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The question is about computing the variance of the random-effect parameters estimated when fitting a linear mixed-effect model when the parameterization of the random-effect parameters changes. This link describes how to use the delta rule to compute the variance of the estimated random effects. I would like to extend it to compute the variance of the covariance parameters but I am not obtaining the expected result. The post is long because it includes all the derivations as well as a concrete example with R.

Let $\Sigma_S$ be the variance-covariance matrix of two random effects (e.g. the random intercept and random slope of a linear mixed-effect model): $$ \begin{bmatrix} b_{0i} \\\\ b_{1i} \end{bmatrix} \sim \mathcal{N}(0, \Sigma_S) $$ where $$ \Sigma_S = \begin{bmatrix} \sigma_{11} & \sigma_{12} \\\\ \sigma_{12} & \sigma_{22} \end{bmatrix} $$ Let $S_{ij}$ be the estimated values of the random effect parameters $\sigma_{ij}$. The $S_{ij}$ are random variables and we can define the 3 by 3 variance-covariance matrix $\Psi_S$ of estimated values $S_{ij}$ of the random-effect parameters: $$ \Psi_S = \text{Var} \left( \begin{bmatrix} S_{ii} \\\\ S_{jj} \\\\ S_{ij} \end{bmatrix} \right) = \begin{bmatrix} \text{Var}(S_{ii}) & \text{Cov}(S_{ii},S_{jj}) & \text{Cov}(S_{ii},S_{ij}) \\\\ \text{Cov}(S_{ii}, S_{jj})& \text{Var}(S_{jj}) & \text{Cov}(S_{jj},S_{ij}) \\\\ \text{Cov}(S_{ii}, S_{ij})& \text{Cov}(S_{jj},S_{ij}) & \text{Var}(S_{ij}) \end{bmatrix} $$ The diagonal elements of this matrix $\text{Var}(S_{ii})$, $\text{Var}(S_{jj})$ and $\text{Var}(S_{ij})$ are the variances of the estimated values of the random-effect parameters $\sigma_{ij}$ in matrix $\Sigma_S$.

Now, let's assume that a different parameterization of the variance and covariance parameters $\sigma_{ij}$ is used (e.g., the so-called "natural" parameterization in Pinheiro, p. 93):
$$ \begin{aligned} \eta_i &= \log(\sqrt{\sigma_{ii}}) \\\\ \eta_{ij} &= \text{logit} \left( \rho_{ij} \right) \end{aligned} $$ where $\rho_{ij} = \frac{\sigma_{ij}}{\sqrt{\sigma_{ii} \sigma_{jj}}}$ and $\text{logit}(x) = \log \left( \frac{x+1}{x-1} \right)$. Let $T_i$ and $T_{ij}$ be the estimated values of $\eta_i$ and $\eta_{ij}$ obtained when fitting the mixed-effect model using this parameterization.

The inverse transformation can be used to get the parameters of $\Sigma_S$ from their natural parameterization $$ \begin{aligned} \sigma_{ii} &= g_1(\eta_{ii}) = \exp(\eta_i)^2 \\\\ \sigma_{ij} &= g_2(\eta_{ij}, \eta_i, \eta_j) = \text{logit}^{-1}( \eta_{ij} ) \exp(\eta_i) \exp(\eta_j) \end{aligned} \tag{1}\label{1} $$ where $\rho_{ij}=\text{logit}^{-1}(\eta_{ij}) = \frac{\exp(\eta_{ij})-1}{\exp(\eta_{ij})+1}$ is the correlation of coefficient between the random effects and $\exp(\eta_i) = \sqrt{\sigma_{ii}}$ is the standard devation of the random effects.

Now, assume that we have estimated the random effect parameters $T_i$, $T_j$ and $T_{ij}$ in the so-called natural scale and that we have computed the corresponding variance-covariance matrix: $$ \Psi_T = \text{Var} \left( \begin{bmatrix} T_i \\\\ T_j \\\\ T_{ij} \end{bmatrix} \right) = \begin{bmatrix} \text{Var}(T_{ii}) & \text{Cov}(T_{ii},T_{jj}) & \text{Cov}(T_{ii},T_{ij}) \\\\ \text{Cov}(T_{ii}, T_{jj})& \text{Var}(T_{jj}) & \text{Cov}(T_{jj},T_{ij}) \\\\ \text{Cov}(T_{ii}, T_{ij})& \text{Cov}(T_{jj},T_{ij}) & \text{Var}(T_{ij}) \end{bmatrix} $$

The general question is how to transform $\Psi_T$ into $\Psi_S$. To start,let's focus on the diagonal terms, i.e the variances $\text{Var}(T_i)$, $\text{Var}(T_j)$ and $\text{Var}(T_{ij})$.

It is quite simple to use the delta-rule to transform the first two terms, i.e. the variance of the variance of the estimated random-effect parameters $\text{Var}(T_i)$ into $\text{Var}(S_{ij})$ (see also here). The delta rule is based on first-order Taylor approximation of the transformation $Y = g(X)$ $$ Y \approx g(\mu_X) + g'(X)( X - \mu_X) $$ where $X$ and $Y$ are two random variables and $\mu_x = E[X]$. It is easy to see that $\text{Var}(Y) \approx g'(\mu_X)^2 \text{Var}(X)$ since $\mu_X$ and $g(\mu_X)$ are constants.

In this case, the derivative of the transformation $S_{ii} = g_1(T_i)$ is $$ g_1'(T_i) = \frac{d}{dT_i} \exp(T_i)^2 = 2 \exp(T_i)^2 $$ (see eq. $\eqref{1}$). Therefore, the transformation from $\text{Var}(T_j)$ to $\text{Var}(S_{ij})$ is: $$ \text{Var}(S_{ii}) \approx \left(2 \exp(T_i)^2 \right)^2 \text{Var}(T_i) \tag{2}\label{2} $$

The delta rule can also be used to transform the variance of correlation/covariance term $\text{Var}(T_{ij})$ but, in this case, the transformation $g$ has three arguments (see eq. $\eqref{1}$).

The first-order Taylor approximation of $Y=g(X_1, X_2, X_3)$ is $$ Y \approx g(\mu_{X_1}, \mu_{X_2}, \mu_{X_3}) + \sum_i \frac{dg}{dX_i}(\mu_{X_1}, \mu_{X_2}, \mu_{X_3}) (X_i-\mu_{X_i}) = \nabla_g(\mu_X)^T (X-\mu_X) $$ The variance of $Y$ is $$ \text{Var}(Y) \approx \sum_i \left( \frac{dg}{dX_i}(\mu_{X_1}, \mu_{X_2}, \mu_{X_3}) \right) ^2 \text{Var}(X_i) = \left( \nabla_g(\mu_X) ^2 \right)^T \text{Var}(X) $$ where $\text{Var}(X) = \left[ \text{Var}(X_1), \text{Var}(X_2), \text{Var}(X_3) \right]$.

In this case, the gradient is $$ \nabla_{g} = \begin{bmatrix} \frac{\delta g_2}{\delta T_{ij}} \\\\ \frac{\delta g_2}{\delta T_i} \\\\ \frac{\delta g_2}{\delta T_j} \end{bmatrix} = \begin{bmatrix} \left[ \frac{d}{d T_{ij}}\text{logit}^{-1}(T_{ij}) \right] \exp(T_i) \exp(T_j) \\\\ \text{logit}^{-1}(T_{ij}) \left[ \frac{d}{d T_i} \exp(T_i) \right] \exp{T_j} \\\\ \text{logit}^{-1}(T_{ij}) \exp(T_i) \left[ \frac{d}{d T_j} \exp(T_j) \right] \end{bmatrix}= \begin{bmatrix} \frac{2 \exp(T_{ij})}{(\exp(T_{ij})+1)^2} \exp(T_i) \exp(T_j) \\\\ S_{ij} \\\\ S_{ij} \end{bmatrix} $$ since $\frac{d}{dx}\exp(x)=\exp(x)$ and $S_{ij} = R_{ij} \sqrt{S_{ii} S_{jj}}$ where $R_{ij} = \text{logit}^{-1}(T_{ij})$ is the sample correlation of $\rho_{ij}$, and $\sqrt{S_{ii}} = \exp(T_i)$ is the sample standard deviation $\sqrt{\sigma_{ii}}$.

Therefore, the variance of the sample covariance $S_{ij}$ is $$ \text{Var}(S_{ij}) = \left( \frac{2 \exp(T_{ij})}{(\exp(T_{ij})+1)^2} \exp{T_i} \exp{T_j} \right)^2 \text{Var}(T_{ij}) + S_{ij}^2 \text{Var}(T_i) + S_{ij}^2 \text{Var}(T_j) \tag{3}\label{3} $$

I include an example to be concrete and check the formulae

library(nlme)
library(lmeInfo)

# data set
data("Orthodont",package="MEMSS")
colnames(Orthodont) <- tolower(colnames(Orthodont))
contrasts(Orthodont$sex) <- "contr.sum"
# Lme model 
fit.lme <- nlme::lme(fixed = distance ~ age*sex, random=~age|subject, data=Orthodont, method="REML")

The function VarCorr and the function lmeInfo::extract_varcomp returns the estimated values of the random effects $T_{ij}$ (matrix $\Sigma_S$):

>  VarCorr(fit.lme)
subject = pdLogChol(age) 
            Variance   StdDev    Corr  
(Intercept) 5.78643480 2.4055009 (Intr)
age         0.03252449 0.1803455 -0.668
Residual    1.71620366 1.3100396 
> extract_varcomp(fit.lme)$Tau
$subject
  subject.var((Intercept)) subject.cov(age,(Intercept))             subject.var(age)
                5.78643480                  -0.28962733                   0.03252449 

Internally, lme use the natural parameterization of the variance-covariance matrix. The slot apVar of the fitted model contains both the estimates $T_i$ and $T_{ij}$ of the random effect parameters (in the attribute Pars) and their covariance matrix $\Psi_T$ (the matrix aPVar).

> (aV <- fit.lme$apVar)
                  reStruct.subject1 reStruct.subject2 reStruct.subject3       lSigma
reStruct.subject1        0.19534695        0.20547534       -0.47083968 -0.017153480
reStruct.subject2        0.20547534        0.32692918       -0.57907986 -0.024253069
reStruct.subject3       -0.47083968       -0.57907986        1.35401393  0.044474133
lSigma                  -0.01715348       -0.02425307        0.04447413  0.009241268
attr(,"Pars")
reStruct.subject1 reStruct.subject2 reStruct.subject3            lSigma 
        0.8777582        -1.7128810        -1.6128707         0.2700573 
attr(,"natural")
[1] TRUE

The function ranef_par implements the transformation from the natural parameterization to the standard parameterization of the random effect parameters (eq. $\eqref{1}$) as well as the transformations between the variances of their estimates (eq. $\eqref{2}$ and $\eqref{3}$).

ranef_params <- function(lmeObject) {
  aV <- lmeObject$apVar
  # number of random effect parameters including covariance terms
  k <- ncol(aV)-1 
  # number of random effects (k = q(q+1)/2)
  q <-  0.5*(sqrt(8*k+1)-1)  
  #
  # standard deviation terms
  # 
  mu.eta.sd <- attr(aV,"Pars")[1:q]
  var.eta.sd <- diag(aV)[1:q]
  # inverse transformation
  mu.sigma.ii <- exp(mu.eta.sd)^2
  g.exp2 <- function(x) 2*exp(x)^2 # derivative
  var.sigma.ii <- g.exp2(mu.eta.sd)^2 * var.eta.sd
  #
  # correlation terms
  #
  mu.eta.r <- attr(aV,"Pars")[(q+1):k]  
  var.eta.r <- diag(aV)[(q+1):k]
  # inverse transformation
  invlogit <- function(x) (exp(x)-1)/(exp(x)+1)
  mu.sigma.ij <- invlogit(mu.eta.r)*prod(exp(mu.eta.sd))  
  g.invlogit <- function(x) 2*exp(x)/(exp(x)+1)^2 
  grad.g <- rbind(g.invlogit(mu.eta.r)*prod(exp(mu.eta.sd)), mu.sigma.ij, mu.sigma.ij) 
  var.sigma.ij <- t(grad.g^2) %*% c(var.eta.r, var.eta.sd)
  #
  res <- cbind(
    eta       = c(mu.eta.sd, mu.eta.r),
    var.eta   = c(var.eta.sd, var.eta.r),
    sigma     = c(mu.sigma.ii, mu.sigma.ij),
    var.sigma = c(var.sigma.ii, var.sigma.ij))
  aux <- outer(1:q,1:q,paste0)
  rownames(res) <- c(paste0("re.",diag(aux)),paste0("re.",aux[upper.tri(aux)])) 
  attr(res,"rho.ij") <- unname(invlogit(mu.eta.r))
  attr(res,"drho.ij_deta.ij") <- unname(g.invlogit(mu.eta.r))
  attr(res,"grad.rhoij") <- as.vector(grad.g)
  res
}
> ranef_params(fit.lme)
             eta   var.eta       sigma    var.sigma
re.11  0.8777582 0.1953469  5.78643480 26.163072722
re.22 -1.7128810 0.3269292  0.03252449  0.001383358
re.12 -1.6128707 1.3540139 -0.28962733  0.063383293
attr(,"rho.ij")
[1] -0.6676191
attr(,"drho.ij_deta.ij")
[1] 0.2771423
attr(,"grad")
[1]  0.1202302 -0.2896273 -0.2896273

The columns corresponds to

  • eta: estimated values of the random-effect parameters in the natural scale
  • var.eta: variances of the random effect parameters in the natural scale (diagonal of $\Psi_T$)
  • sigma: estimates values of random-effect parameters (elements of matrix $\Sigma_S$)
  • var.sigma: variances of the random effect parameters in the natural scale (diagonal of $\Psi_S$)

The two first rows refer to the variances of the random effects while the third row refers to their covariance (note: the function is limited to models with only two random effects). The attributes give some intermediary results in the computation of the sample correlation of $\rho_{ij}$ and variance of the sample covariance $\text{Var}(S_{ij})$.

The function lmerInfo::varcomp_vcovcomputes the variance-covariance matrix $\Psi_S$ of the sample random-effect parameters and can be used to check the results (I excluded the fourth dimension that corresponds to the residual error and reordered the terms):

> # variance-covariance matrix
> re.vv <- varcomp_vcov(fit.lme, type="expected")
> # variance terms (diagonal elements)
> diag(re.vv)[c(1,3,2)]
    Tau.subject.var((Intercept))             Tau.subject.var(age) Tau.subject.cov(age,(Intercept)) 
                    26.370535502                      0.001392966                      0.172430864 
> # full matrix                  
> round(re.vv[c(1,3,2),c(1,3,2)], 4)
                                 Tau.subject.var((Intercept)) Tau.subject.var(age) Tau.subject.cov(age,(Intercept))
Tau.subject.var((Intercept))                          26.3705               0.1561                          -2.0160
Tau.subject.var(age)                                   0.1561               0.0014                          -0.0147
Tau.subject.cov(age,(Intercept))                      -2.0160              -0.0147                           0.1724

The diagonal show the variances of the random effects ($\text{Var}(S_{ii})$ and $\text{Var}(S_{jj})$) and the of the covariance $\text{Var}(S_{ij})$. Comparing the results show a discrepancy with the values returned by the function ranef_params for the variance of the covriance term (0.1724 vs 0.0634).

I would appreciate help the error or reason for this discrepancy. I also wonder how to obtain the off-diagonal terms of $\Psi_S$ from $\Psi_T$.

Update

After the answer, it is easy the correct the line in the function above function that computes $\text{Var}(S_{ij})$:

> var.sigma.ij <- t(grad.g) %*% aV[3:1,3:1] %*% grad.g
> var.sigma.ij
                  reStruct.subject3
reStruct.subject3          0.170976

The rows and column of the matrix aV need to be reordered to be consistent with the gradient definition.

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Your derivation of $\text{Var}(S_{ij})$ is incorrect because it treats $T_i$, $T_j$, and $T_{ij}$ as uncorrelated. The correct delta method formula would use

$$\text{Var}(S_{ij}) \approx \left(\nabla_g(\mathbf{T})\right)' \boldsymbol\Psi_T \left(\nabla_g(\mathbf{T})\right)$$

This blog post provides more background on the multivariate delta method approximation. You can use the formulas there to get the off-diagonal terms also.

One last note: The fit.lme$apVar matrix is a numerical approximation (using numerical derivative approximations) to the observed information matrix, whereas varcomp_vcov() calculates the expected information matrix using analytic derivatives. Thus, even with the correct formulas, you will not get an exact match between the fit.lme$apVar matrix and the (suitably transformed) lmeInfo::varcomp_vcov() matrix.

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  • $\begingroup$ Thank you for the answer and the link toward blog post. I get now the correct result. I have update the question to include the change in the code. $\endgroup$
    – gavril
    Nov 18, 2023 at 22:15

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