Package 'personnelSelectionUtility'

Description

Computes selection rates and adverse-impact ratios by group. If no reference group is supplied, the highest selection-rate group is used as reference.

Usage

adverse_impact_ratio(selected, group, reference = NULL)

Arguments

Value

A data frame with selection rates and ratios.

References

De Corte, W., Lievens, F., & Sackett, P. R. (2007). Combining predictors to achieve optimal trade-offs between selection quality and adverse impact. Journal of Applied Psychology, 92, 1380-1393.

Pyburn, K. M., Ployhart, R. E., & Kravitz, D. A. (2008). The diversity- validity dilemma: Overview and legal context. Personnel Psychology, 61, 143-151.

Examples

# Literature: Pyburn, Ployhart, and Kravitz (2008); De Corte et al. (2007).
adverse_impact_ratio(c(1, 0, 1, 1, 0, 0), c("A", "A", "A", "B", "B", "B"))

Description

Returns the package's recommended argument names and the notation they map to in the utility-analysis literature. The glossary is intended to make the API explicit and to avoid mixing compact statistical notation with readable R argument names.

Usage

argument_glossary()

Value

A data frame with argument names, literature notation, and usage notes.

References

Sturman, M. C. (2001). Utility analysis for multiple selection devices and multiple outcomes. Journal of Human Resource Costing and Accounting, 6(2), 9-28.

Holling, H. (1998). Utility analysis of personnel selection: An overview and empirical study based on objective performance measures. Methods of Psychological Research Online, 3(1), 5-24.

Examples

# Literature: Holling (1998); Sturman (2001).
argument_glossary()
subset(argument_glossary(), argument %in% c("base_rate", "selection_ratio", "sdy"))

Description

Converts AUC to Cohen's d using $d = \sqrt{2}\Phi^{-1}(AUC)$. This conversion assumes two normal distributions with equal variances and should therefore be interpreted as a model-based effect-size conversion, not as a universal transformation from classifier accuracy to personnel-selection validity.

Usage

auc_to_d_equal_variance(auc)

Arguments

Value

Numeric vector of Cohen's d values.

References

Hanley, J. A., & McNeil, B. J. (1982). The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology, 143(1), 29-36.

Rice, M. E., & Harris, G. T. (2005). Comparing effect sizes in follow-up studies: ROC area, Cohen's d, and r. Law and Human Behavior, 29(5), 615-620.

Salgado, J. F. (2018). Transforming the area under the normal curve (AUC) into Cohen's d, Pearson's r_pb, odds-ratio, and natural log odds-ratio: Two conversion tables. The European Journal of Psychology Applied to Legal Context, 10(1), 35-47.

Examples

# Minimal example based on the equal-variance binormal conversion.
auc_to_d_equal_variance(.75)

# Direction is preserved: AUC below .50 implies a negative effect.
auc_to_d_equal_variance(.40)

Description

Converts AUC to Cohen's d under the equal-variance binormal model and then converts d to a point-biserial correlation for a user-specified base rate. This is the preferred correlation-like conversion when a utility-analysis function requires a validity input but the available evidence is reported as AUC.

Usage

auc_to_point_biserial(auc, base_rate = 0.5)

Arguments

Value

Numeric vector of point-biserial correlations.

References

Hanley, J. A., & McNeil, B. J. (1982). The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology, 143(1), 29-36.

Rice, M. E., & Harris, G. T. (2005). Comparing effect sizes in follow-up studies: ROC area, Cohen's d, and r. Law and Human Behavior, 29(5), 615-620.

Salgado, J. F. (2018). Transforming the area under the normal curve (AUC) into Cohen's d, Pearson's r_pb, odds-ratio, and natural log odds-ratio: Two conversion tables. The European Journal of Psychology Applied to Legal Context, 10(1), 35-47.

Examples

# Minimal example: AUC to d, then to r_pb for a balanced binary criterion.
auc_to_point_biserial(.75)

# Substantive example: examine how base rate affects the implied r_pb.
auc_to_point_biserial(.75, base_rate = c(.50, .30, .20, .10))

Description

Converts the area under the ROC curve to the rank-biserial correlation, $r_{rb} = 2 AUC - 1$. This is a distribution-free dominance summary: it rescales the probability that a randomly chosen successful applicant is ranked above a randomly chosen unsuccessful applicant from the ⁠[0, 1]⁠ AUC scale to the ⁠[-1, 1]⁠ correlation-like scale.

Usage

auc_to_rank_biserial(auc)

Arguments

Value

Numeric vector of rank-biserial correlations.

References

Hanley, J. A., & McNeil, B. J. (1982). The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology, 143(1), 29-36.

Kerby, D. S. (2014). The simple difference formula: An approach to teaching nonparametric correlation. Comprehensive Psychology, 3, 11.IT.3.1.

Rice, M. E., & Harris, G. T. (2005). Comparing effect sizes in follow-up studies: ROC area, Cohen's d, and r. Law and Human Behavior, 29(5), 615-620.

Examples

# Minimal example: AUC = .50 implies no dominance.
auc_to_rank_biserial(.50)

# AUC = .75 means 75% favorable pairwise ordering; r_rb = .50.
auc_to_rank_biserial(.75)

Description

Computes classical Brogden-Cronbach-Gleser utility. By default the baseline is random selection (baseline_validity = 0), but an operating baseline can be supplied using baseline_validity and, optionally, baseline_selection_ratio.

Usage

bcg_utility(
  validity,
  selection_ratio,
  sdy,
  n_selected,
  tenure,
  cost = 0,
  baseline_validity = 0,
  baseline_selection_ratio = NULL
)

Arguments

Value

A psu_bcg object.

References

Cronbach, L. J., & Gleser, G. C. (1965). Psychological tests and personnel decisions (2nd ed.). University of Illinois Press.

Brogden, H. E. (1946). On the interpretation of the correlation coefficient as a measure of predictive efficiency. Journal of Educational Psychology, 37, 65-76.

Brogden, H. E. (1949). When testing pays off. Personnel Psychology, 2, 171-183.

Sturman, M. C. (2001). Utility analysis for multiple selection devices and multiple outcomes. Journal of Human Resource Costing and Accounting, 6(2), 9-28.

Examples

# Literature: Brogden (1946, 1949); Cronbach and Gleser (1965); Sturman (2001).
# 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 (Brogden (1946, 1949); Cronbach and Gleser (1965); Sturman (2001)).
bcg_utility(validity = .35, selection_ratio = .20, sdy = 50000,
            n_selected = 100, tenure = 3, cost = 75000)

# Substantive example (Brogden, 1946, 1949;
# Cronbach and Gleser, 1965; Sturman, 2001).
# Use an operating baseline rather than random selection.
naive <- bcg_utility(.35, .20, 50000, n_selected = 100, tenure = 3, cost = 75000)
incremental <- bcg_utility(.35, .20, 50000, n_selected = 100, tenure = 3,
                           cost = 75000, baseline_validity = .20)
c(naive = naive$net_utility, incremental = incremental$net_utility)

Description

Computes discounted multi-period utility with optional value, tax, and cost adjustments. The expected standardized criterion gain can be supplied directly as delta_z_y, or computed from validity and selection-ratio parameters.

Usage

boudreau_utility(
  delta_z_y = NULL,
  validity = NULL,
  selection_ratio = NULL,
  baseline_validity = 0,
  baseline_selection_ratio = NULL,
  sdy,
  n_by_period = NULL,
  variable_value = 0,
  contribution_margin = NULL,
  variable_value_convention = c("paper_plus", "cost_rate"),
  tax_rate = 0,
  discount_rate = 0,
  cost_by_period = NULL,
  discount_costs = TRUE,
  n_t = NULL,
  cost_t = NULL
)

Arguments

Value

A psu_boudreau object.

References

Boudreau, J. W. (1983). Economic considerations in estimating the utility of human resource productivity improvement programs. Personnel Psychology, 36, 551-576.

Boudreau, J. W. (1991). Utility analysis for decisions in human resource management. In M. D. Dunnette & L. M. Hough (Eds.), Handbook of industrial and organizational psychology (Vol. 2, pp. 621-745). Consulting Psychologists Press.

Sturman, M. C. (2001). Utility analysis for multiple selection devices and multiple outcomes. Journal of Human Resource Costing and Accounting, 6(2), 9-28.

Holling, H. (1998). Utility analysis of personnel selection: An overview and empirical study based on objective performance measures. Methods of Psychological Research Online, 3(1), 5-24.

Examples

# Literature: Boudreau (1983, 1991); Sturman (2001); Holling (1998).
# 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 (Boudreau (1983, 1991); Sturman (2001); Holling (1998)).
boudreau_utility(validity = .35, selection_ratio = .20, sdy = 50000,
                 n_by_period = c(100, 90, 80), discount_rate = .08,
                 cost_by_period = c(75000, 10000, 10000))

# Substantive example (Boudreau, 1983, 1991;
# Sturman, 2001; Holling, 1998).
# Use an explicit contribution margin and operating baseline.
boudreau_utility(
  validity = .35,
  baseline_validity = .20,
  selection_ratio = .20,
  sdy = 50000,
  n_by_period = c(100, 90, 80, 70),
  contribution_margin = .30,
  tax_rate = .25,
  discount_rate = .08,
  cost_by_period = c(75000, 10000, 10000, 10000)
)

Description

Solves the validity needed to obtain zero net utility under the BCG model.

Usage

break_even_validity(
  selection_ratio,
  sdy,
  n_selected,
  tenure,
  cost = 0,
  baseline_validity = 0
)

Arguments

Value

Required focal validity.

References

Boudreau, J. W. (1991). Utility analysis for decisions in human resource management. In M. D. Dunnette & L. M. Hough (Eds.), Handbook of industrial and organizational psychology (Vol. 2, pp. 621-745). Consulting Psychologists Press.

Holling, H. (1998). Utility analysis of personnel selection: An overview and empirical study based on objective performance measures. Methods of Psychological Research Online, 3(1), 5-24.

Examples

# Literature: Boudreau (1991); Holling (1998).
# 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 (Boudreau (1991); Holling (1998)).
break_even_validity(.20, 50000, 100, 3, cost = 75000)

# Substantive example (Boudreau, 1991; Holling, 1998).
# Required incremental validity under different costs.
costs <- c(25000, 75000, 150000)
setNames(
  break_even_validity(.20, 50000, 100, 3, cost = costs, baseline_validity = .15),
  paste0('cost_', costs)
)

Description

Computes the squared validity coefficient.

Usage

coefficient_of_determination(validity)

Arguments

Value

Numeric vector with validity^2.

References

Taylor, H. C., & Russell, J. T. (1939). The relationship of validity coefficients to the practical effectiveness of tests in selection. Journal of Applied Psychology, 23, 565-578.

Holling, H. (1998). Utility analysis of personnel selection: An overview and empirical study based on objective performance measures. Methods of Psychological Research Online, 3(1), 5-24.

Examples

# Literature: Taylor and Russell (1939); Holling (1998).
coefficient_of_determination(.30)

Description

Computes analytic compensatory expected performance and simulated multiple-hurdle expected performance using the same predictor/criterion correlation structure.

Usage

compare_selection_systems(
  predictor_cor,
  validities,
  compensatory_weights = NULL,
  compensatory_selection_ratio,
  hurdle_selection_ratios,
  n_sim = 1e+05,
  seed = NULL,
  n_applicants = NA_real_,
  compensatory_cost_per_applicant = 0,
  hurdle_cost_per_stage = 0,
  sdy = NULL,
  applicant_n = NULL
)

Arguments

Value

A list with compensatory, multiple-hurdle, and difference summaries.

References

Ock, J., & Oswald, F. L. (2018). The utility of personnel selection decisions: Comparing compensatory and multiple-hurdle selection models. Journal of Personnel Psychology, 17(4), 172-182.

Examples

# Literature: Ock and Oswald (2018).
# Minimal example (Ock and Oswald (2018)).
Rxx <- matrix(c(1, .30, .30, 1), 2, 2)
compare_selection_systems(Rxx, c(.40, .30), hurdle_selection_ratios = c(.50, .50),
                          compensatory_selection_ratio = .25, n_sim = 1000, seed = 1)

# Substantive example with monetary utility.
compare_selection_systems(
  predictor_cor = Rxx,
  validities = c(.40, .30),
  compensatory_selection_ratio = .25,
  hurdle_selection_ratios = c(.50, .50),
  n_sim = 5000,
  seed = 123,
  n_applicants = 400,
  compensatory_cost_per_applicant = 800,
  hurdle_cost_per_stage = c(100, 300),
  sdy = 50000
)

Description

Compares a compensatory top-down composite against a staged multiple-hurdle system in which stages can be composites.

Usage

compare_selection_systems_staged(
  predictor_cor,
  validities,
  compensatory_weights = NULL,
  compensatory_selection_ratio,
  stage_predictors,
  stage_selection_ratios,
  stage_weights = NULL,
  n_sim = 1e+05,
  seed = NULL,
  n_applicants = NA_real_,
  compensatory_cost_per_applicant = 0,
  hurdle_cost_per_stage = 0,
  sdy = NULL,
  applicant_n = NULL
)

Arguments

Value

A list with compensatory, staged multiple-hurdle, and difference summaries.

References

Ock, J., & Oswald, F. L. (2018). The utility of personnel selection decisions: Comparing compensatory and multiple-hurdle selection models. Journal of Personnel Psychology, 17(4), 172-182.

Examples

# Literature: Ock and Oswald (2018).
# 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 (Ock and Oswald (2018)).
Rxx <- diag(4); Rxx[lower.tri(Rxx)] <- Rxx[upper.tri(Rxx)] <- .20
compare_selection_systems_staged(Rxx, validities = c(.40, .35, .20, .30),
  compensatory_selection_ratio = .20, stage_predictors = list(1:3, 4),
  stage_selection_ratios = c(.25, .80), n_sim = 1000, seed = 1)

# Substantive Ock-Oswald-style staged comparison.
compare_selection_systems_staged(
  predictor_cor = Rxx,
  validities = c(.40, .35, .20, .30),
  compensatory_weights = rep(1, 4),
  compensatory_selection_ratio = .20,
  stage_predictors = list(c(1, 3, 4), 2),
  stage_selection_ratios = c(.25, .80),
  n_sim = 5000,
  seed = 123,
  n_applicants = 500,
  compensatory_cost_per_applicant = 1000,
  hurdle_cost_per_stage = c(100, 900),
  sdy = 60000
)

Description

Computes the expected standardized criterion performance of applicants selected on a weighted predictor composite. This is the compensatory cell of the package taxonomy: scores on stronger predictors can offset lower scores on weaker ones.

Usage

compensatory_selection(
  predictor_cor,
  validities,
  weights = NULL,
  selection_ratio,
  n_applicants = NA_real_,
  cost_per_applicant = 0,
  sdy = NULL,
  applicant_n = NULL
)

Arguments

Value

A psu_comparison object.

References

Naylor, J. C., & Shine, L. C. (1965). A table for determining the increase in mean criterion score obtained by using a selection device. Journal of Industrial Psychology, 3, 33-42.

Ock, J., & Oswald, F. L. (2018). The utility of personnel selection decisions: Comparing compensatory and multiple-hurdle selection models. Journal of Personnel Psychology, 17(4), 172-182.

Examples

# Literature: Naylor and Shine (1965); Ock and Oswald (2018).
# 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 (Naylor and Shine (1965); Ock and Oswald (2018)).
Rxx <- matrix(c(1, .30, .30, 1), 2, 2)
compensatory_selection(Rxx, validities = c(.40, .30), selection_ratio = .20)

# Substantive example with costs and SDy.
Rxx <- matrix(c(
  1.00, .30, .20,
  .30, 1.00, .25,
  .20, .25, 1.00
), 3, 3, byrow = TRUE)
compensatory_selection(
  predictor_cor = Rxx,
  validities = c(.45, .35, .25),
  weights = c(1, 1, 1),
  selection_ratio = .20,
  n_applicants = 500,
  cost_per_applicant = 250,
  sdy = 60000
)

Description

Computes Sackett-Ellingson-style composite d for a weighted battery.

Usage

composite_d(d, weights = NULL, predictor_cor = NULL, sd = NULL)

Arguments

Value

Composite effect size.

References

Sackett, P. R., & Ellingson, J. E. (1997). The effects of forming multi- predictor composites on group differences and adverse impact. Personnel Psychology, 50, 707-721.

Examples

# Literature: Sackett and Ellingson (1997).
composite_d(d = c(.80, .30), weights = c(.7, .3),
            predictor_cor = matrix(c(1, .30, .30, 1), 2, 2))

Description

Uses the common two-group approximation ⁠d = 2r / sqrt(1 - r^2)⁠.

Usage

cor_to_d(r)

Arguments

Value

Cohen's d.

References

Schmidt, F. L., Hunter, J. E., & Pearlman, K. (1982). Assessing the economic impact of personnel programs on workforce productivity. Personnel Psychology, 35, 333-347.

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Erlbaum.

Examples

# Literature: Cohen (1988); Schmidt, Hunter, and Pearlman (1982).
cor_to_d(.30)

Description

Corrects a restricted validity coefficient for direct range restriction on the predictor using the standard Thorndike Case II expression.

Usage

correct_r_direct_range_restriction(
  r_restricted,
  range_restriction_ratio = NULL,
  u = NULL
)

Arguments

Value

Corrected validity coefficient.

References

Sackett, P. R., Laczo, R. M., & Arvey, R. D. (2002). The effects of range restriction on estimates of criterion interrater reliability: Implications for validation research. Personnel Psychology, 55, 807-825.

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.

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

Examples

# Literature: Lawley (1943); Sackett, Laczo, and Arvey (2002); Ree et al. (1994).
correct_r_direct_range_restriction(.25, range_restriction_ratio = 1.40)
correct_r_direct_range_restriction(.25, u = 1.40)

Description

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

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.

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).

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)

Description

Uses $r = d / \sqrt{d^2 + 4}$.

Usage

d_to_cor(d)

Arguments

Value

Correlation coefficient.

References

Schmidt, F. L., Hunter, J. E., & Pearlman, K. (1982). Assessing the economic impact of personnel programs on workforce productivity. Personnel Psychology, 35, 333-347.

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Erlbaum.

Examples

# Literature: Cohen (1988); Schmidt, Hunter, and Pearlman (1982).
d_to_cor(.50)

Description

Converts a standardized mean difference to the point-biserial correlation implied by a dichotomous criterion with base rate $p$. The implemented formula is $r_{pb} = d\sqrt{p(1-p)} / \sqrt{1 + d^2p(1-p)}$. When base_rate = .50, this reduces to the common equal-group conversion $r = d / \sqrt{d^2 + 4}$.

Usage

d_to_point_biserial(d, base_rate = 0.5)

Arguments

Value

Numeric vector of point-biserial correlations.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Erlbaum.

Rice, M. E., & Harris, G. T. (2005). Comparing effect sizes in follow-up studies: ROC area, Cohen's d, and r. Law and Human Behavior, 29(5), 615-620.

Salgado, J. F. (2018). Transforming the area under the normal curve (AUC) into Cohen's d, Pearson's r_pb, odds-ratio, and natural log odds-ratio: Two conversion tables. The European Journal of Psychology Applied to Legal Context, 10(1), 35-47.

Examples

# Minimal example: equal base-rate conversion equals d_to_cor().
d_to_point_biserial(.50, base_rate = .50)
d_to_cor(.50)

# Unequal base rates reduce the attainable point-biserial correlation.
d_to_point_biserial(.50, base_rate = c(.50, .20, .10))

Description

Corrects an observed correlation for unreliability in either or both variables.

Usage

disattenuate_correlation(r_observed, reliability_x = 1, reliability_y = 1)

Arguments

Value

Disattenuated correlation. Capped at +/- 1 with a warning when the algebraic value exceeds 1 in magnitude (typically a sign of unreliable reliability inputs).

References

Spearman, C. (1904). The proof and measurement of association between two things. American Journal of Psychology, 15, 72-101.

Examples

disattenuate_correlation(0.30, reliability_x = 0.80, reliability_y = 0.70)

Description

Implements Budescu's (1993) dominance analysis to decompose the coefficient of determination of a multiple regression into contributions attributable to each predictor. Three dominance summaries are returned:

Usage

dominance_analysis(predictor_cor, predictor_criterion_cor)

Arguments

Details

• Complete dominance: predictor i completely dominates j if ⁠R^2(S \cup {i}) > R^2(S \cup {j})⁠ for every subset S not containing i or j. Reported as a pairwise dominance matrix.

• Conditional dominance: average increment of predictor i to R^2 across subsets of size k, for ⁠k = 0, ..., p-1⁠.

• General dominance: the average of conditional dominance values; equivalent to the Shapley value of R^2.

Value

A list with components:

r_squared_full

The full-model R^2.

general_dominance

Vector of length p whose entries sum to r_squared_full.

conditional_dominance

⁠p x p⁠ matrix; row i gives the average contribution of predictor i at subset sizes ⁠0, 1, ..., p-1⁠.

complete_dominance

⁠p x p⁠ logical matrix where entry ⁠[i,j]⁠ is TRUE if i completely dominates j, FALSE if j completely dominates i, NA otherwise.

References

Azen, R., & Budescu, D. V. (2003). The dominance analysis approach for comparing predictors in multiple regression. Psychological Methods, 8, 129-148.

Budescu, D. V. (1993). Dominance analysis: A new approach to the problem of relative importance of predictors in multiple regression. Psychological Bulletin, 114, 542-551.

Examples

Rxx <- matrix(c(1, .30, .20,
                .30, 1, .25,
                .20, .25, 1), 3, 3)
rxy <- c(.40, .30, .25)
dominance_analysis(Rxx, rxy)

Description

Computes retained headcount after hires and losses: N_t = initial + cumsum(hired - lost).

Usage

employee_flow(hired, lost, initial = 0)

Arguments

Value

Numeric vector of headcount by period.

References

Boudreau, J. W., & Berger, C. J. (1985). Decision-theoretic utility analysis applied to employee separations and acquisitions. Journal of Applied Psychology, 70, 581-612.

Boudreau, J. W. (1991). Utility analysis for decisions in human resource management. In M. D. Dunnette & L. M. Hough (Eds.), Handbook of industrial and organizational psychology (Vol. 2, pp. 621-745). Consulting Psychologists Press.

Examples

# Literature: Boudreau and Berger (1985); Boudreau (1991).
employee_flow(hired = c(100, 20, 20), lost = c(0, 30, 25))

Description

Computes the proportional reduction in the standard error of prediction: 1 - sqrt(1 - validity^2).

Usage

forecasting_efficiency(validity)

Arguments

Value

Numeric vector with forecasting efficiency values.

References

Holling, H. (1998). Utility analysis of personnel selection: An overview and empirical study based on objective performance measures. Methods of Psychological Research Online, 3(1), 5-24.

Examples

# Literature: Holling (1998).
forecasting_efficiency(.30)

Description

Given a stack of items and a weight matrix W whose columns are composite-specific weight vectors, computes the correlation matrix between the resulting composites under the standard Lord-Novick formula.

Usage

fuse_composite_cor(weights_matrix, item_cor)

Arguments

Value

⁠m x m⁠ correlation matrix among composites.

Examples

R <- diag(4); R[lower.tri(R)] <- R[upper.tri(R)] <- .25
W <- cbind(c(1, 1, 0, 0), c(0, 0, 1, 1))
fuse_composite_cor(W, R)

Description

Reliability of a weighted composite (Mosier, 1943; Lord & Novick, 1968)

Usage

fuse_reliability(weights, item_cor, item_reliabilities = NULL)

Arguments

Value

The reliability of the weighted composite.

References

Lord, F. M., & Novick, M. R. (1968). Statistical theories of mental test scores. Addison-Wesley.

Mosier, C. I. (1943). On the reliability of a weighted composite. Psychometrika, 8, 161-168.

Examples

R <- matrix(c(1, .3, .3, 1), 2, 2)
fuse_reliability(c(.5, .5), R, item_reliabilities = c(.80, .85))

Description

Implements the standard formula $r_{C,Y} = (w' \rho_{XY}) / \sqrt{w' R_{XX} w}$ for the correlation between a weighted composite of items and an external criterion Y, where the items have correlations R_XX and individual validities ⁠\rho_{XY}⁠ (Lord & Novick, 1968, Ch. 4).

Usage

fuse_validity(weights, item_cor, item_validities)

Arguments

Value

Scalar correlation.

Examples

R <- matrix(c(1, .3, .3, 1), 2, 2)
fuse_validity(c(.5, .5), R, item_validities = c(.30, .25))

Description

Applies tr_multivariate() separately by group. This is useful for sensitivity analyses in which base rates or correlation matrices differ across demographic groups. It does not by itself establish legal compliance or fairness.

Usage

group_tr_multivariate(
  selection_ratios,
  base_rates,
  R_list,
  group_names = NULL,
  group_proportions = NULL
)

Arguments

Value

A list with group-level Taylor-Russell summaries and optional weighted overall metrics.

References

De Corte, W., Lievens, F., & Sackett, P. R. (2007). Combining predictors to achieve optimal trade-offs between selection quality and adverse impact. Journal of Applied Psychology, 92, 1380-1393.

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.

Examples

# Literature: Thomas, Owen, and Gunst (1977); De Corte et al. (2007).
R <- matrix(c(1, .30, .40, .30, 1, .35, .40, .35, 1), 3, 3)
group_tr_multivariate(c(.50, .50), base_rates = c(.50, .40),
                      R_list = list(R, R), group_names = c("A", "B"))

Description

Computes the difference in restricted canonical validity between a baseline predictor set and an expanded predictor set.

Usage

incremental_validity(
  predictor_cor,
  predictor_criterion_cor,
  criterion_cor,
  criterion_weights,
  baseline_predictors,
  added_predictors = NULL,
  focal_predictors = NULL
)

Arguments

Value

A psu_incremental_validity object.

References

Sturman, M. C. (2001). Utility analysis for multiple selection devices and multiple outcomes. Journal of Human Resource Costing and Accounting, 6(2), 9-28.

Examples

# Literature: Sturman (2001).
# 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 (Sturman (2001)).
Rxx <- matrix(c(1, .30, .20, .30, 1, .25, .20, .25, 1), 3, 3)
Rxy <- matrix(c(.30, .20, .25, .15, .10, .35), 3, 2, byrow = TRUE)
Ryy <- matrix(c(1, .40, .40, 1), 2, 2)
incremental_validity(Rxx, Rxy, Ryy, c(.6, .4), baseline_predictors = 1:2,
                     added_predictors = 3)

# Substantive example (Sturman (2001)): compare two possible additions to the same baseline.
add_2 <- incremental_validity(Rxx, Rxy, Ryy, c(.6, .4),
                              baseline_predictors = 1, added_predictors = 2)
add_3 <- incremental_validity(Rxx, Rxy, Ryy, c(.6, .4),
                              baseline_predictors = 1, added_predictors = 3)
c(add_predictor_2 = add_2$incremental_validity,
  add_predictor_3 = add_3$incremental_validity)

Description

Computes i_a = i + f + i*f.

Usage

inflation_adjusted_rate(discount_rate, inflation_rate)

Arguments

Value

Inflation-adjusted discount rate.

References

Tziner, A., Meir, E. I., Dahan, M., & Birati, A. (1994). An investigation of the predictive validity and economic utility of the assessment center for the high- management level. Canadian Journal of Behavioural Science, 26, 228-245.

Holling, H. (1998). Utility analysis of personnel selection: An overview and empirical study based on objective performance measures. Methods of Psychological Research Online, 3(1), 5-24.

Examples

# Literature: Tziner et al. (1994); Holling (1998).
inflation_adjusted_rate(.08, .025)

Description

Returns the package's working taxonomy: criterion scale crossed with selection structure. The taxonomy is designed to keep the Taylor-Russell, Brogden-Cronbach-Gleser, Sturman, Ock-Oswald, and Thomas-Owen-Gunst formulations distinct.

Usage

model_taxonomy()

Value

A data frame with model families, decision structures, and package functions.

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.

Ock, J., & Oswald, F. L. (2018). The utility of personnel selection decisions: Comparing compensatory and multiple-hurdle selection models. Journal of Personnel Psychology, 17(4), 172-182.

Examples

# Literature: Thomas, Owen, and Gunst (1977); Ock and Oswald (2018).
model_taxonomy()

Description

Computes additive multi-attribute utility sum(weights * utilities) for one or more alternatives.

Usage

multiattribute_utility(values, weights, utility_functions = NULL)

Arguments

Value

Numeric utility score per alternative.

References

Keeney, R. L., & Raiffa, H. (1976). Decisions with multiple objectives: Preferences and value tradeoffs. Wiley.

Roth, P. L., & Bobko, P. (1997). A research agenda for multi-attribute utility analysis in human resource management. Human Resource Management Review, 7, 341-368.

Roth, P. L. (1994). Multi-attribute utility analysis using the PROMES approach. Journal of Business and Psychology, 9, 69-80.

Examples

# Literature: Keeney and Raiffa (1976); Roth (1994); Roth and Bobko (1997).
multiattribute_utility(matrix(c(80, .90, 70, .95), nrow = 2, byrow = TRUE),
                       weights = c(.7, .3))

Description

Simulates expected standardized criterion performance under conjunctive multiple-hurdle selection. Predictors are first in R; criterion is last. Candidates pass only if they exceed all marginal cutoffs.

Usage

multiple_hurdle_selection(
  selection_ratios,
  R,
  n_sim = 1e+05,
  seed = NULL,
  n_applicants = NA_real_,
  cost_per_stage = 0,
  sdy = NULL,
  applicant_n = NULL
)

Arguments

Value

A psu_comparison object.

References

Ock, J., & Oswald, F. L. (2018). The utility of personnel selection decisions: Comparing compensatory and multiple-hurdle selection models. Journal of Personnel Psychology, 17(4), 172-182.

Examples

# Literature: Sackett and Roth (1996); Ock and Oswald (2018).
# Minimal example (Sackett and Roth (1996); Ock and Oswald (2018)).
R <- matrix(c(1, .30, .40, .30, 1, .30, .40, .30, 1), 3, 3)
multiple_hurdle_selection(c(.50, .50), R, n_sim = 1000, seed = 1)

# Substantive example with two marginal hurdles and costs.
multiple_hurdle_selection(
  selection_ratios = c(.40, .50),
  R = R,
  n_sim = 5000,
  seed = 123,
  n_applicants = 500,
  cost_per_stage = c(100, 400),
  sdy = 60000
)

Description

Simulates a sequential multiple-hurdle design in which each stage can be one predictor or a composite of predictors. This matches Ock-Oswald-style designs: an inexpensive first-stage composite can screen applicants before a later, more expensive stage such as a structured interview.

Usage

multiple_hurdle_selection_staged(
  stage_predictors,
  stage_selection_ratios,
  R,
  stage_weights = NULL,
  n_sim = 1e+05,
  seed = NULL,
  n_applicants = NA_real_,
  cost_per_stage = 0,
  sdy = NULL,
  applicant_n = NULL
)

Arguments

Value

A psu_comparison object.

References

Ock, J., & Oswald, F. L. (2018). The utility of personnel selection decisions: Comparing compensatory and multiple-hurdle selection models. Journal of Personnel Psychology, 17(4), 172-182.

Examples

# Literature: Sackett and Roth (1996); Ock and Oswald (2018).
# 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 (Sackett and Roth (1996); Ock and Oswald (2018)).
R <- diag(5)
R[lower.tri(R)] <- R[upper.tri(R)] <- .20
diag(R) <- 1
multiple_hurdle_selection_staged(list(1:3, 4), c(.25, .80), R,
                                 n_sim = 1000, seed = 1)

# Substantive example (Sackett and Roth, 1996;
# Ock and Oswald, 2018).
# Use an inexpensive first-stage composite, then an interview.
R <- matrix(c(
  1.00, .41, .04, .46, .37,
  .41, 1.00, .18, .22, .35,
  .04, .18, 1.00, .66, .16,
  .46, .22, .66, 1.00, .23,
  .37, .35, .16, .23, 1.00
), 5, 5, byrow = TRUE)
multiple_hurdle_selection_staged(
  stage_predictors = list(c(1, 3, 4), 2),
  stage_selection_ratios = c(.25, .80),
  R = R,
  n_sim = 5000,
  seed = 123,
  n_applicants = 500,
  cost_per_stage = c(100, 900),
  sdy = 60000
)

Description

Computes expected standardized criterion gain among selected applicants and, optionally, converts it to utility using sdy, n_selected, tenure, and cost. The expected standardized criterion gain is validity * selected_mean_z(selection_ratio).

Usage

naylor_shine(
  validity,
  selection_ratio,
  sdy = 1,
  n_selected = 1,
  tenure = 1,
  cost = 0
)

Arguments

Value

A psu_ns object.

References

Naylor, J. C., & Shine, L. C. (1965). A table for determining the increase in mean criterion score obtained by using a selection device. Journal of Industrial Psychology, 3, 33-42.

Examples

# Literature: Naylor and Shine (1965).
# 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: standardized criterion gain only.
naylor_shine(validity = .35, selection_ratio = .20)

# Substantive example (Naylor and Shine (1965)): standardized gain translated to monetary utility.
naylor_shine(
  validity = .35,
  selection_ratio = .20,
  sdy = 50000,
  n_selected = 100,
  tenure = 3,
  cost = 75000
)

Description

Adjusts the expected standardized criterion score of accepted hires when offer recipients can decline. When the probability of accepting an offer is negatively correlated with candidate quality (top candidates have more outside options), the realized mean criterion of accepted hires is below the mean of selected (offered) candidates.

Usage

offer_rejection_adjustment(
  expected_z_offered,
  mode = c("uniform", "selective", "correlated"),
  acceptance_rate = 1,
  rho_quality_acceptance = 0,
  logit_intercept = NULL,
  logit_slope = NULL,
  n_offered = NULL
)

Arguments

Details

Three modes are supported:

• mode = "uniform": a fixed acceptance probability p independent of candidate quality. The expected criterion of accepted hires equals the expected criterion of those offered, but the realized headcount is scaled by p.

• mode = "selective": the probability of acceptance depends on candidate standardized quality z through a logit link logit(p) = a + b * z with b < 0 for adverse selection. The adjusted mean criterion is computed by integrating the standard normal weighted by the acceptance probability.

• mode = "correlated": a closed-form approximation under the assumption that quality and acceptance are jointly normal with correlation rho_quality_acceptance. The adjustment is $\bar{z}_{accepted} \approx \bar{z}_{offered} + \rho \cdot (\lambda(z_p) - \bar{z}_{offered})$ for an acceptance threshold z_p derived from the expected acceptance rate.

Value

A list with expected_z_accepted, acceptance_rate, effective_validity_loss (the difference between offered and accepted means), and optionally expected_n_accepted.

References

Hogarth, R. M., & Einhorn, H. J. (1976). Optimal strategies for personnel selection when candidates can reject job offers. Journal of Business, 49, 479-495.

Murphy, K. R. (1986). When your top choice turns you down: Effect of rejected offers on the utility of selection tests. Psychological Bulletin, 99, 133-138.

Examples

z_offered <- selected_mean_z(0.20)

# Uniform 70% acceptance rate, no quality dependence:
offer_rejection_adjustment(z_offered, mode = "uniform",
                           acceptance_rate = 0.70, n_offered = 100)

# Adverse selection: top candidates more likely to decline.
offer_rejection_adjustment(z_offered, mode = "correlated",
                           acceptance_rate = 0.70,
                           rho_quality_acceptance = -0.20,
                           n_offered = 100)

Description

Identifies non-dominated alternatives for objectives to be maximized or minimized.

Usage

pareto_frontier(objectives, maximize = TRUE)

Arguments

Value

Logical vector indicating Pareto-efficient rows.

References

De Corte, W., Lievens, F., & Sackett, P. R. (2007). Combining predictors to achieve optimal trade-offs between selection quality and adverse impact. Journal of Applied Psychology, 92, 1380-1393. De Corte, W., Sackett, P. R., & Lievens, F. (2011). Designing Pareto-optimal selection systems: Formalizing the decisions required for selection system development. Journal of Applied Psychology, 96, 907-926.

Examples

# Literature: De Corte, Lievens, and Sackett (2007); De Corte, Sackett, and Lievens (2011).
pareto_frontier(data.frame(validity = c(.30, .35, .32), diversity = c(.80, .70, .85)))

Description

Computes the mean of a standard normal criterion among employees surviving a probation rule Y >= probation_cutoff_z.

Usage

probation_adjustment(probation_cutoff_z)

Arguments

Value

Expected standardized criterion score among survivors.

References

De Corte, W. (1994). Utility analysis for the one-cohort selection-retention decision with a probationary period. Journal of Applied Psychology, 79, 402-411.

Sturman, M. C. (2001). Utility analysis for multiple selection devices and multiple outcomes. Journal of Human Resource Costing and Accounting, 6(2), 9-28.

Examples

# Literature: De Corte (1994); Sturman (2001).
probation_adjustment(-1)

Description

A compact helper for selection systems where year 1 utility follows BCG and later periods include an additional survivor-performance gain caused by a probation cutoff.

Usage

probation_utility(
  validity,
  selection_ratio,
  sdy,
  n_selected,
  tenure,
  probation_cutoff_z,
  cost = 0
)

Arguments

Value

A psu_bcg object.

References

De Corte, W. (1994). Utility analysis for the one-cohort selection-retention decision with a probationary period. Journal of Applied Psychology, 79, 402-411.

Sturman, M. C. (2001). Utility analysis for multiple selection devices and multiple outcomes. Journal of Human Resource Costing and Accounting, 6(2), 9-28.

Examples

# Literature: De Corte (1994); Sturman (2001).
probation_utility(.35, .20, 50000, 100, tenure = 3, probation_cutoff_z = -1)

Description

Computes approximate relative weights for correlated predictors in a multiple regression with one criterion.

Usage

relative_weights(predictor_cor, criterion_cor)

Arguments

Value

A data frame with raw and rescaled relative weights.

References

Johnson, J. W. (2000). A heuristic method for estimating the relative weight of predictor variables in multiple regression. Multivariate Behavioral Research, 35, 1-19.

Examples

# Literature: Johnson (2000).
relative_weights(matrix(c(1, .30, .30, 1), 2, 2), c(.40, .30))

Description

Computes Sturman-style restricted canonical validity. Predictor weights are optimized, but criterion weights are fixed by the analyst.

Usage

restricted_canonical_validity(
  predictor_cor,
  predictor_criterion_cor,
  criterion_cor,
  criterion_weights
)

Arguments

Value

A psu_incremental_validity object with restricted canonical validity and optimized standardized predictor weights.

References

Sturman, M. C. (2001). Utility analysis for multiple selection devices and multiple outcomes. Journal of Human Resource Costing and Accounting, 6(2), 9-28.

Examples

# Literature: Sturman (2001).
# 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 (Sturman (2001)).
S11 <- matrix(c(1, .30, .30, 1), 2, 2)
S12 <- matrix(c(.30, .20, .25, .15), 2, 2)
S22 <- matrix(c(1, .40, .40, 1), 2, 2)
restricted_canonical_validity(S11, S12, S22, criterion_weights = c(.6, .4))

# Substantive example (Sturman (2001)): change criterion weights and compare restricted validity.
task_weighted <- restricted_canonical_validity(S11, S12, S22, c(.8, .2))
balanced <- restricted_canonical_validity(S11, S12, S22, c(.5, .5))
c(task_weighted = task_weighted$validity, balanced = balanced$validity)

Description

Computes a mean-variance risk-adjusted utility score. With risk_aversion = 0, the score equals expected utility. Larger values penalize uncertainty more strongly.

Usage

risk_adjusted_utility(expected_utility, utility_sd, risk_aversion = 0)

Arguments

Value

Risk-adjusted utility score.

References

Bhattacharya, M., & Wright, P. M. (2005). Managing human assets in an uncertain world: Applying real options theory to HRM. International Journal of Human Resource Management, 16, 929-948.

Cronshaw, S. F., Alexander, R. A., Wiesner, W. H., & Barrick, M. R. (1987). Incorporating risk into selection utility. Organizational Behavior and Human Decision Processes, 40, 270-286.

Examples

# Literature: Cronshaw et al. (1987); Bhattacharya and Wright (2005).
risk_adjusted_utility(expected_utility = 100000, utility_sd = 25000,
                      risk_aversion = 1e-6)

Description

Computes individual criterion values from production units and unit values, then returns the standard deviation of those values.

Usage

sdy_cost_accounting(units, unit_values, na.rm = TRUE)

Arguments

Value

A list with individual criterion values and sdy.

References

Cronbach, L. J., & Gleser, G. C. (1965). Psychological tests and personnel decisions (2nd ed.). University of Illinois Press.

Holling, H. (1998). Utility analysis of personnel selection: An overview and empirical study based on objective performance measures. Methods of Psychological Research Online, 3(1), 5-24.

Cascio, W. F. (1982). Costing human resources: The financial impact of behavior in organizations. Kent.

Examples

# Literature: Cronbach and Gleser (1965); Cascio (1982); Holling (1998).
sdy_cost_accounting(matrix(c(10, 12, 8, 11), ncol = 2), unit_values = c(100, 200))

Description

Computes a monetary performance index by weighting activity ratings with activity time/frequency and importance weights.

Usage

sdy_crepid(
  activities,
  ratings,
  salary,
  time_col = "time_frequency",
  importance_col = "importance",
  activity_names = NULL,
  na.rm = TRUE
)

Arguments

Value

A list with activity weights, individual criterion values, and sdy.

References

Cascio, W. F., & Ramos, R. A. (1986). Development and application of a new method for assessing job performance in behavioral/economic terms. Journal of Applied Psychology, 71, 20-28.

Examples

# Literature: Cascio and Ramos (1986).
activities <- data.frame(time_frequency = c(.4, .6), importance = c(2, 3))
ratings <- matrix(c(3, 4, 2, 5, 4, 4), ncol = 2, byrow = TRUE)
sdy_crepid(activities, ratings, salary = 80000)

Description

Computes the observed standard deviation of job performance in monetary or productivity units. This is the direct empirical counterpart to subjective SDy estimation methods.

Usage

sdy_observed(y, na.rm = TRUE)

Arguments

Value

Observed SDy.

References

Holling, H. (1998). Utility analysis of personnel selection: An overview and empirical study based on objective performance measures. Methods of Psychological Research Online, 3(1), 5-24.

Examples

# Literature: Holling (1998).
sdy_observed(c(100, 120, 80, 150))

Description

Implements the percentile approximation SDy = (P85 - P15) / 2.

Usage

sdy_percentile(p15, p85)

Arguments

Value

Estimated standard deviation of job performance in monetary units.

References

Bobko, P., Karren, R., & Parkington, J. J. (1983). Estimation of standard deviations in utility analyses: An empirical test. Journal of Applied Psychology, 68, 170-176.

Schmidt, F. L., Hunter, J. E., McKenzie, R. C., & Muldrow, T. W. (1979). Impact of valid selection procedures on work-force productivity. Journal of Applied Psychology, 64, 609-626.

Examples

# Literature: Schmidt et al. (1979); Bobko, Karren, and Parkington (1983).
sdy_percentile(p15 = 60000, p85 = 140000)

Description

Computes a salary- or value-based SDy estimate using a multiplier such as .40 or .70.

Usage

sdy_proportional(mean_pay, multiplier = 0.4)

Arguments

Value

Estimated SDy.

References

Schmidt, F. L., Hunter, J. E., & Pearlman, K. (1982). Assessing the economic impact of personnel programs on workforce productivity. Personnel Psychology, 35, 333-347.

Hunter, J. E., & Schmidt, F. L. (1982). Fitting people to jobs: The impact of personnel selection on national productivity. In M. D. Dunnette & E. A. Fleishman (Eds.), Human performance and productivity (Vol. 1, pp. 233-284). Erlbaum.

Examples

# Literature: Schmidt, Hunter, and Pearlman (1982); Hunter and Schmidt (1982).
sdy_proportional(mean_pay = 80000, multiplier = .40)
sdy_proportional(mean_pay = 80000, multiplier = .70)

Description

A compact implementation of the Raju-Burke-Normand logic: SDy = CV * mean_pay. Use this function when the coefficient of variation is theoretically or empirically justified for the job family.

Usage

sdy_rbn(mean_pay, coefficient_variation)

Arguments

Value

Estimated SDy.

References

Raju, N. S., Burke, M. J., & Normand, J. (1990). A new approach for utility analysis. Journal of Applied Psychology, 75, 3-12.

Examples

# Literature: Raju, Burke, and Normand (1990).
sdy_rbn(mean_pay = 80000, coefficient_variation = .35)

Description

Computes SDy from a judged monetary difference between a superior and a typical performer, divided by the standardized distance assumed to separate them.

Usage

sdy_superior_equivalents(superior_value, typical_value, z_difference = 1)

Arguments

Value

Estimated SDy.

References

Eaton, N. K., Wing, H., & Mitchell, K. J. (1985). Alternate methods of estimating the dollar value of performance. Personnel Psychology, 38, 27-40.

Burke, M. J., & Frederick, J. T. (1984). Two modified procedures for estimating standard deviations in utility analyses. Journal of Applied Psychology, 69, 482-489.

Burke, M. J., & Frederick, J. T. (1986). A comparison of economic utility estimates for alternative SDy estimation procedures. Journal of Applied Psychology, 71, 334-339.

Examples

# Literature: Eaton, Wing, and Mitchell (1985); Burke and Frederick (1984, 1986).
sdy_superior_equivalents(superior_value = 140000, typical_value = 100000)

Description

Computes the mean of a standard normal predictor after top-down selection at a given selection ratio: dnorm(qnorm(1 - selection_ratio)) / selection_ratio.

Usage

selected_mean_z(selection_ratio)

Arguments

Value

Numeric vector with expected standardized predictor scores.

References

Naylor, J. C., & Shine, L. C. (1965). A table for determining the increase in mean criterion score obtained by using a selection device. Journal of Industrial Psychology, 3, 33-42.

Examples

# Literature: Naylor and Shine (1965).
selected_mean_z(c(.10, .20, .50))

Description

Computes a 2x2 classification table from observed selected/success outcomes.

Usage

selection_table(selected, success)

Arguments

Value

A list with table and classification metrics.

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.

Taylor, H. C., & Russell, J. T. (1939). The relationship of validity coefficients to the practical effectiveness of tests in selection. Journal of Applied Psychology, 23, 565-578.

Examples

# Literature: Taylor and Russell (1939); Thomas, Owen, and Gunst (1977).
selection_table(c(1, 1, 0, 0), c(1, 0, 1, 0))

Description

Computes the Ock-Oswald/BCG-style utility expression expected_criterion_z * sdy * n_selected - n_applicants * cost_per_applicant - fixed_cost from an expected standardized criterion score among selected applicants.

Usage

selection_utility_from_z(
  expected_criterion_z,
  sdy,
  n_selected,
  n_applicants = n_selected,
  cost_per_applicant = 0,
  fixed_cost = 0,
  applicant_n = NULL
)

Arguments

Value

A psu_comparison object.

References

Cronbach, L. J., & Gleser, G. C. (1965). Psychological tests and personnel decisions (2nd ed.). University of Illinois Press.

Naylor, J. C., & Shine, L. C. (1965). A table for determining the increase in mean criterion score obtained by using a selection device. Journal of Industrial Psychology, 3, 33-42.

Brogden, H. E. (1946). On the interpretation of the correlation coefficient as a measure of predictive efficiency. Journal of Educational Psychology, 37, 65-76.

Examples

# Literature: Naylor and Shine (1965); Brogden (1946); Cronbach and Gleser (1965).
# Minimal example: expected performance converted to monetary utility.
selection_utility_from_z(1.25, sdy = 50000, n_selected = 20,
                         n_applicants = 100, cost_per_applicant = 200)

# Substantive example: compare two systems from expected criterion gains.
compensatory <- selection_utility_from_z(1.25, 50000, n_selected = 20,
                                         n_applicants = 100,
                                         cost_per_applicant = 1000)
hurdle <- selection_utility_from_z(.55, 50000, n_selected = 20,
                                   n_applicants = 100,
                                   cost_per_applicant = 300)
compensatory$net_utility - hurdle$net_utility

Description

Computes BCG net utility for all combinations of selected parameter values.

Usage

sensitivity_grid(validity, selection_ratio, sdy, n_selected, tenure, cost = 0)

Arguments

Value

A data frame with one row per scenario.

References

Cronshaw, S. F., Alexander, R. A., Wiesner, W. H., & Barrick, M. R. (1987). Incorporating risk into selection utility. Organizational Behavior and Human Decision Processes, 40, 270-286.

Ock, J., & Oswald, F. L. (2018). The utility of personnel selection decisions: Comparing compensatory and multiple-hurdle selection models. Journal of Personnel Psychology, 17(4), 172-182.

Boudreau, J. W. (1991). Utility analysis for decisions in human resource management. In M. D. Dunnette & L. M. Hough (Eds.), Handbook of industrial and organizational psychology (Vol. 2, pp. 621-745). Consulting Psychologists Press.

Examples

# Literature: Cronshaw et al. (1987); Boudreau (1991); Ock and Oswald (2018).
# 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 (Cronshaw et al. (1987); Boudreau (1991); Ock and Oswald (2018)).
sensitivity_grid(validity = c(.20, .30), selection_ratio = c(.10, .20),
                 sdy = c(40000, 60000), n_selected = 100, tenure = 3)

# Substantive example (Cronshaw et al., 1987; Boudreau, 1991;
# Ock and Oswald, 2018). Find the best sensitivity scenario.
grid <- sensitivity_grid(validity = seq(.20, .40, .10),
                         selection_ratio = c(.10, .20, .40),
                         sdy = c(30000, 60000),
                         n_selected = 100, tenure = 3, cost = 75000)
grid[which.max(grid$net_utility), ]

Description

Computes utility from an intervention effect size rather than from a selection validity coefficient. This is appropriate for training or intervention designs where the key input is a standardized mean difference.

Usage

shp_utility(effect_size_d, sdy, n_treated, tenure, cost = 0)

Arguments

Value

A psu_shp object.

References

Schmidt, F. L., Hunter, J. E., & Pearlman, K. (1982). Assessing the economic impact of personnel programs on workforce productivity. Personnel Psychology, 35, 333-347.

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Erlbaum.

Examples

# Literature: Schmidt, Hunter, and Pearlman (1982); Cohen (1988).
# 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 (Schmidt, Hunter, and Pearlman (1982); Cohen (1988)).
shp_utility(effect_size_d = .30, sdy = 50000, n_treated = 100,
            tenure = 2, cost = 40000)

# Substantive example (Schmidt, Hunter, and Pearlman, 1982;
# Cohen, 1988). Compare two training designs.
short_training <- shp_utility(.20, 50000, n_treated = 120, tenure = 1, cost = 30000)
intensive_training <- shp_utility(.35, 50000, n_treated = 120, tenure = 1,
                                  cost = 95000)
c(short = short_training$net_utility, intensive = intensive_training$net_utility)

Description

Composes the six adjustments of Sturman's (2001) comprehensive model (§15 of the package's accompanying review) into a single call. The returned object includes both the integrated comprehensive estimate and a stepwise cascade that documents the contribution of each adjustment.

Usage

sturman_comprehensive(
  validity,
  baseline_validity = 0,
  selection_ratio,
  sdy,
  n_year_one,
  tenure = 5,
  fixed_cost = 0,
  hires_per_period = NULL,
  losses_per_period = NULL,
  tax_rate = 0,
  discount_rate = 0,
  variable_value = 0,
  maintenance_cost_per_period = NULL,
  predictor_cor = NULL,
  predictor_criterion_cor = NULL,
  criterion_cor = NULL,
  criterion_weights = NULL,
  probation_cutoff_z = NULL,
  acceptance_rate = 1,
  quality_acceptance_correlation = 0
)

Arguments

Details

The six adjustments combined here are: (1) baseline correction (Sturman, 2000, 2001), (2) restricted canonical validity for a multidimensional criterion, (3) multi-period employee flows, (4) Boudreau-style economic adjustments (taxes, variable costs, discount rate), (5) De Corte (1994) probation-period truncation, and optionally (6) Murphy's (1986) offer-rejection adjustment. See bcg_utility(), boudreau_utility(), restricted_canonical_validity(), probation_adjustment(), employee_flow(), and offer_rejection_adjustment() for the underlying components.

Value

Object of class c("psu_sturman", "psu_utility") with components: net_utility (final comprehensive estimate), cascade (a data frame documenting each step), effective_validity, effective_baseline_validity, and the relevant intermediate objects.

References

De Corte, W. (1994). Utility analysis for the one-cohort selection-retention decision with a probationary period. Journal of Applied Psychology, 79, 402-411.

Murphy, K. R. (1986). When your top choice turns you down: Effect of rejected offers on the utility of selection tests. Psychological Bulletin, 99, 133-138.

Sturman, M. C. (2000). Implications of utility analysis adjustments for estimates of human resource intervention value. Journal of Management, 26, 281-299.

Sturman, M. C. (2001). Utility analysis for multiple selection devices and multiple outcomes. Journal of Human Resource Costing and Accounting, 6(2), 9-28.

Examples

Rxx <- matrix(c(1, .30, .30, 1), 2, 2)
Rxy <- matrix(c(.30, .10, .15, .25), 2, 2, byrow = TRUE)
Ryy <- matrix(c(1, .40, .40, 1), 2, 2)

sturman_comprehensive(
  validity = .35, baseline_validity = .20, selection_ratio = .20,
  sdy = 50000, n_year_one = 100, tenure = 5, fixed_cost = 75000,
  tax_rate = .25, discount_rate = .08,
  predictor_cor = Rxx, predictor_criterion_cor = Rxy,
  criterion_cor = Ryy, criterion_weights = c(.7, .3),
  probation_cutoff_z = -1,
  acceptance_rate = 0.70, quality_acceptance_correlation = -0.20
)

Description

Converts a Taylor-Russell success ratio into finite-sample probabilities. This follows the finite-sampling logic discussed by Thomas, Owen, and Gunst: once a conditional probability of success is known, the number of successful selected applicants in a finite cohort can be modeled with a binomial distribution.

Usage

tr_binomial_success_probability(n_selected, ppv, at_least = NULL)

Arguments

Value

A data frame with the full binomial distribution and, if requested, the cumulative upper-tail probability.

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.

Examples

# Literature: Thomas, Owen, and Gunst (1977).
tr_binomial_success_probability(n_selected = 20, ppv = .91, at_least = 18)

Description

Computes the Taylor-Russell classification table for one normally distributed predictor and one dichotomized criterion.

Usage

tr_classic(base_rate, selection_ratio, validity, digits = 3)

Arguments

Value

A list with thresholds, TP, FP, FN, TN, PPV, sensitivity, specificity, and incremental success over the base rate.

References

Taylor, H. C., & Russell, J. T. (1939). The relationship of validity coefficients to the practical effectiveness of tests in selection. Journal of Applied Psychology, 23, 565-578.

Cascio, W. F. (1980). Responding to the demand for accountability: A critical analysis of three utility models. Organizational Behavior and Human Performance, 25, 32-45.

Examples

# Literature: Taylor and Russell (1939); Cascio (1980).
# 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 (Taylor and Russell (1939); Cascio (1980)).
tr_classic(base_rate = .50, selection_ratio = .20, validity = .35)

# Substantive example (Taylor and Russell, 1939; Cascio, 1980).
# Examine how selectivity changes the success ratio.
low_sr <- tr_classic(base_rate = .50, selection_ratio = .10, validity = .35)
high_sr <- tr_classic(base_rate = .50, selection_ratio = .50, validity = .35)
c(low_selection_ratio = low_sr$ppv, high_selection_ratio = high_sr$ppv)

Description

Implements the Thomas-Owen-Gunst multivariate extension of the Taylor-Russell model. Candidates are selected if and only if they exceed all predictor cutoffs. The correlation matrix must include the predictors first and the criterion last.

Usage

tr_multivariate(selection_ratios, base_rate, R, digits = 3)

Arguments

Value

A psu_tr object with TP, FP, FN, TN, joint selection ratio, PPV, sensitivity, specificity, and cutoffs.

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.

Taylor, H. C., & Russell, J. T. (1939). The relationship of validity coefficients to the practical effectiveness of tests in selection. Journal of Applied Psychology, 23, 565-578.

Genz, A., & Bretz, F. (2009). Computation of multivariate normal and t probabilities. Springer.

Examples

# Literature: Taylor and Russell (1939); Thomas, Owen, and Gunst
# (1977); Genz and Bretz (2009).
# 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 (Taylor and Russell, 1939;
# Thomas, Owen, and Gunst, 1977; Genz and Bretz, 2009).
R <- matrix(c(1, .30, .40,
              .30, 1, .35,
              .40, .35, 1), nrow = 3, byrow = TRUE)
tr_multivariate(selection_ratios = c(.50, .50), base_rate = .50, R = R)

# Substantive example (Taylor and Russell, 1939;
# Thomas, Owen, and Gunst, 1977; Genz and Bretz, 2009).
# Compare two validity patterns under the same marginal cutoffs.
R_stronger <- matrix(c(1, .30, .60,
                       .30, 1, .55,
                       .60, .55, 1), nrow = 3, byrow = TRUE)
weak <- tr_multivariate(c(.50, .50), base_rate = .50, R = R)
strong <- tr_multivariate(c(.50, .50), base_rate = .50, R = R_stronger)
c(weak_ppv = weak$ppv, strong_ppv = strong$ppv)

Description

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

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)

# 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)

Description

Solves for one missing Taylor-Russell parameter among base rate, selection ratio, validity, and PPV. Exactly one of the four arguments must be NULL. The default validity interval is non-negative to match the classical Taylor-Russell table convention and the defensive behavior of the TaylorRussell::TR() implementation.

Usage

tr_solve(
  base_rate = NULL,
  selection_ratio = NULL,
  validity = NULL,
  ppv = NULL,
  interval = NULL,
  tol = 1e-08,
  allow_negative_validity = FALSE
)

Arguments

Value

A psu_tr object containing the solved parameter and the resulting Taylor-Russell table.

References

Taylor, H. C., & Russell, J. T. (1939). The relationship of validity coefficients to the practical effectiveness of tests in selection. Journal of Applied Psychology, 23, 565-578.

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

Examples

# Literature: Taylor and Russell (1939); 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: solve validity from desired PPV.
tr_solve(base_rate = .50, selection_ratio = .20, validity = NULL, ppv = .70)

# Substantive example (Taylor and Russell, 1939; Waller, 2024).
# Solve the selection ratio needed for a desired PPV.
tr_solve(base_rate = .50, selection_ratio = NULL, validity = .35, ppv = .70)

Description

Convenience wrapper around pareto_frontier() for selection-system alternatives evaluated on utility, fairness, and optionally validity.

Usage

utility_fairness_frontier(utility, fairness, validity = NULL)

Arguments

Value

A data frame with frontier membership.

References

De Corte, W., Lievens, F., & Sackett, P. R. (2007). Combining predictors to achieve optimal trade-offs between selection quality and adverse impact. Journal of Applied Psychology, 92, 1380-1393.

Tippins, N. T., Oswald, F. L., & McPhail, S. M. (2021). Scientific, legal, and ethical concerns about AI-based personnel selection tools: A call to action. Personnel Assessment and Decisions, 7(2), Article 1.

Examples

# Literature: De Corte, Lievens, and Sackett (2007); Tippins, Oswald, and McPhail (2021).
utility_fairness_frontier(utility = c(100, 120, 90), fairness = c(.80, .70, .95))

Description

Samples validity and SDy and propagates them through the BCG model. This is a simple decision-support approximation, not a full Bayesian model.

Usage

utility_monte_carlo(
  n_sim = 10000,
  validity_mean,
  validity_se,
  sdy_mean,
  sdy_sd,
  selection_ratio,
  n_selected,
  tenure,
  cost = 0,
  baseline_validity = 0,
  seed = NULL
)

Arguments

Value

A psu_monte_carlo object with draws and quantiles.

References

Alexander, R. A., & Barrick, M. R. (1987). Estimating the standard error of projected dollar gains in utility analysis. Journal of Applied Psychology, 72, 475-479.

Rich, J. R., & Boudreau, J. W. (1987). The effects of variability and risk on selection utility analysis. Personnel Psychology, 40, 55-84.

Ock, J., & Oswald, F. L. (2018). The utility of personnel selection decisions: Comparing compensatory and multiple-hurdle selection models. Journal of Personnel Psychology, 17(4), 172-182.

Examples

# Literature: Alexander and Barrick (1987); Rich and Boudreau (1987); Ock and Oswald (2018).
# 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 (Alexander and Barrick (1987); Rich and Boudreau (1987); Ock and Oswald (2018)).
utility_monte_carlo(n_sim = 1000, validity_mean = .35, validity_se = .05,
                    sdy_mean = 50000, sdy_sd = 10000, selection_ratio = .20,
                    n_selected = 100, tenure = 3, cost = 75000, seed = 1)

# Substantive example (Alexander and Barrick, 1987;
# Rich and Boudreau, 1987; Ock and Oswald, 2018).
# Quantify the probability that net utility is positive.
mc <- utility_monte_carlo(n_sim = 2000, validity_mean = .30, validity_se = .06,
                          sdy_mean = 50000, sdy_sd = 15000,
                          selection_ratio = .20, n_selected = 100,
                          tenure = 3, cost = 75000,
                          baseline_validity = .15, seed = 123)
mc$probability_positive

Description

Fits a simple linear model and returns empirical inputs and normality checks relevant to linear utility analysis.

Usage

utility_regression_diagnostics(x, y)

Arguments

Value

A list with sample size, validity, SDy, regression coefficients, residual summaries, optional Shapiro-Wilk tests, and the fitted model.

References

Holling, H. (1998). Utility analysis of personnel selection: An overview and empirical study based on objective performance measures. Methods of Psychological Research Online, 3(1), 5-24.

Examples

# Literature: Holling (1998).
utility_regression_diagnostics(1:10, c(2, 3, 3, 5, 4, 6, 7, 8, 8, 10))