brmultinom {brglm2}  R Documentation 
Bias reduction for multinomial response models using the Poisson trick.
Description
brmultinom()
is a wrapper of brglmFit()
that fits multinomial
regression models using implicit and explicit bias reduction
methods. See Kosmidis & Firth (2011) for details.
Usage
brmultinom(
formula,
data,
weights,
subset,
na.action,
contrasts = NULL,
ref = 1,
model = TRUE,
x = TRUE,
control = list(...),
...
)
Arguments
formula 
a formula expression as for regression models, of the form

data 
an optional data frame in which to interpret the variables occurring
in 
weights 
optional case weights in fitting. 
subset 
expression saying which subset of the rows of the data should be used in the fit. All observations are included by default. 
na.action 
a function to filter missing data. 
contrasts 
a list of contrasts to be used for some or all of the factors appearing as variables in the model formula. 
ref 
the reference category to use for multinomial
regression. Either an integer, in which case

model 
logical. If true, the model frame is saved as component 
x 
should the model matrix be included with in the result
(default is 
control 
a list of parameters for controlling the fitting
process. See 
... 
arguments to be used to form the default 
Details
The models brmultinom()
handles are also known as
baselinecategory logit models (see, Agresti, 2002, Section 7.1),
because they model the logodds of every category against a
baseline category. The user can control which baseline (or
reference) category is used via the ref
. By default
brmultinom()
uses the first category as reference.
The maximum likelihood estimates for the parameters of baselinecategory logit models have infinite components with positive probability, which can result in problems in their estimation and the use of inferential procedures (e.g. Wad tests). Albert and Andreson (1984) have categorized the possible data patterns for such models into the exclusive and exhaustive categories of complete separation, quasicomplete separation and overlap, and showed that infinite maximum likelihood estimates result when complete or quasicomplete separation occurs.
The adjusted score approaches to bias reduction that brmultinom()
implements for type = "AS_mean"
and type = "AS_median"
are
alternatives to maximum likelihood that result in estimates with
smaller asymptotic mean and median bias, respectively, that are
also always finite, even in cases of complete or quasicomplete
separation.
brmultinom()
is a wrapper of brglmFit()
that fits multinomial
logit regression models through the 'Poisson trick' (see, for
example, Palmgren, 1981; Kosmidis & Firth, 2011).
The implementation relies on the construction of an extended model
matrix for the loglinear model and constraints on the sums of the
Poisson means. Specifically, a loglinear model is fitted on a
Kronecker product of the
original model matrix X
implied by the formula, augmented by
nrow(X)
dummy variables.
The extended model matrix is sparse, and the Matrix package is used for its effective storage.
While brmultinom()
can be used for analyses using multinomial
regression models, the current implementation is more of a proof of
concept and is not expected to scale well with either of nrow(X)
,
ncol(X)
or the number of levels in the categorical response.
Author(s)
Ioannis Kosmidis [aut, cre]
ioannis.kosmidis@warwick.ac.uk
References
Kosmidis I, Kenne Pagui E C, Sartori N (2020). Mean and median bias reduction in generalized linear models. Statistics and Computing, 30, 4359. doi:10.1007/s11222019098606.
Agresti A (2002). Categorical data analysis (2nd edition). Wiley Series in Probability and Statistics. Wiley.
Albert A, Anderson J A (1984). On the Existence of Maximum Likelihood Estimates in Logistic Regression Models. Biometrika, 71 1–10. doi:10.2307/2336390.
Kosmidis I, Firth D (2011). Multinomial logit bias reduction via the Poisson loglinear model. Biometrika, 98, 755759. doi:10.1093/biomet/asr026.
Palmgren, J (1981). The Fisher Information Matrix for Log Linear Models Arguing Conditionally on Observed Explanatory Variables. Biometrika, 68, 563566. doi:10.1093/biomet/68.2.563.
See Also
nnet::multinom()
, bracl()
for adjacent category logit models for ordinal responses
Examples
## The housing data analysis from ?MASS::housing
data("housing", package = "MASS")
# Maximum likelihood using nnet::multinom
houseML1nnet < nnet::multinom(Sat ~ Infl + Type + Cont, weights = Freq,
data = housing)
# Maximum likelihood using brmultinom with baseline category 'Low'
houseML1 < brmultinom(Sat ~ Infl + Type + Cont, weights = Freq,
data = housing, type = "ML", ref = 1)
# The estimates are numerically the same as houseML0
all.equal(coef(houseML1nnet), coef(houseML1), tolerance = 1e04)
# Maximum likelihood using brmultinom with 'High' as baseline
houseML3 < brmultinom(Sat ~ Infl + Type + Cont, weights = Freq,
data = housing, type = "ML", ref = 3)
# The fitted values are the same as houseML1
all.equal(fitted(houseML3), fitted(houseML1), tolerance = 1e10)
# Bias reduction
houseBR3 < update(houseML3, type = "AS_mean")
# Bias correction
houseBC3 < update(houseML3, type = "correction")
## Reproducing Bull et al. (2002, Table 3)
data("hepatitis", package = "brglm2")
# Construct a variable with the multinomial categories according to
# the HCV and nonABC columns
hepat < hepatitis
hepat$type < with(hepat, factor(1  HCV * nonABC + HCV + 2 * nonABC))
hepat$type < factor(hepat$type, labels = c("noDisease", "C", "nonABC"))
contrasts(hepat$type) < contr.treatment(3, base = 1)
# Maximum likelihood estimation fails to converge because some estimates are infinite
hepML < brmultinom(type ~ group * time, data = hepat, weights = counts, type = "ML", slowit = 0.1)
# Mean bias reduction returns finite estimates
hep_meanBR < brmultinom(type ~ group * time, data = hepat, weights = counts, type = "AS_mean")
# The estimates in Bull et al. (2002, Table 3, DOI: 10.1016/S01679473(01)000482)
coef(hep_meanBR)
# Median bias reduction also returns finite estimates, which are a bit larger in absolute value
hep_medianBR < brmultinom(type ~ group * time, data = hepat, weights = counts, type = "AS_median")
coef(hep_medianBR)