contrast {stats}R Documentation

(Possibly Sparse) Contrast Matrices


Return a matrix of contrasts.


contr.helmert(n, contrasts = TRUE, sparse = FALSE)
contr.poly(n, scores = 1:n, contrasts = TRUE, sparse = FALSE)
contr.sum(n, contrasts = TRUE, sparse = FALSE)
contr.treatment(n, base = 1, contrasts = TRUE, sparse = FALSE)
contr.SAS(n, contrasts = TRUE, sparse = FALSE)



a vector of levels for a factor, or the number of levels.


a logical indicating whether contrasts should be computed.


logical indicating if the result should be sparse (of class dgCMatrix), using package Matrix.


the set of values over which orthogonal polynomials are to be computed.


an integer specifying which group is considered the baseline group. Ignored if contrasts is FALSE.


These functions are used for creating contrast matrices for use in fitting analysis of variance and regression models. The columns of the resulting matrices contain contrasts which can be used for coding a factor with n levels. The returned value contains the computed contrasts. If the argument contrasts is FALSE a square indicator matrix (the dummy coding) is returned except for contr.poly (which includes the 0-degree, i.e. constant, polynomial when contrasts = FALSE).

contr.helmert returns Helmert contrasts, which contrast the second level with the first, the third with the average of the first two, and so on. contr.poly returns contrasts based on orthogonal polynomials. contr.sum uses ‘sum to zero contrasts’.

contr.treatment contrasts each level with the baseline level (specified by base): the baseline level is omitted. Note that this does not produce ‘contrasts’ as defined in the standard theory for linear models as they are not orthogonal to the intercept.

contr.SAS is a wrapper for contr.treatment that sets the base level to be the last level of the factor. The coefficients produced when using these contrasts should be equivalent to those produced by many (but not all) SAS procedures.

For consistency, sparse is an argument to all these contrast functions, however sparse = TRUE for contr.poly is typically pointless and is rarely useful for contr.helmert.


A matrix with n rows and k columns, with k=n-1 if contrasts is TRUE and k=n if contrasts is FALSE.


Chambers, J. M. and Hastie, T. J. (1992) Statistical models. Chapter 2 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

See Also

contrasts, C, and aov, glm, lm.


(cH <- contr.helmert(4))
apply(cH, 2, sum) # column sums are 0
crossprod(cH) # diagonal -- columns are orthogonal
contr.helmert(4, contrasts = FALSE) # just the 4 x 4 identity matrix

(cT <- contr.treatment(5))
all(crossprod(cT) == diag(4)) # TRUE: even orthonormal

(cT. <- contr.SAS(5))
all(crossprod(cT.) == diag(4)) # TRUE

zapsmall(cP <- contr.poly(3)) # Linear and Quadratic
zapsmall(crossprod(cP), digits = 15) # orthonormal up to fuzz

[Package stats version 3.5.0 Index]