Taylor-Russell Tables — Selection Utility of a Test

Dr. R. Düsing · Osnabrück University

Help — Taylor-Russell Tables

Does / Does NOT

Does: compute what proportion of those selected with a test later turn out to be successful/qualified — as a function of validity, selection ratio, and base rate. Shows the table and the nomogram, and makes visible that the table values are grid points of a continuous function. Does NOT: no cost-benefit analysis in monetary terms (that would be Brogden-Cronbach-Gleser), and no correction for bias — as validity, the correlation corrected for range restriction should be used.

What Is This About?

Taylor & Russell (1939) answer the practical question: "Is a selection test worth it?" A test with validity ρ is more useful the more selectively one selects, and depending on the base rate of suitability. The utility is the success rate of those selected minus the base rate (= what you'd get without a test).

The Running Example

A clinic (or HR department) selects applicants via an aptitude test. Validity = correlation between test and later success. Selection ratio = proportion of applicants taken. Base rate = proportion who would succeed without the test. Result: the success rate of those selected = positive predictive value.

How to Read the Table

A table always applies to one base rate (selectable above via the slider). Rows = validity (ascending), columns = selection ratio. The cell value is the success rate in percent. The currently set combination is outlined. Rules of thumb: under strict selection (small column) and a medium base rate, even moderate validity gains a lot; near a base rate of 0 or 1, there's hardly anything to gain.

The Nomogram + Target Crosshair

The nomogram shows the same information continuously: x-axis = validity, y-axis = success rate, each curve a selection ratio. The horizontal line is the base rate (start of all curves at validity 0). Click into the chart to set a target crosshair: validity is read from the x-position, the selection ratio snaps to the nearest curve. Dashed lines drop onto both axes; the boxes below show the exact values.

The Symbols

ρ — validity (correlation test ↔ criterion).
SR — selection ratio = proportion selected.
BR — base rate = proportion in the population that meets the success criterion.
EQ — success rate of those selected = positive predictive value (PPV).
Lift — gain EQ − BR (utility relative to "selecting without a test").

Formula

Bivariate normal distribution of predictor X and criterion Y with correlation ρ. Criterion threshold yc = Φ⁻¹(1−BR), selection threshold xc = Φ⁻¹(1−SR). Then EQ = P(Y > yc | X > xc) = P(X>xc, Y>yc) / SR. At ρ = 0, EQ = BR (test useless); at ρ = 1, EQ = min(BR/SR, 1).

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(5), 565–578.

📋 Example — Is the Selection Test Worth It?
Without a test, 50% of applicants would succeed (base rate). With a test of validity ρ = 0.50, taking the best 20%, the success rate of those selected rises to —% — a gain of — percentage points over "selecting blindly".
What Do the Symbols Mean?
ρValidity — correlation between the test and the later success criterion
SRSelection ratio — proportion of applicants selected (small = strict)
BRBase rate — proportion who meet the criterion without the test
EQSuccess rate of those selected = positive predictive value (PPV)
LiftEQ − BR — the incremental utility of the test in percentage points
Φ⁻¹Quantile of the standard normal distribution — translates proportions into thresholds
Base Rate BR
—
without test
ρ Validity
—
test ↔ criterion
Selection Ratio SR
—
proportion selected
Success Rate EQ
—
PPV of the selected
Lift = EQ − BR
—
Incremental utility
Taylor-Russell Table — Success Rate (%) at Base Rate 0.50
↓ Rows: validity ρ (ascending) → Columns: selection ratio SR  ·  outlined = current setting
Nomogram — the Same Table as a Continuous Function

Clicking into the chart sets the target crosshair: x → validity, selection ratio snaps to the nearest curve. Dashed lines show the projection onto both axes.

Reading: —
Concepts
The Basic Idea (Taylor & Russell 1939)
The utility of a selection test cannot be read off from its validity alone. What matters is the interplay with two contextual quantities: the selection ratio (how strictly is one selecting?) and the base rate (how many would succeed anyway?). Under the assumption of a bivariate normal distribution, Taylor and Russell provided tables that give the success rate of those selected.
Why the Selection Ratio Has This Effect
The stricter the selection (small SR), the further right the threshold sits on the test — and the more strongly even a moderate validity comes through. At SR = 1 (everyone is taken), the success rate is necessarily equal to the base rate: no selection, no utility. In the nomogram this shows up as all curves starting at the base-rate line at validity 0, with lower-SR curves rising more steeply.
The Role of the Base Rate
The maximum room for improvement lies at a medium base rate (≈ .50). If the base rate is very high (almost everyone qualified), a test can hardly improve anything — you'd be right anyway. If it's very low, absolute improvement is hard, but the relative improvement can be large. Set the base rate to .20 vs. .80 and watch how the whole table and nomogram shift.
Success Rate = Positive Predictive Value
The Taylor-Russell success rate is nothing other than the PPV: among those selected as "qualified", the proportion who actually meet the criterion. That makes this tool the continuous counterpart to the confusion matrix from sensitivity/specificity — just via a continuous bivariate normal distribution instead of discrete classification. → Sensitivity & Specificity
Connection: Range Restriction
Important: the ρ used here must be the true validity in the applicant population. But validity studies are often conducted on people already selected → the observed correlation is biased downward by range restriction. Using the uncorrected value systematically underestimates the utility. Correct via Thorndike first, then plug into the Taylor-Russell table. → Range Restriction
Connection: Diagnostic Validity
Taylor-Russell is one building block in the utility analysis of selection procedures. The Diagnostic Validity tool places it within construct, criterion, and content validity, and shows the bridge to utility analysis (Brogden-Cronbach-Gleser), which additionally expresses selection utility in monetary units. → Diagnostic Validity
Limitations & Assumptions
The table assumes (1) a bivariate normal distribution of predictor and criterion, (2) a dichotomous success criterion (qualified/unqualified via a cutoff), (3) a linear relationship and constant spread (homoscedasticity). Real criteria are often skewed or multi-level; then the values are approximations. For continuous utility measures without an artificial criterion cutoff, the Brogden-Cronbach-Gleser model is more suitable.
Table vs. Nomogram
The classic printed table is a discrete grid-point matrix — practical for lookup. The nomogram makes visible that a smooth surface EQ(ρ, SR, BR) lies behind it: every table cell is just one point on a curve. The target crosshair allows stepless reading between table rows — and shows that interpolation in the printed table is valid.