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
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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 |
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