Solve equal marginal cutoffs for a target joint selection ratio — tr_multivariate_equal

Thomas, Owen, and Gunst's printed tables are indexed by the overall proportion selected under two equal cutoffs. This helper solves the common marginal selection ratio that yields a target conjunctive selection ratio for any predictor correlation matrix, then calls tr_multivariate().

Usage

tr_multivariate_equal_cutoff(
  joint_selection_ratio,
  base_rate,
  R,
  interval = NULL,
  tol = 1e-08,
  digits = 3
)

Arguments

joint_selection_ratio: Target conjunctive selection ratio, P(X_1 >= c, ..., X_k >= c).
base_rate: Population proportion of successful applicants.
R: Correlation matrix with predictors first and criterion last.
interval: Optional search interval for the common marginal selection ratio. Defaults to (joint_selection_ratio, 1).
tol: Numerical tolerance passed to optimize().
digits: Number of digits used for printed summaries.

Value

A psu_tr object from tr_multivariate() with the solved marginal selection ratio, the target joint selection ratio, the computed joint selection ratio, and the numerical joint-selection error added.

References

Thomas, J. G., Owen, D. B., & Gunst, R. F. (1977). Improving the use of educational tests as selection tools. Journal of Educational Statistics, 2(1), 55-77.

Waller, N. G. (2024). TaylorRussell: A Taylor-Russell function for multiple predictors (R package version 1.2.1). CRAN.

Examples

# Literature: Thomas, Owen, and Gunst (1977); Waller (2024).
# Use the first call as a minimal example; the longer block illustrates
# how to interpret the function in the substantive setting discussed in the literature.
# Minimal example (Thomas, Owen, and Gunst (1977); Waller (2024)).
R <- matrix(c(1, .50, .70,
              .50, 1, .70,
              .70, .70, 1), 3, 3, byrow = TRUE)
tr_multivariate_equal_cutoff(joint_selection_ratio = .20, base_rate = .60, R = R)
#> <psu_tr>
#>   base_rate: 0.6
#>   joint_selection_ratio: 0.2
#>   criterion_cutoff_z: -0.253347
#>   true_positive: 0.194408
#>   false_positive: 0.00559223
#>   false_negative: 0.405592
#>   true_negative: 0.394408
#>   ppv: 0.972039
#>   success_ratio: 0.972039
#>   incremental_success: 0.372039
#>   sensitivity: 0.324013
#>   specificity: 0.986019
#>   digits: 3
#>   target_joint_selection_ratio: 0.2
#>   computed_joint_selection_ratio: 0.199984
#>   solved_marginal_selection_ratio: 0.354321
#>   joint_selection_error: -1.61346e-05

# Substantive example (Thomas, Owen, and Gunst, 1977;
# Waller, 2024). Reproduce the Example 1 pattern.
tog <- tr_multivariate_equal_cutoff(.20, .60, R)
c(marginal_sr = tog$solved_marginal_selection_ratio, ppv = tog$ppv)
#> marginal_sr         ppv 
#>   0.3543206   0.9720477