
betas.0 <- 0
betas.t <- 1
# use a list of length 1 so that betas are the same across Y
betas <- list(seq(0.5, 1.5, 0.25))
# interaction between ordinal and Poisson variable, becomes 
# another group-level variable
int.var <- matrix(c(1, 1, 4, 2, 1, 4, 3, 1, 4), 3, 3, byrow = TRUE)
betas.int <- list(0.5)
# continuous non-mixture, continuous mixture, and NB variables are 
# subject-level variables
subj.var <- matrix(c(1, 2, 1, 3, 1, 5, 2, 2, 2, 3, 2, 5, 3, 2, 3, 3, 3, 5), 
  nrow = 9, ncol = 2, byrow = TRUE)
# there are 3 subject-level variables and 3 group-level variables forming 
# 9 group-subject interactions
betas.subj <- list(seq(0.5, 0.5 + (9 - 1) * 0.1, 0.1))
# only ordinal and Poisson variable interact with time (excluding the 
# ordinal-Poisson interaction variable)
tint.var <- matrix(c(1, 1, 1, 4, 2, 1, 2, 4, 3, 1, 3, 4), 6, 2, byrow = TRUE)
betas.tint <- list(c(0.25, 0.5))
```

### Step 2: Check parameter inputs {-}

```{r}
checkpar(M, method, error_type, means, vars, skews, skurts, fifths, sixths, 
  Six, mix_pis, mix_mus, mix_sigmas, mix_skews, mix_skurts, mix_fifths, 
  mix_sixths, mix_Six, marginal, support, lam, p_zip, pois_eps = list(), 
  size, prob, mu, p_zinb, nb_eps = list(), corr.x, corr.yx = list(), corr.e, 
  same.var, subj.var, int.var, tint.var, betas.0, betas, betas.subj, betas.int, 
  betas.t, betas.tint, quiet = TRUE)
```

### Step 3: Generate system {-}

Note that `use.nearPD = FALSE` and `adjgrad = FALSE` so that negative eigen-values will be replaced with `eigmin` (default $0$) instead of using the nearest positive-definite matrix (found with @Matrix's `Matrix::nearPD` function by @Higham's algorithm) or the adjusted gradient updating method via `adj_grad` [@YinZhang1;@YinZhang2;@Maree].    

```{r}
Sys1 <- corrsys(n, M, Time, method, error_type, means, vars,
  skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas, mix_skews,
  mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal, support, lam, p_zip,
  size, prob, mu, p_zinb, corr.x, corr.e, same.var, subj.var, int.var,
  tint.var, betas.0, betas, betas.subj, betas.int, betas.t, betas.tint,
  seed = seed, use.nearPD = FALSE, quiet = TRUE)
```
```{r}
knitr::kable(Sys1$constants[[1]], booktabs = TRUE, 
  caption = "PMT constants for Y_1")
Sys1$valid.pdf
```

### Step 4: Describe results {-}

```{r}
Sum1 <- summary_sys(Sys1$Y, Sys1$E, E_mix = NULL, Sys1$X, Sys1$X_all, M, 
  method, means, vars, skews, skurts, fifths, sixths, mix_pis, mix_mus, 
  mix_sigmas, mix_skews, mix_skurts, mix_fifths, mix_sixths, marginal, 
  support, lam, p_zip, size, prob, mu, p_zinb, corr.x, corr.e)
names(Sum1)
```
```{r}
knitr::kable(Sum1$cont_sum_y, digits = 3, booktabs = TRUE, 
  caption = "Simulated Distributions of Outcomes")
```
```{r}
knitr::kable(Sum1$target_sum_e, digits = 3, booktabs = TRUE, 
  caption = "Target Distributions of Error Terms")
```
```{r}
knitr::kable(Sum1$cont_sum_e, digits = 3, booktabs = TRUE, 
  caption = "Simulated Distributions of Error Terms")
```
```{r}
knitr::kable(Sum1$target_sum_x, digits = 3, booktabs = TRUE, 
  caption = "Target Distributions of Continuous Non-Mixture and Components of 
  Mixture Variables")
```
```{r}
knitr::kable(Sum1$cont_sum_x, digits = 3, booktabs = TRUE, 
  caption = "Simulated Distributions of Continuous Non-Mixture and Components 
  of Mixture Variables")
```
```{r}
knitr::kable(Sum1$target_mix_x, digits = 3, booktabs = TRUE, 
  caption = "Target Distributions of Continuous Mixture Variables")
```
```{r}
knitr::kable(Sum1$mix_sum_x, digits = 3, booktabs = TRUE, 
  caption = "Simulated Distributions of Continuous Mixture Variables")
```

```{r}
Nplot <- plot_simpdf_theory(sim_y = Sys1$X_all[[1]][, 3], ylower = -10, 
  yupper = 10, 
  title = "PDF of X_mix(21): N(-5, 2) and N(3, 1) Mixture",
  fx = function(x) mix_pis[[1]][[1]][1] * dnorm(x, mix_mus[[1]][[1]][1], 
    mix_sigmas[[1]][[1]][1]) + mix_pis[[1]][[1]][2] * 
    dnorm(x, mix_mus[[1]][[1]][2], mix_sigmas[[1]][[1]][2]), 
  lower = -Inf, upper = Inf)
Nplot
```

Summary of Ordinal Variable: (for $Y_1$)       

```{r}
knitr::kable(Sum1$ord_sum_x[[1]][1:2, ], digits = 3, row.names = FALSE,
             booktabs = TRUE, caption = "Simulated Distribution of X_ord(1)")
```

Summary of Poisson Variable:    

```{r}
knitr::kable(Sum1$pois_sum_x, digits = 3, row.names = FALSE,
             booktabs = TRUE, caption = "Simulated Distribution of X_pois(1)")
Pplot <- plot_simpdf_theory(sim_y = Sys1$X_all[[1]][, 4], 
  title = "PMF of X_pois(1): Zero-Inflated Poisson Distribution", 
  Dist = "Poisson", params = c(lam[[1]][1], p_zip), cont_var = FALSE)
Pplot
```

Summary of Negative Binomial Variables $X_{nb(11)}, X_{nb(21)},$ and $X_{nb(31)}$:    

```{r}
knitr::kable(Sum1$nb_sum_x, digits = 3, row.names = FALSE,
             booktabs = TRUE, caption = "Simulated Distributions")
NBplot <- plot_simtheory(sim_y = Sys1$X_all[[1]][, 5], binwidth = 0.5, 
  title = "Simulated Values for X_nb(11)", Dist = "Negative_Binomial", 
  params = c(size[[1]][1], mu[[1]][1], p_zinb), cont_var = FALSE)
NBplot
```

Maximum Correlation Errors for X Variables by Outcome:   

```{r}
maxerr <- do.call(rbind, Sum1$maxerr)
rownames(maxerr) <- colnames(maxerr) <- paste("Y", 1:M, sep = "")
knitr::kable(as.data.frame(maxerr), digits = 5, booktabs = TRUE, 
  caption = "Maximum Correlation Errors for X Variables")
```

### Linear model {-}

A linear model will be fit to the data using `glm` in order to see if the slope coefficients can be recovered [@Stats].  First, the data is reshaped into long format using `reshape2::melt` [@Reshape2].  Note that since $X_{ord(1)}$ and $X_{pois(1)}$ are the same for each outcome, they will be used as factors (`id.vars`) and are only needed once.      

```{r}
data1 <- as.data.frame(cbind(factor(1:n), Sys1$Y, Sys1$X_all[[1]][, 1:5],
  Sys1$X_all[[2]][, c(2, 3, 5)], Sys1$X_all[[3]][, c(2, 3, 5)]))
colnames(data1)[1] <- "Subject"
data1.a <- melt(data1[, c("Subject", "ord1_1", "pois1_1", "Y1", "Y2", "Y3")], 
  id.vars = c("Subject", "ord1_1", "pois1_1"),
  measure.vars = c("Y1", "Y2", "Y3"), variable.name = "Time", value.name = "Y")
data1.b <- melt(data1[, c("Subject", "cont1_1", "cont2_1", "cont3_1")],
  id.vars = c("Subject"), variable.name = "Time", value.name = "cont1")
data1.c <- melt(data1[, c("Subject", "mix1_1", "mix2_1", "mix3_1")],
  id.vars = c("Subject"), variable.name = "Time", value.name = "mix1")
data1.d <- melt(data1[, c("Subject", "nb1_1", "nb2_1", "nb3_1")],
  id.vars = c("Subject"), variable.name = "Time", value.name = "nb1")
data1.a$Time <- data1.b$Time <- data1.c$Time <- data1.d$Time <- 
  c(rep(1, n), rep(2, n), rep(3, n))
data1 <- merge(merge(merge(data1.a, data1.b, by = c("Subject", "Time")), 
  data1.c, by = c("Subject", "Time")), data1.d, by = c("Subject", "Time"))
```

Errors $E_1, E_2,$ and $E_3$ modeled as having Normal distributions:   

```{r}
fm1 <- glm(Y ~ ord1_1 + cont1 + mix1 + pois1_1 + nb1 + ord1_1:pois1_1 + 
  ord1_1:cont1 + pois1_1:cont1 + ord1_1:pois1_1:cont1 + 
  ord1_1:mix1 + pois1_1:mix1 + ord1_1:pois1_1:mix1 + 
  ord1_1:nb1 + pois1_1:nb1 + ord1_1:pois1_1:nb1 + 
  Time + ord1_1:Time + pois1_1:Time, data = data1)
summary(fm1)
```

Each effect in the model was found to be statistically significant at the $\alpha = 0.001$ level.  Now, compare betas used in simulation to those returned by `glm`:  

```{r}
fm1.coef <- fm1$coefficients[c("(Intercept)", "ord1_1", "cont1", "mix1", 
  "pois1_1", "nb1", "ord1_1:pois1_1", "Time", "ord1_1:cont1", "cont1:pois1_1", 
  "ord1_1:cont1:pois1_1", "ord1_1:mix1", "mix1:pois1_1", 
  "ord1_1:mix1:pois1_1", "ord1_1:nb1", "pois1_1:nb1", 
  "ord1_1:pois1_1:nb1", "ord1_1:Time", "pois1_1:Time")]
coef <- rbind(c(betas.0, betas[[1]], betas.int[[1]], betas.t, 
  betas.subj[[1]], betas.tint[[1]]), fm1.coef)
colnames(coef) <- names(fm1.coef)
rownames(coef) <- c("Simulated", "Estimated")
knitr::kable(as.data.frame(coef[, 1:6]), digits = 3, booktabs = TRUE, 
  caption = "Beta Coefficients for Repeated Measures Model 1")
knitr::kable(as.data.frame(coef[, 7:12]), digits = 3, booktabs = TRUE)
knitr::kable(as.data.frame(coef[, 13:19]), digits = 3, booktabs = TRUE)
```

All of the slope coefficients are estimated well.

## Example 2: System from Example 1 with no independent variables for $Y_2$ {-}     

This example uses the `corrsys2` function which employs **correlation method 2**.  It requires the additional parameters `pois_eps` and `nb_eps`, which default to $0.0001$ for each variable.    

### Step 1: Set up parameter inputs {-}

```{r}
seed <- 137
n <- 10000
M <- 3

# Ordinal variable
marginal <- list(list(c(1/3, 2/3)), NULL, list(c(1/3, 2/3)))
support <- list(list(c(0, 1, 2)), NULL, list(c(0, 1, 2)))

# Non-mixture continuous variables
skews <- list(c(Stcum1[3], Error1[3]), Error2[3], 
  c(Stcum3[3], Error3[3]))
skurts <- list(c(Stcum1[4], Error1[4]), Error2[4],
  c(Stcum3[4], Error3[4]))
fifths <- list(c(Stcum1[5], Error1[5]), Error2[5],
  c(Stcum3[5], Error3[5]))
sixths <- list(c(Stcum1[6], Error1[6]), Error2[6], 
  c(Stcum3[6], Error3[6]))
Six <- list(list(0.08, 0.06), NULL, list(0.36, 0.15))

# Mixture continuous variable
mix_pis <- list(list(c(0.3, 0.7)), NULL, list(c(0.3, 0.7)))
mix_mus <- list(list(c(-5, 3)), NULL, list(c(-5, 3)))
mix_sigmas <- list(list(c(2, 1)), NULL, list(c(2, 1)))
mix_skews <- mix_skurts <- mix_fifths <- mix_sixths <- 
  list(list(c(0, 0)), NULL, list(c(0, 0)))
mix_Six <- list()

means <- list(c(Stcum1[1], Nstcum[1], Error1[1]), Error2[1], 
  c(Stcum3[1], Nstcum[1], Error3[1]))
vars <- list(c(Stcum1[2]^2, Nstcum[2]^2, Error1[2]^2), Error2[2]^2,
  c(Stcum3[2]^2, Nstcum[2]^2, Error3[2]^2))

# Poisson variable
lam <- list(15, NULL, 15)
p_zip <- 0.10

# Negative Binomial variables
size <- list(10, NULL, 10)
mu <- list(3, NULL, 5)
prob <- list()
p_zinb <- 0

# X_ord(1) and X_pois(1) are the same for Y_1 and Y_3
same.var <- matrix(c(1, 1, 3, 1, 1, 5, 3, 5), 2, 4, byrow = TRUE)

# set up X correlation matrix
corr.x <- list()
corr.x[[1]] <- list(matrix(0.4, 6, 6), NULL, matrix(0.25, 6, 6))
diag(corr.x[[1]][[1]]) <- 1
# set correlations between components of X_mix(11) to 0
corr.x[[1]][[1]][3:4, 3:4] <- diag(2)
# set correlations between time-varying covariates of Y1 and Y3
corr.x[[1]][[3]][2, 2] <- 0.5^2
corr.x[[1]][[3]][3:4, 3:4] <- matrix(0.4^2, 2, 2)
corr.x[[1]][[3]][6, 6] <- 0.3^2
# set correlations for the same variables equal across outcomes
corr.x[[1]][[3]][, c(1, 5)] <- corr.x[[1]][[1]][, c(1, 5)]

corr.x[[3]] <- list(t(corr.x[[1]][[3]]), NULL, matrix(0.3, 6, 6))
diag(corr.x[[3]][[3]]) <- 1
# set correlations between components of X_mix(31) to 0
corr.x[[3]][[3]][3:4, 3:4] <- diag(2)
# set correlations for the same variables equal across outcomes
corr.x[[3]][[3]][c(1, 5), ] <- corr.x[[1]][[3]][c(1, 5), ]
corr.x[[3]][[3]][, c(1, 5)] <- t(corr.x[[3]][[3]][c(1, 5), ])

Time <- 1:M
betas.0 <- 0
betas.t <- 1
betas <- list(seq(0.5, 1.5, 0.25), NULL, seq(0.5, 1.5, 0.25))
# interaction between ordinal and Poisson variable, becomes 
# another group-level variable
int.var <- matrix(c(1, 1, 4, 3, 1, 4), 2, 3, byrow = TRUE)
betas.int <- list(0.5, NULL, 0.5)
# continuous non-mixture, continuous mixture, and NB variables are 
# subject-level variables
subj.var <- matrix(c(1, 2, 1, 3, 1, 5, 3, 2, 3, 3, 3, 5), 
  nrow = 6, ncol = 2, byrow = TRUE)
# there are 3 subject-level variables and 3 group-level variables forming 
# 9 group-subject interactions
betas.subj <- list(seq(0.5, 0.5 + (9 - 1) * 0.1, 0.1), NULL, 
  seq(0.5, 0.5 + (9 - 1) * 0.1, 0.1))
# only ordinal and Poisson variable interact with time (excluding the 
# ordinal-Poisson interaction variable)
tint.var <- matrix(c(1, 1, 1, 4, 3, 1, 3, 4), 4, 2, byrow = TRUE)
betas.tint <- list(c(0.25, 0.5), NULL, c(0.25, 0.5))
```

### Step 2: Check parameter inputs {-}         

```{r}
checkpar(M, method, error_type, means, vars, skews, skurts, fifths, sixths, 
  Six, mix_pis, mix_mus, mix_sigmas, mix_skews, mix_skurts, mix_fifths, 
  mix_sixths, mix_Six, marginal, support, lam, p_zip, pois_eps = list(), 
  size, prob, mu, p_zinb, nb_eps = list(), corr.x, corr.yx = list(), corr.e, 
  same.var, subj.var, int.var, tint.var, betas.0, betas, betas.subj, betas.int, 
  betas.t, betas.tint, quiet = TRUE)
```

### Step 3: Generate system {-}

```{r}
Sys2 <- corrsys2(n, M, Time, method, error_type, means, vars,
  skews, skurts, fifths, sixths, Six, mix_pis, mix_mus, mix_sigmas, mix_skews,
  mix_skurts, mix_fifths, mix_sixths, mix_Six, marginal, support, lam, p_zip,
  pois_eps = list(), size, prob, mu, p_zinb, nb_eps = list(), corr.x, corr.e, 
  same.var, subj.var, int.var, tint.var, betas.0, betas, betas.subj, betas.int, 
  betas.t, betas.tint, seed = seed, use.nearPD = FALSE, quiet = TRUE)
```

### Step 4: Describe results {-}

```{r}
Sum2 <- summary_sys(Sys2$Y, Sys2$E, E_mix = NULL, Sys2$X, Sys2$X_all, M, 
  method, means, vars, skews, skurts, fifths, sixths, mix_pis, mix_mus, 
  mix_sigmas, mix_skews, mix_skurts, mix_fifths, mix_sixths, marginal, 
  support, lam, p_zip, size, prob, mu, p_zinb, corr.x, corr.e)
names(Sum2)
```
```{r}
knitr::kable(Sum2$cont_sum_y, digits = 3, booktabs = TRUE, 
  caption = "Simulated Distributions of Outcomes")
```
```{r}
knitr::kable(Sum2$target_sum_e, digits = 3, booktabs = TRUE, 
  caption = "Target Distributions of Error Terms")
```
```{r}
knitr::kable(Sum2$cont_sum_e, digits = 3, booktabs = TRUE, 
  caption = "Simulated Distributions of Error Terms")
```
```{r}
knitr::kable(Sum2$target_sum_x, digits = 3, booktabs = TRUE, 
  caption = "Target Distributions of Continuous Non-Mixture and Components of 
  Mixture Variables")
```
```{r}
knitr::kable(Sum2$cont_sum_x, digits = 3, booktabs = TRUE, 
  caption = "Simulated Distributions of Continuous Non-Mixture and Components 
  of Mixture Variables")
```
```{r}
knitr::kable(Sum2$target_mix_x, digits = 3, booktabs = TRUE, 
  caption = "Target Distributions of Continuous Mixture Variables")
```
```{r}
knitr::kable(Sum2$mix_sum_x, digits = 3, booktabs = TRUE, 
  caption = "Simulated Distributions of Continuous Mixture Variables")
```

Summary of Ordinal Variable: (for $Y_1$)       

```{r}
knitr::kable(Sum2$ord_sum_x[[1]][1:2, ], digits = 3, row.names = FALSE,
  booktabs = TRUE, caption = "Simulated Distribution of X_ord(1)")
```

Summary of Poisson Variable:    

```{r}
knitr::kable(Sum2$pois_sum_x, digits = 3, row.names = FALSE,
  booktabs = TRUE, caption = "Simulated Distribution of X_pois(1)")
```

Summary of Negative Binomial Variables $X_{nb(11)}, X_{nb(21)},$ and $X_{nb(31)}$:    

```{r}
knitr::kable(Sum2$nb_sum_x, digits = 3, row.names = FALSE,
  booktabs = TRUE, caption = "Simulated Distributions")
```

Maximum Correlation Errors for X Variables by Outcome:   

```{r}
maxerr <- rbind(Sum2$maxerr[[1]][-2], Sum2$maxerr[[3]][-2])
rownames(maxerr) <- colnames(maxerr) <- c("Y1", "Y3")
knitr::kable(as.data.frame(maxerr), digits = 5, booktabs = TRUE, 
  caption = "Maximum Correlation Errors for X Variables")
```

## Example 3: System of 4 equations with random intercept and random slope for time {-}     

\begin{equation}   
\begin{split}    
Y_1 &= \beta_0 + \beta_1 * X_{O1} + \beta_2 * X_{C11} +\beta_{\rm{subj1}} * X_{O1} * X_{C11} + \beta_{\rm{tint1}} * X_{O1} * Time_1 + \beta_{\rm{t}} * Time_1 + U_0 + U_1 * Time_1 + E_1 \\
Y_2 &= \beta_0 + \beta_1 * X_{O1} + \beta_2 * X_{C21} +\beta_{\rm{subj1}} * X_{O1} * X_{C21} + \beta_{\rm{tint1}} * X_{O1} * Time_2 + \beta_{\rm{t}} * Time_2 + U_0 + U_1 * Time_2 + E_2 \\
Y_3 &= \beta_0 + \beta_1 * X_{O1} + \beta_2 * X_{C31} +\beta_{\rm{subj1}} * X_{O1} * X_{C31} + \beta_{\rm{tint1}} * X_{O1} * Time_3 + \beta_{\rm{t}} * Time_3 + U_0 + U_1 * Time_3 + E_3 \\
Y_4 &= \beta_0 + \beta_1 * X_{O1} + \beta_2 * X_{C41} +\beta_{\rm{subj1}} * X_{O1} * X_{C41} + \beta_{\rm{tint1}} * X_{O1} * Time_4 + \beta_{\rm{t}} * Time_4 + U_0 + U_1 * Time_4 + E_4
\end{split}   
(\#eq:System2)
\end{equation}      

### Description of Variables {-}

1) Ordinal variable $X_{O1}$, where $\Pr[X_{O1} = 0] = 0.2$, $\Pr[X_{O1} = 1] = 0.35$, and $\Pr[X_{O1} = 2] = 0.45$, is a group-level variable and is static across equations.
2) Continuous non-mixture variable $X_{C1}$ is a subject-level variable with a Logistic(0, 1) distribution, which requires a sixth cumulant correction of 1.75.
3) $X$ terms are correlated at 0.1 within an equation and have an AR(1) structure across equations.  The correlations for the static variable are held constant across equations.
4) Random intercept $U_0$ and time slope $U_1$ with Normal(0, 1) distributions.  Correlation between random effects is 0.3.
5) The error terms have $t$(10) distributions (mean 0, variance 1) and an AR(1, 0.4) correlation structure.

In this example, the random intercept and time slope have continuous non-mixture distributions for all $Y$.  However, the functions `corrsys` and `corrsys2` permit a combination of none, non-mixture, and mixture distributions across the $Y$ (i.e., if `rand.int = c("non_mix", "mix", "none")` then the random intercept for $Y_1$ has a non-mixture, and the random intercept for $Y_2$ has a mixture distribution; there is no random intercept for $Y_3$).  In addition, the distributions themselves can vary across outcomes.  This is also true for random effects assigned to independent variables as specified in `rand.var`.

### Step 1: Set up parameter inputs {-}        

```{r}
seed <- 1
n <- 10000
M <- 4

# Binary variable
marginal <- lapply(seq_len(M), function(x) list(c(0.2, 0.55)))
support <- lapply(seq_len(M), function(x) list(0:2))

same.var <- 1
subj.var <- matrix(c(1, 2, 2, 2, 3, 2, 4, 2), 4, 2, byrow = TRUE)

# create list of X correlation matrices
corr.x <- list()

rho1 <- 0.1
rho2 <- 0.5
rho3 <- rho2^2
rho4 <- rho2^3
# Y_1
corr.x[[1]] <- list(matrix(rho1, 2, 2), matrix(rho2, 2, 2), matrix(rho3, 2, 2),
  matrix(rho4, 2, 2))
diag(corr.x[[1]][[1]]) <- 1
# set correlations for the same variables equal across outcomes
corr.x[[1]][[2]][, same.var] <- corr.x[[1]][[3]][, same.var] <-
  corr.x[[1]][[4]][, same.var] <- corr.x[[1]][[1]][, same.var]

# Y_2
corr.x[[2]] <- list(t(corr.x[[1]][[2]]), matrix(rho1, 2, 2),
  matrix(rho2, 2, 2), matrix(rho3, 2, 2))
diag(corr.x[[2]][[2]]) <- 1
# set correlations for the same variables equal across outcomes
corr.x[[2]][[2]][same.var, ] <- corr.x[[1]][[2]][same.var, ]
corr.x[[2]][[2]][, same.var] <- corr.x[[2]][[3]][, same.var] <-
  corr.x[[2]][[4]][, same.var] <- t(corr.x[[1]][[2]][same.var, ])
corr.x[[2]][[3]][same.var, ] <- corr.x[[1]][[3]][same.var, ]
corr.x[[2]][[4]][same.var, ] <- corr.x[[1]][[4]][same.var, ]

# Y_3
corr.x[[3]] <- list(t(corr.x[[1]][[3]]), t(corr.x[[2]][[3]]),
  matrix(rho1, 2, 2), matrix(rho2, 2, 2))
diag(corr.x[[3]][[3]]) <- 1
# set correlations for the same variables equal across outcomes
corr.x[[3]][[3]][same.var, ] <- corr.x[[1]][[3]][same.var, ]
corr.x[[3]][[3]][, same.var] <- t(corr.x[[3]][[3]][same.var, ])
corr.x[[3]][[4]][same.var, ] <- corr.x[[1]][[4]][same.var, ]
corr.x[[3]][[4]][, same.var] <- t(corr.x[[1]][[3]][same.var, ])

# Y_4
corr.x[[4]] <- list(t(corr.x[[1]][[4]]), t(corr.x[[2]][[4]]),
  t(corr.x[[3]][[4]]), matrix(rho1, 2, 2))
diag(corr.x[[4]][[4]]) <- 1
# set correlations for the same variables equal across outcomes
corr.x[[4]][[4]][same.var, ] <- corr.x[[1]][[4]][same.var, ]
corr.x[[4]][[4]][, same.var] <- t(corr.x[[4]][[4]][same.var, ])

# create error term correlation matrix
corr.e <- matrix(c(1, 0.4, 0.4^2, 0.4^3,
                   0.4, 1, 0.4, 0.4^2,
                   0.4^2, 0.4, 1, 0.4,
                   0.4^3, 0.4^2, 0.4, 1), M, M, byrow = TRUE)

Log <- calc_theory("Logistic", c(0, 1))
t10 <- calc_theory("t", 10)

# Continuous variables: 1st non-mixture, 2nd error terms
means <- lapply(seq_len(M), function(x) c(Log[1], 0))
vars <- lapply(seq_len(M), function(x) c(Log[2]^2, 1))
skews <- lapply(seq_len(M), function(x) c(Log[3], t10[3]))
skurts <- lapply(seq_len(M), function(x) c(Log[4], t10[4]))
fifths <- lapply(seq_len(M), function(x) c(Log[5], t10[5]))
sixths <- lapply(seq_len(M), function(x) c(Log[6], t10[6]))
Six <- lapply(seq_len(M), function(x) list(1.75, NULL))

## RANDOM EFFECTS
rand.int <- "non_mix" # random intercept
rand.tsl <- "non_mix" # random time slope
rand.var <- NULL # no additional random effects

rmeans <- rskews <- rskurts <- rfifths <- rsixths <- c(0, 0)
rvars <- c(1, 1)
rSix <- list(NULL, NULL)

# append parameters for random effect distributions to parameters for
# continuous fixed effects and error terms
means <- append(means, list(rmeans))
vars <- append(vars, list(rvars))
skews <- append(skews, list(rskews))
skurts <- append(skurts, list(rskurts))
fifths <- append(fifths, list(rfifths))
sixths <- append(sixths, list(rsixths))
Six <- append(Six, list(rSix))

# use a list of length 1 so that betas are the same across Y
betas <- list(c(1, 1))
betas.subj <- list(0.5)
betas.tint <- list(0.75)

# set up correlation matrix for random effects
corr.u <- matrix(c(1, 0.3, 0.3, 1), 2, 2)
```

### Step 2: Check parameter inputs {-}         

```{r}
checkpar(M, "Polynomial", "non_mix", means, vars, skews, skurts, fifths,
  sixths, Six, marginal = marginal, support = support, corr.x = corr.x,
  corr.e = corr.e, same.var = same.var, subj.var = subj.var, betas = betas,
  betas.subj = betas.subj, betas.tint = betas.tint, rand.int = rand.int,
  rand.tsl = rand.tsl, corr.u = corr.u, quiet = TRUE)
```

### Step 3: Generate system {-}

```{r}
Sys3 <- corrsys(n, M, Time = NULL, "Polynomial", "non_mix", means, vars,
  skews, skurts, fifths, sixths, Six, marginal = marginal, support = support,
  corr.x = corr.x, corr.e = corr.e, same.var = same.var, subj.var = subj.var,
  betas = betas, betas.subj = betas.subj, betas.tint = betas.tint,
  rand.int = rand.int, rand.tsl = rand.tsl, corr.u = corr.u, seed = seed,
  use.nearPD = FALSE, quiet = TRUE)
```

### Step 4: Describe results {-}

```{r}
Sum3 <- summary_sys(Sys3$Y, Sys3$E, E_mix = NULL, Sys3$X,
  Sys3$X_all, M, "Polynomial", means, vars, skews, skurts, fifths,
  sixths, marginal = marginal, support = support, corr.x = corr.x,
  corr.e = corr.e, U = Sys3$U, U_all = Sys3$U_all, rand.int = rand.int,
  rand.tsl = rand.tsl, corr.u = corr.u, rmeans2 = Sys3$rmeans2,
  rvars2 = Sys3$rvars2)
names(Sum3)
```
```{r}
knitr::kable(Sum3$cont_sum_y, digits = 3, booktabs = TRUE, 
  caption = "Simulated Distributions of Outcomes")
```
```{r}
knitr::kable(Sum3$target_sum_u, digits = 3, booktabs = TRUE, 
  caption = "Target Distributions of Random Effects")
```
```{r}
knitr::kable(Sum3$sum_uall, digits = 3, booktabs = TRUE, 
  caption = "Simulated Distributions of Random Effects")
```

Maximum Correlation Error for Random Effects:
```{r}
Sum3$maxerr_u
```

### Linear mixed model {-}

A linear mixed model will be fit to the data using `lme` from package **nlme** in order to see if the random effects are estimated according to the simulation parameters [@Nlme].  The data is again reshaped into long format using `reshape2::melt`.      

```{r}
data3 <- as.data.frame(cbind(factor(1:n), Sys3$Y,
  Sys3$X_all[[1]][, c(1:2, 5)], Sys3$X_all[[2]][, c(2, 5)],
  Sys3$X_all[[3]][, c(2, 5)], Sys3$X_all[[4]][, c(2, 5)]))
colnames(data3)[1] <- "Subject"
data3.a <- melt(data3[, c("Subject", "ord1_1", "Y1", "Y2", "Y3", "Y4")],
  id.vars = c("Subject", "ord1_1"),
  measure.vars = c("Y1", "Y2", "Y3", "Y4"), variable.name = "Time",
  value.name = "Y")
data3.b <- melt(data3[, c("Subject", "cont1_1", "cont2_1", "cont3_1",
                          "cont4_1")],
  id.vars = c("Subject"), variable.name = "Time", value.name = "cont1")
data3.a$Time <- data3.b$Time <- c(rep(1, n), rep(2, n), rep(3, n), rep(4, n))
data3 <- merge(data3.a, data3.b, by = c("Subject", "Time"))
```

Errors modeled as having Gaussian distributions with an AR(1) correlation structure:   

```{r}
fm3 <- lme(Y ~ ord1_1 * Time + ord1_1 * cont1,
  random = ~ Time | Subject, correlation = corAR1(), data = data3)
sum_fm3 <- summary(fm3)
```

Each effect in the model was again found to be statistically significant at the $\alpha = 0.001$ level.      

Now, compare betas used in simulation to those returned by `lme`:   

```{r}
fm3.coef <- as.data.frame(sum_fm3$tTable[c("(Intercept)",
  "ord1_1", "cont1", "Time", "ord1_1:cont1", "ord1_1:Time"), ])
coef <- cbind(c(betas.0, betas[[1]], betas.t, betas.subj[[1]], 
  betas.tint[[1]]), fm3.coef)
colnames(coef)[1] <- "Simulated"
knitr::kable(as.data.frame(coef), digits = 3, booktabs = TRUE, 
  caption = "Beta Coefficients for Repeated Measures Model 2")
```

Estimated standard deviation and AR(1) parameter for error terms:
```{r}
sum_fm3$sigma
coef(fm3$modelStruct$corStruct, unconstrained = FALSE)
```

Summary of estimated random effects:
```{r}
varcor <- VarCorr(fm3)
fm3.ranef <- data.frame(Cor = as.numeric(varcor[2, 3]),
  SD_int = as.numeric(varcor[1, 2]), SD_Tsl = as.numeric(varcor[2, 2]))
knitr::kable(fm3.ranef, digits = 3, booktabs = TRUE)
```

# References {-}

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