VGAM (version 0.7-1)

multinomial: Multinomial Logit Model

Description

Fits a multinomial logit model to an unordered factor response.

Usage

multinomial(zero = NULL, parallel = FALSE, nointercept = NULL)

Arguments

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The values must be from the set {1,2,...,$M$}. The default value means none are modelled as intercept-only terms.

parallel

A logical, or formula specifying which terms have equal/unequal coefficients.

nointercept

An integer-valued vector specifying which linear/additive predictors have no intercepts. The values must be from the set {1,2,...,$M$}.

Value

• An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, rrvglm and vgam.

Warning

The arguments zero and nointercept can be inputted with values that fail. For example, multinomial(zero=2, nointercept=1:3) means the second linear/additive predictor is identically zero, which will cause a failure.

Be careful about the use of other potentially contradictory constraints, e.g., multinomial(zero=2, parallel = TRUE ~ x3). If in doubt, apply constraints() to the fitted object to check.

No check is made to verify that the response is nominal.

Details

The model can be written $$\eta_j = \log(P[Y=j]/ P[Y=M+1])$$ where $\eta_j$ is the $j$th 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.

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 (except for the intercept) equal to a vector of $M$ 1's. 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.

References

Yee, T. W. and Hastie, T. J. (2003) Reduced-rank vector generalized linear models. Statistical Modelling, 3, 15--41.

McCullagh, P. and Nelder, J. A. (1989) Generalized Linear Models, 2nd ed. London: Chapman & Hall.

Agresti, A. (2002) Categorical Data Analysis, 2nd ed. New York: Wiley.

Simonoff, J. S. (2003) Analyzing Categorical Data, New York: Springer-Verlag.

Anderson, J. A. (1984) Regression and ordered categorical variables. Journal of the Royal Statistical Society, Series B, Methodological, 46, 1--30.

Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.

See Also

acat, cumulative, cratio, sratio, dirichlet, dirmultinomial, rrvglm, Multinomial, iris.

Examples

# Example 1: fit a multinomial logit model to Edgar Anderson's iris data
data(iris)
fit = vglm(Species ~ ., multinomial, iris)
coef(fit, matrix=TRUE)


# Example 2a: a simple example 
y = t(rmultinom(10, size = 20, prob=c(0.1,0.2,0.8))) # Counts
fit = vglm(y ~ 1, multinomial)
fitted(fit)[1:4,]   # Proportions
[e-mail] # Not recommended for extraction of prior weights
weights(fit, type="prior", matrix=FALSE) # The better method
fit@y   # Sample proportions
constraints(fit)   # Constraint matrices

# Example 2b: Different input to Example 2a but same result
w = apply(y, 1, sum) # Prior weights
yprop = y / w    # Sample proportions
fitprop = vglm(yprop ~ 1, multinomial, weights=w)
fitted(fitprop)[1:4,]   # Proportions
weights(fitprop, type="prior", matrix=FALSE)
fitprop@y # Same as the input


# Example 3: Fit a rank-1 stereotype model 
data(car.all)
fit = rrvglm(Country ~ Width + Height + HP, multinomial, car.all, Rank=1)
coef(fit)   # Contains the C matrix
constraints(fit)$HP     # The A matrix 
coef(fit, matrix=TRUE)  # The B matrix
Coef(fit)@C             # The C matrix 
ccoef(fit)              # Better to get the C matrix this way
Coef(fit)@A             # The A matrix 
svd(coef(fit, matrix=TRUE)[-1,])$d    # This has rank 1; = C %*% t(A) 


# Example 4: The use of the xij argument (conditional logit model)
set.seed(111)
n = 100  # Number of people who travel to work
M = 3  # There are M+1 models of transport
ymat = matrix(0, n, M+1)
ymat[cbind(1:n, sample(x=M+1, size=n, replace=TRUE))] = 1
dimnames(ymat) = list(NULL, c("bus","train","car","walk"))
transport = data.frame(cost.bus=runif(n), cost.train=runif(n),
                       cost.car=runif(n), cost.walk=runif(n),
                       durn.bus=runif(n), durn.train=runif(n),
                       durn.car=runif(n), durn.walk=runif(n))
transport = round(transport, dig=2) # For convenience
transport = transform(transport,
                      Cost.bus   = cost.bus   - cost.walk,
                      Cost.car   = cost.car   - cost.walk,
                      Cost.train = cost.train - cost.walk,
                      Durn.bus   = durn.bus   - durn.walk,
                      Durn.car   = durn.car   - durn.walk,
                      Durn.train = durn.train - durn.walk)
fit = vglm(ymat ~ Cost.bus + Cost.train + Cost.car + 
                  Durn.bus + Durn.train + Durn.car,
           fam = multinomial,
           xij = list(Cost ~ Cost.bus + Cost.train + Cost.car,
                      Durn ~ Durn.bus + Durn.train + Durn.car),
           data=transport)
model.matrix(fit, type="lm")[1:7,]   # LM model matrix
model.matrix(fit, type="vlm")[1:7,]  # Big VLM model matrix
coef(fit)
coef(fit, matrix=TRUE)
coef(fit, matrix=TRUE, compress=FALSE)
summary(fit)