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.

Masters, G. N., Adams, R. J., & Lokan, J. (1994). Mapping student achievement. *International Journal of Educational Research, 21*(6), 595-610.

Mead, R. J. (2009). The ISR: Intelligent Student Reports. *Journal of Applied Measurement, 10*(2), 208-224.

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.

Smith, E. V., Jr. (2000). Metric development and score reporting in Rasch measurement. *Journal of Applied Measurement, 1*(3), 303-26.

Smith, R. M., & Taylor, P. (2004). Equating rehabilitation outcome scales: Developing common metrics. *Journal of Applied Measurement, 5*(3), 229-42.

Solloway, S., & Fisher, W. P., Jr. (2007). Mindfulness in measurement: Reconsidering the measurable in mindfulness. *International Journal of Transpersonal Studies, 26*, 58-81 [http://www.transpersonalstudies.org/volume_26_2007.html].

Stenner, A. J. (2001). The Lexile Framework: A common metric for matching readers and texts. *California School Library Journal, 25*(1), 41-2.

Wolfe, E. W., Ray, L. M., & Harris, D. C. (2004, October). A Rasch analysis of three measures of teacher perception generated from the School and Staffing Survey. *Educational and Psychological Measurement, 64*(5), 842-860.

Wolfe, F., Hawley, D., Goldenberg, D., Russell, I., Buskila, D., & Neumann, L. (2000, Aug). The assessment of functional impairment in fibromyalgia (FM): Rasch analyses of 5 functional scales and the development of the FM Health Assessment Questionnaire. *Journal of Rheumatology, 27*(8), 1989-99.

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endt, A., & Tatum, D. S. (2005). Credentialing health care professionals. In N. Bezruczko (Ed.), *Rasch measurement in health sciences* (pp. 161-75). Maple Grove, MN: JAM Press.