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Covariance Structures comparison (lme4/glmmTMB)
In lme4: Linear Mixed-Effects Models using 'Eigen' and S4
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 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$set(eval = FALSE)
} else {
library(glmmTMB)
library(lme4)
}
## Often want to ignore attributes and class.
## Set a fairly large tolerance for convenience.
all.equal.nocheck <- function(x, y, tolerance = 3e-5, ..., check.attributes = FALSE, check.class = FALSE) {
require("Matrix", quietly = TRUE)
## working around mode-matching headaches
if (is(x, "Matrix")) x <- matrix(x)
if (is(y, "Matrix")) y <- matrix(y)
all.equal(x, y, ..., tolerance = tolerance, check.attributes = check.attributes, check.class = check.class)
}
get.cor1 <- function(x) {
v <- VarCorr(x)
vv <- if (inherits(x, "merMod")) v$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)$cond
all.equal.nocheck(fixef(lme4.us), fixef(glmmTMB.us)$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.
## 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
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.
## 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:
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
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:
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)
## 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 $\rho = 0.7$.
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, 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.
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)
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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 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$set(eval = FALSE) } else { library(glmmTMB) library(lme4) } ## Often want to ignore attributes and class. ## Set a fairly large tolerance for convenience. all.equal.nocheck <- function(x, y, tolerance = 3e-5, ..., check.attributes = FALSE, check.class = FALSE) { require("Matrix", quietly = TRUE) ## working around mode-matching headaches if (is(x, "Matrix")) x <- matrix(x) if (is(y, "Matrix")) y <- matrix(y) all.equal(x, y, ..., tolerance = tolerance, check.attributes = check.attributes, check.class = check.class) } get.cor1 <- function(x) { v <- VarCorr(x) vv <- if (inherits(x, "merMod")) v$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)$cond all.equal.nocheck(fixef(lme4.us), fixef(glmmTMB.us)$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.
## 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
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.
## 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:
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 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:
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) ## 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 $\rho = 0.7$.
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, 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.
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)
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