# Continuous and monetary utility: Naylor-Shine, BCG, Boudreau, and Sturman

## From success ratios to continuous performance

Taylor-Russell utility dichotomises the criterion. Naylor and Shine (1965) keep the criterion continuous and compute the expected criterion gain in standard-deviation units. This is usually a better match when performance is itself a continuous construct, which is the modal case in industrial-organisational research on job-performance taxonomies (Borman & Motowidlo, 1993; Campbell, 1990; Rotundo & Sackett, 2002).

    library(personnelSelectionUtility)

    naylor_shine(validity = .35, selection_ratio = .20)
    #> <psu_ns>
    #>   validity: 0.35
    #>   selection_ratio: 0.2
    #>   selected_mean_z: 1.39981
    #>   expected_criterion_z: 0.489933
    #>   sdy: 1
    #>   n_selected: 1
    #>   tenure: 1
    #>   cost: 0
    #>   gross_utility: 0.489933
    #>   net_utility: 0.489933

The expected gain has two components. First, `selected_mean_z()` returns the expected standardised predictor score among selected applicants, which under the normal model is the inverse Mills ratio. Second, this is multiplied by the validity coefficient to obtain the expected standardised criterion gain:

\[ \bar{Z}_{x_s} \;=\; \frac{\varphi(z_c)}{1 - \Phi(z_c)} \;=\; \frac{\lambda(SR)}{SR}, \qquad \bar{Z}_{y_s} \;=\; r_{xy} \cdot \bar{Z}_{x_s}, \]

where \(z_c = \Phi^{-1}(1 - SR)\) is the cutoff in standardised units and \(\lambda(\cdot)\) denotes the standard normal density.

    selected_mean_z(.20)
    #> [1] 1.39981
    .35 * selected_mean_z(.20)
    #> [1] 0.4899334

Naylor-Shine is the natural starting point when the analysis stops at expected criterion gain in standard-deviation units. The transition to monetary utility is governed by Brogden’s (1949) demonstration that, under linearity and normality, the expected criterion gain in dollars equals \(r_{xy} \cdot SD_y \cdot \bar{Z}_{x_s}\).

## Brogden-Cronbach-Gleser monetary utility

Brogden (1946, 1949) and Cronbach and Gleser (1965) convert standardised criterion gain into monetary utility by multiplying by \(SD_y\), the standard deviation of job performance value. The expected incremental utility for \(N_s\) applicants selected with validity \(r_{xy}\) at selection ratio \(SR\) over a tenure horizon \(T\), net of total cost \(C\), is

\[ \Delta U \;=\; N_s \cdot T \cdot r_{xy} \cdot SD_y \cdot \bar{Z}_{x_s} \;-\; C. \]

When the comparator is an operating procedure with validity \(r_{\text{baseline}}\) rather than random selection, the focal validity is replaced by the difference \(r_{xy} - r_{\text{baseline}}\) (Sturman, 2000, 2001). Schmidt, Hunter, McKenzie, and Muldrow (1979) popularised the organisational implications of this formulation in their analysis of the Programmer Aptitude Test (PAT). Their estimates of multi-million-dollar utility gains generated the modern utility-analysis literature, and a stripped-down version of their calculation is the canonical pedagogical example.

    bcg_utility(
      validity = .35,
      selection_ratio = .20,
      sdy = 50000,
      n_selected = 100,
      tenure = 3,
      cost = 75000
    )
    #> <psu_bcg>
    #>   validity: 0.35
    #>   selection_ratio: 0.2
    #>   baseline_validity: 0
    #>   baseline_selection_ratio: 0.2
    #>   selected_mean_z: 1.39981
    #>   baseline_selected_mean_z: 1.39981
    #>   focal_expected_criterion_z: 0.489933
    #>   baseline_expected_criterion_z: 0
    #>   incremental_criterion_z: 0.489933
    #>   sdy: 50000
    #>   n_selected: 100
    #>   tenure: 3
    #>   cost: 75000
    #>   gross_utility: 7349000
    #>   net_utility: 7274000

The Schmidt et al. (1979) PAT calculation uses different inputs (notably \(SD_y\) in 1979 dollars, a much larger cohort, and a longer tenure horizon) and is reproduced in detail in the Reproducing canonical examples vignette. The simplified call above illustrates the structure: the gross utility is the product of \(N\), \(T\), \(r_{xy}\), \(SD_y\), and the inverse-Mills selection-intensity term \(\bar{Z}_{x_s}\), minus total cost.

## The baseline problem

The classical Brogden-Cronbach-Gleser expression compares the focal selection procedure against random selection. Sturman (2000, 2001) argued forcefully that the realistic baseline is rarely random choice. Almost every organisation already operates with some procedure: reference checks (used by \(97\%\) of organisations according to Gatewood and Feild, 2001), unstructured interviews (\(81\%\)), or biodata. The package implements this critique through the `baseline_validity` and `baseline_selection_ratio` arguments. Setting `baseline_validity` to the validity of the operating procedure shifts the comparison from “test versus random” to “test versus current system”, which is the operationally relevant difference.

    random_baseline <- bcg_utility(
      validity = .35,
      selection_ratio = .20,
      sdy = 50000,
      n_selected = 100,
      tenure = 3,
      cost = 75000
    )

    operating_baseline <- bcg_utility(
      validity = .35,
      baseline_validity = .20,
      selection_ratio = .20,
      sdy = 50000,
      n_selected = 100,
      tenure = 3,
      cost = 75000
    )

    c(random   = random_baseline$net_utility,
      operating = operating_baseline$net_utility)
    #>    random operating
    #>   7274000   3074572

The shift from random to operating baseline reduces the estimated utility by approximately \(40\%\) in this example, consistent with the average reduction of \(59\%\) that Sturman (2000) documented across the published utility-analysis literature. The decision to adopt the new procedure may be unchanged, since the incremental utility remains positive, but the expected return communicated to organisational decision-makers is materially different.

## Estimating \(SD_y\): the four families

Because \(SD_y\) enters the Brogden-Cronbach-Gleser expression linearly, any error in its magnitude propagates one-for-one into the final \(\Delta U\) figure: doubling \(SD_y\) exactly doubles the reported utility, all else equal. As Holling (1998) documents empirically, plausible alternative methods can yield \(SD_y\) estimates that differ by a factor of two or more (with the corresponding doubling of \(\Delta U\)), so the choice of estimation method is not a technicality but a substantive decision that should be justified and triangulated.

Holling (1998) classifies the methods for estimating \(SD_y\) into four families. The package implements representative members of each.

### Cost accounting

The cost-accounting approach assigns a monetary value to each unit of measurable production, originating in Roche (1961, cited in Cronbach & Gleser, 1965). It is conceptually the most defensible method when production is genuinely measurable, but is operationally restricted to jobs with quantifiable output.

    # 5 employees, 2 output types (e.g., transactions completed and customer-service
    # hours); unit_values gives the monetary value of one unit of each type.
    sdy_cost_accounting(
      units = matrix(c(2400,  18,
                       3100,  22,
                       1800,  14,
                       2700,  25,
                       2200,  20),
                     ncol = 2, byrow = TRUE),
      unit_values = c(25, 80)
    )
    #> $y
    #> [1] 61440 79260 46120 69500 56600
    #>
    #> $sdy
    #> [1] 12590.67

### Global percentile

Schmidt et al. (1979) proposed the percentile-judgement method: experts estimate the dollar value of performance at the \(15\)th, \(50\)th, and \(85\)th percentiles, and \(SD_y\) is approximated as \((P_{85} - P_{15}) / 2\) under the normal-model assumption. This method has been the most widely applied historically but has been criticised for between-judge dispersion, anchoring effects, and cognitive difficulty (Bobko, Karren, & Parkington, 1983).

    sdy_percentile(p15 = 60000, p85 = 140000)
    #> [1] 40000

### Proportional rules

Schmidt, Hunter, and Pearlman (1982) proposed proportional rules tying \(SD_y\) to mean salary: approximately \(40\%\) of mean pay for jobs of low to medium complexity, and approximately \(70\%\) for jobs of high complexity or when output value is the relevant criterion. Raju, Burke, and Normand (1990) reformulated the model in terms of the coefficient of variation of performance ratings, which is implemented in `sdy_rbn()`.

    sdy_proportional(mean_pay = 80000, multiplier = .40)
    #> [1] 32000
    sdy_proportional(mean_pay = 80000, multiplier = .70)
    #> [1] 56000

    sdy_rbn(mean_pay = 80000, coefficient_variation = .20)
    #> [1] 16000

### Individualised methods

CREPID (Cascio & Ramos, 1986) decomposes the job into weighted activities and estimates the dollar value of each, summing across activities to obtain \(SD_y\). Variants appear in Janz and Dunnette (1977), Edwards, Frederick, and Burke (1988), and the superior-equivalents technique of Eaton, Wing, and Mitchell (1985) and Burke and Frederick (1984, 1986).

    # CREPID weights activities by time/frequency and importance, distributes the
    # average salary across activities, and computes individual-level monetary value.
    activities <- data.frame(
      activity        = c("Strategic planning", "Team supervision", "Reporting"),
      time_frequency  = c(.40, .35, .25),
      importance      = c(3, 2, 2)
    )
    ratings <- matrix(c(
      4, 3, 3,
      5, 4, 4,
      3, 4, 3,
      4, 5, 4,
      5, 5, 5
    ), nrow = 5, byrow = TRUE)
    sdy_crepid(activities, ratings, salary = 80000)
    #> $activity_weights
    #>     activity time_frequency importance raw_weight final_weight dollar_value
    #> 1 activity_1           0.40          3        1.2    0.5000000     40000.00
    #> 2 activity_2           0.35          2        0.7    0.2916667     23333.33
    #> 3 activity_3           0.25          2        0.5    0.2083333     16666.67
    #>
    #> $y
    #> [1] 280000.0 360000.0 263333.3 343333.3 400000.0
    #>
    #> $sdy
    #> [1] 56833.09

    # Superior-equivalents: SDy = (superior - typical) / z_difference, with z_difference
    # the standardised distance the analyst assumes separates the two anchors.
    sdy_superior_equivalents(superior_value = 140000, typical_value = 100000)
    #> [1] 40000

### Triangulation

The empirical comparisons of Bobko, Karren, and Parkington (1983), Becker and Huselid (1992), and Hakstian, Wooley, Woolsey, and Kryger (1991) indicate that CREPID and the \(40\%\) rule converge to broadly similar values, while the global percentile method tends to produce values approximately \(1.8\) times larger. Triangulation across at least two methods is the practice supported by these comparisons, and the analyst should report a sensitivity range rather than a single value. Direct estimation from observed monetary criterion data, when available, is the most defensible anchor.

    sdy_observed(c(90000, 110000, 85000, 150000, 125000))
    #> [1] 26598.87

## Schmidt-Hunter-Pearlman intervention utility

A common conceptual error is to apply Brogden-Cronbach-Gleser to evaluation problems for which it is not designed. Schmidt, Hunter, and Pearlman (1982) developed the parallel utility model for interventions such as training programmes, where the appropriate effect size is Cohen’s \(d\) between treated and control groups rather than a validity coefficient:

\[ \Delta U_{SHP} \;=\; N \cdot T \cdot d \cdot SD_y \;-\; C, \qquad d \;=\; \frac{\bar{Y}_{\text{treated}} - \bar{Y}_{\text{control}}}{SD_{\text{pooled}}}. \]

The function `shp_utility()` implements this formulation.

    shp_utility(
      effect_size_d = .50,
      sdy           = 50000,
      n_treated     = 100,
      tenure        = 3,
      cost          = 25000
    )
    #> <psu_shp>
    #>   effect_size_d: 0.5
    #>   approximate_r: 0.242536
    #>   sdy: 50000
    #>   n_treated: 100
    #>   tenure: 3
    #>   cost: 25000
    #>   gross_utility: 7500000
    #>   net_utility: 7475000

The conversion between \(r\) and \(d\) under the equal-variance binormal assumption is provided by `cor_to_d()` and `d_to_cor()`. Mathieu and Leonard (1987) and Cascio (1989) document representative applications to training programmes, while Burke and Day (1986) and Morrow, Jarrett, and Rupinski (1997) provide cumulative meta-analytic and longitudinal estimates respectively. The substantive point is that selection utility (\(r\)-based) and intervention utility (\(d\)-based) require different inputs and different study designs, even when the outcome metric (\(\Delta U\) in dollars) is identical.

## Boudreau-style extensions

Boudreau (1983, 1991) extended the basic monetary utility model along several economically relevant dimensions: discounting future benefits to net present value, separating fixed from variable costs, applying tax rates, incorporating contribution margins, and modelling employee flows over multiple periods. Per period, the discounted incremental utility is

\[ \Delta U_t \;=\; \frac{N_t \cdot \Delta\bar{Z}_y \cdot SD_y \cdot (1 + V)(1 - TAX)}{(1 + i)^t} \;-\; C_t, \]

with \(N_t\) the active headcount in period \(t\), \(V\) the variable-value (or contribution-margin) multiplier, \(TAX\) the tax rate, \(i\) the discount rate, and \(C_t\) the period cost. Total utility is the sum of \(\Delta U_t\) over the horizon. The function `boudreau_utility()` accepts these inputs through the arguments `n_by_period`, `cost_by_period`, `contribution_margin`, `tax_rate`, and `discount_rate`.

    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)
    )
    #> <psu_boudreau>
    #>   delta_z_y: 0.209971
    #>   sdy: 50000
    #>   variable_value: 0
    #>   contribution_margin: 0.3
    #>   effective_margin: 0.3
    #>   tax_rate: 0.25
    #>   discount_rate: 0.08
    #>   net_present_value: 579234

If the expected incremental standardised gain comes from an external model, it can be passed directly through `delta_z_y`, bypassing the internal computation from `validity`, `baseline_validity`, and `selection_ratio`.

    boudreau_utility(
      delta_z_y = .25,
      sdy = 50000,
      n_by_period = c(100, 90, 80),
      discount_rate = .08,
      cost_by_period = c(75000, 10000, 10000)
    )
    #> <psu_boudreau>
    #>   delta_z_y: 0.25
    #>   sdy: 50000
    #>   variable_value: 0
    #>   effective_margin: 1
    #>   tax_rate: 0
    #>   discount_rate: 0.08
    #>   net_present_value: 2829790

When inflation is non-trivial, the discount rate should be adjusted accordingly. Tziner, Meir, Dahan, and Birati (1994) show that the inflation-adjusted rate is \(i_a = i + f + i \cdot f\), where \(i\) is the nominal discount rate and \(f\) is the inflation rate. The package exposes this transformation as `inflation_adjusted_rate()`.

    inflation_adjusted_rate(discount_rate = .08, inflation_rate = .025)
    #> [1] 0.107

The combined `boudreau_utility()` framework is closer to the standard capital-budgeting practice of corporate finance, but it inherits the well-documented difficulties that Hunter, Schmidt, and Coggin (1988) identified from inside the Schmidt-Hunter programme: discount-rate selection, period definition, and the boundary between fixed and variable costs are all decisions that materially affect the estimate and that admit no purely statistical resolution.

## Composite formation

When multiple predictors are combined into a single composite score, the validity, reliability, and intercorrelation of the composite follow from the well-known formulae in Lord and Novick (1968). The package implements these as the `fuse_*` family.

    weights            <- c(.5, .3, .2)
    item_validities    <- c(.40, .30, .25)
    item_reliabilities <- c(.85, .80, .75)
    item_cor <- matrix(c(
      1.00, .30, .20,
      .30, 1.00, .25,
      .20, .25, 1.00
    ), 3, 3, byrow = TRUE)

    fuse_validity(weights, item_cor, item_validities)
    #> [1] 0.4626814
    fuse_reliability(weights, item_cor, item_reliabilities)
    #> [1] 0.8787037

    # fuse_composite_cor() returns the correlation matrix between several composites
    # whose weights are stacked column-wise. The example below contrasts a unit-weighted
    # composite with a validity-weighted composite of the same items.
    W <- cbind(unit = c(1, 1, 1), validity_weighted = item_validities)
    fuse_composite_cor(weights_matrix = W, item_cor = item_cor)
    #>                        unit validity_weighted
    #> unit              1.0000000         0.9900312
    #> validity_weighted 0.9900312         1.0000000

The composite validity is generally smaller than the largest single-predictor validity unless the predictors carry independent variance with respect to the criterion, which connects directly to the incremental-validity reasoning developed below. Disattenuation of the observed correlation for measurement error in the criterion is provided by `disattenuate_correlation()`.

    disattenuate_correlation(r_observed = .35, reliability_x = .85, reliability_y = .70)
    #> [1] 0.4537426

Disattenuation moves the analysis from the observed-score metric to the true-score metric, which is the appropriate input for population-level utility statements but should be reported alongside the observed-score estimate (Schmidt & Hunter, 2015).

## Range-restriction corrections

A second standard correction is for range restriction. Sackett, Laczo, and Arvey (2002) and Sackett, Lievens, Berry, and Landers (2007) distinguish direct range restriction, which operates on the predictor that was used as the selection variable, from incidental range restriction, which operates on variables correlated with the selected variable. The Thorndike Case II correction handles direct restriction; the Lawley (1943) multivariate correction handles incidental restriction in correlated predictors.

    correct_r_direct_range_restriction(
      r_restricted            = .25,
      range_restriction_ratio = 1.40
    )
    #> [1] 0.3399501

The Lawley multivariate correction is more general and covers the case in which selection occurs on a composite or on a different variable than the one being corrected.

    # Three-variable example: selection on X1 (cognitive ability, the predictor used as
    # the selection variable); incidental restriction on X2 (interview) and Y (criterion).
    sigma_star <- matrix(c(
      1.00, .30, .25,
      .30, 1.00, .20,
      .25, .20, 1.00
    ), 3, 3)
    # Unrestricted SD of X1 is 1/.6 times the restricted SD; variance increases by 1/.6^2.
    sigma_ss_unrestricted <- matrix(1 / 0.6^2, 1, 1)
    correct_r_lawley(
      sigma_restricted      = sigma_star,
      selection_indices     = 1,
      sigma_ss_unrestricted = sigma_ss_unrestricted
    )
    #> $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

Ree, Carretta, Earles, and Albert (1994) document that multivariate range-restriction corrections can occasionally produce sign changes when the unrestricted covariance structure is poorly estimated, which is a strong argument for reporting both corrected and uncorrected values and for inspecting the implied unrestricted matrix for plausibility.

## Incremental validity and predictor importance

When a battery already exists and a candidate predictor is being considered for addition, the relevant quantity is not the marginal correlation of the new predictor with the criterion but the incremental validity of the augmented battery over the existing one. Sturman (2001) developed the matrix formulation of this problem as a restricted canonical correlation, in which the predictor-side weights are optimised given a fixed criterion-side composite weighting. With predictor correlation matrix \(\boldsymbol{\Sigma}_{11}\), predictor-criterion correlation matrix \(\boldsymbol{\Sigma}_{12}\), criterion correlation matrix \(\boldsymbol{\Sigma}_{22}\), fixed criterion weights \(\mathbf{b}\), and predictor weights \(\mathbf{a}\) to be determined,

\[ r_{uv} \;=\; \frac{\mathbf{a}^{\top} \boldsymbol{\Sigma}_{12} \mathbf{b}}{\sqrt{\mathbf{a}^{\top} \boldsymbol{\Sigma}_{11} \mathbf{a}} \cdot \sqrt{\mathbf{b}^{\top} \boldsymbol{\Sigma}_{22} \mathbf{b}}}, \]

with the optimal predictor-side weights given by \(\mathbf{a} \propto \boldsymbol{\Sigma}_{11}^{-1} \boldsymbol{\Sigma}_{12} \mathbf{b}\).

    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))
    #> <psu_incremental_validity>
    #>   validity: 0.352614

The function `incremental_validity()` evaluates the gain from adding new predictors to an existing baseline.

    Rxx <- matrix(c(1, .30, .20,
                    .30, 1, .25,
                    .20, .25, 1), 3, 3, byrow = TRUE)
    Rxy <- matrix(c(.30, .20,
                    .25, .15,
                    .10, .35), 3, 2, byrow = TRUE)
    Ryy <- matrix(c(1, .40, .40, 1), 2, 2)

    incremental_validity(
      predictor_cor = Rxx,
      predictor_criterion_cor = Rxy,
      criterion_cor = Ryy,
      criterion_weights = c(.6, .4),
      baseline_predictors = 1:2,
      added_predictors = 3
    )
    #> <psu_incremental_validity>
    #>   baseline_validity: 0.34905
    #>   expanded_validity: 0.379438
    #>   incremental_validity: 0.0303873
    #>   added_predictors: 3

Two contemporary methods for quantifying the relative importance of predictors complement this matrix formulation. Johnson’s (2000) relative-weights method decomposes the model \(R^2\) into approximately uncorrelated contributions, and Budescu’s (1993) dominance analysis evaluates each predictor across all subsets of the other predictors.

    # For predictor-importance methods, the criterion side collapses to a single
    # criterion (e.g., overall job performance):
    rxy_single <- c(.30, .25, .35)
    relative_weights(predictor_cor = Rxx, criterion_cor = rxy_single)
    #>   predictor raw_weight rescaled_weight percent_of_r2
    #> 1         1 0.06156283      0.06156283      32.45758
    #> 2         2 0.03381629      0.03381629      17.82886
    #> 3         3 0.09429252      0.09429252      49.71356
    dominance_analysis(predictor_cor = Rxx, predictor_criterion_cor = rxy_single)
    #> <psu_dominance>
    #>   r_squared_full: 0.189672

The substantive lesson, formalised by Sturman (2001), is that a predictor with high marginal correlation can have negligible or even negative incremental validity if it is highly redundant with predictors already in the battery, and conversely a predictor with low marginal correlation can have high incremental validity through suppression effects. Reporting both relative weights and dominance results, alongside the matrix-restricted canonical validity, provides a multi-method check on the importance of any candidate predictor.

## The integrated Sturman comprehensive model

Sturman (2001) integrates several of the corrections developed above into a single comprehensive utility model: incremental validity over the operating baseline, multidimensional criterion through restricted canonical reweighting, multi-period employee flows, taxes and discounting, probationary survival, and offer rejection. The function `sturman_comprehensive()` composes these adjustments and returns both the integrated estimate and the cumulative cascade.

    S11 <- matrix(c(1, .30, .30, 1), 2, 2)
    S12 <- matrix(c(.30, .10, .15, .25), 2, 2, byrow = TRUE)
    S22 <- matrix(c(1, .40, .40, 1), 2, 2)

    s <- sturman_comprehensive(
      validity                       = .35,
      baseline_validity              = .20,
      selection_ratio                = .20,
      sdy                            = 50000,
      n_year_one                     = 100,
      tenure                         = 5,
      fixed_cost                     = 75000,
      hires_per_period               = c(100, 15, 15, 15, 15),
      losses_per_period              = c(0, 15, 15, 15, 15),
      tax_rate                       = .25,
      discount_rate                  = .08,
      predictor_cor                  = S11,
      predictor_criterion_cor        = S12,
      criterion_cor                  = S22,
      criterion_weights              = c(.7, .3),
      probation_cutoff_z             = -1,
      acceptance_rate                = .70,
      quality_acceptance_correlation = -0.20
    )

    s
    #> <psu_sturman: Sturman (2001) comprehensive utility>
    #>   Comprehensive net utility: 3759820
    #>   Effective validity: 0.3068  (baseline: 0.2 )
    #>
    #>   Cascade:
    #>                            step net_utility pct_of_naive
    #>  1. Naive BCG (random baseline)    12173334    100.00000
    #>         2. + operating baseline     5174286     42.50509
    #>  3. + multidim. criterion (RCV)     3663342     30.09317
    #>     4. + flows + tax + discount     2169473     17.82152
    #>                  5. + probation     5476999     44.99178
    #>            6. + offer rejection     3759816     30.88567

The cumulative cascade is available in `s$cascade`, the effective validity actually used for the calculations after restricted canonical reweighting in `s$effective_validity`, and the active headcount per period after employee flows in `s$n_active_by_period`. The applied reproduction of Sturman’s (2001) cascade, including the published shrinkage to approximately \(8\%\) of the naive Brogden-Cronbach-Gleser estimate, is developed in the Reproducing canonical examples vignette.

## How to proceed in applied work

1. Start with `naylor_shine()` if the analysis stops at expected criterion gain in standard-deviation units; do not introduce \(SD_y\) unless the decision genuinely requires monetary translation.
2. Use `bcg_utility()` for transparent monetary utility, but treat it as a starting point and not as an endpoint.
3. Replace random selection with the actual operating baseline whenever it is identifiable; report the random-baseline result as an upper bound only.
4. Estimate \(SD_y\) using at least two methods and report a sensitivity range; a single point estimate of \(SD_y\) understates the uncertainty in the final utility figure (Holling, 1998).
5. Use `boudreau_utility()` when the analysis spans several periods, when costs and returns occur at different times, or when taxes and contribution margins are non-negligible.
6. Distinguish selection utility (\(r\)-based, `bcg_utility()`) from intervention utility (\(d\)-based, `shp_utility()`); they are calibrated for different study designs.
7. Disattenuate validities with `disattenuate_correlation()` and correct for range restriction with `correct_r_lawley()` or `correct_r_direct_range_restriction()` before utility calculation; report both corrected and uncorrected values.
8. When the battery has multiple predictors, use `restricted_canonical_validity()` and `incremental_validity()`, complemented by `relative_weights()` and `dominance_analysis()` for predictor-importance triangulation.
9. Use `sturman_comprehensive()` as the integrated reporting standard for continuous-monetary, compensatory-selection problems; the cascade output makes the contribution of each adjustment auditable.
10. Report inputs in full; utility estimates are only as credible as the validity, \(SD_y\), cost, baseline, tenure, and discount-rate assumptions on which they rest.

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