Introducing chandwich: Chandler-Bate
Sandwich Loglikelihood Adjustment
Paul Northrop and Richard Chandler
2023-08-25
The chandwich package performs adjustments of an
independence loglikelihood using a robust sandwich estimator of the
parameter covariance matrix, based on the methodology in Chandler and Bate (2007). This can be used for
cluster correlated data when interest lies in the parameter vector \(\theta\) of the marginal distributions or
for performing inferences that are robust to certain types of model
misspecification.
Suppose that we have \(k\) clusters
of observations \(\{y_j, j = 1, \ldots,
k\}\), where \(y_j\) is a vector
of observations for the \(j\)th
cluster. The independence loglikelihood function is given by
\[\ell_I(\theta) = \sum_{j=1}^{k}
\ell_j(\theta; y_j),\] where \(\ell_j(\theta; y_j)\) is the contribution
from the \(j\)th cluster. Suppose that
the independence maximum likelihood estimator (MLE) \(\hat{\theta}\) is the unique root of \[\frac{\partial \ell_I(\theta)}{\partial \theta} =
\sum_{j=1}^{k} \frac{\partial \ell_j(\theta; y_j)}{\partial \theta} =
\sum_{j=1}^{k} U_j(\theta) = 0.\] The adjustments scale \(\ell_I(\theta)\) about \(\hat{\theta}\) so that the Hessian \(\hat{H}_A\) of the adjusted loglikelihood
is consistent with the sandwich estimator (Davison 2003) of the covariance matrix of \(\hat{\theta}\). Specifically, \(\hat{H}_A^{-1} = -\hat{H}^{-1} \hat{V}
\hat{H}^{-1},\) where \(\hat{H}\) is the Hessian of \(l_I(\theta)\) at \(\hat{\theta}\) and \(\hat{V} =\sum_{j=1}^{k} U_j(\hat{\theta})
U^{T}_j(\hat{\theta})\).
There are two types of adjustment. A horizontal adjustment \[\ell_A(\theta) = \ell_I(\hat{\theta} + C(\theta -
\hat{\theta}))\] where the parameter scale is adjusted, and a
vertical adjustment \[\ell_{A2}(\theta) =
\ell_I(\hat{\theta}) +
\{(\theta - \hat{\theta})^T \hat{H}_A (\theta - \hat{\theta})\}
\frac{\ell_I(\theta) - \ell_I(\hat{\theta})}{(\theta - \hat{\theta})^T
\hat{H}_I (\theta - \hat{\theta})},\] where the loglikelihood is
scaled. The differences between these adjustments are discussed in
Section 6 of Chandler and Bate (2007). The
horizontal adjustment involves calculating matrix square roots of \(\hat{H}\) and \(\hat{H}_A\). Chandler and Bate (2007) consider two ways of
doing this, one using Cholesky decomposition another using spectral
decomposition.
Loglikelihood adjustment using the adjust_loglik function
The function adjust_loglik returns an object of class
chandwich: a function that can be used to evaluate \(\ell_I(\theta)\), \(\ell_{A2}(\theta)\) and both choices of
\(\ell_A(\theta)\). Functions that take
a chandwich object as an argument have a vector argument
type, equal to one of
"vertical", "cholesky", "spectral", "none", to select the
type of adjustment. The default is type = "vertical".
We illustrate the loglikelihood adjustments and the use of the
functions in chandwich using some examples, starting with a
simple model that has one parameter.
Binomial model
The rats data (Tarone
1982) contain information about an experiment in which, for each
of 71 groups of rats, the total number of rats in the group and the
numbers of rats who develop a tumor is recorded. We model these data
using a binomial distribution, treating each groups of rats as a
separate cluster. A Bayesian analysis of these data based on a
hierarchical binomial-beta model is presented in Section 5.3 of Gelman et al. (2014).
The following code creates a function that returns a vector of the
loglikelihood contributions from individual groups of rats, calls
adjust_loglik to make the adjustments, produces a basic
summary of the MLE and the unadjusted and adjusted standard errors and
plots the adjusted and unadjusted loglikelihood. There is no need to
supply the argument cluster in this example because the
default, that each observation (group of rats) forms its own cluster
applies here.
library(chandwich)
binom_loglik <- function(prob, data) {
if (prob < 0 || prob > 1) {
return(-Inf)
}
return(dbinom(data[, "y"], data[, "n"], prob, log = TRUE))
}
# Make the adjustments
rat_res <- adjust_loglik(loglik = binom_loglik, data = rats, par_names = "p")
summary(rat_res)
#> MLE SE adj. SE
#> p 0.1535 0.008645 0.01305
plot(rat_res, type = 1:4, legend_pos = "bottom", lwd = 2, col = 1:4)
In the one-dimensional case the (horizontal) Cholesky and spectral
adjustments are identical and, over the range in the plot, the vertical
adjustment is very similar to the horizontal adjustments. Appreciable
differences only becomes apparent towards the edges of the parameter
space, where the behaviour of the vertical adjustment, which approaches
\(-\infty\) as \(p\) approaches 0 and 1, may be preferable
to the behaviour of the horizontal adjustments. As we expect in this
example the adjusted standard error is slightly greater than the
unadjusted standard error.
plot(rat_res, type = 1:4, legend_pos = "bottom", lwd = 2, col = 1:4,
xlim = c(0, 1))
Confidence intervals
The function conf_intervals calculates (profile, if
necessary) likelihood-based confidence intervals for individual
parameters, and also provides symmetric intervals based on a normal
approximation to the sampling distribution of the estimator.
chandwich also has a confint S3 method, which
produces only the likelihood-based intervals.
# 95% confidence intervals, unadjusted and vertically adjusted
conf_intervals(rat_res, type = "none")
#> Waiting for profiling to be done...
#> Model: binom_loglik
#>
#> 95% confidence interval, independence loglikelihood
#>
#> Symmetric:
#> lower upper
#> p 0.1366 0.1705
#>
#> Likelihood-based:
#> lower upper
#> p 0.1372 0.1710
conf_intervals(rat_res)
#> Waiting for profiling to be done...
#> Model: binom_loglik
#>
#> 95% confidence interval, adjusted loglikelihod with type = ''vertical''
#>
#> Symmetric:
#> lower upper
#> p 0.1280 0.1791
#>
#> Likelihood-based:
#> lower upper
#> p 0.1292 0.1802
confint(rat_res)
#> Waiting for profiling to be done...
#> 2.5 % 97.5 %
#> p 0.1292236 0.1802395
Extreme value modelling of maximum temperatures
We consider the example presented in Section 5.2 of Chandler and Bate (2007). The
owtemps data contain annual maximum temperatures in Oxford
and Worthing in the U.K. from 1901 to 1980, which are viewed as 80
clusters of independent observations from a bivariate distribution. We
construct an independence loglikelihood based on generalized extreme
value (GEV) models for the marginal distributions at Oxford and
Worthing. The model is parameterized so that the marginal distribution
at Oxford is GEV(\(\mu_0 + \mu_1, \sigma_0 +
\sigma_1, \xi_0 + \xi_1\)) and at Worthing is GEV(\(\mu_0 - \mu_1, \sigma_0 - \sigma_1, \xi_0 -
\xi_1\)), where GEV(\(\mu, \sigma,
\xi\)) denotes a GEV distribution with location \(\mu\), scale \(\sigma\) and shape \(\xi\).
We perform loglikelihood adjustment for the full six-parameter model
and reproduce the relevant rows of Table 2 in Chandler and Bate (2007).
gev_loglik <- function(pars, data) {
o_pars <- pars[c(1, 3, 5)] + pars[c(2, 4, 6)]
w_pars <- pars[c(1, 3, 5)] - pars[c(2, 4, 6)]
if (isTRUE(o_pars[2] <= 0 | w_pars[2] <= 0)) return(-Inf)
o_data <- data[, "Oxford"]
w_data <- data[, "Worthing"]
check <- 1 + o_pars[3] * (o_data - o_pars[1]) / o_pars[2]
if (isTRUE(any(check <= 0))) return(-Inf)
check <- 1 + w_pars[3] * (w_data - w_pars[1]) / w_pars[2]
if (isTRUE(any(check <= 0))) return(-Inf)
o_loglik <- log_gev(o_data, o_pars[1], o_pars[2], o_pars[3])
w_loglik <- log_gev(w_data, w_pars[1], w_pars[2], w_pars[3])
return(o_loglik + w_loglik)
}
# Initial estimates (method of moments for the Gumbel case)
sigma <- as.numeric(sqrt(6 * diag(var(owtemps))) / pi)
mu <- as.numeric(colMeans(owtemps) - 0.57722 * sigma)
init <- c(mean(mu), -diff(mu) / 2, mean(sigma), -diff(sigma) / 2, 0, 0)
# Perform the log-likelihood adjustment of the full model
par_names <- c("mu[0]", "mu[1]", "sigma[0]", "sigma[1]", "xi[0]", "xi[1]")
large <- adjust_loglik(gev_loglik, data = owtemps, init = init,
par_names = par_names)
# Rows 1, 3 and 4 of Table 2 of Chandler and Bate (2007)
round(attr(large, "MLE"), 4)
#> mu[0] mu[1] sigma[0] sigma[1] xi[0] xi[1]
#> 81.1702 2.6684 3.7291 0.5312 -0.1989 -0.0883
round(attr(large, "SE"), 4)
#> mu[0] mu[1] sigma[0] sigma[1] xi[0] xi[1]
#> 0.3282 0.3282 0.2293 0.2293 0.0494 0.0494
round(attr(large, "adjSE"), 4)
#> mu[0] mu[1] sigma[0] sigma[1] xi[0] xi[1]
#> 0.4036 0.2128 0.2426 0.1911 0.0394 0.0362
Confidence intervals
We use conf_intervals to calculate profile
likelihood-based confidence intervals for the parameters.
# 95% confidence intervals, vertically adjusted
conf_intervals(large)
#> Waiting for profiling to be done...
#> Model: gev_loglik
#>
#> 95% confidence intervals, adjusted loglikelihod with type = ''vertical''
#>
#> Symmetric:
#> lower upper
#> mu[0] 80.37910 81.96121
#> mu[1] 2.25128 3.08552
#> sigma[0] 3.25368 4.20459
#> sigma[1] 0.15660 0.90574
#> xi[0] -0.27623 -0.12166
#> xi[1] -0.15939 -0.01731
#>
#> Profile likelihood-based:
#> lower upper
#> mu[0] 80.37291 81.96021
#> mu[1] 2.24357 3.08287
#> sigma[0] 3.29922 4.25998
#> sigma[1] 0.16118 0.94338
#> xi[0] -0.27410 -0.11572
#> xi[1] -0.16519 -0.02002
Confidence regions
We reproduce Figure 4(b) of Chandler and Bate
(2007), adding a 95% confidence region for the vertical
adjustment.
which_pars <- c("sigma[0]", "sigma[1]")
gev_none <- conf_region(large, which_pars = which_pars, type = "none")
#> Waiting for profiling to be done...
gev_vertical <- conf_region(large, which_pars = which_pars)
#> Waiting for profiling to be done...
gev_cholesky <- conf_region(large, which_pars = which_pars, type = "cholesky")
#> Waiting for profiling to be done...
gev_spectral <- conf_region(large, which_pars = which_pars, type = "spectral")
#> Waiting for profiling to be done...
plot(gev_none, gev_cholesky, gev_vertical, gev_spectral, lwd = 2,
xlim = c(3.0, 4.5), ylim = c(-0.1, 1.25))
The 95% contours of the profile adjusted loglikelihoods are
similar.
Comparing nested models
Suppose that we wish to test the null hypothesis that Oxford and
Worthing share a common GEV shape parameter, that is, that \(\xi_1 = 0\). We call
adjust_loglik again using the argument
fixed_pars to fix \(\xi_1\) to zero and perform loglikelihood
adjustment under the reduced model. Then we use
compare_models to carry out an adjusted likelihood ratio
test. If approx = FALSE (the default) then equation (17) in
Section 3.3 of Chandler and Bate (2007) is
used. If approx = TRUE then the approximation detailed in
equations (18)-(20) is used.
medium <- adjust_loglik(larger = large, fixed_pars = "xi[1]")
compare_models(large, medium)
#> Model: gev_loglik
#> H0: "xi[1]" = 0
#> HA: unrestricted model
#>
#> test statistic = 6.356, df = 1, p-value = 0.0117
compare_models(large, medium, approx = TRUE)
#> Model: gev_loglik
#> H0: "xi[1]" = 0
#> HA: unrestricted model
#>
#> Using using approximation (18) of Chandler and Bate (2007):
#> test statistic = 5.245, df = 1, p-value = 0.022
Alternatively, we can call compare_models directly
without first creating medium, using the argument
fixed_pars to fix \(\xi_1\) to zero.
compare_models(large, fixed_pars = "xi[1]")
#> Model: gev_loglik
#> H0: "xi[1]" = 0
#> HA: unrestricted model
#>
#> test statistic = 6.356, df = 1, p-value = 0.0117
In this case whether or not we use the approximate approach has a
non-negligible effect on the \(p\)-value but we would probably reject the
null in either case.
chandwich also has an anova S3 method to
compare nested models of class "chandwich". We perform
loglikelihood adjustment of the models in which Oxford and Worthing
share common GEV scale and shape parameters (small) and
share all GEV parameters (tiny). Then we perform pairwise
comparisons of neighbouring nest models using anova().
small <- adjust_loglik(larger = large, fixed_pars = c("sigma[1]", "xi[1]"))
tiny <- adjust_loglik(larger = large, fixed_pars = c("mu[1]", "sigma[1]", "xi[1]"))
anova(large, medium, small, tiny)
#> Analysis of (Adjusted) Deviance Table
#>
#> Model.Df Df ALRTS Pr(>ALRTS)
#> large 6
#> medium 5 1 6.356 0.01170 *
#> small 4 1 4.251 0.03924 *
#> tiny 3 1 81.714 < 2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(large, medium, small, tiny, approx = TRUE)
#> Analysis of (Adjusted) Deviance Table
#>
#> Model.Df Df ALRTS Pr(>ALRTS)
#> large 6
#> medium 5 1 5.245 0.02200 *
#> small 4 1 4.184 0.04081 *
#> tiny 3 1 112.822 < 2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Misspecified Poisson model
We repeat part of an example from Section 5.1 of the Object-Oriented
Computation of Sandwich Estimators vignette of the sandwich
package (Zeileis 2006), using the same
simulated dataset. This example illustrates that sandwich estimators may
result in inferences that are robust against certain types model
misspecification. We simulate data from a log-linear negative binomial
regression model and fit a misspecified log-quadratic Poisson regression
model.
set.seed(123)
x <- rnorm(250)
y <- rnbinom(250, mu = exp(1 + x), size = 1)
fm_pois <- glm(y ~ x + I(x^2), family = poisson)
round(summary(fm_pois)$coefficients, 3)
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 1.063 0.041 25.709 0.000
#> x 0.996 0.054 18.606 0.000
#> I(x^2) -0.049 0.023 -2.122 0.034
Although the conditional mean of y is linear in
x overdispersion of the responses relative to the Poisson
distribution contributes to the spurious significance of the quadratic
term. We adjust the independence log-likelihood, with each observation
forming its own cluster.
pois_glm_loglik <- function(pars, y, x) {
log_mu <- pars[1] + pars[2] * x + pars[3] * x ^ 2
return(dpois(y, lambda = exp(log_mu), log = TRUE))
}
pars <- c("alpha", "beta", "gamma")
pois_quad <- adjust_loglik(pois_glm_loglik, y = y, x = x, par_names = pars)
summary(pois_quad)
#> MLE SE adj. SE
#> alpha 1.06300 0.04136 0.08378
#> beta 0.99610 0.05354 0.10520
#> gamma -0.04913 0.02315 0.03628
The adjusted standard errors agree with those in the
sandwich vignette and are sufficiently larger than the
unadjusted standard errors to alleviate the spurious significance of the
quadratic term: the \(p\)-value changes
from 0.034 to 0.18. For full details please see the sandwich
vignette.
Confidence intervals
We use conf_intervals to calculate profile
likelihood-based confidence intervals for the parameters.
# 95% confidence intervals, vertically adjusted
conf_intervals(pois_quad)
#> Waiting for profiling to be done...
#> Model: pois_glm_loglik
#>
#> 95% confidence intervals, adjusted loglikelihod with type = ''vertical''
#>
#> Symmetric:
#> lower upper
#> alpha 0.89907 1.22747
#> beta 0.78986 1.20231
#> gamma -0.12024 0.02199
#>
#> Profile likelihood-based:
#> lower upper
#> alpha 0.8954 1.2232
#> beta 0.7877 1.1991
#> gamma -0.1198 0.0222
Confidence regions
We call adjust_loglik again to fix the quadratic
coefficient at zero, producing a model with two free parameters, and use
conf_region to calculate the vertically adjusted and
unadjusted loglikelihoods over a grid for plotting. The plot below
illustrates the way in which the independence loglikelihood for \((\alpha, \beta)\) has been scaled.
pois_lin <- adjust_loglik(larger = pois_quad, fixed_pars = "gamma")
pois_vertical <- conf_region(pois_lin)
#> Waiting for profiling to be done...
pois_none <- conf_region(pois_lin, type = "none")
#> Waiting for profiling to be done...
plot(pois_none, pois_vertical, conf = c(50, 75, 95, 99), col = 2:1, lwd = 2, lty = 1)
Comparing nested models
We could examine the significance of the quadratic term using an
adjusted likelihood ratio test.
compare_models(pois_quad, pois_lin)
#> Model: pois_glm_loglik
#> H0: "gamma" = 0
#> HA: unrestricted model
#>
#> test statistic = 1.82, df = 1, p-value = 0.1773
compare_models(pois_quad, pois_lin, approx = TRUE)
#> Model: pois_glm_loglik
#> H0: "gamma" = 0
#> HA: unrestricted model
#>
#> Using using approximation (18) of Chandler and Bate (2007):
#> test statistic = 1.921, df = 1, p-value = 0.1658
The \(p\)-value based on equation
(17) of Chandler and Bate (2007) agrees
with the value in the sandwich vignette, which is based on a
Wald test, and the \(p\)-value based on
equations (18)-(20) is only slightly different.
References
Chandler, R. E., and S. Bate. 2007.
“Inference for Clustered Data
Using the Independence Loglikelihood.” Biometrika 94
(1): 167–183. doi:
10.1093/biomet/asm015.
Davison, A. C. 2003.
Statistical Models. Cambridge Series in
Statistical and Probabilistic Mathematics. Cambridge University Press.
doi:
10.1017/CBO9780511815850.
Gelman, A., J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D.
B. Rubin. 2014. Bayesian Data Analysis. third. Florida, USA:
Chapman & Hall / CRC.
Tarone, Robert E. 1982. “The Use of Historical Control Information
in Testing for a Trend in Proportions.” Biometrics 38
(1). [Wiley, International Biometric Society]: 215–220.
Zeileis, Achim. 2006.
“Object-Oriented Computation of Sandwich
Estimators.” Journal of Statistical Software 16 (9):
1–16. doi:
10.18637/jss.v016.i09.