Stochastic Resonance and Rasch Measurement

Stochastic resonance occurs when faint and hard to detect signals are amplified by additional input received from surrounding background noise, paradoxically making them more easily detected than if the background was free of noise. In astronomy, faint stars are more easily observed when the sky has a slight background glow than when it is black. In mechanics, light taps on the box of a hand-held pinball game gets the balls rolling with less tilt. They move more slowly and are easier to control. The taps increase the random motion of the balls, resulting in greater sensitivity and responsiveness. In dentistry, the pain some patients experience is enhanced by the mental noise of fear, apprehension, and agitation. When these phenomena are reduced, so is the pain. (Examples from Ivars Peterson, The Signal Value of Noise, Science News, 2/23/91, and 5/18/91).

Duncan (1984, p. 220) observes that "It is curious that the stochastic model of Rasch, which might be said to involve weaker assumptions than Guttman uses [in his deterministic models], actually leads to a stronger measurement model." Stochastic measurement models provide more useful and meaningful analysis of psycho-social data than deterministic models because stochastic models allow noise to amplify the signal instead of confusing it. But Rasch's model is even stronger because it not only allows noise to amplify signals, but also specifies the optimal amount and kind of noise.

Guttman determinism:
Guttman (1950) follows the requirement that "no scale can really be called a scale unless one can tell from a given attitude [rating or score on an item] that an individual will maintain every attitude falling to the right or to the left of that point" (Murphy et al., 1937 p.897). By allowing no noise, such scales enable the reconstruction of individual response sets from the raw score total alone. Guttman data exhibits this structure:
1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0

Since there is no random noise, there is no way to construct a linear measurement system for person measures or item difficulties. A person has an ability measure greater than the difficulty of the last item answered correctly and less than the difficulty of the first item answered incorrectly, but there is no information about the distance between those two items. The items could have almost the same difficulty or be very different. Further, there is no information as to where best to estimate the person's ability between the two items.

In practice, Guttman's requirement is impossible to meet because people do not share absolutely uniform ability and attitude structures. There is always some further source of variation in abilities and attitudes that often results in coefficients of reproducibility too low for the data to be accepted as meaningful according to Guttman's criteria.

Rasch Stochasticism:
Rasch transforms Guttman's deterministic sense of order into a stochastic one:
"A person having greater ability than another should have the greater probability of solving any item of the type in question, and similarly, one item being more difficult than another one means that for any person the probability of solving the second item correctly is the greater one" (Rasch, 1960, p. 117).

By requiring the probabilities of response, rather than the responses themselves, to maintain an invariant order, Rasch not only brought rigorous measurement within the reach of the human sciences, but also made it more meaningful. The increase in meaningfulness arises because probabilistic models enable the researcher to focus the instrument on individual abilities and attitudes, at the juncture between the known and the unknown, the agreeable and the disagreeable, where the probability of response is 50-50. Data fitting a Rasch model has a structure like this:
1 1 1 1 1 1 1 1 0 1 0 1 0 0 0 0 0

There is maximum randomness about the point at which a person's measure is located. In contrast to Guttman's specifications, a harder item can be answered correctly (or can be found agreeable) after an easier item has been answered incorrectly (or found disagreeable). The size of the stochastic element in the response patterns provides a shared quantitative frame of reference that permits estimation of differences among item difficulties and person abilities. Thus the unavoidable noise in the data contributes information essential to construct quantitative comparisons.

Stochastic Resonance in Educational Measurement:
The meaningfulness and resolution provided by stochastic resonance follows from the relation between part and whole, or text and context, that enables background noise to amplify an otherwise faint signal. Some idea concerning the contextual whole enters into and transforms the perception and interpretation of any part of a text, no matter whether the text read is data from a scientific instrument or a love song. Conversely, understanding of any particular part enters into and transforms the reading of the whole, as well. Thus there develops a circle of interpretation between any part and the whole: a hermeneutic circle.

Qualitative modes of inquiry are aimed at making science by such hermeneutic means. Those using these methods often assume that quantitative work is inherently unhermeneutic, that it always and everywhere takes its data as given, never as constituted out of the interplay of part and whole, text and context. Stochastic resonance challenges these prejudices. Rasch's separability hypothesis refutes them by providing a provocative connection between stochastic resonance and conjoint measurement that opens the door to thoughtful and effective quantitative work in the social and human sciences.

Duncan OD 1984. Measurement and structure: Strategies for the design and analysis of subjective survey data. In CF Turner & E Martin, Surveying Subjective Phenomena, Vol. 1. New York: Russell Sage Foundation.

Guttman L 1950. The basis for scalogram analysis. In SA Stouffer et al. Measurement and Prediction. The American Soldier Vol. IV. New York: Wiley.

Murphy G, Murphy, LB, Newcombe TM 1937. Experimental Social Psychology. Westport, CT: Greenwood Press.

Rasch G 1980. Probabilistic Models for Some Intelligence and Attainment Tests. Chicago: University of Chicago Press.

"Moderately messy systems use resources more efficiently, yield better solutions, and are harder to break than neat ones."
A Perfect Mess: The Hidden Benefits of Disorder--How Crammed Closets, Cluttered Offices, and On-the-Fly Planning Make the World a Better Place. Eric Abrahamson and David H. Freedman. Back Bay Books. 2007

Stochastic resonance and Rasch measurement. Fisher WP Jr. … Rasch Measurement Transactions, 1992, 5:4 p.186

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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