# Reliability, Separation, Strata Statistics

Reliabilities are often reported as though they were invariable characteristics of tests. Of course, they are not. They depend not only on the construction of the test, but also on the distribution of the examinee sample tested. Conventionally, only person separation reliability is reported, but item separation statistics are also useful indicators. They tell how well this sample of examinees have spread out the items along the measure of the test, and so defined a meaningful variable.

The Table below aids interpreting and predicting reliabilities. Its underlying components are:

Observed SD = the observed standard deviation of reported measures, for examinees or for items.

Root Mean-Square Error (RMSE) = "average" measurement error of reported measures.

True SD = standard deviation of reported measures corrected for measurement error inflation.

Observed SD and RMSE are calculated directly from the reported measures and their standard errors. Then,
(True SD)^2 = (Observed SD)^2 - (RMSE)^2

Separation Ratio:
G = (True SD)/(RMSE) is a ratio scale index comparing the "true" spread of the measures with their measurement error. It indicates the measure of spread of this sample of examinees (or test items) in units of the test error in their measures.

Separation Reliability:
G^2/(1+G^2) = (True SD)^2/(Observed SD)^2 = KR-20 or Alpha. This is a correlation coefficient, the ratio of true measure variance to observed measure variance. When G=1, True SD = RMSE, and reliability is 0.5. A reliability less than 0.5 implies that the differences between measures are mainly due to measurement error.

Discernible Strata: (4G+1)/3
The functional range of measures is around 4 True SD. Inflate this by 1 RMSE to allow for the error in the observed measures. Set a significant difference between two measures at 3 RMSE. Then there are (4 True SD + RMSE)/(3 RMSE) = (4G+1)/3
significantly different levels of measures in the functional range. See discussion at RMT 16:3 p. 888

```-----------------------------------------------------------------
Separation  KR-20, Alpha:    % Variance:         Distinct Strata:
Ratio: G    G^2/(1+G^2)   Not Due Error/Due Error   (4G+1)/3
-----------------------------------------------------------------
0          .00          0/100                    1
1          .50          50/50                    1
1.5        .70          70/30                    2
2          .80          80/20                    3
3          .90          90/10                    4
4          .94          94/6                     5
5          .96          96/4                     7
6          .97          97/3                     8
7          .98          98/2                     9
-----------------------------------------------------------------
```

William P. Fisher, Jr.

Wright, B. D., & Masters, G. N. (1982, pp. 92, 105-106). Rating scale analysis: Rasch measurement. Chicago, Illinois: MESA Press.

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

Reliability, Separation, Strata Statistics, W Fisher Jr … Rasch Measurement Transactions, 1992, 6:3 p. 238

Please help with Standard Dataset 4: Andrich Rating Scale Model

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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