multinomial: Multinomial Logit Model

Description

Fits a multinomial logit model to an unordered factor response.

Usage

multinomial(zero = NULL, parallel = FALSE, nointercept = NULL,
            refLevel = "last")

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,...,$M$}.

refLevel

Either a single positive integer or a value of a factor. If an integer then it specifies which column of the response matrix is the reference or baseline level. The default is the last one (the $(M+1)$th one). If used, this argument will be often

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 default 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. The reference or baseline level can be chosen by using 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 (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.

Examples

Run this code

# 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
fit@prior.weights # 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 reference level used as the baseline 
fit2 = vglm(y ~ 1, multinomial(refLevel=2))
coef(fit2, matrix=TRUE)
coef(fit , matrix=TRUE) # Easy to reconcile this output with fit2

# Example 2c: 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: The response is a factor.
nn = 10
yfactor = gl(3, nn, labels = c("Control", "Treatment1", "Treatment2"))
x = runif(3 * nn)
fit3a = vglm(yfactor ~ x, multinomial(refLevel=yfactor[12]))
fit3b = vglm(yfactor ~ x, multinomial(refLevel=2))
coef(fit3a, matrix=TRUE)  # "Treatment1" is the reference level
coef(fit3b, matrix=TRUE)  # "Treatment1" is the reference level


# Example 4: Fit a rank-1 stereotype model 
data(car.all)
fit4 = rrvglm(Country ~ Width + Height + HP, multinomial, car.all, Rank=1)
coef(fit4)   # Contains the C matrix
constraints(fit4)$HP     # The A matrix 
coef(fit4, matrix=TRUE)  # The B matrix
Coef(fit4)@C             # The C matrix 
ccoef(fit4)              # Better to get the C matrix this way
Coef(fit4)@A             # The A matrix 
svd(coef(fit4, matrix=TRUE)[-1,])$d    # This has rank 1; = C %*% t(A) 


# Example 5: 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)

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