stats (version 3.5.0)

model.matrix: Construct Design Matrices


model.matrix creates a design (or model) matrix, e.g., by expanding factors to a set of dummy variables (depending on the contrasts) and expanding interactions similarly.


model.matrix(object, …)

# S3 method for default model.matrix(object, data = environment(object), contrasts.arg = NULL, xlev = NULL, …)



an object of an appropriate class. For the default method, a model formula or a terms object.


a data frame created with model.frame. If another sort of object, model.frame is called first.


A list, whose entries are values (numeric matrices or character strings naming functions) to be used as replacement values for the contrasts replacement function and whose names are the names of columns of data containing factors.


to be used as argument of model.frame if data is such that model.frame is called.

further arguments passed to or from other methods.


The design matrix for a regression-like model with the specified formula and data.

There is an attribute "assign", an integer vector with an entry for each column in the matrix giving the term in the formula which gave rise to the column. Value 0 corresponds to the intercept (if any), and positive values to terms in the order given by the term.labels attribute of the terms structure corresponding to object.

If there are any factors in terms in the model, there is an attribute "contrasts", a named list with an entry for each factor. This specifies the contrasts that would be used in terms in which the factor is coded by contrasts (in some terms dummy coding may be used), either as a character vector naming a function or as a numeric matrix.


model.matrix creates a design matrix from the description given in terms(object), using the data in data which must supply variables with the same names as would be created by a call to model.frame(object) or, more precisely, by evaluating attr(terms(object), "variables"). If data is a data frame, there may be other columns and the order of columns is not important. Any character variables are coerced to factors. After coercion, all the variables used on the right-hand side of the formula must be logical, integer, numeric or factor.

If contrasts.arg is specified for a factor it overrides the default factor coding for that variable and any "contrasts" attribute set by C or contrasts.

In an interaction term, the variable whose levels vary fastest is the first one to appear in the formula (and not in the term), so in ~ a + b + b:a the interaction will have a varying fastest.

By convention, if the response variable also appears on the right-hand side of the formula it is dropped (with a warning), although interactions involving the term are retained.


Chambers, J. M. (1992) Data for models. Chapter 3 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

See Also

model.frame, model.extract, terms

sparse.model.matrix from package Matrix for creating sparse model matrices, which may be more efficient in large dimensions.


Run this code
ff <- log(Volume) ~ log(Height) + log(Girth)
utils::str(m <- model.frame(ff, trees))
mat <- model.matrix(ff, m)

dd <- data.frame(a = gl(3,4), b = gl(4,1,12)) # balanced 2-way
model.matrix(~ a + b, dd)
model.matrix(~ a + b, dd, contrasts = list(a = "contr.sum"))
model.matrix(~ a + b, dd, contrasts = list(a = "contr.sum", b = "contr.poly"))
m.orth <- model.matrix(~a+b, dd, contrasts = list(a = "contr.helmert"))
crossprod(m.orth) # m.orth is  ALMOST  orthogonal
# }

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