# Reliability and Separation

"The reliability of any set of measurements is logically defined as the proportion of their variance that is true variance... We think of the total variance of a set of measures as being made up of two sources of variance: true variance and error variance... The true measure is assumed to be the genuine value of whatever is being measured... The error components occur independently and at random" (Guilford 1965, p.439-40).

Observed Variance = "True" Variance + Error Variance

Reliability = "True" Variance / Observed Variance

In Rasch terms, "True" valiance is the "adjusted" variance (observed variance adjusted for measurement error). Error Variance is a mean-square error (derived from the model) inflated by misfit to the model encountered in the data.

Kubiszyn and Borich (1993, p.353) say "an acceptable standardized test should have reliability coefficients of about .95 for internal consistency". But "reliability depends upon the population measured as well as the measuring instrument... One should speak of the reliability of a certain instrument applied to a certain population under certain conditions" (Guilford p.439). This is because the "true" variance is a characteristic of the sample tested and the "error" variance is a characteristic of the measuring instrument.

Since reliability is restricted to the range 0 to 1, it is convenient to express it as a separation coefficient with range 0 to infinity:

G = Separation = sqrt(Rel./(1-Rel.)) =
"True" Standard Deviation / Error Standard Deviation

In Rasch literature, the "True" S.D. is often called the "Adjusted" (for measurement error) S.D.

Separation is the number of statistically different performance strata that the test can identify in the sample. This can be pictured by placing an error distribution in each stratum (see Examples). A separation of "2" implies that only two levels of performance can be consistently identified by the test for samples like the one tested. Kubiszyn & Borich's 0.95 corresponds to a separation of 4.5, i.e., 4 consistently identifiable strata. (See RMT 6:3 p. 238 for a Table of Reliability - Separation equivalences). Compare the Separation plot with with a different definition of "ability strata" at Number of Person or Item Strata (4G+1)/3 (RMT 16:3).

Example 1: Separation = 2.0, "True" S.D. = 2.0, Error S.D. = 1.0
Reliability = (2.0*2.0) / (2.0*2.0 + 1.0*.1.0) = 0.8

Example 2:Separation = 3.0, "True" S.D. = 3.0, Error S.D. = 1.0
Reliability = (3.0*3.0) / (3.0*3.0 + 1.0*.1.0) = 0.9
In the graph below, notice that the middle error strata is from -1.5 Error S.D. to +1.5 Error S.D.

Guilford J. P. (1965) Fundamental Statistics in Psychology and Education. 4th Edn. New York: McGraw-Hill.

Kubiszyn T., Borich G. (1993) Educational Testing and Measurement. New York: Harper Collins.

Note:

• For separation G, the levels in the true distribution are 3*"True S.D."/G apart, centered on the sample mean.
The levels in the observed distribution are 3*"Observed S.D."/G apart, centered on the sample mean.

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 and separation. Wright BD. … Rasch Measurement Transactions, 1996, 9:4 p.472

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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