Multivariate range-restriction correction (Lawley, 1943)

Corrects an observed (restricted) correlation matrix for direct selection on a subset of variables and incidental selection on the remaining variables, using Lawley's (1943) multivariate formulae.

Usage

correct_r_lawley(
  sigma_restricted,
  selection_indices,
  sigma_ss_unrestricted,
  standardize = TRUE
)

Arguments

sigma_restricted

Observed (restricted) covariance or correlation matrix in the selected sample. Must be symmetric and positive semi-definite.

selection_indices

Integer vector indicating which rows/columns of sigma_restricted correspond to variables on which selection was applied.

sigma_ss_unrestricted

Unrestricted covariance submatrix for the selection variables (the same dimension as sigma_restricted[selection_indices, selection_indices]). Typically estimated from applicant-pool data.

standardize

Logical. If TRUE (default), the corrected covariance matrix is converted to a correlation matrix via stats::cov2cor().

Value

A list with components:

sigma_corrected

The corrected (unrestricted) covariance or correlation matrix of the same dimension as sigma_restricted.

sigma_restricted

The input restricted matrix (echoed).

selection_indices

Indices treated as direct-selection variables.

incidental_indices

Indices treated as incidentally restricted.

u

Vector of sd_restricted / sd_unrestricted per selection variable, one of the standard summaries of restriction severity.

sign_changes

Integer count of off-diagonal entries whose sign differs between corrected and observed matrices, flagged in the spirit of Ree, Carretta, Earles & Albert (1994).

Details

Let S index the variables on which selection was applied and U index the remaining (incidentally restricted) variables. Given the observed restricted covariance matrix Sigma_star and the unrestricted covariance submatrix Sigma_SS_unrestricted for the selection variables, Lawley's correction recovers the unrestricted covariance matrix:

$$\Sigma_{UV} = \Sigma_{UV}^{*} + \Sigma_{US}^{*}\, (\Sigma_{SS}^{*})^{-1}\, (\Sigma_{SS} - \Sigma_{SS}^{*})\, (\Sigma_{SS}^{*})^{-1}\, \Sigma_{SV}^{*}$$ $$\Sigma_{UU} = \Sigma_{UU}^{*} + \Sigma_{US}^{*}\, (\Sigma_{SS}^{*})^{-1}\, (\Sigma_{SS} - \Sigma_{SS}^{*})\, (\Sigma_{SS}^{*})^{-1}\, \Sigma_{SU}^{*}$$

for any partitioning into selection variables S and other variables U,V.

Sign changes flagged in sign_changes are not necessarily errors but should be inspected: Ree et al. (1994) documented that legitimate Lawley corrections can flip the sign of small predictor-criterion correlations when the restriction matrix is large.

References

Lawley, D. N. (1943). A note on Karl Pearson's selection formulae. Proceedings of the Royal Society of Edinburgh, Section A, 62, 28-30.

Mendoza, J. L., & Mumford, M. D. (1987). Corrections for attenuation and range restriction on the predictor. Journal of Educational and Behavioral Statistics, 12, 282-293.

Ree, M. J., Carretta, T. R., Earles, J. A., & Albert, W. (1994). Sign changes when correcting for range restriction: A note on Pearson's and Lawley's selection formulas. Journal of Applied Psychology, 79, 298-301.

Sackett, P. R., Lievens, F., Berry, C. M., & Landers, R. N. (2007). A cautionary note on the effects of range restriction on predictor intercorrelations. Journal of Applied Psychology, 92, 538-544.

Examples

# Three-variable example: selection on X1 (cognitive ability),
# incidental restriction on X2 (interview) and Y (criterion).
sigma_star <- matrix(c(
  1.00, 0.30, 0.25,
  0.30, 1.00, 0.20,
  0.25, 0.20, 1.00
), 3, 3)
# Unrestricted SD of X1 is larger; var increases by factor 1/u^2 = 1/.6^2
sigma_ss <- matrix(1 / 0.6^2, 1, 1)
correct_r_lawley(sigma_star, selection_indices = 1,
                 sigma_ss_unrestricted = sigma_ss)
#> $sigma_corrected
#>           [,1]      [,2]      [,3]
#> [1,] 1.0000000 0.4642383 0.3952847
#> [2,] 0.4642383 1.0000000 0.2936101
#> [3,] 0.3952847 0.2936101 1.0000000
#> 
#> $sigma_restricted
#>      [,1] [,2] [,3]
#> [1,] 1.00  0.3 0.25
#> [2,] 0.30  1.0 0.20
#> [3,] 0.25  0.2 1.00
#> 
#> $selection_indices
#> [1] 1
#> 
#> $incidental_indices
#> [1] 2 3
#> 
#> $u
#> [1] 0.6
#> 
#> $sign_changes
#> [1] 0
#>