Check Whether a Set of Contrasts Is Mutually Orthogonal

Tests whether every pair of columns in a contrast-coefficient matrix is orthogonal under either the equal-\(n\) convention \(\sum_i c_{ik} c_{ij} = 0\) or the unequal-\(n\) convention \(\sum_i c_{ik} c_{ij} / n_i = 0\) (Maxwell, Delaney, & Kelley, 2027, Sec. 4.10; Kirk, 2013). Also checks that each column sums to zero (the contrast property).

Usage

is_orthogonal_set(contrasts, n = NULL, tol = 1e-08)

Arguments

contrasts

A numeric \(a \times m\) matrix or data.frame, where \(a\) is the number of groups and \(m\) is the number of contrasts.

n

Optional integer vector of length \(a\) giving the per- group sample sizes. If supplied, the unequal-\(n\) convention is used; otherwise the equal-\(n\) convention is assumed.

tol

Numerical tolerance for declaring orthogonality. Default 1e-8.

Value

A data.frame with rows for the overall orthogonality flag (1 = all pairs orthogonal, 0 = not), the contrast-sum-to-zero flag, the number of contrasts tested, and one row per pairwise dot-product, named by contrast pair.

Details

Equal-\(n\). Two contrasts \(\mathbf c, \mathbf d\) on \(a\) groups of equal size are orthogonal iff \(\sum_{i=1}^{a} c_i d_i = 0\).

Unequal-\(n\). With sample sizes \(n_1, \ldots, n_a\), the orthogonality condition that yields uncorrelated sample contrasts is \(\sum_{i=1}^{a} c_i d_i / n_i = 0\).

Useful for design checks. Before performing planned comparisons or partitioning the omnibus sums of squares, the user typically wants confirmation that the chosen contrast set is orthogonal so that its component SS sum to the omnibus SS.

References

Kirk, R. E. (2013). Experimental design: Procedures for the behavioral sciences (4th ed.). Sage.

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Sec. 4.10.)

See also

effects_coding, helmert_coding, ci_scheffe

Other design utilities: design_consequences(), design_effect(), effects_coding(), helmert_coding(), orthogonal_polynomial()

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Two orthogonal contrasts on a 4-group design (equal n):
cmat <- cbind(
  c_linear  = c(-3, -1,  1,  3),
  c_quad    = c( 1, -1, -1,  1)
)
is_orthogonal_set(cmat)
#>  term                      value
#>  all_orthogonal            1    
#>  all_contrasts_sum_to_zero 1    
#>  n_contrasts               2    
#>  dot[c_linear . c_quad]    0    

# 2. Same contrasts under unequal sample sizes:
is_orthogonal_set(cmat, n = c(10, 8, 12, 9))
#>  term                      value
#>  all_orthogonal            0    
#>  all_contrasts_sum_to_zero 1    
#>  n_contrasts               2    
#>  dot[c_linear . c_quad]    0.075

# 3. Non-orthogonal pair:
cmat_bad <- cbind(
  c_diff_1  = c( 1, -1,  0,  0),
  c_diff_2  = c( 1,  0, -1,  0)
)
is_orthogonal_set(cmat_bad)
#>  term                      value
#>  all_orthogonal            0    
#>  all_contrasts_sum_to_zero 1    
#>  n_contrasts               2    
#>  dot[c_diff_1 . c_diff_2]  1