---
title: "Taylor-Russell and the Thomas-Owen-Gunst multivariate extension"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Taylor-Russell and the Thomas-Owen-Gunst multivariate extension}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
```


## The classificatory problem

Taylor and Russell (1939) ask a classificatory question: **what proportion of selected applicants will be successful, given a base rate, a selection ratio, and a validity coefficient?** In modern diagnostic language, the central quantity is a positive predictive value (PPV): $P(\text{success} \mid \text{selected})$. Under bivariate normality of predictor $X$ and criterion $Y$ at correlation $\rho = r_{xy}$, with cutoffs $x_c$ on the predictor and $y_c$ on the dichotomised criterion, the PPV has the closed form

$$
PPV \;=\; \frac{P(X \geq x_c,\, Y \geq y_c)}{P(X \geq x_c)} \;=\; \frac{\displaystyle\int_{x_c}^{\infty}\!\!\int_{y_c}^{\infty} \phi_2(x, y;\, \rho)\, dy\, dx}{\displaystyle\int_{x_c}^{\infty} \phi_1(x)\, dx},
$$

where $\phi_2$ and $\phi_1$ are the standard bivariate and univariate normal densities. The Taylor and Russell (1939) utility metric is the increment of $PPV$ over the population base rate $\phi$, i.e. $\Delta P_S = PPV - \phi$. Cascio (1980) showed that the Taylor-Russell model is a special case of the Brogden-Cronbach-Gleser framework when the criterion is dichotomised at a fixed cutoff, but the success-ratio metric remains uniquely interpretable when the practical decision is binary (e.g., probationary pass/fail, certification, retention beyond a fixed horizon).

```{r}
library(personnelSelectionUtility)

tr_classic(base_rate = .50, selection_ratio = .20, validity = .35)
```

The output includes the four cells of the $2 \times 2$ classification table, the success ratio among selected applicants (`ppv`), and the increment over the base rate. Sensitivity and specificity are reported as additional diagnostic indices. The increment $\Delta P_S = PPV - BR$ is the original Taylor and Russell (1939) utility metric.

## Solving unknown Taylor-Russell inputs

The original tables in Taylor and Russell (1939) are forward-looking: given the base rate, selection ratio, and validity, they return the success ratio. In practice, the analyst frequently asks the inverse question: **what validity is required to achieve a target PPV?** The function `tr_solve()` follows this logic and is intentionally similar to the flexible solver implemented in Waller's (2024) `TaylorRussell` package.

```{r}
tr_solve(base_rate = .50, selection_ratio = .20, validity = NULL, ppv = .70)
```

The same function solves for the selection ratio implied by a desired PPV and a known validity, which is the operationally relevant inversion when the validity is fixed by the available battery and the analyst chooses the cutoff.

```{r}
tr_solve(base_rate = .50, selection_ratio = NULL, validity = .35, ppv = .70)
```

These inversions are useful for sensitivity reasoning: rather than asking "what PPV will my $r = .35$ test deliver", the analyst can ask "what validity floor would I need to achieve the PPV target the organisation cares about", which connects directly to the break-even logic of Cronshaw, Alexander, Wiesner, and Barrick (1987).

## Why the univariate model is not enough

The univariate Taylor-Russell model handles a single predictor or a composite that has already been collapsed into one score. Many real systems do not operate as a single composite. Applicants may need to pass multiple independent cutoffs: for instance, a minimum cognitive score *and* a minimum interview score *and* a minimum integrity score. Forcing such a multiple-hurdle conjunctive design into a univariate model loses two structurally important features: the *joint* selection ratio (which is materially smaller than the product of marginals when predictors are positively correlated) and the differential restriction of range that each predictor experiences (Sackett, Lievens, Berry, & Landers, 2007).

Thomas, Owen, and Gunst (1977) generalised the Taylor-Russell logic to multiple cutoffs. Their derivation shows that the multiple-cutoff problem is, in principle, a natural extension of the single-test case; the obstacle was historically computational, since the integrals require multivariate normal probabilities that were tractable only for $k \leq 3$ predictors using the tables of Owen (1956, 1962). For $k$ predictors with marginal cutoffs $\mathbf{c} = (c_1, \ldots, c_k)$ and a dichotomised criterion with cutoff $y_0$, the joint conjunctive selection ratio and the joint true-positive rate are

$$
SR_{\text{conj}} \;=\; \int_{c_1}^{\infty}\!\!\!\cdots\!\int_{c_k}^{\infty} \phi_k(\mathbf{x};\, \mathbf{R}_{XX})\, d\mathbf{x},
$$

$$
TP \;=\; \int_{c_1}^{\infty}\!\!\!\cdots\!\int_{c_k}^{\infty}\!\int_{y_0}^{\infty} \phi_{k+1}(\mathbf{x},\, y;\, \mathbf{R})\, dy\, d\mathbf{x},
$$

with $\mathbf{R}$ the full $(k+1) \times (k+1)$ correlation matrix containing predictor intercorrelations and predictor-criterion validities. The multivariate positive predictive value follows as $PPV = TP / SR_{\text{conj}}$. Modern implementations rely on numerical integration of the multivariate normal density via the Genz-Bretz quasi-Monte Carlo algorithm (Genz, 1992; Genz & Bretz, 2009), which the package accesses through `mvtnorm::pmvnorm()`. The implementation in this package closely parallels Waller's (2024) `TaylorRussell::TaylorRussell()`, which provided the first widely available rehabilitation of the Thomas-Owen-Gunst integral after several decades of computational neglect (Ren & Waller, 2024).

## Multivariate Taylor-Russell with specified marginal cutoffs

The matrix `R` must include the predictors first and the criterion last. The following example has two predictors and one dichotomised criterion.

```{r}
R <- matrix(c(
  1.00, .30, .40,
  .30, 1.00, .35,
  .40, .35, 1.00
), nrow = 3, byrow = TRUE)

tr_multivariate(selection_ratios = c(.50, .50), base_rate = .50, R = R)
```

The output reports both the marginal selection ratios supplied by the user and the implied `joint_selection_ratio`. These are not the same. If each predictor has a marginal selection ratio of $.50$, the joint selected proportion is materially smaller than $.50$ because applicants must pass both cutoffs. The exact reduction depends on the predictor intercorrelation: independent predictors yield a joint rate near the product of marginals, while strongly correlated predictors yield a joint rate closer to the smaller marginal.

## Equal-cutoff tables and target joint selection ratio

The historical Thomas-Owen-Gunst tables are indexed by the **joint** proportion selected under equal cutoffs, not by the marginal cutoffs. The function `tr_multivariate_equal_cutoff()` solves for the common marginal cutoff that yields a desired joint conjunctive probability.

```{r}
R_tog <- matrix(c(
  1.00, .50, .70,
  .50, 1.00, .70,
  .70, .70, 1.00
), nrow = 3, byrow = TRUE)

tog <- tr_multivariate_equal_cutoff(
  joint_selection_ratio = .20,
  base_rate = .60,
  R = R_tog
)

tog
```

The same function exposes the solved marginal selection ratio.

```{r}
c(
  marginal_selection_ratio = tog$solved_marginal_selection_ratio,
  joint_selection_ratio    = tog$joint_selection_ratio,
  ppv                      = tog$ppv
)
```

This is the canonical Thomas-Owen-Gunst (1977) example: when the population base rate is $.60$, the predictor intercorrelation is $.50$, and both predictor-criterion validities are $.70$, selecting the joint top $20\%$ through equal cutoffs yields a success ratio close to $.97$ and a marginal pass rate near $.35$ on each test. The substantive lesson is that conjunctive selection can produce very high success ratios, but at the cost of a small selection ratio per predictor that may be operationally infeasible if the applicant pool is limited or if early stages cannot screen out enough candidates.

## Group-specific multivariate Taylor-Russell

When base rates or predictor-criterion correlations differ across demographic groups, applying a single matrix `R` to the whole population conceals systematic differential prediction. The function `group_tr_multivariate()` evaluates the model separately by group, which is the natural primitive for adverse-impact reasoning under conjunctive selection. The package also provides `adverse_impact_ratio()` for the four-fifths comparison and `utility_fairness_frontier()` for the joint utility-fairness Pareto frontier discussed by De Corte, Lievens, and Sackett (2007).

```{r}
# Group-specific evaluation: same predictor structure but different base rates
# across two demographic groups (e.g., focal and reference). The marginal
# selection ratios are common; the base rates and, optionally, the correlation
# matrices may differ.
group_tr_multivariate(
  selection_ratios = c(.35, .35),
  base_rates       = c(.60, .45),
  R_list           = list(R_tog, R_tog),
  group_names      = c("Group A", "Group B")
)
```

The substantive value of this disaggregation is that the same marginal cutoffs can produce very different group-specific success ratios when base rates differ, even when the predictor-criterion correlations are identical across groups. This is one of the mechanisms behind the validity-diversity dilemma analysed by Pyburn, Ployhart, and Kravitz (2008) and is the diagnostic input for Pareto-optimal selection systems (De Corte, Lievens, & Sackett, 2007; De Corte, Sackett, & Lievens, 2011).

## Finite sampling

The Thomas-Owen-Gunst (1977) framework returns expected proportions in the population. In a finite cohort of selected applicants, the realised count of successes is a binomial random variable with parameter equal to the PPV. The function `tr_binomial_success_probability()` returns this distribution.

```{r}
finite <- tr_binomial_success_probability(n_selected = 20, ppv = .91, at_least = 18)
finite
attr(finite, "probability_at_least")
```

Reporting this finite-sample probability is particularly relevant for small selection cohorts, where the difference between an expected success ratio of $.91$ and a $90\%$ probability of at least $18$ out of $20$ successes can be operationally meaningful. The same logic underlies the recommendation, in Cronshaw et al. (1987), to combine point utility estimates with risk-simulation summaries.

## Reproducing Thomas-Owen-Gunst (1977), Table 6

The package reproduces, digit-for-digit at the precision allowed by the Genz-Bretz integration tolerance, the canonical example from Thomas, Owen, and Gunst (1977). Two predictors correlate $.50$ with each other and have validities of $.70$ against a dichotomised criterion with base rate $.60$. The original table is indexed by target joint selection ratio.

```{r}
R_tog <- matrix(c(
  1.00, .50, .70,
  .50, 1.00, .70,
  .70, .70, 1.00
), nrow = 3, byrow = TRUE)

joint_targets <- c(.20, .50)
tog_grid <- lapply(joint_targets, function(jsr) {
  tr_multivariate_equal_cutoff(
    joint_selection_ratio = jsr,
    base_rate = .60,
    R = R_tog
  )
})

tog_table <- data.frame(
  joint_sr        = joint_targets,
  marginal_sr     = vapply(tog_grid,
                           function(o) o$solved_marginal_selection_ratio,
                           numeric(1)),
  ppv             = vapply(tog_grid, function(o) o$ppv, numeric(1)),
  sensitivity     = vapply(tog_grid, function(o) o$sensitivity, numeric(1)),
  specificity     = vapply(tog_grid, function(o) o$specificity, numeric(1))
)
tog_table
```

The pattern is the one Thomas, Owen, and Gunst (1977) emphasise: as the target joint selection ratio increases from $.20$ to $.50$, the marginal cutoff per predictor relaxes substantially, and the PPV decreases monotonically from a value near $.97$ to a value near $.85$. Selectivity and base rate jointly determine the success ratio, exactly as in the univariate Taylor-Russell logic, but the multivariate generalisation makes explicit the role of predictor intercorrelation. The same reproduction is documented in the help file of `TaylorRussell::TaylorRussell()` (Waller, 2024), and the figures returned by the two implementations agree to integration tolerance.

## How to proceed in applied work

1. Use `tr_classic()` when the system is genuinely one-dimensional or when a defensible composite has already been formed.
2. Use `tr_solve()` to invert the model and obtain the validity or selection ratio implied by a target PPV; this anchors goal-setting and break-even reasoning in operational terms.
3. Use `tr_multivariate()` when the actual decision is conjunctive and applicants must pass multiple simultaneous cutoffs.
4. Use `tr_multivariate_equal_cutoff()` when the overall joint selected proportion is fixed by the organisation and the marginal equal cutoff is what the analyst must solve.
5. Always inspect whether the joint selected proportion is operationally realistic; multiple marginal cutoffs can become extremely selective, and a joint $SR$ of $.05$ may be incompatible with the available applicant pool.
6. Treat the multivariate normal assumption as a modelling assumption, not a fact. If the empirical distributions are skewed or truncated, report sensitivity analyses and consider non-Gaussian extensions or simulation alternatives.
7. Use `group_tr_multivariate()` when group-specific base rates or correlation matrices are available; the disaggregated reporting is a precondition for any defensible adverse-impact analysis.
8. If success is not naturally dichotomous, consider Naylor-Shine, Brogden-Cronbach-Gleser, or Boudreau-style continuous utility instead of forcing a dichotomous criterion.

## References

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.

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

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.

Genz, A. (1992). Numerical computation of multivariate normal probabilities. *Journal of Computational and Graphical Statistics*, *1*, 141--149.

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

Owen, D. B. (1956). Tables for computing bivariate normal probabilities. *Annals of Mathematical Statistics*, *27*, 1075--1090.

Owen, D. B. (1962). *Handbook of statistical tables*. Addison-Wesley.

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

Ren, Z., & Waller, N. G. (2024). An extended Taylor-Russell model for multiple predictors. *Multivariate Behavioral Research*, *59*(3), 654--655. https://doi.org/10.1080/00273171.2024.2310427

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.

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.

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). https://CRAN.R-project.org/package=TaylorRussell
