Diagnostic Intervals — Single-Case Assessment

Dr. R. Düsing · Osnabrück University
Running Example
PersonChild undergoing IQ testing (Wechsler scale, μ=100, σ=15) XObserved test score = 115 (single measurement, 1 SD above μ) r_xxSubtest reliability = 0.85 SEMStandard error of measurement = σ·√(1−r_xx) = 15·√0.15 ≈ 5.81
In what range does this child's true IQ lie with 95% probability — and why do the three methods yield different intervals even though all use the same test score?
Distributions & Intervals
Population distribution Prior (background knowledge) Likelihood Posterior / CrI
Results Table
Method Point Estimate Lower Bound Upper Bound Width
95% Intervals
CI Equivalence Hypothesis — — — —
CI Regression Hypothesis — — — —
Bayesian CrI — — — —
Concepts
Why an Interval Instead of a Point Value?
Every measurement contains measurement error — the observed value X is not the true value τ. The size of the uncertainty is given by the standard error of measurement: SEM = σ · √(1 − r_xx). The less reliable the test (small r_xx), the larger the SEM and the wider the interval. So instead of a single point value ("the IQ is 115"), one states a range of uncertainty — but what exactly this range means depends on the method (frequentist CI vs. Bayesian CrI). That's exactly the difference this tool shows.
CI per the Equivalence Hypothesis
The classical CTT confidence interval, centered on the observed value X: CI = X ± 1.96 · SEM. Assumption: X is an unbiased estimator of τ (observed = true value "on average"). Frequentist interpretation: over infinitely many repetitions, this interval would contain the true value in 95% of cases.
CI per the Regression Hypothesis
Accounts for regression to the mean: extreme test scores contain more measurement error, so the best estimator sits closer to the mean. τ = μ + r_xx · (X − μ) always lies between X and μ; SE_reg = σ · √(r_xx · (1 − r_xx)). Compared to the equivalence CI, the interval is shifted toward the mean and somewhat narrower. → Regression to the Mean
Bayesian CrI (Credible Interval)
Prior N(μ, SD_prior) + likelihood N(X, SEM) → posterior via precision weighting. The estimator τ is a weighted mean of μ and X (weight = 1/variance). With SD_prior = σ·√r_xx, the CrI exactly matches the regression CI; with a flat prior, it approaches the equivalence CI. The advantage: direct probability statements P(τ > T | X).
Which Interval — and How to Interpret It?
Equivalence: when you consider X unbiased and only want to represent measurement uncertainty. Regression: when you need the best corrected estimate of the true value (single-case assessment, progress monitoring). Bayes: when you want to incorporate prior knowledge and answer cutoff questions directly. Caution: the frequentist CI says something about repetitions, the CrI about credibility for the specific case.
Context & Related Tools
SEM and reliability come from Classical Test Theory; the regression hypothesis is the application of regression to the mean to the single case; the CrI brings in the Bayesian perspective (prior knowledge, no multiple-testing problem with flexible cutoffs). → CTT Foundations · → Regression to the Mean
Diagnostic Intervals — Help
Example

A child scores an IQ of X=115 (Wechsler, μ=100, σ=15, r_xx=0.85). The standard error of measurement is SEM = 15·√(1−0.85) ≈ 5.81. Question: in what range does the true IQ lie — and what statement does the chosen interval allow?

CI Equivalence Hypothesis

The interval is centered on the observed test score X. Assumption: X is an unbiased estimator of the true value τ. The classical CTT confidence interval.

CI = X ± 1.96 · SEM   (SEM = σ · √(1 − r_xx))

In the example: 115 ± 1.96 · 5.81 → [103.6, 126.4]. Interpretation: under repeated testing (n → ∞), this interval would contain the true value in 95% of cases.

CI Regression Hypothesis

The interval is centered on the regressed estimate τ̂, not on the observed X. Accounts for regression to the mean: extreme test scores have more measurement error than values near the mean — that's why τ̂ always lies between X and μ.

τ̂ = μ + r_xx · (X − μ)   SE_reg = σ · √(r_xx · (1 − r_xx))

In the example: τ̂ = 100 + 0.85 · (115−100) = 112.75. SE_reg = 15·√(0.85·0.15) ≈ 5.36. CI_reg = 112.75 ± 1.96 · 5.36 → [102.2, 123.3]. The location and width sit closer to μ than the equivalence-hypothesis CI.

Bayesian CrI (Credible Interval)

Prior N(μ, SD_prior) + likelihood N(X, SEM) → posterior via precision weighting. The posterior is a weighted mean of prior mean μ and observed X, where the weight is inversely proportional to the variance.

w_data = 1/SEM²   w_prior = 1/SD_prior² τ̂ = (μ·w_prior + X·w_data) / (w_prior + w_data)   SD_post = √(1/(w_prior+w_data))

What does SD_prior control? SD_prior indicates how far the true value τ is allowed to deviate from the population mean μ according to prior knowledge. Large SD_prior: weak, broad prior knowledge — the prior barely pulls, and the location and width of the CrI are governed almost entirely by the data (X and SEM). Small SD_prior: strong prior knowledge — τ̂ and the CrI are pulled noticeably toward μ. Recommendation: SD_prior = σ·√r_xx is the Bayesian equivalent of the regression hypothesis and, given known reliability, a well-justified choice. SD_prior = σ (population SD) is a neutral, population-based starting point.

In the example (SD_prior=σ=15, X=115, r_xx=0.85): w_prior=1/225, w_data=1/33.75 → τ̂ ≈ 113.1, SD_post ≈ 5.42. CrI ≈ [102.5, 123.7]. The CrI allows direct probability statements: P(τ > 110 | X=115) can be read directly.

Special cases:
— SD_prior = σ·√r_xx: the CrI matches the regression-hypothesis CI exactly (same location and width).
— SD_prior = σ (default): the CrI's location (τ̂ ≈ 113) and width sit close to the regression-hypothesis CI (τ̂=112.75), because the SEM is small under good reliability and the data dominate the weighting. Not to be confused with the equivalence-hypothesis CI, which is centered on X=115.
— SD_prior → ∞ (flat prior): CrI → equivalence-hypothesis CI (location at X, width SEM-based).
The Bayesian advantage: direct probability statements about any cutoff without frequentist detours.

Bayesian Cutoff — a Flexible Decision Question

Once the posterior is computed, one can ask directly for any cutoff T: how likely is it that the true value τ lies above (or below) T?

P(τ > T | X) = 1 − Φ((T − τ̂) / SD_post)

In the example (τ̂≈113, SD_post≈5.42): at cutoff T=112, P(τ > 112 | X=115) ≈ 58% — even though the observed X=115 is above T, considerable uncertainty remains due to measurement imprecision. Moving T to 100 (the average) gives P(τ > 100 | X=115) ≈ 99%. These values can be read directly in the tool by dragging the cutoff.

Interpretation: P(τ > T | X) = 87% means: given the observed test score and the prior knowledge, the true value lies above T with 87% probability. This statement is direct and substantively interpretable — unlike a p-value, which refers to hypothetical repetitions under H₀.

No Multiple Testing: a frequentist test is tied to an α. If you ask sequentially "Is τ > T₁?", "Is τ > T₂?", "Is τ > T₃?", the probability of a Type I error rises — and an α correction (Bonferroni etc.) becomes necessary. The posterior, by contrast, is computed once: all cutoff questions draw on the same distribution, without further questions increasing the error risk. The cutoff can therefore be chosen freely — clinically, for support planning, or administratively — without a statistical price for that flexibility.

Wagenmakers, E.-J., Morey, R. D., & Lee, M. D. (2016). Bayesian benefits for the pragmatic researcher. Current Directions in Psychological Science, 25, 169–176.
Moosbrugger, H. & Kelava, A. (Eds.) (2020). Testtheorie und Fragebogenkonstruktion (3rd ed.). Springer — chapter on norming, percentile ranks, and norm scores.