Rank-ordered Raw Scores imply the Rasch Model

Stout (1990) discusses a reading comprehension test in which there are several passages and a set of questions relating to each passage. Traditionalists, he asserts, would consider each passage a separate dimension. [Though a far stronger tradition counts correct responses to accumulate a reading ability score.] He then comments "The example, by displaying a test that should be psychometrically labelled as `unidimensional', illustrates our view that minor or idiosyncratic dimensions should be ignored in assessing test dimensionality from the applications viewpoint" (p. 302). This practical view of unidimensionality Stout labels "essential unidimensionality". But Stout's view coincides exactly with Rasch's: "In the first place we must admit that on the whole the [Rasch] model gives a satisfactory description of the [BPP-N] data. The deviations found should therefore not lead us to give up the model at once; we should rather look for technical explanations of them" (Rasch 1992, p.91).

Stout discovers that "essential dimensionality" leads to one unique ordinal scale for each latent variable (p. 311), which is given by the raw scores. He then remarks: "Indeed unless there are solid psychometric grounds for preferring one interval scale over the rest, the situation is one of a unique ordinal scale with the choice of a convenient interval scale left up to the practitioner to be decided on pragmatic grounds" (p. 313).

What intervals scale choices are available to the practitioner? Earlier, Stout quoted Lord and Novick, "A major problem of mental test theory is to determine a good interval scaling to impose when the supporting theory implies only ordinal properties (1968)" (p. 304).

Suppose that a unique rank ordering supports two different linearizations. These linearizations must maintain the rank order, but on their different equal-interval metrics. Rasch (p. 121) uses cubes as an example. Cubes can be rank-ordered in size by volume or by length of a side. These always yield the same rank order. Both volume and side-length are expressed as numbers on linear-appearing scales. Do both linearizations produce interval scales of cubeness? Is the choice of scale merely a pragmatic decision for the practitioner? The crux of the matter is the definition of the latent variable. Let us add a one-unit cube to another cube (in such a way as to maintain cubeness). On which "linear" scale will cube-size increment by 1? Clearly the volume scale, not the side-length scale. This was Norman Campbell's (1920) argument. Interval scaling requires a concatenation rule. The concatenation rule depends on the definition of the variable, e.g., "cubeness". Then there is only one linear scale, which is defined by the concatenation. When any linear scaling (which supports concatenation) has been constructed all other scalings are merely linear transformations. The Rasch model constructs linear scales from raw scores (RMT 3:2, p. 62). It also constructs scales that support concatenation (RMT 2:1 p. 16). So the practitioner's pragmatic choice of a good interval scale is reduced to selecting the local origin and unit size for that Rasch scale.

Stout's attempt to eliminate "extraneous requirements such as Rasch's specific objectivity" (p.304), has served to confirm that specific objectivity is fundamental to measurement. Stout's "essential unidimensionality" is seen to be equivalent to "reasonable fit to the Rasch model."

John Michael Linacre

Campbell N.R. (1920) Physics: the elements. Cambridge: Cambridge University Press

Lord F.M., Novick M.R. (1968) Statistical Theories of Mental Test Scores. Reading MA: Addison-Wesley.

Stout W.F. (1990) A new item response theory modeling approach with applications to unidimensionality assessment and ability estimation. Psychometrika 55, 2, 293-325.

Rank-ordered raw scores imply the Rasch model.Wright B.D. … Rasch Measurement Transactions, 1998, 12:2 p. 637-8.

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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