VGAM (version 1.1-6)

multinomial: Multinomial Logit Model


Fits a multinomial logit model (MLM) to a (preferably unordered) factor response.


multinomial(zero = NULL, parallel = FALSE, nointercept = NULL,
            refLevel = "(Last)", imethod = 1, imu = NULL,
            byrow.arg = FALSE, whitespace = FALSE)



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,…,\(M\)}. The default value means none are modelled as intercept-only terms. See CommonVGAMffArguments for more information.


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

nointercept, whitespace

See CommonVGAMffArguments for details.

imu, byrow.arg

See CommonVGAMffArguments for details.


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 \((M+1)\)th one). If used, this argument will be usually assigned the value 1. If inputted as a value of a factor then beware of missing values of certain levels of the factor (drop.unused.levels = TRUE or drop.unused.levels = FALSE). See the example below. If inputted as a character string then this should be equal to (A) one of the levels of the factor response, else (B) one of the column names of the matrix response of counts; e.g., vglm(cbind(normal, mild, severe) ~ let, multinomial(refLevel = "severe"), data = pneumo) if it was (incorrectly because the response is ordinal) applied to the pneumo data set. Another example is vglm(ethnicity ~ age, multinomial(refLevel = "European"), data = if it was applied to the data set.


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 CommonVGAMffArguments for more details.


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


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

See CommonVGAMffArguments for more warnings.


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 \(j\)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 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).


Yee, T. W. (2010). The VGAM package for categorical data analysis. Journal of Statistical Software, 32, 1--34. 10.18637/jss.v032.i10.

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

Yee, T. W. and Ma, C. (2021). Generally--altered, --inflated, --truncated and --altered count regression. In preparation.

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

Agresti, A. (2013). Categorical Data Analysis, 3rd ed. Hoboken, NJ, USA: Wiley.

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.

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

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

Tutz, G. (2012). Regression for Categorical Data, Cambridge University Press.

See Also

multilogitlink, margeff, cumulative, acat, cratio, sratio, dirichlet, dirmultinomial, rrvglm, fill1, Multinomial, gaitdpoisson, Gaitdpois, iris.


Run this code
# Example 1: fit a multinomial logit model to Edgar Anderson's iris data
# }
 fit <- vglm(Species ~ ., multinomial, iris)
coef(fit, matrix = TRUE) 
# }
# 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 extraction of 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

# Example 4: Fit a rank-1 stereotype model
fit4 <- rrvglm(Country ~ Width + Height + HP, multinomial, data = 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  # This has rank 1; = C %*% t(A)
# Classification (but watch out for NAs in some of the variables):
apply(fitted(fit4), 1, which.max)  # Classification
colnames(fitted(fit4))[apply(fitted(fit4), 1, which.max)]  # Classification
apply(predict(fit4, car.all, type = "response"), 1, which.max)  # Ditto

# Example 5: The use of the xij argument (aka conditional logit model)
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),
               = runif(nn),  = 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.walk,
                      Cost.train = cost.train - cost.walk,
                      Cost       = cost.train - cost.walk,  # for labelling
                      Time.bus   = time.bus   - time.walk,
               =   - 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 +,
                       Time ~ Time.bus + Time.train +,
            form2 =  ~ Cost + Cost.bus + Cost.train + +
                       Time + Time.bus + Time.train +,
            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, matrix = TRUE)
max(abs(predict(fit) - predict(fit, new = gotowork)))  # Should be 0
# }

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