
---
title: "Covariance Structures"
output: rmarkdown::html_vignette
bibliography: lmer.bib
vignette: >
  %\VignetteIndexEntry{Covariance Structures}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
  %\VignetteDepends{glmmTMB}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```


## Introduction

The current version of `lme4` offers four covariance classes/structures: `Covariance.us` (unstructured), `Covariance.diag` (diagonal), `Covariance.cs` (compound symmetry), and `Covariance.ar1` (autoregressive order 1). The syntax for use is somewhat similar to the `glmmTMB` package (see [covariance structures in glmmTMB](https://glmmtmb.github.io/glmmTMB/articles/covstruct.html)), although the results are slightly different.

The first half of this vignette provide a more detailed explanation of how the 
new machinery works, and the second half of the vignette will compare `lme4` 
and `glmmTMB` implementations and results.

### Background

For exact details of this structure, refer to the `lmer` [vignette](https://cran.r-project.org/package=lme4/vignettes/lmer.pdf) [@Bates_JSS]. We provide
a quick summary in this vignette.

In matrix notation, a linear mixed model can be represented as:
\[
\mathbf{y} = \mathbf{X} \boldsymbol \beta + \mathbf Z \mathbf{b} + \boldsymbol{\epsilon}
\]
Where $\mathbf{b}$ represents an unknown vector of random effects. The 
\code{Covariance} class helps define the structure of the variance-covariance 
matrix as specified as $\textrm{Var}(\mathbf{b})$. Typically, we denote the 
variance-covariance matrix as $\mathbf \Sigma$.

<!-- can't use \usepackage{bm}: https://stackoverflow.com/a/33226711/190277 -->
First, we create the relative co-factor $\mathbf \Lambda_{\mathbf \theta}$ which is a 
$q \times q$ block diagonal  matrix that depends on the variance-component 
parameter $\theta$. Let $\sigma$ be the scale parameter of the variance of a 
linear mixed model. In `lme4`, the variance-covariance matrix is constructed by:
\[
{\mathbf \Sigma}_{\mathbf{\theta}} = 
\sigma^2 {\mathbf \Lambda}_{\mathbf \theta} {\mathbf \Lambda}_{\mathbf \theta}^{\top},
\]

For generalized linear mixed models, $\mathbf \Lambda$ instead represents the *unscaled* Cholesky factor; that is,
the scaling term $\sigma^2$ is omitted from the equation above.

The major difference between the four covariance classes (`Covariance.us` (unstructured), `Covariance.diag` (diagonal), `Covariance.cs` (compound symmetry), and `Covariance.ar1` (autoregressive order 1)) is the construction of the the relative Cholesky factor $\mathbf \Lambda_{\mathbf \theta}$.

### Covariance Structures

Suppose there are $p$ number of random effect terms for a particular
grouping variable. The *unstructured covariance*, which is the default 
in `lme4`, of size $p \times p$ has the following form:
\[
\mathbf{\Sigma}
= \begin{bmatrix}
\sigma^{2}_{1} & \sigma_{12} & \dots & \sigma_{1p} \\
\sigma_{21} & \sigma^{2}_{2} & \dots & \sigma_{2p} \\
\vdots & \vdots & \ddots & \vdots \\
\sigma_{p1} & \sigma_{p2} & \dots & \sigma^{2}_{p} \\
\end{bmatrix}	
\]

The next three covariance structures can either be heterogeneous or homogeneous.
If we have a homogeneous covariance structure (`hom = TRUE`), then we assume 
$\sigma_{1} = \sigma_{2} = \dots = \sigma_{p}$.

The *diagonal covariance* has the following form:
\[
\mathbf{\Sigma}
= \begin{bmatrix}
\sigma^{2}_{1} & 0 & \dots & 0 \\
0 & \sigma^{2}_{2} & \dots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \dots & \sigma^{2}_{p} \\
\end{bmatrix}	
\]
By default, we assume a heterogeneous diagonal covariance structure.

The *compound symmetric covariance* has the following form:
\[
\mathbf{\Sigma}
= \begin{bmatrix}
\sigma^{2}_{1}              & \sigma_{1}\sigma_{2}\rho  & \sigma_{1}\sigma_{3}\rho  & \dots  & \sigma_{1}\sigma_{p}\rho \\
\sigma_{2}\sigma_{1} \rho   & \sigma^{2}_{2}             & \sigma_{2}\sigma_{3}\rho  & \dots  & \sigma_{2}\sigma_{p}\rho \\
\sigma_{3}\sigma_{1} \rho   & \sigma_{3}\sigma_{2}\rho   & \sigma^{2}_{3}            & \dots  & \sigma_{3}\sigma_{p}\rho \\
\vdots                      & \vdots                     & \vdots                    & \ddots & \vdots \\
\sigma_{p}\sigma_{1} \rho   & \sigma_{p}\sigma_{2}\rho   & \sigma_{p}\sigma_{3}\rho  & \dots  & \sigma^{2}_{p} 
\end{bmatrix}	
\]
By default, we assume a heterogeneous compound symmetric covariance structure.

The *AR1 (auto-regressive order 1) covariance* has the following form:
\[
\mathbf{\Sigma}
= \begin{bmatrix}
\sigma^{2}_{1}                  & \sigma_{1}\sigma_{2} \rho      & \sigma_{1}\sigma_{3}\rho^{2}   & \dots  & \sigma_{1}\sigma_{p}\rho^{p-1} \\
\sigma_{2}\sigma_{1} \rho       & \sigma^{2}_{2}                 & \sigma_{2}\sigma_{3}\rho       & \dots  & \sigma_{2}\sigma_{p}\rho^{p-2} \\
\sigma_{3}\sigma_{1} \rho^{2}   & \sigma_{3}\sigma_{2}\rho       & \sigma^{2}_{3}                 & \dots  & \sigma_{3}\sigma_{p}\rho^{p-3} \\
\vdots                          & \vdots                         & \vdots                         & \ddots & \vdots \\
\sigma_{p}\sigma_{1} \rho^{p-1} & \sigma_{p}\sigma_{2}\rho^{p-2} & \sigma_{p}\sigma_{3}\rho^{p-3} & \dots  & \sigma^{2}_{p} 
\end{bmatrix}	
\]
Unlike the diagonal and compound symmetric structures, by default we assume a
homogeneous ar1 covariance structure.

### Construction of the Relative Co-factor

For the unstructured covariance matrix, `lme4` estimates the following parameters in `par`:
$\mathbf{\theta} = (\theta_{1}, \theta_{2}, \dots, \theta_{p(p+1)/2})$ to construct the relative
co-factor ${\mathbf \Lambda}_{\mathbf{\theta}}$ (this is the same as in previous versions):
\[
\Lambda_{\mathbf{\theta}} = \begin{bmatrix}
\theta_1 & 0 & 0 & \dots & 0 \\
\theta_2 & \theta_3 & 0 & \dots & 0 \\
\theta_4 & \theta_5 & \theta_6 & \dots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\theta_{...} & \theta_{...} & \theta_{...} & \dots & \theta_{p(p+1)/2}
\end{bmatrix}
\]

The definition of the parameter vectors differs for the other covariance structures. In the diagonal covariance matrix 
case, $\mathbf{\theta}$ (or `par`) only contains the standard deviations. 
The relative co-factor ${\mathbf \Lambda}_{\mathbf{\theta}}$ is:
\[
{\mathbf Lambda}_{\mathbf{\theta}} = \begin{bmatrix}
\theta_1 & 0 & 0 & \dots & 0 \\
0 & \theta_2 & 0 & \dots & 0 \\
0 & 0 & \theta_3 & \dots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \dots & \theta_{p}
\end{bmatrix}
\]

For the compound symmetry covariance structure, the parameter 
vector `par` contains the $p$ standard deviations $(\sigma_1, \sigma_2, \ldots, \sigma_p)$ 
and the common correlation $\rho$. In contrast to `glmmTMB`, the correlation is estimated on its original scale (bounded between -1 and 1), rather than on an unconstrained, transformed scale.

The relative co-factor ${\mathbf \Lambda}_{\mathbf{\theta}}$ is a lower triangular $p \times p$ matrix. Consider the form:
\[
{\mathbf \Lambda}_{\mathbf{\theta}} = \begin{bmatrix}
\theta_{11} & 0 & 0 & \cdots & 0 \\
\theta_{21} & \theta_{22} & 0 & \cdots & 0 \\
\theta_{31} & \theta_{32} & \theta_{33} & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
\theta_{p1} & \theta_{p2} & \theta_{p3} & \cdots & \theta_{pp}
\end{bmatrix}
\]

Its elements $\theta_{ij}$ are constructed as follows. First, define the 
sequence $\{a_j\}$ recursively:
\[
a_1 = 0, \quad a_{j+1} = a_j + \frac{(1 - \rho \cdot a_j)^2}{1 - \rho^2 \cdot a_j} \quad \text{for } j = 1, \ldots, p-1
\]

Then the elements of ${\mathbf \Lambda}$ are given by:
\[
\theta_{ij} = \begin{cases}
 \sqrt{1 - \rho^2 a_j} & \text{if } i = j \text{ (diagonal)} \\[0.5em]
 \dfrac{\rho - \rho^2 a_j}{\sqrt{1 - \rho^2 a_j}} & \text{if } i > j \text{ (below diagonal)} \\[0.5em]
0 & \text{if } i < j \text{ (above diagonal)}
\end{cases}
\]
In the code, one can extract the values $\theta_{ij}$ via the `getTheta()` function 
of the `Covariance.cs` object, in which `theta` will be a vector in the column-wise 
elements of ${\mathbf \Lambda}$.
 
The setup is similar for the autoregressive order 1 (AR1) covariance structure. 
Again, the parameter vector contains the $p$ standard deviations 
$(\sigma_1, \sigma_2, \ldots, \sigma_p)$ and the autocorrelation parameter 
$\rho$. The relative co-factor ${\mathbf \Lambda}_{\mathbf{\theta}}$ is a lower 
triangular $p \times p$ matrix whose form is similar to the compound symmetric case.

The elements $\theta_{ij}$ are given by:
\[
\theta_{ij} = \begin{cases}
 \rho^{i-1} & \text{if } j = 1 \text{ (first column)} \\[0.5em]
 \rho^{i-j} \sqrt{1 - \rho^2} & \text{if } 1 < j \leq i \text{ (below diagonal)} \\[0.5em]
0 & \text{if } i < j \text{ (above diagonal)}
\end{cases}
\]
Again, these values can be extracted using `getTheta()` function 
of the `Covariance.ar1` object, in which `theta` will be still be a vector in 
the column-wise elements of ${\mathbf \Lambda}$.

### Extracting model components

This section illustrates how to extract
`par`, `theta`, `Lambda`, as described in the previous section, as well as
the variance covariance matrices of a model, from a `merMod` object.

We'll fit the standard `sleepstudy` example, except that we will use a
model with a compound symmetric covariance structure. Because this model has only two varying effects (intercept and slope with respect to day) per subject, and
 hence the covariance matrix is $2 \times 2$, there is no difference in overall model fit between the compound-symmetric and the unstructured covariance matrices. Howevever, the models are parameterized differently, so this example will highlight the differences between `par` and `theta`.

```{r cs-fit}
library(lme4)
fm1.cs <- lmer(Reaction ~ Days + cs(1 + Days | Subject), sleepstudy)
```

Extracting the covariance structure:

```{r getReCovs}
print(fm1.cs_cov <- getReCovs(fm1.cs))
```

The result is a list with only one element as we only have one random-effects term (`cs(1 + Days | Subject)`). To 
see the values of `par` and `theta` for this object, we can call:

```{r getInfo}
getME(fm1.cs, "par")
getME(fm1.cs, "theta")
```

The ${\mathbf \Lambda}$ matrix is large, so we'll view it instead of printing:
```{r lambda-mat}
library(Matrix)
image(getME(fm1.cs, "Lambda"))
```

To most users, the most crucial information is simply the variance-covariance matrices. Extracte these via `VarCorr.merMod()` (the list has one element per random-effect term in the model --- in this case, only one):

```{r varcorr}
vc_mat <- VarCorr(fm1.cs)
vc_mat$Subject
```

## Comparisons with glmmTMB

### Reason for Computational Differences

When fitting linear mixed models, `lme4`  parameterizes the random-effects variance–covariance matrix on an unconstrained scale, using box-constrained optimization algorithms to ensure that the variance-covariance matrix is positive semidefinite. For unstructured covariance matrices, this means that the elements of $\mathbf \theta$ that parameterize the diagonal elements of $\mathbf \Lambda$ are constrained to be $\ge 0$ (for diagonal models, all of the elements of $\mathbf \theta$ fill the diagonal of $\mathbf \Lambda$ and hence are $\ge 0$; for models such as the compound-symmetric or AR1 models that use correlation parameters, we constrain $|\rho| \le 1$. As discussed in @Bates_JSS, this constrained parameterization works well for handling model where the 
estimated covariance matrix is *singular* (i.e. $\mathbf \Sigma$ is only positive semidefinite, not positive definite). In addition, for linear mixed models `lme4` profiles the fixed-effect parameters out of the objective function [@Bates_JSS]; finally, the scale parameter $\sigma$ is not estimated directly, but is derived from the residual variance or deviance of the fitted model.

In contrast, `glmmTMB` uses direct maximum likelihood estimation via Template Model Builder (TMB), fitting to the full parameter vector $\{\mathbf \theta, \mathbf \beta, \sigma^2\}$. Covariance parameters are fitted on a transformed (unconstrained) scale: log scale for standard deviations and various scales for correlation parameters (see the `glmmTMB` [covariance structures vignette](https://cran.r-project.org/package=glmmTMB/vignettes/covstruct.html) for details). This parameterization simplifies fitting (a box-constrained algorithm isn't necessary), but is less convenient in singular fits and other cases where the maximum likelihood estimate is infinite on the unconstrained scale.

Despite these differences, we will show examples where `lme4` and `glmmTMB` provide similar estimates when they both use maximum likelihood estimation. By default, `lme4` uses the restricted maximum likelihood; hence in the following examples, we use `lmer(..., REML = FALSE)` to compare against `glmmTMB`.

### Comparison Setup

```{r comparison-setup}
if (!requireNamespace("glmmTMB", quietly = TRUE)) {
  knitr::opts_chunk$set(eval = FALSE)
} else {
  library(glmmTMB)
}

## Often want to ignore attributes and class.
## Set a fairly large tolerance for convenience.
all.equal.nocheck <- function(x, y, tolerance = 3e-5, ..., check.attributes = FALSE, check.class = FALSE) {
  require("Matrix", quietly = TRUE)
  ## working around mode-matching headaches
  if (is(x, "Matrix")) x <- matrix(x)
  if (is(y, "Matrix")) y <- matrix(y)
  all.equal(x, y, ..., tolerance = tolerance, check.attributes = check.attributes, check.class = check.class)
}

get.cor1 <- function(x) {
  v <- VarCorr(x)
  vv <- if (inherits(x, "merMod")) v$group else v$cond$group
  attr(vv, "correlation")[1,2]
}
```

### Unstructured (General Positive Definite)

This is the default setting for both `lme4` and `glmmTMB`. Below we simulate a dataset with known `beta`, `theta` and `sigma` values.

```{r unstruc-sim}
n_groups <- 20
n_per_group <- 20
n <- n_groups * n_per_group

set.seed(1)
dat.us <- data.frame(
  group = rep(1:n_groups, each = n_per_group),
  x1 = rnorm(n),
  x2 = rnorm(n)
)
## Constructing a similar dataset for the other examples
gdat.us <- dat.diag <- gdat.diag <- dat.us

form <- y ~ 1 + x1 * x2 + us(1 + x1|group)
dat.us$y <- simulate(form[-2], 
                    newdata = dat.us,
                    family = gaussian,
                    newparams = list(beta = c(-7, 5, -100, 20),
                                     theta = c(2.5, 1.4, 6.3),
                                     sigma = 2))[[1]]

form2 <- y ~ 1 + x1 + us(1 + x1|group)
gdat.us$y <- simulate(
  form2[-2],
  newdata = gdat.us,
  family = binomial,
  newparams = list(
    beta  = c(-1.5, 0.5),     
    theta = c(0.3, 0.1, 0.3)
  ))[[1]]
```


### Linear Mixed Model

```{r compare-lmm}
lme4.us <- lmer(form, data = dat.us, REML = "FALSE")
glmmTMB.us <- glmmTMB(form, dat = dat.us)

## Fixed effects
fixef(lme4.us); fixef(glmmTMB.us)$cond
all.equal.nocheck(fixef(lme4.us), fixef(glmmTMB.us)$cond)

## Sigma
sigma(lme4.us); sigma(glmmTMB.us)
all.equal.nocheck(sigma(lme4.us), sigma(glmmTMB.us))

## Log likelihoods
logLik(lme4.us); logLik(glmmTMB.us)
all.equal.nocheck(logLik(lme4.us), logLik(glmmTMB.us))
```

As expected, calculations related to the random-effects term differ slightly beyond this point.

```{r compare-cov}
## Variance-Covariance Matrix
vcov(lme4.us); vcov(glmmTMB.us)$cond
all.equal.nocheck(vcov(lme4.us), vcov(glmmTMB.us)$cond)

## Variance and Covariance Components
all.equal.nocheck(VarCorr(lme4.us)$group,
          VarCorr(glmmTMB.us)$cond$group)

## Conditional Modes of the Random Effects
all.equal.nocheck(ranef(lme4.us)$group, ranef(glmmTMB.us)$cond$group)
```

### Generalized Linear Mixed Model

```{r compare-glmm}
glme4.us <- glmer(form2, data = gdat.us, family = binomial)
gglmmTMB.us <- glmmTMB(form2, dat = gdat.us, family = binomial)

## Fixed effects
fixef(glme4.us); fixef(gglmmTMB.us)$cond
all.equal.nocheck(fixef(glme4.us), fixef(gglmmTMB.us)$cond)

## Sigma
all.equal.nocheck(sigma(glme4.us), sigma(gglmmTMB.us))

## Log likelihoods
logLik(glme4.us); logLik(gglmmTMB.us)
all.equal.nocheck(logLik(glme4.us), logLik(gglmmTMB.us))
```

As expected, calculations related to the random-effects term differ slightly beyond this point.

```{r compare-glmm-cov}
## Variance-Covariance Matrix
vcov(glme4.us); vcov(gglmmTMB.us)$cond
all.equal.nocheck(vcov(glme4.us), vcov(gglmmTMB.us)$cond)

## Variance and Covariance Components
all.equal.nocheck(VarCorr(glme4.us)$group,
          VarCorr(gglmmTMB.us)$cond$group)

## Conditional Modes of the Random Effects
all.equal.nocheck(ranef(glme4.us)$group, ranef(gglmmTMB.us)$cond$group)
```

### Diagonal

The syntax is the same for fitting a heterogeneous diagonal covariance structure for `lme4` and `glmmTMB`. It changes when we want to fit a *homogeneous* diagonal covariance structure.

To fit a homogeneous diagonal covariance structure we would write:

```{r, fit-homog-diag, eval = FALSE}
lme4.us <- lmer(Reaction ~ Days + diag(Days | Subject, hom = TRUE), sleepstudy)
glmmTMB.us <- glmmTMB(Reaction ~ Days + homdiag(Days | Subject), sleepstudy)
```

We will focus on comparisons of an estimated *heterogeneous* diagonal covariance structure.

```{r sim-het-diag}
form <- y ~ 1 + x1 * x2 + diag(1|group)
dat.diag$y <- simulate(form[-2], 
                       newdata = dat.diag,
                       family = gaussian,
                       newparams = list(beta = c(-7, 5, -100, 20),
                                        theta = c(2.5),
                                        sigma = 2))[[1]]
```


```{r fit-het-diag}
lme4.diag <- lmer(form, data = dat.diag, REML = "FALSE")
glmmTMB.diag <- glmmTMB(form, dat = dat.diag)

## Fixed effects
fixef(lme4.diag); fixef(glmmTMB.diag)$cond
all.equal.nocheck(fixef(lme4.diag), fixef(glmmTMB.diag)$cond)

## Sigma
sigma(lme4.diag); sigma(glmmTMB.diag)
all.equal.nocheck(sigma(lme4.diag), sigma(glmmTMB.diag))

## Log likelihoods
logLik(lme4.diag); logLik(glmmTMB.diag)
all.equal.nocheck(logLik(lme4.diag), logLik(glmmTMB.diag))

## Variance-Covariance Matrix
vcov(lme4.diag); vcov(glmmTMB.diag)$cond
all.equal.nocheck(vcov(lme4.diag), vcov(glmmTMB.diag)$cond)

## Variance and Covariance Components
all.equal.nocheck(VarCorr(lme4.diag)[[1]], 
          VarCorr(glmmTMB.diag)$cond$group)

## Conditional Modes of the Random Effects
all.equal.nocheck(ranef(lme4.diag)$group, ranef(glmmTMB.diag)$cond$group)
```

### Compound Symmetry

Similar to the diagonal case, the syntax is the same for fitting a heterogeneous compound symmetry covariance structure for `lme4` and `glmmTMB`: 

```{r cs-fit-2, eval = FALSE}
lme4.us <- lmer(Reaction ~ Days + cs(Days | Subject, hom = TRUE), sleepstudy)
glmmTMB.us <- glmmTMB(Reaction ~ Days + cs(Days | Subject), sleepstudy)
```

Again, it differs when we want to fit a *homogeneous* compound symmetry covariance structure, which we will use for our comparisons.

```{r simgroup}
simGroup <- function(g, n=6, phi=0.6) {
  x <- MASS::mvrnorm(mu = rep(0,n),
                     Sigma = phi^as.matrix(dist(1:n)) )  
  y <- x + rnorm(n)                              
  times <- factor(1:n)
  group <- factor(rep(g,n))
  data.frame(y, times, group)
}

set.seed(1)
dat.cs <- do.call("rbind", lapply(1:2000, simGroup))
```

<!-- this is the slow one ... cache? -->
```{r more-comp, cache=TRUE}
lme4.cs <- lmer(y ~ times + cs(0 + times | group, hom = TRUE), data = dat.cs, REML = FALSE)
glmmTMB.cs <- glmmTMB(y ~ times + homcs(0 + times | group), data = dat.cs)

## Fixed effects
fixef(lme4.cs); fixef(glmmTMB.cs)$cond
all.equal.nocheck(fixef(lme4.cs), fixef(glmmTMB.cs)$cond)

## Sigma
sigma(lme4.cs); sigma(glmmTMB.cs)
all.equal.nocheck(sigma(lme4.cs), sigma(glmmTMB.cs))

## Log likelihoods
logLik(lme4.cs); logLik(glmmTMB.cs)
all.equal.nocheck(logLik(lme4.cs), logLik(glmmTMB.cs))

## Variance-Covariance Matrix
all.equal.nocheck(vcov(lme4.cs), vcov(glmmTMB.cs)$cond)

## Variance and Covariance Components
all.equal.nocheck(VarCorr(lme4.cs)[[1]], 
          VarCorr(glmmTMB.cs)$cond$group)

## Conditional Modes of the Random Effects
all.equal.nocheck(ranef(lme4.cs)$group, ranef(glmmTMB.cs)$cond$group)

## Comparing against the predicted rho value
lme4.rho <- get.cor1(lme4.cs)
glmmTMB.rho <- get.cor1(glmmTMB.cs)
lme4.rho; glmmTMB.rho
all.equal.nocheck(lme4.rho, glmmTMB.rho)
```

### Autoregressive Order 1

For this comparison, we focus on a simulated data set with $\rho = 0.7$.

```{r ar1-sim}
set.seed(1)
dat.ar1 <- do.call("rbind", lapply(1:2000, function(g) simGroup(g, phi = 0.7)))
```

Unlike the diagonal and compound symmetry case, the syntax differs for fitting either a heterogeneous or a homogeneous AR1 model for `lme4` and `glmmTMB`.

For a *heterogeneous* AR1 covariance structure we would write the following:

```{r het-ar1-fit, eval = FALSE}
lme4.ar1 <- lmer(y ~ times + ar1(0 + times | group, hom = FALSE),
                 data = dat.ar1, REML = FALSE)
glmmTMB.ar1 <- glmmTMB(y ~ times + hetar1(0 + times | group), data = dat.ar1)
```

We will instead focus on comparisons for a *homogeneous* AR1 covariance structure.

```{r ar1-fit, cache=TRUE}
lme4.ar1 <- lmer(y ~ times + ar1(0 + times | group, hom = TRUE), data = dat.ar1, REML = FALSE)
glmmTMB.ar1 <- glmmTMB(y ~ times + ar1(0 + times | group), data = dat.ar1)

## Fixed effects
fixef(lme4.ar1); fixef(glmmTMB.ar1)$cond
all.equal.nocheck(fixef(lme4.ar1), fixef(glmmTMB.ar1)$cond)

## Sigma
sigma(lme4.ar1); sigma(glmmTMB.ar1)
all.equal.nocheck(sigma(lme4.ar1), sigma(glmmTMB.ar1))

## Log likelihoods
logLik(lme4.ar1); logLik(glmmTMB.ar1)
all.equal.nocheck(logLik(lme4.ar1), logLik(glmmTMB.ar1))

## Variance-Covariance Matrix
all.equal.nocheck(vcov(lme4.ar1), vcov(glmmTMB.ar1)$cond)

## Variance and Covariance Components
all.equal.nocheck(VarCorr(lme4.ar1)$group, 
                  VarCorr(glmmTMB.ar1)$cond$group)

## Conditional Modes of the Random Effects
all.equal.nocheck(ranef(lme4.ar1)$group, ranef(glmmTMB.ar1)$cond$group)

## Comparing against the predicted rho value
lme4.ar1.rho <- get.cor1(lme4.ar1)
glmmTMB.ar1.rho <- get.cor1(glmmTMB.ar1)
lme4.ar1.rho; glmmTMB.ar1.rho
all.equal.nocheck(lme4.ar1.rho, glmmTMB.ar1.rho)
```

## References

