What is Information?

The immediate quantification of precision is the standard error (SE) of estimation. The calculation of an estimate and its SE are specified by the estimation model and use the same data. This SE estimates the standard deviation of innumerable independent replications of this data collecting process when the only disturbances encountered are those modelled.

The convenience of the SE is that it is in the units of the estimate and can be used directly to specify regions of confidence, 2 SE's, or margins of error, 1 SE.

The inconvenience of the SE is that, when several pieces of independent data bearing on a common quantity are brought together to form a "better" estimate or when improvement of "precision" is tracked during data collecting, the SE's are not additive. R A Fisher devised a cure for this inconvenience in 1920. While the SE's of independently obtained but commonly bearing estimates are not additive, their inverse squares are. Fisher called 1/SE^2 the "information" (I) in an estimate.

When estimating a measure from a sample of independent observations, the information obtained from each observation (1/SE^2) combine to give the same information as the standard error of the estimated measure (SEM): sum(1/SE^2) = I = 1/SEM^2. For Rasch-modelled dichotomous data, the maximum possible information in one observation, 4, is obtained when a person encounters a perfectly targeted item. This means that 4/SEM^2 is the minimum number of perfectly targeted items it would take to produce the SEM estimated from the data. We will call units of (4/SEM^2), "EQUITS", (EQUivalent on-target ITemS).

The algebraic definition of SE for one Rasch modelled dichotomous datum is SE^2 = 1/[P(1-P)], where P = exp(b-d)/[1+exp(b-d)], the probability of a right answer. For a test, 1/SEM^2 = sum[P(1-P)]. When an item is perfectly targeted, P = 1/2 and P(1-P) = 1/4, so that 4 * sum[P(1-P)] = 4/SEM^2 is the number of perfectly targeted responses necessary to obtain this SEM. We can compare the information values of measures by calculating the EQUITS of information in each one.

The relative information in any pair of measures can be determined by the "Relative Efficiency" (RE) of one measure with respect to the other. The inverse ratio of their error variances, RE21 = SE1^2/SE2^2, gives the "information" provided by the second measure, b2, in units of the "information" provided by the first, b1.

What is Information?, B Wright … Rasch Measurement Transactions, 1990, 4:2 p. 109

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
Rasch Books and Publications: Winsteps and Facets
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	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang	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
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

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