| multinomial {VGAM} | R Documentation |
Multinomial Logit Model
Description
Fits a multinomial logit model (MLM) to a (preferably unordered) factor response.
Usage
multinomial(zero = NULL, parallel = FALSE, nointercept = NULL,
refLevel = "(Last)", ynames = FALSE,
imethod = 1, imu = NULL, byrow.arg = FALSE,
Thresh = NULL, Trev = FALSE,
Tref = if (Trev) "M" else 1, whitespace = FALSE)
Arguments
zero |
Can be an integer-valued vector specifying which
linear/additive predictors are modelled as intercepts only.
Any values must be from the set {1,2,..., |
parallel |
A logical, or formula specifying which terms have equal/unequal coefficients. |
ynames |
Logical.
If |
nointercept, whitespace |
See |
imu, byrow.arg |
See |
refLevel |
Either a (1) single positive integer or (2) a value of
the factor or (3) a character string.
If inputted as an integer then it specifies which
column of the response matrix is the reference or baseline level.
The default is the last one (the |
imethod |
Choosing 2 will use the mean sample proportions of each
column of the response matrix, which corresponds to
the MLEs for intercept-only models.
See |
Thresh, Trev, Tref |
Same as |
Details
In this help file the response Y is
assumed to be a factor with unordered values
1,2,\dots,M+1, so
that M is the number of linear/additive
predictors \eta_j.
The default model can be written
\eta_j = \log(P[Y=j]/ P[Y=M+1])
where \eta_j is the jth
linear/additive predictor.
Here, j=1,\ldots,M, and
\eta_{M+1}
is 0 by definition. That is, the last level
of the factor,
or last column of the response matrix, is
taken as the
reference level or baseline—this is for
identifiability
of the parameters. The reference or
baseline level can
be changed with the refLevel argument.
In almost all the literature, the constraint matrices associated with
this family of models are known. For example, setting parallel
= TRUE will make all constraint matrices (including the intercept)
equal to a vector of M 1's; to suppress the intercepts from
being parallel then set parallel = FALSE ~ 1. If the
constraint matrices are unknown and to be estimated, then this can be
achieved by fitting the model as a reduced-rank vector generalized
linear model (RR-VGLM; see rrvglm). In particular, a
multinomial logit model with unknown constraint matrices is known as a
stereotype model (Anderson, 1984), and can be fitted with
rrvglm.
The above details correspond to the ordinary MLM where all the levels are altered (in the terminology of GAITD regression).
Value
An object of class "vglmff"
(see vglmff-class).
The object is used by modelling functions
such as vglm,
rrvglm and vgam.
Warning
No check is made to verify that the response is nominal.
See CommonVGAMffArguments for more warnings.
Note
The response should be either a matrix of counts
(with row sums that are all positive), or a
factor. In both cases, the y slot returned by
vglm/vgam/rrvglm
is the matrix of sample proportions.
The multinomial logit model is more appropriate for a nominal
(unordered) factor response than for an
ordinal (ordered) factor
response.
Models more suited for the latter include those based on
cumulative probabilities, e.g., cumulative.
multinomial is prone to numerical difficulties if
the groups are separable and/or the fitted probabilities
are close to 0 or 1. The fitted values returned
are estimates of the probabilities P[Y=j] for
j=1,\ldots,M+1. See safeBinaryRegression
for the logistic regression case.
Here is an example of the usage of the parallel
argument. If there are covariates x2, x3
and x4, then parallel = TRUE ~ x2 + x3 -
1 and parallel = FALSE ~ x4 are equivalent. This
would constrain the regression coefficients for x2
and x3 to be equal; those of the intercepts and
x4 would be different.
In Example 4 below, a conditional logit model is
fitted to an artificial data set that explores how
cost and travel time affect people's decision about
how to travel to work. Walking is the baseline group.
The variable Cost.car is the difference between
the cost of travel to work by car and walking, etc. The
variable Time.car is the difference between
the travel duration/time to work by car and walking,
etc. For other details about the xij argument see
vglm.control and fill1.
The multinom function in the
nnet package uses the first level of the factor as
baseline, whereas the last level of the factor is used
here. Consequently the estimated regression coefficients
differ.
Author(s)
Thomas W. Yee
References
Agresti, A. (2013). Categorical Data Analysis, 3rd ed. Hoboken, NJ, USA: Wiley.
Anderson, J. A. (1984). Regression and ordered categorical variables. Journal of the Royal Statistical Society, Series B, Methodological, 46, 1–30.
Hastie, T. J., Tibshirani, R. J. and Friedman, J. H. (2009). The Elements of Statistical Learning: Data Mining, Inference and Prediction, 2nd ed. New York, USA: Springer-Verlag.
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.
Tutz, G. (2012). Regression for Categorical Data, Cambridge: Cambridge University Press.
Yee, T. W. and Hastie, T. J. (2003). Reduced-rank vector generalized linear models. Statistical Modelling, 3, 15–41.
Yee, T. W. (2010). The VGAM package for categorical data analysis. Journal of Statistical Software, 32, 1–34. doi:10.18637/jss.v032.i10.
Yee, T. W. and Ma, C. (2024). Generally altered, inflated, truncated and deflated regression. Statistical Science, 39 (in press).
See Also
multilogitlink,
margeff,
cumulative,
acat,
cratio,
sratio,
CM.equid,
CommonVGAMffArguments,
dirichlet,
dirmultinomial,
rrvglm,
fill1,
Multinomial,
gaitdpoisson,
Gaitdpois,
iris.
Examples
# Example 1: Regn spline VGAM: marital status versus age
data(marital.nz)
ooo <- with(marital.nz, order(age))
om.nz <- marital.nz[ooo, ]
fit1 <- vglm(mstatus ~ sm.bs(age), multinomial, om.nz)
coef(fit1, matrix = TRUE) # Mostly meaningless
## Not run: with(om.nz,
matplot(age, fitted(fit1), type = "l", las = 1, lwd = 2))
legend("topright", leg = colnames(fitted(fit1)),
lty = 1:4, col = 1:4, lwd = 2)
## End(Not run)
# Example 2a: a simple example
ycounts <- t(rmultinom(10, size = 20, prob = c(0.1, 0.2, 0.8)))
fit <- vglm(ycounts ~ 1, multinomial)
head(fitted(fit)) # Proportions
fit@prior.weights # NOT recommended for the prior weights
weights(fit, type = "prior", matrix = FALSE) # The better method
depvar(fit) # Sample proportions; same as fit@y
constraints(fit) # Constraint matrices
# Example 2b: Different reference level used as the baseline
fit2 <- vglm(ycounts ~ 1, multinomial(refLevel = 2))
coef(fit2, matrix = TRUE)
coef(fit , matrix = TRUE) # Easy to reconcile this output with fit2
# Example 3: The response is a factor.
nn <- 10
dframe3 <- data.frame(yfac = gl(3, nn, labels = c("Ctrl",
"Trt1", "Trt2")),
x2 = runif(3 * nn))
myrefLevel <- with(dframe3, yfac[12])
fit3a <- vglm(yfac ~ x2, multinomial(refLevel = myrefLevel), dframe3)
fit3b <- vglm(yfac ~ x2, multinomial(refLevel = 2), dframe3)
coef(fit3a, matrix = TRUE) # "Trt1" is the reference level
coef(fit3b, matrix = TRUE) # "Trt1" is the reference level
margeff(fit3b)
# Example 4: Fit a rank-1 stereotype model
fit4 <- rrvglm(Country ~ Width + Height + HP, multinomial, car.all)
coef(fit4) # Contains the C matrix
constraints(fit4)$HP # The A matrix
coef(fit4, matrix = TRUE) # The B matrix
Coef(fit4)@C # The C matrix
concoef(fit4) # Better to get the C matrix this way
Coef(fit4)@A # The A matrix
svd(coef(fit4, matrix = TRUE)[-1, ])$d # Has rank 1; = C %*% t(A)
# Classification (but watch out for NAs in some of the variables):
apply(fitted(fit4), 1, which.max) # Classification
# Classification:
colnames(fitted(fit4))[apply(fitted(fit4), 1, which.max)]
apply(predict(fit4, car.all, type = "response"),
1, which.max) # Ditto
# Example 5: Using the xij argument (aka conditional logit model)
set.seed(111)
nn <- 100 # Number of people who travel to work
M <- 3 # There are M+1 models of transport to go to work
ycounts <- matrix(0, nn, M+1)
ycounts[cbind(1:nn, sample(x = M+1, size = nn, replace = TRUE))] = 1
dimnames(ycounts) <- list(NULL, c("bus","train","car","walk"))
gotowork <- data.frame(cost.bus = runif(nn), time.bus = runif(nn),
cost.train= runif(nn), time.train= runif(nn),
cost.car = runif(nn), time.car = runif(nn),
cost.walk = runif(nn), time.walk = runif(nn))
gotowork <- round(gotowork, digits = 2) # For convenience
gotowork <- transform(gotowork,
Cost.bus = cost.bus - cost.walk,
Cost.car = cost.car - cost.walk,
Cost.train = cost.train - cost.walk,
Cost = cost.train - cost.walk, # for labelling
Time.bus = time.bus - time.walk,
Time.car = time.car - time.walk,
Time.train = time.train - time.walk,
Time = time.train - time.walk) # for labelling
fit <- vglm(ycounts ~ Cost + Time,
multinomial(parall = TRUE ~ Cost + Time - 1),
xij = list(Cost ~ Cost.bus + Cost.train + Cost.car,
Time ~ Time.bus + Time.train + Time.car),
form2 = ~ Cost + Cost.bus + Cost.train + Cost.car +
Time + Time.bus + Time.train + Time.car,
data = gotowork, trace = TRUE)
head(model.matrix(fit, type = "lm")) # LM model matrix
head(model.matrix(fit, type = "vlm")) # Big VLM model matrix
coef(fit)
coef(fit, matrix = TRUE)
constraints(fit)
summary(fit)
max(abs(predict(fit) - predict(fit, new = gotowork))) # Should be 0