# Covariance Structures

## 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), 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 (Bates et al. 2015). 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 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 \(\mathbf \Sigma\).

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`.

    library(lme4)
    fm1.cs <- lmer(Reaction ~ Days + cs(1 + Days | Subject), sleepstudy)

Extracting the covariance structure:

    print(fm1.cs_cov <- getReCovs(fm1.cs))
    #> [[1]]
    #> An object of class "Covariance.cs"
    #> Slot "hom":
    #> [1] FALSE
    #>
    #> Slot "nc":
    #> [1] 2
    #>
    #> Slot "par":
    #> [1] 0.96679232 0.23140420 0.06561148

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:

    getME(fm1.cs, "par")
    #> Subject.(Intercept)        Subject.Days         Subject.rho
    #>          0.96679232          0.23140420          0.06561148
    getME(fm1.cs, "theta")
    #>      Subject.(Intercept) Subject.Days.(Intercept)             Subject.Days
    #>               0.96679232               0.01518277               0.23090558

The \({\mathbf \Lambda}\) matrix is large, so we’ll view it instead of printing:

    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):

    vc_mat <- VarCorr(fm1.cs)
    vc_mat$Subject
    #>             (Intercept)      Days
    #> (Intercept)  612.160273  9.613533
    #> Days           9.613533 35.070443
    #> attr(,"class")
    #> [1] "vcmat_cs" "matrix"   "array"
    #> attr(,"stddev")
    #> (Intercept)        Days
    #>    24.74187     5.92203
    #> attr(,"correlation")
    #>             (Intercept)       Days
    #> (Intercept)  1.00000000 0.06561148
    #> Days         0.06561148 1.00000000

## 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 et al. (2015), 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 et al. 2015); 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 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

    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.

    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

    lme4.us <- lmer(form, data = dat.us, REML = "FALSE")
    glmmTMB.us <- glmmTMB(form, dat = dat.us)

    ## Fixed effects
    fixef(lme4.us); fixef(glmmTMB.us)$cond
    #> (Intercept)          x1          x2       x1:x2
    #>   -7.280739    5.795443 -100.070859   19.958369
    #> (Intercept)          x1          x2       x1:x2
    #>   -7.280736    5.795388 -100.070860   19.958369
    all.equal.nocheck(fixef(lme4.us), fixef(glmmTMB.us)$cond)
    #> [1] TRUE

    ## Sigma
    sigma(lme4.us); sigma(glmmTMB.us)
    #> [1] 2.049705
    #> [1] 2.049702
    all.equal.nocheck(sigma(lme4.us), sigma(glmmTMB.us))
    #> [1] TRUE

    ## Log likelihoods
    logLik(lme4.us); logLik(glmmTMB.us)
    #> 'log Lik.' -971.3046 (df=8)
    #> 'log Lik.' -971.3046 (df=8)
    all.equal.nocheck(logLik(lme4.us), logLik(glmmTMB.us))
    #> [1] TRUE

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

    ## Variance-Covariance Matrix
    vcov(lme4.us); vcov(glmmTMB.us)$cond
    #> 4 x 4 Matrix of class "dpoMatrix"
    #>               (Intercept)            x1            x2         x1:x2
    #> (Intercept)  1.2014218984 -0.1733456044  0.0008614804 -0.0004201174
    #> x1          -0.1733456044 12.4138871011 -0.0008710106  0.0026046078
    #> x2           0.0008614804 -0.0008710106  0.0101254746 -0.0018079198
    #> x1:x2       -0.0004201174  0.0026046078 -0.0018079198  0.0117364803
    #>               (Intercept)            x1            x2         x1:x2
    #> (Intercept)  1.2014959308 -0.1733102259  0.0008614825 -0.0004201922
    #> x1          -0.1733102259 12.4138803796 -0.0008713988  0.0026042335
    #> x2           0.0008614825 -0.0008713988  0.0101257004 -0.0018074214
    #> x1:x2       -0.0004201922  0.0026042335 -0.0018074214  0.0117377999
    all.equal.nocheck(vcov(lme4.us), vcov(glmmTMB.us)$cond)
    #> [1] TRUE

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

    ## Conditional Modes of the Random Effects
    all.equal.nocheck(ranef(lme4.us)$group, ranef(glmmTMB.us)$cond$group)
    #> [1] TRUE

### Generalized Linear Mixed Model

    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)$cond
    #> (Intercept)          x1
    #>  -1.5400786   0.4547925
    #> (Intercept)          x1
    #>  -1.5403288   0.4545996
    all.equal.nocheck(fixef(glme4.us), fixef(gglmmTMB.us)$cond)
    #> [1] "Mean relative difference: 0.0002221048"

    ## Sigma
    all.equal.nocheck(sigma(glme4.us), sigma(gglmmTMB.us))
    #> [1] TRUE

    ## Log likelihoods
    logLik(glme4.us); logLik(gglmmTMB.us)
    #> 'log Lik.' -191.8811 (df=5)
    #> 'log Lik.' -191.8809 (df=5)
    all.equal.nocheck(logLik(glme4.us), logLik(gglmmTMB.us))
    #> [1] TRUE

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

    ## Variance-Covariance Matrix
    vcov(glme4.us); vcov(gglmmTMB.us)$cond
    #> 2 x 2 Matrix of class "dpoMatrix"
    #>              (Intercept)           x1
    #> (Intercept)  0.028221233 -0.001848413
    #> x1          -0.001848413  0.038084198
    #>              (Intercept)           x1
    #> (Intercept)  0.028453808 -0.001749237
    #> x1          -0.001749237  0.038270693
    all.equal.nocheck(vcov(glme4.us), vcov(gglmmTMB.us)$cond)
    #> [1] "Mean relative difference: 0.008820021"

    ## Variance and Covariance Components
    all.equal.nocheck(VarCorr(glme4.us)$group,
              VarCorr(gglmmTMB.us)$cond$group)
    #> [1] "Mean relative difference: 0.001268453"

    ## Conditional Modes of the Random Effects
    all.equal.nocheck(ranef(glme4.us)$group, ranef(gglmmTMB.us)$cond$group)
    #> [1] "Component \"(Intercept)\": Mean relative difference: 0.0009258143"
    #> [2] "Component \"x1\": Mean relative difference: 0.0007570482"

### 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:

    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.

    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]]

    lme4.diag <- lmer(form, data = dat.diag, REML = "FALSE")
    glmmTMB.diag <- glmmTMB(form, dat = dat.diag)

    ## Fixed effects
    fixef(lme4.diag); fixef(glmmTMB.diag)$cond
    #> (Intercept)          x1          x2       x1:x2
    #>   -6.484337    5.111952 -100.045319   20.037624
    #> (Intercept)          x1          x2       x1:x2
    #>   -6.484329    5.111952 -100.045319   20.037624
    all.equal.nocheck(fixef(lme4.diag), fixef(glmmTMB.diag)$cond)
    #> [1] TRUE

    ## Sigma
    sigma(lme4.diag); sigma(glmmTMB.diag)
    #> [1] 2.156913
    #> [1] 2.156913
    all.equal.nocheck(sigma(lme4.diag), sigma(glmmTMB.diag))
    #> [1] TRUE

    ## Log likelihoods
    logLik(lme4.diag); logLik(glmmTMB.diag)
    #> 'log Lik.' -915.8749 (df=6)
    #> 'log Lik.' -915.8749 (df=6)
    all.equal.nocheck(logLik(lme4.diag), logLik(glmmTMB.diag))
    #> [1] TRUE

    ## Variance-Covariance Matrix
    vcov(lme4.diag); vcov(glmmTMB.diag)$cond
    #> 4 x 4 Matrix of class "dpoMatrix"
    #>               (Intercept)            x1            x2         x1:x2
    #> (Intercept)  0.6899487512 -0.0006448103  0.0008535087 -0.0005702566
    #> x1          -0.0006448103  0.0135575383 -0.0008171565  0.0025591747
    #> x2           0.0008535087 -0.0008171565  0.0107715407 -0.0021874630
    #> x1:x2       -0.0005702566  0.0025591747 -0.0021874630  0.0115262417
    #>               (Intercept)            x1            x2         x1:x2
    #> (Intercept)  0.6899487512 -0.0006448095  0.0008535098 -0.0005702545
    #> x1          -0.0006448095  0.0135576594 -0.0008169500  0.0025595384
    #> x2           0.0008535098 -0.0008169500  0.0107718878 -0.0021868490
    #> x1:x2       -0.0005702545  0.0025595384 -0.0021868490  0.0115273226
    all.equal.nocheck(vcov(lme4.diag), vcov(glmmTMB.diag)$cond)
    #> [1] TRUE

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

    ## Conditional Modes of the Random Effects
    all.equal.nocheck(ranef(lme4.diag)$group, ranef(glmmTMB.diag)$cond$group)
    #> [1] TRUE

### Compound Symmetry

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

    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.

    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))

    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)
    #> Warning in finalizeTMB(TMBStruc, obj, fit, h, data.tmb.old): Model convergence
    #> problem; non-positive-definite Hessian matrix. See vignette('troubleshooting')

    ## Fixed effects
    fixef(lme4.cs); fixef(glmmTMB.cs)$cond
    #>  (Intercept)       times2       times3       times4       times5       times6
    #> -0.003601028  0.024047964  0.004877225  0.052026751  0.049632430  0.058705282
    #>  (Intercept)       times2       times3       times4       times5       times6
    #> -0.003623487  0.024074739  0.004896515  0.052053537  0.049659692  0.058730881
    all.equal.nocheck(fixef(lme4.cs), fixef(glmmTMB.cs)$cond)
    #> [1] "Mean relative difference: 0.00076817"

    ## Sigma
    sigma(lme4.cs); sigma(glmmTMB.cs)
    #> [1] 1.006676
    #> [1] 1.041603
    all.equal.nocheck(sigma(lme4.cs), sigma(glmmTMB.cs))
    #> [1] "Mean relative difference: 0.03469535"

    ## Log likelihoods
    logLik(lme4.cs); logLik(glmmTMB.cs)
    #> 'log Lik.' -20850.22 (df=18)
    #> 'log Lik.' NA (df=9)
    all.equal.nocheck(logLik(lme4.cs), logLik(glmmTMB.cs))
    #> [1] "'is.NA' value mismatch: 1 in current 0 in target"

    ## Variance-Covariance Matrix
    all.equal.nocheck(vcov(lme4.cs), vcov(glmmTMB.cs)$cond)
    #> [1] TRUE

    ## Variance and Covariance Components
    all.equal.nocheck(VarCorr(lme4.cs)[[1]],
              VarCorr(glmmTMB.cs)$cond$group)
    #> [1] "Mean relative difference: 0.0255908"

    ## Conditional Modes of the Random Effects
    all.equal.nocheck(ranef(lme4.cs)$group, ranef(glmmTMB.cs)$cond$group)
    #> Warning in sqrt(as.numeric(object$diag.cov.random)): NaNs produced
    #> [1] "Component \"times1\": Mean relative difference: 0.07526183"
    #> [2] "Component \"times2\": Mean relative difference: 0.07067207"
    #> [3] "Component \"times3\": Mean relative difference: 0.06907347"
    #> [4] "Component \"times4\": Mean relative difference: 0.06932043"
    #> [5] "Component \"times5\": Mean relative difference: 0.07089525"
    #> [6] "Component \"times6\": Mean relative difference: 0.07424953"

    ## Comparing against the predicted rho value
    lme4.rho <- get.cor1(lme4.cs)
    glmmTMB.rho <- get.cor1(glmmTMB.cs)
    lme4.rho; glmmTMB.rho
    #> [1] 0.3642212
    #> [1] 0.3925523
    all.equal.nocheck(lme4.rho, glmmTMB.rho)
    #> [1] "Mean relative difference: 0.07778538"

### Autoregressive Order 1

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

    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:

    lme4.ar1 <- lmer(y ~ times + ar1(0 + times | group), 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.

    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)$cond
    #> (Intercept)      times2      times3      times4      times5      times6
    #> -0.01323035  0.03297898  0.02689120  0.07389340  0.05907372  0.05783090
    #> (Intercept)      times2      times3      times4      times5      times6
    #> -0.01322832  0.03297395  0.02686705  0.07391147  0.05907415  0.05782987
    all.equal.nocheck(fixef(lme4.ar1), fixef(glmmTMB.ar1)$cond)
    #> [1] "Mean relative difference: 0.0001922882"

    ## Sigma
    sigma(lme4.ar1); sigma(glmmTMB.ar1)
    #> [1] 0.9807867
    #> [1] 0.9807956
    all.equal.nocheck(sigma(lme4.ar1), sigma(glmmTMB.ar1))
    #> [1] TRUE

    ## Log likelihoods
    logLik(lme4.ar1); logLik(glmmTMB.ar1)
    #> 'log Lik.' -20427.08 (df=18)
    #> 'log Lik.' -20427.08 (df=9)
    all.equal.nocheck(logLik(lme4.ar1), logLik(glmmTMB.ar1))
    #> [1] TRUE

    ## Variance-Covariance Matrix
    all.equal.nocheck(vcov(lme4.ar1), vcov(glmmTMB.ar1)$cond)
    #> [1] TRUE

    ## Variance and Covariance Components
    all.equal.nocheck(VarCorr(lme4.ar1)$group,
                      VarCorr(glmmTMB.ar1)$cond$group)
    #> [1] "Mean relative difference: 4.596753e-05"

    ## Conditional Modes of the Random Effects
    all.equal.nocheck(ranef(lme4.ar1)$group, ranef(glmmTMB.ar1)$cond$group)
    #> [1] TRUE

    ## 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
    #> [1] 0.6865827
    #> [1] 0.6866054
    all.equal.nocheck(lme4.ar1.rho, glmmTMB.ar1.rho)
    #> [1] "Mean relative difference: 3.318898e-05"

## References

Bates, Douglas, Martin Mächler, Ben Bolker, and Steve Walker. 2015. “Fitting Linear Mixed-Effects Models Using lme4.” Journal of Statistical Software 67 (1): 1–48. https://doi.org/10.18637/jss.v067.i01.