Satterthwaite’s method for degrees of freedom has many applications with the Welch-Satterthwaite two-sample \(t\)-test allowing for different variances as the most well-known example. Confidence intervals for linear functions of mean squares is another example as is \(t\) and \(F\)-tests for contrasts of the mean-value parameters or so-called fixed-effects in linear mixed models which is the topic of this note.
In applications of linear mixed models to balanced datasets it is often possible to derive known integer denominator degrees of freedom for \(t\)-statistics for individual parameters (or one-dimensional contrasts) and \(F\)-statistics for model terms (or multi-dimensional contrasts). These statistics are convenient since they follow exact \(t\) and \(F\)-distributions thus exempting the need for asymptotic Wald or likelihood ratio tests whose small-sample behavior is dubious and often anti-conservative – even for rather large samples. [We can think of the Wald test as an \(F\)-test with infinite denominator df so in experiments which are balanced except for one or a few missing observations the denominator df from the corresponding balanced experiment provides an upper bound for the denominator df with the unbalanced dataset. The likelihood ratio test can be viewed as a small adjustment to the Wald test but otherwise with the same caveats.] But balance is a fragile aspect which is easily lost due to a missing observation even in well-designed experiments. In such cases it is concerning and inconvenient having to resort to the asymptotic tests when intuitively we would expect an approximate \(F\)-test to be much more accurate. Essentially, Satterthwaite’s method produces such an approximate \(t\) or \(F\)-test and if in fact an exact \(F\)-test exists this is produced. Simulations [1, 2] have shown these approximate \(F\)-tests to be an important improvement over the asymptotic tests in many situations.
In this note we describe the development of Satterthwaite’s method [3] for estimation of denominator degrees of freedom in \(t\) and \(F\)-test for contrasts of mean-value parameters in linear mixed models. It is worth noting that the Welch-Satterthwaite two-sample \(t\)-test is in fact a special case of the more general case presented here. We also exemplify how the actual computations can be done using the implementation of linear mixed models in the lme4 package [4] for \(\mathsf R\).
In the remainder of this section we briefly introduce a class of multivariate normal models and the class of linear mixed models. We also briefly introduce the \(t\)-test for one-dimensional contrasts of the mean-value parameters.
In the next section we describe the development of Satterthwaite’s method for estimation of denominator degrees of freedom in \(t\)-tests of one-dimensional contrasts. After describing how we address the computational challenges involved we present an example where \(t\)-tests of the mean-value parameters are evaluated in an unbalanced dataset.
The following section extends the application of Satterthwaite’s method for denominator degrees of freedom from one-dimensional contrasts leading to \(t\)-tests on one numerator degree of freedom to multi-dimensional contrasts leading to \(F\)-tests on multiple numerator degrees of freedom. Also here is an example given on an unbalanced dataset.
Satterthwaite’s method for denominator degrees of freedom as presented here is not only applicable to linear mixed models, but also to the more general class of multivariate normal models of which linear mixed models are a special case. The multivariate normal model can be written \[Y = X\beta + \varepsilon\] with \(\varepsilon \sim N(0, V_\tau)\), \(X\) being a \(n\times p\) design matrix, \(V_\tau\) being the variance-covariance matrix of the residuals \(\varepsilon\) (as well as of the observations \(Y\)), and \(\tau\) is the (usually small) \(k\)-vector of unique variance-covariance parameters. The \(p\)-vector of mean-value parameters \(\beta\) can be profiled out of the likelihood function since the estimate \(\hat\beta_\tau\) given a value for \(\tau\) can be expressed by the generalized least squares (GLS) estimator as the solution to \[X^\top V_\tau^{-1} X \hat\beta = X^\top V^{-1}_\tau y \] where \(y\) is the observed value of \(Y\). The asymptotic variance-covariance matrix of \(\hat\beta\) is \[\mathsf{V}_{\tau} (\hat\beta) = \left( X^\top V_\tau^{-1} X \right)^{-1}\] which is a function of \(\tau\) (only) and denoted \(\mathsf{V}_{\hat\tau} (\hat\beta)\) when evaluated at the (ML or REML) estimate \(\hat\tau\). Note that the distribution of \(Y\) and therefore the likelihood function for the model, only depends on the, usually small, parameter vector, \(\tau\).
The linear mixed model may be written \[Y = X\beta + Zb + \epsilon\] with random-effects \(b \sim N(0, G_\tau)\) and residuals \(\epsilon \sim N(0, R_\tau)\). Noting that \(\mathsf E(Y)=X\beta\) and \(\mathsf V(Y) \equiv V_\tau = Z G_\tau Z^\top + R_\tau\) we essentially have the multivariate normal model with some structure imposed on \(V_\tau\) with the same GLS estimator of \(\beta\) and its asymptotic variance-covariance matrix.
In this context we are interested in assessing a hypothesis concerning a linear function of the mean-value or fixed-effect parameters: \[H_0: L^\top \beta = 0\] where \(L^\top\) is a known contrast vector, we will consider the \(t\)-statistic \[t = \frac{L^\top \hat\beta}{\sqrt{L^\top \mathsf V_{\hat\tau}(\hat\beta) L} }\]
This statistic will in general not follow an exact \(t\)-distribution, but we may choose the degrees of freedom, \(\nu\), so that it follows approximately a \(t\)-distribution. The next section describes a method known as Satterthwaite’s method for choosing or approximating \(\nu\).
The method of finding \(\nu\) attributed to Satterthwaite (1946) [3] begins by matching the \(t\)-statistic above with a variable following an exact \(t\)-distribution, so we begin with the definition of the Student’s \(t\)-distribution (see for example this wiki page):
The random variable \[t_\nu = \frac{Z}{\sqrt{V/\nu }} \] is \(t_\nu\)-distributed with \(\nu\) degrees of freedom if \(Z \sim N(0, 1)\), \(V \sim \chi^2_\nu\) and \(Z\) and \(V\) are independent.
Considering the \(t\)-variable above, we divide both numerator and denominator with the (true, but unknown) variance of \(L^\top \hat\beta\): \[t = \frac{L^\top \hat\beta}{\sqrt{L^\top \mathsf V_{\hat\tau}(\hat\beta) L} } = \frac{L^\top \hat\beta / \sqrt{L^\top \mathsf V_{\tau}(\hat\beta) L}}{\sqrt{L^\top \mathsf V_{\hat\tau}(\hat\beta) L}\Big / \sqrt{L^\top \mathsf V_{\tau}(\hat\beta) L} } = \frac{L^\top \hat\beta / \sqrt{L^\top \mathsf V_{\tau}(\hat\beta) L}}{\sqrt{L^\top \mathsf V_{\hat\tau}(\hat\beta) L/ L^\top \mathsf V_{\tau}(\hat\beta) L} }\]
Now assuming that \(L^\top \hat\beta\) is normally distributed with expectation \(\mathsf E[L^\top \hat\beta]=0\) and (true, unknown) variance \(\mathsf V[L^\top \hat\beta] = L^\top \mathsf V_{\tau}(\hat\beta) L\) leads to \[L^\top \hat\beta / \sqrt{L^\top \mathsf V_{\tau}(\hat\beta) L} \sim N(0, 1)\] under the null hypothesis. Matching the \(t\)-variable with the definition of the \(t\)-distribution leads us to assume that \[\frac{\nu (L^\top \mathsf V_{\hat\tau}(\hat\beta) L)}{ L^\top \mathsf V_{\tau}(\hat\beta) L} \sim \chi^2_\nu\] is approximately \(\chi^2\)-distributed with \(\nu\) degrees of freedom. Note that this statistic has the familiar form \(\nu S^2 / \sigma^2\) which is \(\chi_\nu^2\)-distributed if \((\nu + 1)\) independent normally distributed random variables has variance \(\sigma^2\) and sum-of-squares \(S^2\).
Denoting \(S^2 = L^\top \mathsf V_{\hat\tau}(\hat\beta) L\) and \(\sigma^2 = L^\top \mathsf V_{\tau}(\hat\beta) L\) we have that (approximately) \[S^2 \sim \frac{\sigma^2}{\nu}\chi^2_\nu\] with expectation \[\mathsf{E}(S^2) = \mathsf{E}\left(\frac{\sigma^2}{\nu}\chi^2_\nu\right) = \frac{\sigma^2}{\nu} \mathsf{E}(\chi^2_\nu) = \frac{\sigma^2}{\nu} \nu = \sigma^2\] and variance \[\mathsf{V}(S^2) = \left(\frac{\sigma^2}{\nu}\right)^2 \mathsf{V}(\chi^2_\nu) = \left(\frac{\sigma^2}{\nu}\right)^2 2\nu = \frac{2(\sigma^2)^2}{\nu} \] The expectation doesn’t bring anything new to the table, but isolating \(\nu\) the expression for the variance leads to the following estimator for \(\nu\) (depending on the data through \(\tau\)): \[\hat\nu (\tau) = \frac{2(\sigma^2)^2}{\mathsf{V}(S^2)} = \frac{2(L^\top \mathsf V_{\tau}(\hat\beta) L)^2}{\mathsf{V}(L^\top \mathsf V_{\hat\tau}(\hat\beta) L)} \]
We may obtain the estimate \(\hat\nu (\hat\tau)\) (a value, for \(\hat\nu(\tau)\)) by evaluating at \(\tau=\hat\tau\). This is straight forward for the numerator, but we don’t directly have the denominator so we need an extra step here.
Noting that \(L^\top \mathsf V_{\tau}(\hat\beta) L \equiv f(\tau)\) is a function of \(\tau\) and using the delta method we may approximate the variance of \(f(\tau)\) at \(\tau=\hat\tau\) by \[\mathsf{V}(f(\hat\tau)) \approx f{'}(\hat\tau)^\top \mathsf{V}(\hat\tau) f{'}(\hat\tau)\] where \(f{'}(\hat\tau) = f{'}(\tau) |_{\tau=\hat\tau}\) is the vector of derivatives of \(f(\tau)\) wrt. \(\tau\) evaluated at \(\hat\tau\) and \(\mathsf{V}(\hat\tau)\) is the asymptotic variance-covariance matrix of the \(\tau\)-vector evaluated at \(\hat\tau\) (which can be obtained (numerically) as the inverse Hessian of the negative log-likelihood function).
Finally, we may evaluate the estimator of \(\nu\) at \(\hat\tau\) to obtain \[\hat\nu(\hat\tau) = \frac{2(L^\top \mathsf V_{\hat\tau}(\hat\beta) L)^2}{f{'}(\hat\tau)^\top \mathsf{V}(\hat\tau) f{'}(\hat\tau)} = \frac{2f(\hat\tau)^2}{f{'}(\hat\tau)^\top \mathsf{V}(\hat\tau) f{'}(\hat\tau)} \]
Instead of directly evaluating the gradient of \(f(\hat\tau) = L^\top \mathsf V_{\hat\tau}(\hat\beta) L\) we write the gradient as \[f'(\hat\tau) = L^\top \{ \nabla_\tau \mathsf V_\tau(\hat\beta) \} L \, \big |_{\tau=\hat\tau}\] Here the Jacobian \(\mathcal J \equiv \nabla_\tau \mathsf V_\tau(\hat\beta)\) is the 3-dimensional array with \(k\) faces of size \(p\times p\). Left and right multiplying each face by \(L^\top\) and \(L\) respectively (where \(L\) is \(p\times 1\)) leads to the \(k\)-vector \(f'(\hat\tau)\).
This has the advantage that the gradient is not tied to a particular \(L\) and \(\mathcal J\) may be re-used for other contrast matrices, and, as we shall see later, can also be used for multi-degree of freedom \(F\)-tests.
The computational challenge is essentially to evaluate the denominator in the expression for \(\hat\nu(\hat\tau)\), which amounts to computing the gradient \(f{'}(\hat\tau)\) and variance-covariance matrix of the variance parameter vector \(\mathsf{V}(\hat\tau)\). We will now address these challenges in turn starting with the latter.
The variance-covariance matrix of the variance parameter vector \(\mathsf{V}(\hat\tau)\) can be computed as the inverse of Hessian of the negative log-likelihood function with respect to the variance-parameters. Conveniently the lme4 package provides a simple way to extract a deviance function from a linear mixed model fit. The deviance is a scaled version of the log-likelihood function for the model so numerically computing the Hessian of the deviance function provides an estimate of the covariance matrix of the variance parameters. One minor challenge is that the residual variance, \(\sigma\) is profiled out of the deviance function as implemented in lme4, so the deviance function cannot be used directly as it comes. A variant of the deviance function has to be written that operates on the variance parameters, \(\tau\). Using the hessian function from the numDeriv package [5] conveniently and accurately evaluates the Hessian numerically.
The variance-covariance matrix of \(\beta\) can also be expressed as a function of the variance parameters using the same deviance function. Using the jacobian function from numDeriv allows us to numerically evaluate \(\mathcal J\) at \(\hat\tau\). Having computed \(\mathsf{V}(\hat\tau)\) and \(\mathcal J\) its only a matter of putting the different quantities together to compute the estimate of the denominator degrees of freedom.
lme4::lmerIn this example we will consider the ham dataset from the lmerTest package [1] in which 81 consumers evaluated 4 products twice leading to 648 observations. The dataset is balanced, but we randomly select and use 580 observations (corresponding to approximately 90%) making the dataset unbalanced. An initial fit of the data using lmer from the lme4 package reads:
library(lme4)
## Loading required package: Matrix
data(ham, package="lmerTest")
set.seed(12345)
model <- lmer(Informed.liking ~ Product + (1|Consumer),
data = ham[sample(x=1:nrow(ham), size=580, replace=FALSE), ])
summary(model, corr=FALSE)
## Linear mixed model fit by REML ['lmerMod']
## Formula: Informed.liking ~ Product + (1 | Consumer)
## Data: ham[sample(x = 1:nrow(ham), size = 580, replace = FALSE), ]
##
## REML criterion at convergence: 2560.7
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.4794 -0.6766 0.1447 0.7477 2.4066
##
## Random effects:
## Groups Name Variance Std.Dev.
## Consumer (Intercept) 0.7408 0.8607
## Residual 4.3152 2.0773
## Number of obs: 580, groups: Consumer, 81
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 5.84346 0.19790 29.528
## Product2 -0.66197 0.24431 -2.709
## Product3 0.20102 0.24476 0.821
## Product4 0.08105 0.24656 0.329Suppose now that we are interested in the contrast
L <- c(0, 1, 0, 0) which simply picks out the second coefficient; the estimate of the difference between product 2 and product 1.
The estimate of this contrast \(L^\top \beta\) is then computed with:
(estimate <- drop(t(L) %*% fixef(model)))
## [1] -0.6619655In this model the variance-covariance parameters are collected in the 2-vector \(\tau = [\sigma_c, \sigma]\); the square root of the random-effect variance for consumers and the residual standard deviation. lmer however, profiles out the residual standard deviation of the (restricted) profile likelihood and operates with vector \(\theta\) of relative variance parameters. In this model \(\theta = \sigma_c / \sigma\).
The parameters that characterize the model can be summarized as
(parlist <- list(beta=fixef(model),
theta=getME(model, "theta"),
sigma=sigma(model)))
## $beta
## (Intercept) Product2 Product3 Product4
## 5.84345560 -0.66196549 0.20102196 0.08105489
##
## $theta
## Consumer.(Intercept)
## 0.4143353
##
## $sigma
## [1] 2.077314and the consumer random effect standard deviation can be retrieved with
with(parlist, theta*sigma)
## Consumer.(Intercept)
## 0.8607045A function that directly evaluates the model deviance (REML or ML criterion) can be obtained directly from the model fit:
devfun <- update(model, devFunOnly=TRUE)Being a function of \(\theta\), we can calculate the deviance at \(\hat\theta\) with
devfun(parlist$theta)
## [1] 2563.524To compute the variance-covariance matrix of \(\tau\) by numerically evaluating the Hessian of the log-likelihood function, we need to re-implement the deviance function as a function of the unique variance-covariance parameters (\(\tau\)) – not only the relative variance parameters (\(\theta\)). Utilizing the deviance function as a function of \(\theta\) and some particulars of the implementation of linear mixed models in lmer as described in [4] we can implement this as:
devfun_varpar <- function(varpar, devfun, reml) {
# Computes deviance as a function of 'varpar=c(theta, sigma)'
# devfun: deviance function as a function of theta only.
# reml: TRUE if REML; FALSE if ML
nvarpar <- length(varpar)
sigma2 <- varpar[nvarpar]^2
theta <- varpar[-nvarpar]
df_envir <- environment(devfun)
devfun(theta) # Evaluate deviance function at varpar
n <- nrow(df_envir$pp$V)
# Compute deviance for ML:
dev <- df_envir$pp$ldL2() + (df_envir$resp$wrss() + df_envir$pp$sqrL(1))/sigma2 +
n * log(2 * pi * sigma2)
if(!reml) return(dev)
# Adjust of REML is used:
RX <- df_envir$pp$RX() # X'V^{-1}X ~ crossprod(RX^{-1}) = cov(beta)^{-1} / sigma^2
dev + 2*c(determinant(RX)$modulus) - ncol(RX) * log(2 * pi * sigma2)
}This function returns the same deviance but as a function of a different parameter vector:
is_reml <- getME(model, "is_REML")
varpar_opt <- unname(c(parlist$theta, parlist$sigma))
devfun_varpar(varpar_opt, devfun, reml=is_reml)
## [1] 2563.532We can now obtain the hessian for \(\tau\) of the deviance function with the numerical approximation provided in the numDeriv package:
library(numDeriv)
h <- hessian(func=devfun_varpar, x=varpar_opt, devfun=devfun, reml=is_reml) Before inverting h, we check that it is positive definite by confirming that all its eigenvalues are positive:
eig_h <- eigen(h, symmetric=TRUE)
eig_h$values
## [1] 795.7477 353.1973
stopifnot(all(eig_h$values > 0))Inverting and scaling h provides the asymptotic variance-covariance matrix of the variance parameters at their estimate \(\mathsf{V}(\hat\tau)\):
h_inv <- with(eig_h, vectors %*% diag(1/values) %*% t(vectors))
(cov_varpar <- 2 * h_inv)
## [,1] [,2]
## [1,] 0.003828133 -0.001553014
## [2,] -0.001553014 0.004347785To compute \(f'(\hat\tau) = L^\top \{ \nabla_\tau \mathsf V_\tau(\hat\beta) \} L \, \big |_{\tau=\hat\tau}\) we first implement \(\mathsf V_\tau(\hat\beta)\) as a function of \(\tau\):
get_covbeta <- function(varpar, devfun) {
# Compute cov(beta) as a function of varpar
#
# varpar: c(theta, sigma)
# devfun: deviance function, ie. update(model, devFunOnly=TRUE) - a function of theta
# return: cov(beta) given specified varpar
#
nvarpar <- length(varpar)
sigma <- varpar[nvarpar] # residual std.dev.
theta <- varpar[-nvarpar] # ranef var-par
devfun(theta) # evaluate REML or ML deviance 'criterion'
df_envir <- environment(devfun) # extract model environment
sigma^2 * tcrossprod(df_envir$pp$RXi()) # vcov(beta)
}Then evaluate the gradient (Jacobian), \(\mathcal J = \nabla_\theta \mathsf V_\theta(\hat\beta)\) numerically using the jacobian function from the numDeriv package and organize it as a list (of length \(k\)) of \(p\times p\) matrices:
Jac <- jacobian(func=get_covbeta, x=varpar_opt, devfun=devfun)
Jac_list <- lapply(1:ncol(Jac), function(i)
array(Jac[, i], dim=rep(length(parlist$beta), 2))) # k-list of jacobian matricesLeft and right multiplying each matrix by \(L^\top\) and \(L\) respectively gives the gradient vector (of length \(k\)):
(grad_var_Lbeta <- vapply(Jac_list, function(x)
sum(L * x %*% L), numeric(1L))) # = {L' Jac L}_i
## [1] 0.0008652581 0.0580662870We now have all elements for the denominator of \(\hat\nu(\hat\tau)\). The only element in the numerator, the estimated covariance matrix of the contrast \(L^\top \beta\) is then simply \(\mathsf V(L^\top \hat\beta) = L^\top \mathsf V_{\hat\tau}(\hat\beta) L\) is simple to evaluate since vcov(model) extracts \(\mathsf V_{\hat\tau}(\hat\beta)\):
cov_beta <- as.matrix(vcov(model))
# Compute vcov(Lbeta)
(var_Lbeta <- drop(t(L) %*% cov_beta %*% L))
## [1] 0.05968911
# Alternative: (var_con <- get_covbeta(varpar_opt, devfun))Collecting all elements gives the following coefficient table:
se.estimate <- sqrt(var_Lbeta)
satt_denom <- sum(grad_var_Lbeta * (cov_varpar %*% grad_var_Lbeta)) # g'Ag
ddf <- drop(2 * var_Lbeta^2 / satt_denom) # denominator DF
tstat <- estimate/se.estimate
pvalue <- 2 * pt(abs(tstat), df = ddf, lower.tail = FALSE)
data.frame(estimate, se=se.estimate, tstat, ddf, pvalue)
## estimate se tstat ddf pvalue
## 1 -0.6619655 0.2443135 -2.709492 491.2089 0.006974127Often it is of interest to assess an hypothesis on more than one linear function of the parameters such as \[H_0:\; L\beta = 0\] were the \(q\times p\) matrix \(L\) has rank \(q\) with \(1 < q \leq p\) and \(p\) is the length of the \(\beta\)-vector.
The conventional \(F\)-statistic for this hypothesis reads \[F = \frac{1}{q} (L\hat\beta)^\top (L \mathsf V_{\hat\tau}(\hat\beta)L^\top)^{-1} (L\hat\beta)\] which is seen to reduce to the square of the \(t\)-statistic considered above for \(q=1\). Despite its name, this statistic only follows an exact \(F\)-distribution with \(q\) numerator degrees of freedom and known denominator degrees of freedom in special cases (such as orthogonal or balanced designs.)
In the general case we are interested in estimating an appropriate denominator degrees of freedom for \(F\) assuming \(q\) numerator degrees of freedom.
Following Fai and Cornelius (1996) [6] we eigen-decompose the variance-covariance matrix of \(L\hat\beta\): \[\mathsf{V}_{\hat\tau}(L\hat\beta) = L \mathsf{V}_{\hat\tau}(\hat\beta) L^{\top} = PDP^{\top}\] such that we may write \[\begin{aligned} qF\equiv Q & = (L \hat\beta)^{\top} P D^{-1}P^{\top} (L \hat\beta) & \\ & = \hat\beta^{\top} L^{\top} P D^{-1}P^{\top} L \hat\beta & \\ & = (P^{\top} L \hat\beta)^{\top} D^{-1} (P^{\top} L \hat\beta) & \\ & = \sum_{m=1}^{q} \frac{(P^{\top} L \hat\beta)_{m}^{2}}{d_{m}} = \sum_{m=1}^{q} t_{m}^{2} \\ \end{aligned}\]
where \((P^\top L \hat\beta)_m\) denotes the \(m\)’th element of the \(q\)-vector \(P^\top L \hat\beta\) (\(P^\top\) is an orthonormal \(q\times q\) rotation matrix), and \(d_m\) is the \(m\)’th diagonal element of \(D\); the diagonal matrix of eigenvalues.
Thus \(Q\) is being rewritten as a sum of \(q\) independent variables that have the form of the square of \(t\)-statistics each on one “numerator-degree of freedom”.
In equivalence with (and multidimensional extension of) the 1D case above let \[f(\hat\tau) \equiv P^\top L \mathsf V_{\hat\tau}(\hat\beta) L^\top P = \tilde L \mathsf V_{\hat\tau}(\hat\beta) \tilde L^\top = D, \quad \mathrm{with} \quad \tilde L = P^\top L\] be the diagonal matrix of eigenvalues (of \(\mathsf V_{\hat\tau}(L\hat\beta)\)) and let \(f(\hat\tau)_m = d_m\) denote the \(m\)’th eigenvalue.
Note also that the \(m\)th diagonal element of \(f(\hat\tau)\) can be written \[f(\hat\tau)_m = (\tilde L \mathsf V_{\hat\tau} (\hat\beta)\tilde L)_m = \tilde L_m \mathsf V_{\hat\tau} (\hat\beta)\tilde L_m\]
The \(m\)’th \(t\)-statistic in the summation above can now be written as \[t_m = \frac{\tilde L_m \hat\beta}{\sqrt{\tilde L_m \mathsf V_{\hat\tau}(\hat\beta)\tilde L_m}} \] which clarifies the similarity with the \(t\)-statistic considered in the 1-degree of freedom case above.
We therefore proceed as in the 1D case to estimate the degrees of freedom \(\nu_m\) for the \(q\) \(t_m\)-statistics with \[\hat\nu_m(\hat\tau) = \frac{2f(\hat\tau)_m^2}{f'(\hat\tau)_m^\top \mathsf V(\hat\tau) f'(\hat\tau)_m} = \frac{2d_m^2}{f'(\hat\tau)_m^\top \mathsf V(\hat\tau) f'(\hat\tau)_m}\] in which the gradient vector of length \(k\) reads \[f'(\hat\tau)_m = \tilde L^\top_m \{ \nabla_\tau \mathsf V_\tau(\hat\beta)|_{\tau = \hat\tau} \} \tilde L_m\] as discussed in the 1D case above.
Having determined estimates of \(\nu_m\) for \(m=1,\ldots, q\), the objective is now to determine the denominator degrees of freedom for the \(F\)-statistic above. Since an \(F\)-variable with \(q\) numerator and \(\nu\) denominator degrees of freedom has expectation \[\mathsf E(F_{q,\nu}) = \frac{\nu}{\nu - 2} \quad \mathrm{for} \quad \nu > 2\] and we may for our \(F\)-statistics evaluate this quantity via \[{\mathsf E} (F_{q, \nu}) = \mathsf E(q^{-1}Q) = q^{-1} \mathsf E(Q)\] we may consider as estimator for \(\nu\) the solution to \[q^{-1} \mathsf E(Q) = \frac{\nu}{\nu - 2} \quad \mathrm{for} \quad \nu > 2\] which leads to \[\hat\nu = \frac{2 \mathsf E(Q)}{\mathsf E(Q) - q}\] Note that we need to require that \(\mathsf E(Q) > q\) since only then is \(\hat\nu\) positive.
The only remaining step is to determine \(\mathsf E(Q)\) where again we utilize the expectation of an \(F\)-variable with at least two denominator degrees of freedom: \[\mathsf E(Q) = \sum_{m=1}^q \mathsf E(t_{\nu_m}^2) = \sum_{m=1}^q \mathsf E(F_{1, \nu_m}) = \sum_{m=1}^q \frac{\nu_m}{\nu_m - 2} \quad \mathrm{for} \quad \nu_m > 2\]
lme4::lmerConsidering the example above, we start by defining the contrast matrix, \(L\) and form the contrast estimate \(L\hat\beta\):
L <- rbind(c(0, 1, 0, 0),
c(0, 0, 1, 0))
(Lbeta <- drop(L %*% parlist$beta))
## [1] -0.6619655 0.2010220and compute the variance of the contrasts \(\mathsf{V}_{\hat\tau} (L\hat\beta) = L \mathsf{V}_{\hat\tau} (\hat\beta) L^\top\):
cov_Lbeta <- L %*% cov_beta %*% t(L) # Var(contrast) = Var(Lbeta)We then compute the eigen-decomposition \(\mathsf{V}(L\hat\beta) = P^\top D P\) and extract the rank, eigenvectors and eigenvalues:
# Get eigen decomposition of vcov(Lbeta):
eig_VLbeta <- eigen(cov_Lbeta)
positive <- eig_VLbeta$values > 1e-8
q <- sum(positive) # rank(VLbeta)
(P <- eig_VLbeta$vectors)
## [,1] [,2]
## [1,] 0.7058316 -0.7083797
## [2,] 0.7083797 0.7058316
(d <- eig_VLbeta$values)
## [1] 0.08984076 0.02975399The new contrast vectors are then \(\tilde L = P^\top L\)
PtL <- crossprod(P, L) # q x p matrixfrom which we can compute the \(t^2\) values and \(F\)-statistic:
t2 <- drop(PtL %*% parlist$beta)^2 / d
Fvalue <- sum(t2) / qFor the new contrast vectors \(\tilde L_m = (P^\top L)_m\) for \(m =1,\ldots, q\) we compute the gradient \(f'(\hat\tau)_m\):
grad_PLcov <- lapply(1:q, function(m) {
vapply(Jac_list, function(J) sum(PtL[m, ] * J %*% PtL[m, ]), numeric(1L))
})The \(m\)’th denominator degree of freedom estimate is then \(\hat\nu_m = 2d_m^2 (f'(\hat\tau)_m^\top A f'(\hat\tau)_m^\top)^{-1}\):
nu_m <- vapply(1:2, function(m) {
denom <- sum(grad_PLcov[[m]] * (cov_varpar %*% grad_PLcov[[m]])) # g'Ag
2*(d[m])^2 / denom # 2d_m^2 / g'Ag
}, numeric(1L))
nu_m
## [1] 501.4952 494.6459From \(\mathsf E(Q)\) we then evaluate the estimated denominator degrees of freedom for the \(F\)-statistic:
EQ <- sum(nu_m / (nu_m - 2))
(ddf <- 2 * EQ / (EQ - q)) # nu
## [1] 498.0469In summary we have the following approximate \(F\)-test:
pvalue <- pf(q=Fvalue, df1=q, df2=ddf, lower.tail=FALSE)
data.frame('F value'=Fvalue, ndf=q, ddf=ddf, 'p-value'=pvalue,
check.names = FALSE)
## F value ndf ddf p-value
## 1 6.856825 2 498.0469 0.00115446Remark 1: If \(L\) is rank deficient
If the contrast matrix \(L\) is row-rank deficient with \(< q\) rows where \(q\) is the rank of \(L\), then all that it is needed is to define \(\tilde L\) as the first \(q\) rows of \(P^\top L\). Notice that we can find \(q\) as the number of non-zero (save a tolerance) eigenvalues in \(D\).
Remark 2: Sum of Squares and Mean Squares
The sums of squares (SSQ) associated with a contrast \(L\) can be written \[\mathrm{SSQ}(L\hat\beta) = F q \hat\sigma^2\] with mean square (MS) \[\mathrm{MS}(L\hat\beta) = \mathrm{SSQ}(L\hat\beta) / q = F \hat\sigma^2\] These SSQ and MS do not correspond to those that can be derived for variance-component models but refer directly to the multivariate normal model.
Remark 3: \(q = 1\)
For \(q = 1\) the \(F\)-value can be computed as the square of the \(t\)-statistics with its estimate of the denominator degrees of freedom without evaluation of \(\mathsf E(Q)\).
Remark 4: \(q \geq 2\)
For \(q \geq 2\) we need \(\mathsf E(Q) > q\), but note that \[\mathsf E(Q) = \sum_{m=1}^q \frac{\nu_m}{\nu_m - 2} > q\] if \(\nu_m > 2\) for all \(m\), since \[\frac{\nu_m}{\nu_m - 2} > 1 \quad \mathrm{for} \quad \nu_m > 2\] Thus the requirement reduces to \(\nu_m > 2\) for \(m = 1, \ldots, q\).
Remark 5: \(\nu\) tends to 2
If, for some \(m\), say \(m'\) \(\nu_{m'} \rightarrow 2_+\) (\(\nu_{m'}\) approaches \(2\) from above), then \[\hat\nu = \frac{2 \mathsf E(Q)}{\mathsf E(Q) - q} \rightarrow 2\] since \[\frac{\nu_{m'}}{\nu_{m'} - 2} \rightarrow \infty \] and therefore \[\mathsf E(Q) \rightarrow \infty\]
Remark 6: lower bound on \(\nu\)
A property of the \(F\)-distribution is that if \[X \sim F_{q, \nu} \quad \mathrm{then} \quad X^{-1} \sim F_{\nu, q}\] Thus we may consider \(2\) as a lower bound on \(\nu\) (recall that \(q \geq 2\)) which may be used if any \(\nu_m < 2\).
Remark 7: all \(q\) estimates \(\nu_m\) are the same
If all \(q\) estimates \(\nu_m\) are the same and \(\nu_1 = \ldots = \nu_q\) we have that \[\mathsf E(Q) = q\frac{\nu_m}{\nu_m - 2}\] and therefore \[\nu = \frac{2\mathsf E(Q)}{\mathsf E(Q) - q} = \frac{q\frac{2 \nu_m}{\nu_m - 2}}{q \frac{\nu_m - \nu_m + 2}{\nu_m - 2}} = \frac{2 \nu_m}{\nu_m - 2} \frac{\nu_m - 2}{2} = \nu_m\] thus the estimate of the degrees of freedom \(\nu\) for the \(F\)-statistic is just \(\nu = \nu_1 = \ldots = \nu_q\).
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