Separation, Reliability and Skewed Distributions
Statistically Different Sample-independent Levels of Performance

Implicit in our understanding of the "Reliability of a test for a sample" is that the sample and test somehow match. Typically, we think that the test is targeted on the sample, and that the sample has a reasonably symmetric unimodal "ability" distribution covering the operational range of the test. But what if this isn't so?

Clinical samples can be highly skewed: at one end, many healthy people, at the other, a very few deathly sick. The instrument may be targeting the sick, so it will be far off-target for the majority of the sample. The reported reliability coefficients will be low, but the instrument is clearly doing its job. What to do?

Conventional raw-score reliability estimates employ empirical samples in order to estimate measurement error. Rasch methods do not need to do this. Once the test items are calibrated, the measure and standard error corresponding to every possible raw score can be estimated without further data collection. These standard errors are usually reported as the smallest possible "model" errors. In practice, these standard errors can be inflated about 10% to allow for the unmodeled noise encountered in real data.

Statistically different levels of performance

Once the standard errors have been computed, they can be used to compute how many statistically different levels of performance can be identified. See the Table below (which is based on Winsteps Table 20 or the equivalent with other software). To do this start at one end of the raw score range and work to toward the other. Advance each time by twice the joint standard error (= square-root of sum of squared standard errors) of the current starting and ending measures until there is not room for another level. In the Table, there are 4 statistically distinct levels of performance. This corresponds to a separation of at least 4, i.e,. this test has a sample-independent reliability of at least 4²/(1+4²) = 0.94. The empirical reliability of this test, reported for a highly central sample, was 0.62, a value which grossly under-reports the test's measurement effectiveness.

Benjamin D. Wright

Raw Score on Test	Logit Measure	Standard Error	Minimum measure for next level (must be less than Logit Measure to start the next level)	Statistical Level
0 (0.3†) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 (13.7†)	-6.17 -4.86 -3.94 -3.27 -2.64 -1.97 -1.19 -.23 .80 1.72 2.55 3.37 4.21 5.23 6.60	1.83 1.08 0.85 0.79 0.72 0.68 0.63 0.64 0.69 0.75 0.81 0.89 0.93 1.12 1.84	-6.17 -6.17 + 2√(1.83²+1.08²) = -1.92   (> -4.86) -6.17 + 2√(1.83²+0.85²) = -2.13   (> -3.94) -6.17 + 2√(1.83²+0.79²) = -2.18   (> -3.27) -6.17 + 2√(1.83²+0.72²) = -2.24   (> -2.64) -6.17 + 2√(1.83²+0.68²) = -2.27   (< -1.97) -1.97 + 2√(0.68²+0.63²) = -0.12   (> -1.19) -1.97 + 2√(0.68²+0.64²) = -0.10   (> -0.23) -1.97 + 2√(0.68²+0.69²) = -0.03   (< 0.80)   0.80 + 2√(0.69²+0.75²) = 2.84   (> 1.72)   0.80 + 2√(0.69²+0.81²) = 2.93   (> 2.55)   0.80 + 2√(0.69²+0.89²) = 3.05   (< 3.37)   3.37 + 2√(0.89²+0.93²) = 5.94   (> 4.21)   3.37 + 2√(0.89²+1.12²) = 6.23   (> 5.23)   3.37 + 2√(0.89²+1.84²) = 7.46   (> 6.60)	1 1 1 1 1 2 2 2 3 3 3 4 4 4 4

† extreme scores of 0 and 14 are made more central by a Bayesian adjustment of 0.3 score-points in order to make their corresponding Rasch measures estimable. This is done automatically in Winsteps Table 20 and similar adjustments may be made by other software.

Later notes:
Another possibility would be to sample a normal distribution from the skewed distribution. Here's how to do it:
1. Divide the x-axis of your skewed distribution of persons into equal intervals (interval size depends on the size of your sample).
2. Divide the x-axis of a normal distribution of the same range into the same equal intervals
3. Compute the number of persons in each normal-distribution interval so that total numbers of persons in normal intervals matches the sample total.
4. Sample-with-replacement the persons from their skewed intervals into the corresponding normal intervals until each normal interval has the correct number of persons in it.
5. Compute the reliability from the resulting normalized sample.
In general, skew reduces reliability, so your reliability should have increased a little with the normalized sample.

There are three different Reliability/Separation/Strata options:

1. Spearman "Test" Reliability for the current sample -> Cronbach Alpha in CTT, and "Separation Reliability" for the observed person sample in Rasch.
"Separation ratio" = square-root (Spearman Reliabity / (1-Spearman Reliability)). Normal distributions are assumed.

2. Strata for the current sample. This is the separation ratio, but with the tales of the observed person distribution treated as performance levels.
"Strata reliability" = Strata^2 / ( 1+ Strata^2)

3. Wright's sample-independent method for strata. This shows the maximum number of statistically different strata the test can identify.
"Wright Strata Reliability" = Wright Strata^2 / ( 1+ Wright Strata^2). This is the maximum possible value of the Spearman "Test" reliability.

Choose the one (and make clear which one you have chosen) that makes most sense in your situation.

1. Reliability, separation, strata statistics, Fisher WP Jr. … 6:3 p.238
2. Reliability and separation nomograms, Linacre JM. … 1995, 9:2 p.421
3. Reliability and separation, Wright BD. … 1996, 9:4 p.472
4. Predicting Reliabilities and Separations of Different Length Tests, Linacre, J.M. … 2000, 14:3 p.767
5. Going beyond Unreliable Reliabilities, Mallinson T., Stelmack J. … 2001, 14:4 p.787-8
6. Separation, Reliability and Skewed Distributions: Statistically Different Levels of Performance, Wright B.D. … 2001, 14:4 p.786
7. Number of Person or Item Strata (4G+1)/3, Wright BD, Masters GN. … 2002, 16:3 p.888
8. Cash value of Reliability, WP Fisher … Rasch Measurement Transactions, 2008, 22:1 p. 1160

Separation, Reliability and Skewed Distributions: Statistically Different Sample-independent Levels of Performance. Wright B.D. … Rasch Measurement Transactions, 2001, 14:4 p.786

Rasch Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene	Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver	Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone	Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind	Rasch Measurement: Applications, Khine	Winsteps Tutorials - free Facets Tutorials - free	Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre	Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan
Other Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free	An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse	Rasch Measurement Theory Analysis in R, Wind, Hua	Applying the Rasch Model in Social Sciences Using R, Lamprianou	El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias	Diseño de Mejores Pruebas - free, Spanish Best Test Design	A Course in Rasch Measurement Theory, Andrich, Marais	Rasch Models in Health, Christensen, Kreiner, Mesba	Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen

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