Archive for the ‘Statistics’ Category

Contesting the Claim, Part II: Are Rasch Measures Really as Objective as Physical Measures?

July 22, 2009

When a raw score is sufficient to the task of measurement, the model is the Rasch model, we can estimate the parameters consistently, and we can evaluate the fit of the data to the model. The invariance properties that follow from a sufficient statistic include virtually the entire class of invariant rules (Hall, Wijsman, & Ghosh, 1965; Arnold, 1985), and similar relationships with other key measurement properties follow from there (Fischer, 1981, 1995; Newby, Conner, Grant, & Bunderson, 2009; Wright, 1977, 1997).

What does this all actually mean? Imagine we were able to ask an infinite number of people an infinite number of questions that all work together to measure the same thing. Because (1) the scores are sufficient statistics, (2) the ruler is not affected by what is measured, (3) the parameters separate, and (4) the data fit the model, any subset of the questions asked would give the same measure. This means that any subscore for any person measured would be a function of any and all other subscores. When a sufficient statistic is a function of all other sufficient statistics, it is not only sufficient, it is necessary, and is referred to as a minimally sufficient statistic. Thus, if separable, independent model parameters can be estimated, the model must be the Rasch model, and the raw score is both sufficient and necessary (Andersen, 1977; Dynkin, 1951; van der Linden, 1992).

This means that scores, ratings, and percentages actually stand for something measurable only when they fit a Rasch model.  After all, what actually would be the point of using data that do not support the estimation of independent parameters? If the meaning of the results is tied in unknown ways to the specific particulars of a given situation, then those results are meaningless, by definition (Roberts & Rosenbaum, 1986; Falmagne & Narens, 1983; Mundy, 1986; Narens, 2002; also see Embretson, 1996; Romanoski and Douglas, 2002). There would be no point in trying to learn anything from them, as whatever happened was a one-time unique event that tells us nothing we can use in any future event (Wright, 1977, 1997).

What we’ve done here is akin to taking a narrative stroll through a garden of mathematical proofs. These conceptual analyses can be very convincing, but actual demonstrations of them are essential. Demonstrations would be especially persuasive if there would be some way of showing three things. First, shouldn’t there be some way of constructing ordinal ratings or scores for one or another physical variable that, when scaled, give us measures that are the same as the usual measures we are accustomed to?

This would show that we can use the type of instrument usually found in the social sciences to construct physical measures with the characteristics we expect. There are four available examples, in fact, involving paired comparisons of weights (Choi, 1998), measures of short lengths (Fisher, 1988), ratings of medium-range distances (Moulton, 1993), and a recovery of the density scale (Pelton & Bunderson, 2003). In each case, the Rasch-calibrated experimental instruments produced measures equivalent to the controls, as shown in linear plots of the pairs of measures.

A second thing to build out from the mathematical proofs are experiments in which we check the purported stability of measures and calibrations. We can do this by splitting large data sets, using different groups of items to produce two or more measures for each person, or using different groups of respondents/examinees to provide data for two or more sets of item calibrations. This is a routine experimental procedure in many psychometric labs, and results tend to conform with theory, with strong associations found between increasing sample sizes and increasing reliability coefficients for the respective measures or calibrations. These associations can be plotted (Fisher, 2008), as can the pairs of calibrations estimated from different samples (Fisher, 1999), and the pairs of measures estimated from different instruments (Fisher, Harvey, Kilgore, et al., 1995; Smith & Taylor, 2004). The theoretical expectation of tighter plots for better designed instruments, larger sample sizes, and longer tests is confirmed so regularly that it should itself have the status of a law of nature (Linacre, 1993).

A third convincing demonstration is to compare studies of the same thing conducted in different times and places by different researchers using different instruments on different samples. If the instruments really measure the same thing, there will not only be obvious similarities in their item contents, but similar items will calibrate in similar positions on the metric across samples. Results of this kind have been obtained in at least three published studies (Fisher, 1997a, 1997b; Belyukova, Stone, & Fox, 2004).

All of these arguments are spelled out in greater length and detail, with illustrations, in a forthcoming article (Fisher, 2009). I learned all of this from Benjamin Wright, who worked directly with Rasch himself, and who, perhaps more importantly, was prepared for what he could learn from Rasch in his previous career as a physicist. Before encountering Rasch in 1960, Wright had worked with Feynman at Cornell, Townes at Bell Labs, and Mulliken at the University of Chicago. Taught and influenced not just by three of the great minds of twentieth-century physics, but also by Townes’ philosophical perspectives on meaning and beauty, Wright had left physics in search of life. He was happy to transfer his experience with computers into his new field of educational research, but he was dissatisfied with the quality of the data and how it was treated.

Rasch’s ideas gave Wright the conceptual tools he needed to integrate his scientific values with the demands of the field he was in. Over the course of his 40-year career in measurement, Wright wrote the first software for estimating Rasch model parameters and continuously improved it; he adapted new estimation algorithms for Rasch’s models and was involved in the articulation of new models; he applied the models to hundreds of data sets using his software; he vigorously invested himself in students and colleagues; he founded new professional societies, meetings, and journals;  and he never stopped learning how to think anew about measurement and the meaning of numbers. Through it all, there was always a yardstick handy as a simple way of conveying the basic requirements of measurement as we intuitively understand it in physical terms.

Those of us who spend a lot of time working with these ideas and trying them out on lots of different kinds of data forget or never realize how skewed our experience is relative to everyone else’s. I guess a person lives in a different world when you have the sustained luxury of working with very large databases, as I have had, and you see the constancy and stability of well-designed measures and calibrations over time, across instruments, and over repeated samples ranging from 30 to several million.

When you have that experience, it becomes a basic description of reasonable expectation to read the work of a colleague and see him say that “when the key features of a statistical model relevant to the analysis of social science data are the same as those of the laws of physics, then those features are difficult to ignore” (Andrich, 1988, p. 22). After calibrating dozens of instruments over 25 years, some of them many times over, it just seems like the plainest statement of the obvious to see the same guy say “Our measurement principles should be the same for properties of rocks as for the properties of people. What we say has to be consistent with physical measurement” (Andrich, 1998, p. 3).

And I find myself wishing more people held the opinion expressed by two other colleagues, that “scientific measures in the social sciences must hold to the same standards as do measures in the physical sciences if they are going to lead to the same quality of generalizations” (Bond & Fox, 2001, p. 2). When these sentiments are taken to their logical conclusion in a practical application, the real value of “attempting for reading comprehension what Newtonian mechanics achieved for astronomy” (Burdick & Stenner, 1996) becomes apparent. Rasch’s analogy of the structure of his model for reading tests and Newton’s Second Law can be restated relative to any physical law expressed as universal conditionals among variable triplets; a theory of the variable measured capable of predicting item calibrations provides the causal story for the observed variation (Burdick, Stone, & Stenner, 2006; DeBoeck & Wilson, 2004).

Knowing what I know, from the mathematical principles I’ve been trained in and from the extensive experimental work I’ve done, it seems amazing that so little attention is actually paid to tools and concepts that receive daily lip service as to their central importance in every facet of life, from health care to education to economics to business. Measurement technology rose up decades ago in preparation for the demands of today’s challenges. It is just plain weird the way we’re not using it to anything anywhere near its potential.

I’m convinced, though, that the problem is not a matter of persuasive rhetoric applied to the minds of the right people. Rather, someone, hopefully me, has got to configure the right combination of players in the right situation at the right time and at the right place to create a new form of real value that can’t be created any other way. Like they say, money talks. Persuasion is all well and good, but things will really take off only when people see that better measurement can aid in removing inefficiencies from the management of human, social, and natural capital, that better measurement is essential to creating sustainable and socially responsible policies and practices, and that better measurement means new sources of profitability.  I’m convinced that advanced measurement techniques are really nothing more than a new form of IT or communications technology. They will fit right into the existing networks and multiply their efficiencies many times over.

And when they do, we may be in a position to finally

“confront the remarkable fact that throughout the gigantic range of physical knowledge numerical laws assume a remarkably simple form provided fundamental measurement has taken place. Although the authors cannot explain this fact to their own satisfaction, the extension to behavioral science is obvious: we may have to await fundamental measurement before we will see any real progress in quantitative laws of behavior. In short, ordinal scales (even continuous ordinal scales) are perhaps not good enough and it may not be possible to live forever with a dozen different procedures for quantifying the same piece of behavior, each making strong but untestable and basically unlikely assumptions which result in nonlinear plots of one scale against another. Progress in physics would have been impossibly difficult without fundamental measurement and the reader who believes that all that is at stake in the axiomatic treatment of measurement is a possible criterion for canonizing one scaling procedure at the expense of others is missing the point” (Ramsay, Bloxom, and Cramer, 1975, p. 262).

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Publications Documenting Score, Rating, Percentage Contrasts with Real Measures

July 7, 2009

A few brief and easy introductions to the contrast between scores, ratings, and percentages vs measures include:

Linacre, J. M. (1992, Autumn). Why fuss about statistical sufficiency? Rasch Measurement Transactions, 6(3), 230 [http://www.rasch.org/rmt/rmt63c.htm].

Linacre, J. M. (1994, Summer). Likert or Rasch? Rasch Measurement Transactions, 8(2), 356 [http://www.rasch.org/rmt/rmt82d.htm].

Wright, B. D. (1992, Summer). Scores are not measures. Rasch Measurement Transactions, 6(1), 208 [http://www.rasch.org/rmt/rmt61n.htm].

Wright, B. D. (1989). Rasch model from counting right answers: Raw scores as sufficient statistics. Rasch Measurement Transactions, 3(2), 62 [http://www.rasch.org/rmt/rmt32e.htm].

Wright, B. D. (1993). Thinking with raw scores. Rasch Measurement Transactions, 7(2), 299-300 [http://www.rasch.org/rmt/rmt72r.htm].

Wright, B. D. (1999). Common sense for measurement. Rasch Measurement Transactions, 13(3), 704-5  [http://www.rasch.org/rmt/rmt133h.htm].

Longer and more technical comparisons include:

Andrich, D. (1989). Distinctions between assumptions and requirements in measurement in the social sciences. In J. A. Keats, R. Taft, R. A. Heath & S. H. Lovibond (Eds.), Mathematical and Theoretical Systems: Proceedings of the 24th International Congress of Psychology of the International Union of Psychological Science, Vol. 4 (pp. 7-16). North-Holland: Elsevier Science Publishers.

van Alphen, A., Halfens, R., Hasman, A., & Imbos, T. (1994). Likert or Rasch? Nothing is more applicable than good theory. Journal of Advanced Nursing, 20, 196-201.

Wright, B. D., & Linacre, J. M. (1989). Observations are always ordinal; measurements, however, must be interval. Archives of Physical Medicine and Rehabilitation, 70(12), 857-867 [http://www.rasch.org/memo44.htm].

Zhu, W. (1996). Should total scores from a rating scale be used directly? Research Quarterly for Exercise and Sport, 67(3), 363-372.

The following lists provide some key resources. The lists are intended to be representative, not comprehensive.  There are many works in addition to these that document the claims in yesterday’s table. Many of these books and articles are highly technical.  Good introductions can be found in Bezruczko (2005), Bond and Fox (2007), Smith and Smith (2004), Wilson (2005), Wright and Stone (1979), Wright and Masters (1982), Wright and Linacre (1989), and elsewhere. The www.rasch.org web site has comprehensive and current information on seminars, consultants, software, full text articles, professional association meetings, etc.

Books and Journal Issues

Andrich, D. (1988). Rasch models for measurement. Sage University Paper Series on Quantitative Applications in the Social Sciences, vol. series no. 07-068. Beverly Hills, California: Sage Publications.

Andrich, D., & Douglas, G. A. (Eds.). (1982). Rasch models for measurement in educational and psychological research [Special issue]. Education Research and Perspectives, 9(1), 5-118. [Full text available at www.rasch.org.]

Bezruczko, N. (Ed.). (2005). Rasch measurement in health sciences. Maple Grove, MN: JAM Press.

Bond, T., & Fox, C. (2007). Applying the Rasch model: Fundamental measurement in the human sciences, 2d edition. Mahwah, New Jersey: Lawrence Erlbaum Associates.

Choppin, B. (1985). In Memoriam: Bruce Choppin (T. N. Postlethwaite ed.) [Special issue]. Evaluation in Education: An International Review Series, 9(1).

DeBoeck, P., & Wilson, M. (Eds.). (2004). Explanatory item response models: A generalized linear and nonlinear approach. Statistics for Social and Behavioral Sciences). New York: Springer-Verlag.

Embretson, S. E., & Hershberger, S. L. (Eds.). (1999). The new rules of measurement: What every psychologist and educator should know. Hillsdale, New Jersey: Lawrence Erlbaum Associates.

Engelhard, G., Jr., & Wilson, M. (1996). Objective measurement: Theory into practice, Vol. 3. Norwood, New Jersey: Ablex.

Fischer, G. H., & Molenaar, I. (1995). Rasch models: Foundations, recent developments, and applications. New York: Springer-Verlag.

Fisher, W. P., Jr., & Wright, B. D. (Eds.). (1994). Applications of Probabilistic Conjoint Measurement [Special Issue]. International Journal of Educational Research, 21(6), 557-664.

Garner, M., Draney, K., Wilson, M., Engelhard, G., Jr., & Fisher, W. P., Jr. (Eds.). (2009). Advances in Rasch measurement, Vol. One. Maple Grove, MN: JAM Press.

Granger, C. V., & Gresham, G. E. (Eds). (1993, August). New Developments in Functional Assessment [Special Issue]. Physical Medicine and Rehabilitation Clinics of North America, 4(3), 417-611.

Linacre, J. M. (1989). Many-facet Rasch measurement. Chicago, Illinois: MESA Press.

Liu, X., & Boone, W. (2006). Applications of Rasch measurement in science education. Maple Grove, MN: JAM Press.

Masters, G. N. (2007). Special issue: Programme for International Student Assessment (PISA). Journal of Applied Measurement, 8(3), 235-335.

Masters, G. N., & Keeves, J. P. (Eds.). (1999). Advances in measurement in educational research and assessment. New York: Pergamon.

Osborne, J. W. (Ed.). (2007). Best practices in quantitative methods. Thousand Oaks, CA: Sage.

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests (Reprint, with Foreword and Afterword by B. D. Wright, Chicago: University of Chicago Press, 1980). Copenhagen, Denmark: Danmarks Paedogogiske Institut.

Smith, E. V., Jr., & Smith, R. M. (Eds.) (2004). Introduction to Rasch measurement. Maple Grove, MN: JAM Press.

Smith, E. V., Jr., & Smith, R. M. (2007). Rasch measurement: Advanced and specialized applications. Maple Grove, MN: JAM Press.

Smith, R. M. (Ed.). (1997, June). Outcome Measurement [Special Issue]. Physical Medicine & Rehabilitation State of the Art Reviews, 11(2), 261-428.

Smith, R. M. (1999). Rasch measurement models. Maple Grove, MN: JAM Press.

von Davier, M. (2006). Multivariate and mixture distribution Rasch models. New York: Springer.

Wilson, M. (1992). Objective measurement: Theory into practice, Vol. 1. Norwood, New Jersey: Ablex.

Wilson, M. (1994). Objective measurement: Theory into practice, Vol. 2. Norwood, New Jersey: Ablex.

Wilson, M. (2005). Constructing measures: An item response modeling approach. Mahwah, New Jersey: Lawrence Erlbaum Associates.

Wilson, M., Draney, K., Brown, N., & Duckor, B. (Eds.). (2009). Advances in Rasch measurement, Vol. Two (p. in press). Maple Grove, MN: JAM Press.

Wilson, M., & Engelhard, G. (2000). Objective measurement: Theory into practice, Vol. 5. Westport, Connecticut: Ablex Publishing.

Wilson, M., Engelhard, G., & Draney, K. (Eds.). (1997). Objective measurement: Theory into practice, Vol. 4. Norwood, New Jersey: Ablex.

Wright, B. D., & Masters, G. N. (1982). Rating scale analysis: Rasch measurement. Chicago, Illinois: MESA Press.

Wright, B. D., & Stone, M. H. (1979). Best test design: Rasch measurement. Chicago, Illinois: MESA Press.

Wright, B. D., & Stone, M. H. (1999). Measurement essentials. Wilmington, DE: Wide Range, Inc. [http://www.rasch.org/memos.htm#measess].

Key Articles

Andersen, E. B. (1977). Sufficient statistics and latent trait models. Psychometrika, 42(1), 69-81.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561-73.

Andrich, D. (2002). Understanding resistance to the data-model relationship in Rasch’s paradigm: A reflection for the next generation. Journal of Applied Measurement, 3(3), 325-59.

Andrich, D. (2004, January). Controversy and the Rasch model: A characteristic of incompatible paradigms? Medical Care, 42(1), I-7–I-16.

Beltyukova, S. A., Stone, G. E., & Fox, C. M. (2008). Magnitude estimation and categorical rating scaling in social sciences: A theoretical and psychometric controversy. Journal of Applied Measurement, 9(2), 151-159.

Choppin, B. (1968). An item bank using sample-free calibration. Nature, 219, 870-872.

Embretson, S. E. (1996, September). Item Response Theory models and spurious interaction effects in factorial ANOVA designs. Applied Psychological Measurement, 20(3), 201-212.

Engelhard, G. (2008, July). Historical perspectives on invariant measurement: Guttman, Rasch, and Mokken. Measurement: Interdisciplinary Research & Perspectives, 6(3), 155-189.

Fischer, G. H. (1973). The linear logistic test model as an instrument in educational research. Acta Psychologica, 37, 359-374.

Fischer, G. H. (1981, March). On the existence and uniqueness of maximum-likelihood estimates in the Rasch model. Psychometrika, 46(1), 59-77.

Fischer, G. H. (1989). Applying the principles of specific objectivity and of generalizability to the measurement of change. Psychometrika, 52(4), 565-587.

Fisher, W. P., Jr. (1997). Physical disability construct convergence across instruments: Towards a universal metric. Journal of Outcome Measurement, 1(2), 87-113.

Fisher, W. P., Jr. (2004, October). Meaning and method in the social sciences. Human Studies: A Journal for Philosophy and the Social Sciences, 27(4), 429-54.

Fisher, W. P., Jr. (2009, July). Invariance and traceability for measures of human, social, and natural capital: Theory and application. Measurement (Elsevier), in press.

Grosse, M. E., & Wright, B. D. (1986, Sep). Setting, evaluating, and maintaining certification standards with the Rasch model. Evaluation & the Health Professions, 9(3), 267-285.

Hall, W. J., Wijsman, R. A., & Ghosh, J. K. (1965). The relationship between sufficiency and invariance with applications in sequential analysis. Annals of Mathematical Statistics, 36, 575-614.

Kamata, A. (2001, March). Item analysis by the Hierarchical Generalized Linear Model. Journal of Educational Measurement, 38(1), 79-93.

Karabatsos, G., & Ullrich, J. R. (2002). Enumerating and testing conjoint measurement models. Mathematical Social Sciences, 43, 487-505.

Linacre, J. M. (1997). Instantaneous measurement and diagnosis. Physical Medicine and Rehabilitation State of the Art Reviews, 11(2), 315-324.

Linacre, J. M. (2002). Optimizing rating scale category effectiveness. Journal of Applied Measurement, 3(1), 85-106.

Lunz, M. E., & Bergstrom, B. A. (1991). Comparability of decision for computer adaptive and written examinations. Journal of Allied Health, 20(1), 15-23.

Lunz, M. E., Wright, B. D., & Linacre, J. M. (1990). Measuring the impact of judge severity on examination scores. Applied Measurement in Education, 3/4, 331-345.

Masters, G. N. (1985, March). Common-person equating with the Rasch model. Applied Psychological Measurement, 9(1), 73-82.

Mislevy, R. J., Steinberg, L. S., & Almond, R. G. (2003). On the structure of educational assessments. Measurement: Interdisciplinary Research and Perspectives, 1(1), 3-62.

Pelton, T., & Bunderson, V. (2003). The recovery of the density scale using a stochastic quasi-realization of additive conjoint measurement. Journal of Applied Measurement, 4(3), 269-81.

Rasch, G. (1961). On general laws and the meaning of measurement in psychology. In Proceedings of the fourth Berkeley symposium on mathematical statistics and probability (pp. 321-333 [http://www.rasch.org/memo1960.pdf]). Berkeley, California: University of California Press.

Rasch, G. (1966). An individualistic approach to item analysis. In P. F. Lazarsfeld & N. W. Henry (Eds.), Readings in mathematical social science (pp. 89-108). Chicago, Illinois: Science Research Associates.

Rasch, G. (1966, July). An informal report on the present state of a theory of objectivity in comparisons. Unpublished paper [http://www.rasch.org/memo1966.pdf].

Rasch, G. (1966). An item analysis which takes individual differences into account. British Journal of Mathematical and Statistical Psychology, 19, 49-57.

Rasch, G. (1968, September 6). A mathematical theory of objectivity and its consequences for model construction. [Unpublished paper [http://www.rasch.org/memo1968.pdf]], Amsterdam, the Netherlands: Institute of Mathematical Statistics, European Branch.

Rasch, G. (1977). On specific objectivity: An attempt at formalizing the request for generality and validity of scientific statements. Danish Yearbook of Philosophy, 14, 58-94.

Romanoski, J. T., & Douglas, G. (2002). Rasch-transformed raw scores and two-way ANOVA: A simulation analysis. Journal of Applied Measurement, 3(4), 421-430.

Smith, R. M. (1996). A comparison of methods for determining dimensionality in Rasch measurement. Structural Equation Modeling, 3(1), 25-40.

Smith, R. M. (2000). Fit analysis in latent trait measurement models. Journal of Applied Measurement, 1(2), 199-218.

Stenner, A. J., & Smith III, M. (1982). Testing construct theories. Perceptual and Motor Skills, 55, 415-426.

Stenner, A. J. (1994). Specific objectivity – local and general. Rasch Measurement Transactions, 8(3), 374 [http://www.rasch.org/rmt/rmt83e.htm].

Stone, G. E., Beltyukova, S. A., & Fox, C. M. (2008). Objective standard setting for judge-mediated examinations. International Journal of Testing, 8(2), 180-196.

Stone, M. H. (2003). Substantive scale construction. Journal of Applied Measurement, 4(3), 282-97.

Wilson, M., & Sloane, K. (2000). From principles to practice: An embedded assessment system. Applied Measurement in Education, 13(2), 181-208.

Wright, B. D. (1968). Sample-free test calibration and person measurement. In Proceedings of the 1967 invitational conference on testing problems (pp. 85-101 [http://www.rasch.org/memo1.htm]). Princeton, New Jersey: Educational Testing Service.

Wright, B. D. (1977). Solving measurement problems with the Rasch model. Journal of Educational Measurement, 14(2), 97-116 [http://www.rasch.org/memo42.htm].

Wright, B. D. (1980). Foreword, Afterword. In Probabilistic models for some intelligence and attainment tests, by Georg Rasch (pp. ix-xix, 185-199. http://www.rasch.org/memo63.htm). Chicago, Illinois: University of Chicago Press.

Wright, B. D. (1984). Despair and hope for educational measurement. Contemporary Education Review, 3(1), 281-288 [http://www.rasch.org/memo41.htm].

Wright, B. D. (1985). Additivity in psychological measurement. In E. Roskam (Ed.), Measurement and personality assessment. North Holland: Elsevier Science Ltd.

Wright, B. D. (1996). Comparing Rasch measurement and factor analysis. Structural Equation Modeling, 3(1), 3-24.

Wright, B. D. (1997, June). Fundamental measurement for outcome evaluation. Physical Medicine & Rehabilitation State of the Art Reviews, 11(2), 261-88.

Wright, B. D. (1997, Winter). A history of social science measurement. Educational Measurement: Issues and Practice, 16(4), 33-45, 52 [http://www.rasch.org/memo62.htm].

Wright, B. D. (1999). Fundamental measurement for psychology. In S. E. Embretson & S. L. Hershberger (Eds.), The new rules of measurement: What every educator and psychologist should know (pp. 65-104 [http://www.rasch.org/memo64.htm]). Hillsdale, New Jersey: Lawrence Erlbaum Associates.

Wright, B. D., & Bell, S. R. (1984, Winter). Item banks: What, why, how. Journal of Educational Measurement, 21(4), 331-345 [http://www.rasch.org/memo43.htm].

Wright, B. D., & Linacre, J. M. (1989). Observations are always ordinal; measurements, however, must be interval. Archives of Physical Medicine and Rehabilitation, 70(12), 857-867 [http://www.rasch.org/memo44.htm].

Wright, B. D., & Mok, M. (2000). Understanding Rasch measurement: Rasch models overview. Journal of Applied Measurement, 1(1), 83-106.

Model Applications

Adams, R. J., Wu, M. L., & Macaskill, G. (1997). Scaling methodology and procedures for the mathematics and science scales. In M. O. Martin & D. L. Kelly (Eds.), Third International Mathematics and Science Study Technical Report: Vol. 2: Implementation and Analysis – Primary and Middle School Years. Boston: Center for the Study of Testing, Evaluation, and Educational Policy.

Andrich, D., & Van Schoubroeck, L. (1989, May). The General Health Questionnaire: A psychometric analysis using latent trait theory. Psychological Medicine, 19(2), 469-485.

Beltyukova, S. A., Stone, G. E., & Fox, C. M. (2004). Equating student satisfaction measures. Journal of Applied Measurement, 5(1), 62-9.

Bergstrom, B. A., & Lunz, M. E. (1999). CAT for certification and licensure. In F. Drasgow & J. B. Olson-Buchanan (Eds.), Innovations in computerized assessment (pp. 67-91). Mahwah, New Jersey: Lawrence Erlbaum Associates, Inc., Publishers.

Bond, T. G. (1994). Piaget and measurement II: Empirical validation of the Piagetian model. Archives de Psychologie, 63, 155-185.

Bunderson, C. V., & Newby, V. A. (2009). The relationships among design experiments, invariant measurement scales, and domain theories. Journal of Applied Measurement, 10(2), 117-137.

Cavanagh, R. F., & Romanoski, J. T. (2006, October). Rating scale instruments and measurement. Learning Environments Research, 9(3), 273-289.

Cipriani, D., Fox, C., Khuder, S., & Boudreau, N. (2005). Comparing Rasch analyses probability estimates to sensitivity, specificity and likelihood ratios when examining the utility of medical diagnostic tests. Journal of Applied Measurement, 6(2), 180-201.

Dawson, T. L. (2004, April). Assessing intellectual development: Three approaches, one sequence. Journal of Adult Development, 11(2), 71-85.

DeSalvo, K., Fisher, W. P. Jr., Tran, K., Bloser, N., Merrill, W., & Peabody, J. W. (2006, March). Assessing measurement properties of two single-item general health measures. Quality of Life Research, 15(2), 191-201.

Engelhard, G., Jr. (1992). The measurement of writing ability with a many-faceted Rasch model. Applied Measurement in Education, 5(3), 171-191.

Engelhard, G., Jr. (1997). Constructing rater and task banks for performance assessment. Journal of Outcome Measurement, 1(1), 19-33.

Fisher, W. P., Jr. (1998). A research program for accountable and patient-centered health status measures. Journal of Outcome Measurement, 2(3), 222-239.

Fisher, W. P., Jr., Harvey, R. F., Taylor, P., Kilgore, K. M., & Kelly, C. K. (1995, February). Rehabits: A common language of functional assessment. Archives of Physical Medicine and Rehabilitation, 76(2), 113-122.

Heinemann, A. W., Gershon, R., & Fisher, W. P., Jr. (2006). Development and application of the Orthotics and Prosthetics User Survey: Applications and opportunities for health care quality improvement. Journal of Prosthetics and Orthotics, 18(1), 80-85 [http://www.oandp.org/jpo/library/2006_01S_080.asp].

Heinemann, A. W., Linacre, J. M., Wright, B. D., Hamilton, B. B., & Granger, C. V. (1994). Prediction of rehabilitation outcomes with disability measures. Archives of Physical Medicine and Rehabilitation, 75(2), 133-143.

Hobart, J. C., Cano, S. J., O’Connor, R. J., Kinos, S., Heinzlef, O., Roullet, E. P., C., et al. (2003). Multiple Sclerosis Impact Scale-29 (MSIS-29):  Measurement stability across eight European countries. Multiple Sclerosis, 9, S23.

Hobart, J. C., Cano, S. J., Zajicek, J. P., & Thompson, A. J. (2007, December). Rating scales as outcome measures for clinical trials in neurology: Problems, solutions, and recommendations. Lancet Neurology, 6, 1094-1105.

Lai, J., Fisher, A., Magalhaes, L., & Bundy, A. C. (1996). Construct validity of the sensory integration and praxis tests. Occupational Therapy Journal of Research, 16(2), 75-97.

Lee, N. P., & Fisher, W. P., Jr. (2005). Evaluation of the Diabetes Self Care Scale. Journal of Applied Measurement, 6(4), 366-81.

Ludlow, L. H., & Haley, S. M. (1995, December). Rasch model logits: Interpretation, use, and transformation. Educational and Psychological Measurement, 55(6), 967-975.

Markward, N. J., & Fisher, W. P., Jr. (2004). Calibrating the genome. Journal of Applied Measurement, 5(2), 129-41.

Massof, R. W. (2007, August). An interval-scaled scoring algorithm for visual function questionnaires. Optometry & Vision Science, 84(8), E690-E705.

Massof, R. W. (2008, July-August). Editorial: Moving toward scientific measurements of quality of life. Ophthalmic Epidemiology, 15, 209-211.

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