VGAM (version 1.0-4)

constraints: Constraint Matrices


Extractor function for the constraint matrices of objects in the VGAM package.


constraints(object, ...)
constraints.vlm(object, type = c("lm", "term"), all = TRUE, which,
                matrix.out = FALSE, colnames.arg = TRUE,
                rownames.arg = TRUE, ...)



Some VGAM object, for example, having class vglmff-class.


Character. Whether LM- or term-type constraints are to be returned. The number of such matrices returned is equal to nvar(object, type = "lm") and the number of terms, respectively.

all, which

If all = FALSE then which gives the integer index or a vector of logicals specifying the selection.


Logical. If TRUE then the constraint matrices are cbind()ed together. The result is usually more compact because the default is a list of constraint matrices.

colnames.arg, rownames.arg

Logical. If TRUE then column and row names are assigned corresponding to the variables.

Other possible arguments such as type.


The extractor function constraints() returns a list comprising of constraint matrices---usually one for each column of the VLM model matrix, and in that order. The list is labelled with the variable names. Each constraint matrix has \(M\) rows, where \(M\) is the number of linear/additive predictors, and whose rank is equal to the number of columns. A model with no constraints at all has an order \(M\) identity matrix as each variable's constraint matrix.

For vglm and vgam objects, feeding in type = "term" constraint matrices back into the same model should work and give an identical model. The default are the "lm"-type constraint matrices; this is a list with one constraint matrix per column of the LM matrix. See the constraints argument of vglm, and the example below.


Constraint matrices describe the relationship of coefficients/component functions of a particular explanatory variable between the linear/additive predictors in VGLM/VGAM models. For example, they may be all different (constraint matrix is the identity matrix) or all the same (constraint matrix has one column and has unit values).

VGLMs and VGAMs have constraint matrices which are known. The class of RR-VGLMs have constraint matrices which are unknown and are to be estimated.


Yee, T. W. and Wild, C. J. (1996) Vector generalized additive models. Journal of the Royal Statistical Society, Series B, Methodological, 58, 481--493.

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

See Also

is.parallel, VGLMs are described in vglm-class; RR-VGLMs are described in rrvglm-class.

Arguments such as zero and parallel found in many VGAM family functions are a way of creating/modifying constraint matrices conveniently, e.g., see zero. See CommonVGAMffArguments for more information.


Run this code
# Fit the proportional odds model:
pneumo <- transform(pneumo, let = log(exposure.time))
(fit1 <- vglm(cbind(normal, mild, severe) ~, 3),
              cumulative(parallel = TRUE, reverse = TRUE), data = pneumo))
coef(fit1, matrix = TRUE)
constraints(fit1)  # Parallel assumption results in this
constraints(fit1, type = "term")  # Same as the default ("vlm"-type)

# An equivalent model to fit1 (needs the type "term" constraints):
clist.term <- constraints(fit1, type = "term")  # "term"-type constraints
(fit2 <- vglm(cbind(normal, mild, severe) ~, 3), data = pneumo,
              cumulative(reverse = TRUE), constraints = clist.term))
abs(max(coef(fit1, matrix = TRUE) -
        coef(fit2, matrix = TRUE)))  # Should be zero

# Fit a rank-1 stereotype (RR-multinomial logit) model:
fit <- rrvglm(Country ~ Width + Height + HP, multinomial, data = car.all)
constraints(fit)  # All except the first are the estimated A matrix
# }

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