addreg fits additive (identity-link) Poisson, negative binomial
and binomial regression models using a stable combinatorial EM algorithm.
addreg(formula, mono = NULL, family, data, standard, subset, na.action,
start = NULL, offset, control = list(...), model = TRUE,
method = c("cem", "em"),
accelerate = c("em", "squarem", "pem", "qn"),
control.method = list(), warn = TRUE, ...)an object of class "formula"
(or one that can be coerced into that class): a symbolic
description of the model to be fitted. The details of
model specification are given under "Details". Note that
the model must contain an intercept, and 2nd-order terms
(such as interactions) or above are currently not supported
--- see "Note".
a vector indicating which terms in
formula should be restricted to have a
monotonically non-decreasing relationship with the
outcome. May be specified as names or indices of the
terms.
a description of the error distribution to
be used in the model. This can be a character string
naming a family function, a family function or the result
of a call to a family function (see
family for details of family
functions), but here it is restricted to be poisson,
negbin1 or binomial family with identity link.
an optional data frame, list or environment
(or object coercible by as.data.frame to a
data frame) containing the variables in the model. If not
found in data, the variables are taken from
environment(formula), typically the environment
from which addreg is called.
a numeric vector of length equal to the number of cases, where each element is a positive constant that (multiplicatively) standardises the fitted value of the corresponding element of the response vector. Ignored for binomial family (two-column specification of response should be used instead).
an optional vector specifying a subset of observations to be used in the fitting process.
a function which indicates what should happen when the data
contain NAs. The default is set be the na.action
setting of options, and is na.fail
if that is unset. The `factory-fresh' default is na.omit.
Another possible value is NULL, no action. Value
na.exclude can be useful.
starting values for the parameters in the
linear predictor, also with the starting value for
the scale as the last element when
family = negbin1.
this can be used to specify an a
priori known component to be included in the linear
predictor during fitting. This should be NULL or a
non-negative numeric vector of length equal to the number of cases.
One or more offset terms can be included in
the formula instead or as well, and if more than one is
specified their sum is used. See
model.offset.
Ignored for binomial family; not yet implemented for negative binomial models.
list of parameters for controlling the
fitting process, passed to
addreg.control.
a logical value indicating whether the model frame (and, for binomial models, the equivalent Poisson model) should be included as a component of the returned value.
a character string that determines which algorithm to use to
find the MLE: "cem" for the combinatorial EM algorithm,
which cycles through a sequence of constrained parameter spaces,
or "em" for a single EM algorithm based on an
overparameterised model.
a character string that determines the acceleration
algorithm to be used, (partially) matching one of "em" (no acceleration --- the default),
"squarem", "pem" or "qn". See turboem
for further details. Note that "decme" is not permitted.
a list of control parameters for the acceleration algorithm, which are passed to
the control.method argument of turboem.
If any items are not specified, the defaults are used.
a logical indicating whether or not warnings should be provided for non-convergence or boundary values.
arguments to be used to form the default
control argument if it is not supplied directly.
addreg returns an object of class "addreg",
which inherits from classes "glm" and "lm".
The function summary.addreg can be used
to obtain or print a summary of the results.
The generic accessor functions coefficients,
fitted.values and residuals can be used to
extract various useful features of the value returned by
addreg. Note that effects will not work.
An object of class "addreg" is a list containing the
same components as an object of class "glm" (see the
"Value" section of glm), but without
contrasts, qr, R or effects
components. It also includes:
the maximised log-likelihood.
a small-sample corrected
version of Akaike's An Information Criterion
(Hurvich, Simonoff and Tsai, 1998). This is used by
addreg.smooth to choose the optimal number of
knots for smooth terms.
the minimum and maximum observed values for each of the continuous covariates, to help define the covariate space of the model.
As well as, for Poisson and negative binomial models:
estimated coefficients associated with the non-negative parameterisation corresponding to the MLE.
non-negative model matrix associated with
nn.coefficients.
the standard argument.
Or, for binomial models:
if requested, the addreg object for the associated identity-link
Poisson model.
The scale component of the result is fixed at 1 for Poisson and binomial models, and is the constant overdispersion parameter for negative binomial models (that is, scale = 1+\phi) where Var(\mu) = (1+\phi)\mu).
addreg fits a generalised linear model (GLM) with a
Poisson or binomial error distribution and identity link
function, as well as additive NegBin I models (which are not
GLMs). Predictors are assumed to be continuous, unless
they are of class factor, or are character or
logical (in which case they are converted to
factors). Specifying a predictor as monotonic using
the mono argument means that for continuous terms,
the associated coefficient will be restricted to be
non-negative, and for categorical terms, the coefficients
will be non-decreasing in the order of the factor
levels. This allows semi-parametric monotonic regression
functions, in the form of unsmoothed step-functions. For
smooth regression functions see addreg.smooth.
As well as allowing monotonicity constraints, the function
is useful when a standard GLM routine, such as
glm, fails to converge with an identity-link Poisson
or binomial model. If glm does achieve successful convergence,
and addreg converges to an interior point, then the two
results will be identical. However, glm may still experience convergence
problems even when addreg converges to an interior point.
Note that if addreg converges to a boundary point, then it
may differ slightly from glm even if glm successfully
converges, because of differences in the definition of the parameter
space. addreg produces valid fitted values for covariate
values within the Cartesian product of the observed range of covariate
values, whereas glm produces valid fitted values just
for the observed covariate combinations (assuming it successfully
converges). This issue is only relevant when addreg
converges to a boundary point.
The computational method is a combinatorial EM algorithm (Marschner, 2014), which accommodates the parameter contraints in the model and is more stable than iteratively reweighted least squares. A collection of restricted parameter spaces is defined which covers the full parameter space, and the EM algorithm is applied within each restricted parameter space in order to find a collection of restricted maxima of the log-likelihood function, from which can be obtained the global maximum over the full parameter space. See Marschner (2010), Donoghoe and Marschner (2014) and Donoghoe and Marschner (2016) for further details.
Acceleration of the EM algorithm can be achieved through the
methods of the turboEM package, specified
through the accelerate argument. However, note that these
methods do not have the guaranteed convergence of the standard
EM algorithm, particularly when the MLE is on the boundary of
its (possibly constrained) parameter space.
Donoghoe, M. W. and I. C. Marschner (2014). Stable computational methods for additive binomial models with application to adjusted risk differences. Computational Statistics and Data Analysis 80: 184--196.
Donoghoe, M. W. and I. C. Marschner (2016). Estimation of adjusted rate differences using additive negative binomial regression. Statistics in Medicine 35(18): 3166--3178.
Hurvich, C. M., J. S. Simonoff and C.-L. Tsai (1998). Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 60(2): 271--293.
Marschner, I. C. (2010). Stable computation of maximum likelihood estimates in identity link Poisson regression. Journal of Computational and Graphical Statistics 19(3): 666--683.
Marschner, I. C. (2014). Combinatorial EM algorithms. Statistics and Computing 24(6): 921--940.
# NOT RUN {
require(glm2)
data(crabs)
#============================================================================
# identity-link Poisson model with periodic non-convergence when glm is used
#============================================================================
crabs.boot <- crabs[crabs$Rep1,-c(5:6)]
crabs.boot$width.shifted <- crabs.boot$Width - min(crabs$Width)
fit.glm <- glm(Satellites ~ width.shifted + factor(Dark) + factor(GoodSpine),
family = poisson(identity), data = crabs.boot, start = rep(1,4),
control = glm.control(trace = TRUE))
fit.addreg <- addreg(formula(fit.glm), family = poisson, data = crabs.boot,
trace = 1)
# Speed up convergence by using single EM algorithm
fit.addreg.em <- update(fit.addreg, method = "em")
# Speed up convergence by using acceleration methods
fit.addreg.acc <- update(fit.addreg, accelerate = "squarem")
fit.addreg.em.acc <- update(fit.addreg.em, accelerate = "squarem")
# Usual S3 methods work on addreg objects
summary(fit.addreg)
vcov(fit.addreg)
confint(fit.addreg)
summary(predict(fit.addreg), type = "response")
fit.addreg2 <- addreg(update(formula(fit.glm), ~ . - factor(GoodSpine)),
family = poisson, data = crabs.boot, trace = 1)
anova(fit.addreg2, fit.addreg, test = "LRT")
# Account for overdispersion (use start to speed it up a little)
fit.addreg.od <- addreg(Satellites ~ factor(Dark) + factor(GoodSpine),
family = negbin1, data = crabs.boot, trace = 1,
start = c(4.3423675,-2.4059273,-0.4531984,5.969648))
summary(fit.addreg.od)
# }
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