Measurement System Analysis

Effectiveness with Wilson score CI

effektivitaet — proportion of correct classifications with a wilson-konfidenzintervall.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Basic Statistics > 1 Proportion (Wilson-score interval option)

Description

In MSA Type 5 the effectiveness of a rater is the proportion of individual ratings that agree with the reference (customer or konsens-fallback). For the confidence interval Qprovement uses the Wilson score interval instead of the classical Wald approximation: at proportions near 0 % or 100 % (exactly where attribute inspection outcomes usually sit) Wilson has markedly better coverage — and it never produces the Wald pathology of interval edges leaking below 0 or above 1. Applicable to any rate: effektivitaet, miss-rate, false-alarm-rate, repeatability.

Formulas

Rate

\overset{p}{^} = \frac{k}{n}

Point estimate: number of successes k divided by number of trials n

Zentrum

\tilde{p} = \frac{p ^ + \frac{z ^{2}}{2 n}}{1 + \frac{z ^{2}}{n}}

Shifted centre of the Wilson interval (z = z_{1−α/2})

Halbweite

Δ = \frac{z \frac{p ^ ( 1 - p ^ )}{n} + \frac{z ^{2}}{4 n ^{2}}}{1 + \frac{z ^{2}}{n}}

Half-width of the Wilson CI

95 %-KI

[max (0, \tilde{p} - Δ), min (1, \tilde{p} + Δ)]

Confidence interval, clamped to [0, 1]

Assumptions

Bernoulli trials: each rating is independently either correct or incorrect
Constant rate p across all ratings of the same rater

Limitations

For n < 5 even the Wilson interval is coarse — consider an exact Clopper-Pearson interval instead
Symmetric UI rounding may make p = 0 or p = 1 look like the CI is a point

References

Wilson, E. B. (1927) — Probable inference, the law of succession, and statistical inference, JASA 22(158)
Agresti, A. & Coull, B. A. (1998) — Approximate is better than "exact" for interval estimation of binomial proportions, The American Statistician 52(2)
Brown, L. D., Cai, T. T. & DasGupta, A. (2001) — Interval estimation for a binomial proportion, Statistical Science 16(2)