Factorial Contrasts

Helper function to construct the default contrast coefficients for various coding schemes.

Note that R has a partial matching feature which allows you to specify shortened versions of arguments, such as coding instead of coding.scheme.

Validated using lm() and aov() functions.

factorial.contrasts(factor.levels = c(3, 2),
                    coding.scheme = rep("deviation", length(factor.levels)),
                    base = factor.levels,
                    intercept = FALSE,
                    verbose = TRUE)

Arguments

factor.levels

Integer. Number of levels or groups in each factor. For example, for two factors each having two levels or groups use e.g. c(2, 2), for three factors each having two levels or groups use e.g. c(2, 2, 2)

coding.scheme

Character vector. Coding scheme for each factor. "sum" for deviation or effect coding, "treatment" for dummy coding, "helmert" for Helmert type of coding, and "poly" for polynomial coding. Each factor can have their own coding scheme. If a single character value is provided, it will be copied to other factors

base

Integer vector. Specifies which group is considered the baseline group. Ignored for coding schemes other than "treatment"

intercept

Logical. FALSE by default. If TRUE contrast matrix includes the intercept

verbose

Logical. TRUE by default. If FALSE no output is printed on the console

Value

factor.levels

Number of levels (or groups) in each factor

factor.data

Unique combination of factor levels

model.matrix

Model (design) matrix based on unique combination of factor levels

contrast.matrix

Contrast matrix

Examples

###################################################
############### dummy coding scheme ###############
####################################################

# one factor w/ 3 levels
contrast.object <- factorial.contrasts(factor.levels = 3,
                                       coding = "treatment")
#>    A1 A2 A3
#> A1  1  0 -1
#> A2  0  1 -1
# model matrix
contrast.object$model.matrix
#>   (Intercept) A1 A2
#> 1           1  1  0
#> 2           1  0  1
#> 3           1  0  0
#> attr(,"assign")
#> [1] 0 1 1
#> attr(,"contrasts")
#> attr(,"contrasts")$A
#>   1 2
#> 1 1 0
#> 2 0 1
#> 3 0 0
#> 

# contrast matrix
contrast.object$contrast.matrix
#>    A1 A2 A3
#> A1  1  0 -1
#> A2  0  1 -1

#######################################################
###############  deviation coding scheme ##############
#######################################################

# especially useful for factorial designs
# two factors w/ 2 and 3 levels, respectively
contrast.object <- factorial.contrasts(factor.levels = c(2, 3),
                                       coding = "sum")
#> Assuming the same coding scheme applies to the other factor(s)
#> Elements of `mu.vector`, `sd.vector`, `n.vector` or `p.vector` should follow this specific order:
#> A1:B1  A1:B2  A1:B3  A2:B1  A2:B2  A2:B3  
#>        A1:B1  A1:B2  A1:B3  A2:B1  A2:B2  A2:B3
#> A1     0.167  0.167  0.167 -0.167 -0.167 -0.167
#> B1     0.333 -0.167 -0.167  0.333 -0.167 -0.167
#> B2    -0.167  0.333 -0.167 -0.167  0.333 -0.167
#> A1:B1  0.333 -0.167 -0.167 -0.333  0.167  0.167
#> A1:B2 -0.167  0.333 -0.167  0.167 -0.333  0.167

# model matrix
contrast.object$model.matrix
#>   (Intercept) A1 B1 B2 A1:B1 A1:B2
#> 1           1  1  1  0     1     0
#> 2           1  1  0  1     0     1
#> 3           1  1 -1 -1    -1    -1
#> 4           1 -1  1  0    -1     0
#> 5           1 -1  0  1     0    -1
#> 6           1 -1 -1 -1     1     1
#> attr(,"assign")
#> [1] 0 1 2 2 3 3
#> attr(,"contrasts")
#> attr(,"contrasts")$A
#>   [,1]
#> 1    1
#> 2   -1
#> 
#> attr(,"contrasts")$B
#>   [,1] [,2]
#> 1    1    0
#> 2    0    1
#> 3   -1   -1
#> 

# contrast matrix
contrast.object$contrast.matrix
#>            A1:B1      A1:B2      A1:B3      A2:B1      A2:B2      A2:B3
#> A1     0.1666667  0.1666667  0.1666667 -0.1666667 -0.1666667 -0.1666667
#> B1     0.3333333 -0.1666667 -0.1666667  0.3333333 -0.1666667 -0.1666667
#> B2    -0.1666667  0.3333333 -0.1666667 -0.1666667  0.3333333 -0.1666667
#> A1:B1  0.3333333 -0.1666667 -0.1666667 -0.3333333  0.1666667  0.1666667
#> A1:B2 -0.1666667  0.3333333 -0.1666667  0.1666667 -0.3333333  0.1666667


######################################################
###############  Helmert coding scheme ###############
######################################################

# one factor w/ 3 levels
contrast.object <- factorial.contrasts(factor.levels = 3,
                                       coding = "helmert")
#>        A1     A2    A3
#> A1 -0.500  0.500 0.000
#> A2 -0.167 -0.167 0.333

# model matrix
contrast.object$model.matrix
#>   (Intercept) A1 A2
#> 1           1 -1 -1
#> 2           1  1 -1
#> 3           1  0  2
#> attr(,"assign")
#> [1] 0 1 1
#> attr(,"contrasts")
#> attr(,"contrasts")$A
#>   [,1] [,2]
#> 1   -1   -1
#> 2    1   -1
#> 3    0    2
#> 

# contrast matrix
contrast.object$contrast.matrix
#>            A1         A2        A3
#> A1 -0.5000000  0.5000000 0.0000000
#> A2 -0.1666667 -0.1666667 0.3333333

#########################################################
###############  polynomial coding scheme ###############
#########################################################

# one factor w/ 3 levels
contrast.object <- factorial.contrasts(factor.levels = 3,
                                       coding = "poly")
#>         A1     A2    A3
#> A.L -0.707  0.000 0.707
#> A.Q  0.408 -0.816 0.408

# model matrix
contrast.object$model.matrix
#>   (Intercept)           A.L        A.Q
#> 1           1 -7.071068e-01  0.4082483
#> 2           1 -9.073800e-17 -0.8164966
#> 3           1  7.071068e-01  0.4082483
#> attr(,"assign")
#> [1] 0 1 1
#> attr(,"contrasts")
#> attr(,"contrasts")$A
#>              .L         .Q
#> 1 -7.071068e-01  0.4082483
#> 2 -9.073800e-17 -0.8164966
#> 3  7.071068e-01  0.4082483
#> 

# contrast matrix
contrast.object$contrast.matrix
#>             A1         A2        A3
#> A.L -0.7071068  0.0000000 0.7071068
#> A.Q  0.4082483 -0.8164966 0.4082483