brmultinom is a wrapper of brglmFit that fits
multinomial regression models using implicit and explicit bias
reduction methods. See Kosmidis & Firth (2011) for details.
brmultinom(
formula,
data,
weights,
subset,
na.action,
contrasts = NULL,
ref = 1,
model = TRUE,
x = TRUE,
control = list(...),
...
)a formula expression as for regression models, of the form
response ~ predictors. The response should be a factor or a
matrix with K columns, which will be interpreted as counts for each of
K classes.
A log-linear model is fitted, with coefficients zero for the first
class. An offset can be included: it should be a numeric matrix with K columns
if the response is either a matrix with K columns or a factor with K >= 2
classes, or a numeric vector for a response factor with 2 levels.
See the documentation of formula() for other details.
an optional data frame in which to interpret the variables occurring
in formula.
optional case weights in fitting.
expression saying which subset of the rows of the data should be used in the fit. All observations are included by default.
a function to filter missing data.
a list of contrasts to be used for some or all of the factors appearing as variables in the model formula.
the reference category to use for multinomial regression. Either an integer, in which case levels(response)[ref] is used as a baseline, or a character string. Default is 1.
logical. If true, the model frame is saved as component model
of the returned object.
should the model matrix be included with in the result
(default is TRUE).
a list of parameters for controlling the fitting
process. See brglmControl for details.
arguments to be used to form the default 'control' argument if it is not supplied directly.
Ioannis Kosmidis ioannis.kosmidis@warwick.ac.uk
The models brmultinom handles are also known as
baseline-category logit models (see, Agresti, 2002, Section 7.1),
because they model the log-odds 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 baseline-category 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, quasi-complete separation and overlap, and showed that infinite maximum likelihood estimates result when complete or quasi-complete 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 quasi-complete 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 log-linear model and constraints on the sums
of the Poisson means. Specifically, a log-linear model is fitted on
a Kronecker product
(https://en.wikipedia.org/wiki/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.
Kosmidis I, Kenne Pagui E C, Sartori N (2020). Mean and median bias reduction in generalized linear models. *Statistics and Computing*, **30**, 43-59 tools:::Rd_expr_doi("10.1007/s11222-019-09860-6")
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, tools:::Rd_expr_doi("10.2307/2336390")
Kosmidis I, Firth D (2011). Multinomial logit bias reduction via the Poisson log-linear model. *Biometrika*, **98**, 755-759 tools:::Rd_expr_doi("10.1093/biomet/asr026")
Palmgren, J (1981). The Fisher Information Matrix for Log Linear Models Arguing Conditionally on Observed Explanatory Variables. *Biometrika*, **68**, 563-566 tools:::Rd_expr_doi("10.1093/biomet/68.2.563")
multinom, bracl for adjacent category logit models for ordinal responses
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 = 1e-04)
# 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 = 1e-10)
# Bias reduction
houseBR3 <- update(houseML3, type = "AS_mean")
# Bias correction
houseBC3 <- update(houseML3, type = "correction")
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