Offer-rejection adjustment for selection utility (Murphy, 1986) — offer_rejection

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

expected_z_offered: Expected standardized score of offered candidates (e.g., selected_mean_z(selection_ratio)).
mode: One of "uniform", "selective", or "correlated".
acceptance_rate: Expected proportion of offers accepted (used in all three modes for the headcount-scaling output).
rho_quality_acceptance: Correlation between standardized candidate quality and acceptance propensity (used for mode = "correlated"). Negative values reflect adverse selection (top candidates more likely to decline).
logit_intercept, logit_slope: Logit link coefficients for mode = "selective". The slope is typically negative for adverse selection.
n_offered: Optional integer; if supplied, the function also returns the expected number of accepted hires.

Value

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

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.

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)
#> <psu_offer_rejection>
#>   expected_z_offered: 1.39981
#>   expected_z_accepted: 1.39981
#>   acceptance_rate: 0.7
#>   effective_validity_loss: 0
#>   expected_n_accepted: 70

# 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)
#> <psu_offer_rejection>
#>   expected_z_offered: 1.39981
#>   expected_z_accepted: 1.30047
#>   acceptance_rate: 0.7
#>   effective_validity_loss: 0.0993407
#>   expected_n_accepted: 70