Determine from the selected CCA and Correlation objects the Stewart-Love redundancy for the selected canonical variates.

The Stewart-Love redundancy for a single canonical variate is the fraction of variance explained by the selected canonical variate in a set times the fraction of shared variance between the corresponding canonical variates in the two sets.

The Stewart-Love redundancy for a canonical variate range is the sum of the individual redundancies.

Settings

X or Y

determines whether you select the dependent (y) or the independent (x) set.

Canonical variate range

determines the canonical variates (or canonical variables).

Algorithm

The formulas can be found on page 170 of Cooley & Lohnes (1971).

For example, the redundancy of the dependent set (y) given the independent set (x) for the i^th canonical variate can be expressed as:

where varianceFraction_i(y) is the variance fraction explained by the i^th canonical variate of the dependent set, and ρ_i is the i^th canonical correlation coefficient.

The redundancy for the selected canonical variate in the dependent set shows what fraction of the variance in the dependent set is already "explained" by the variance in the independent set, i.e. this fraction could be considered as redundant.

In the same way we can measure the redundancy of the independent (x) set giving the dependent set (y).

Remark

In general R_i(y) ≠ R_i(x).