"Subjects in repeated-measures designs provide more than one score" (Introduction to Analysis of Variance, J. R. Turner, Sage, 2001). This motivates well-intentioned statisticians to proclaim that "since the repeated measures cannot be independent, the Rasch model is not appropriate." But this statement can be paradoxical.
If the analysis or repeated-measures is based on raw scores, then the analysis is treating the raw scores as though they represent all the relevant information in the underlying observations. In other words, the raw scores are the "sufficient statistics" for the underlying observations. Further, if a raw score is to represent one quantitative ability, then it must not matter which pattern of observations on a given set of items generated that raw score. If we apply these considerations to a set of responses, we discover that they require the raw scores to fit the Rasch model. See "Rasch Model from Raw Scores as Sufficient Statistics", RMT 3:2, p. 62,
If the reviewers doubt that the Rasch model is applicable, they are also doubting that the raw score is an accurate summary of the observations.
This suggests that the statisticians have their logic backwards. If we have raw scores from a repeated-measures design, we need to submit their underlying observations to Rasch analysis in order to discover whether the raw scores are locally independent enough that they can form the basis for valid statistical analysis.
What if the Rasch analysis does indicate that the raw scores are defective? One approach is to select a subset of the observations that do fit the Rasch model. Then the dependency among the repeated measures will not distort the Rasch measures of the subjects.
For instance, select at random the observations for one time-point for each subject. Use this subset of observations to generate definitive item measures and definitive rating-scale structures (Rasch-Andrich thresholds). Then perform an analysis of the entire dataset with the items and thresholds anchored (fixed) at their definitive values.
Overly predictable data, appearing as overfit to a Rasch model, are rarely a problem because the data become redundant (not needed), but the measures do correspond to the data. Underfit (noisy misfit) to the Rasch model is a problem because the measures do not accurately correspond to the data.
If the time-point data are dependent then
1a) choose one point as definitive
1b) select at random across time points so that each person is in the selection only once.
2) Rasch analyze the data from 1a) or 1b), then output the item difficulties and Rasch-Andrich thresholds 3) Anchor (fix) the item difficulties and Rasch-Andrich thresholds at their values from 2) and analyze all the data. The anchored values prevent the dependency from distorting the estimated measures.
(Suggested by Tsair-Wei Chien, Chi-Mei Medical Center, Tainan, Taiwan.)
Repeated Measure Designs (Time Series) and Rasch T.-W. Chien, Rasch Measurement Transactions, 2008, 22:3, 1171
|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|
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