Beyond Partial Credit: Rasch Success and Failure Models

The extraction of measures from rating scale observations (Andrich 1978) and from items scored for partial credit (Masters 1982) was a major conceptual and statistical advance. Now a wide variety of measurement models for attitude and intermediate levels of performance have followed (Glas, Verhelst 1991; Verhelst, Glas, de Vries, 1997; Linacre 1991). Here are two of these compared with their Andrich/Masters progenitors.

Glas-Verhelst "Success" models growth. Higher steps are administered only after success on lower ones. Persons who fail ("0") on Step 1, do not qualify for Step 2 or Step 3, and have no opportunity to respond to them. This results in missing ("M") responses, rather than pre-determined failures. Each rating level implies a particular count of dichotomous item-step successes based on the administration of a particular number of steps. Introducing higher levels does not alter the calibration or success probability of lower levels. Higher levels can be added, eliminated, combined or redefined without affecting lower level structure. Examples are high-jumping and weight-lifting competitions.

Linacre "Failure" models mastery, as in answer-until-correct and where later tasks are presented only when earlier tasks are failed. The "Failure" model is the reverse of the "Success" model. Lower steps are administered only after failure on higher steps. Step 2 is administered only when Step 3 has been failed. A success on Step 3 means that Steps 1 and 2 are not administered. This results in missing, "M", rather than pre-determined successes. Introducing lower levels does not alter the calibration or success probability of the higher steps. Examples are answer-until-correct and a baseball at-bat.

Andrich "Rating Scale" and Masters "Partial Credit" model attitude and other closed rating scales. The selection of a level implies a position along a complete continuum. Since the scale can be understood as ascent from the bottom, descent from the top, or digression from the center, the levels specify a conceptual hierarchy, but do not require a path that must be followed. The algebraic representation as ascent is only a mathematical convenience. Adding an extra level redefines the rating scale and alters the calibrations and probability of attainment of every step. Examples are attitude surveys and employee performance ratings.

Later note: My experience is that it is better to analyze Success and Failure attempts as a standard rating scale. There are usually too many extraneous factors for pure Success or Failure models to function effectively.

Andrich D 1978. A rating formulation for ordered response categories. Psychometrika 43 561-573

Masters GN 1982. A Rasch model for partial credit scoring. Psychometrika 47 149-174

Glas CAW & Verhelst ND 1991. Using the Rasch model for dichotomous data for analyzing polytomous responses. Measurement & Research Dept Report 91-3. Arnhem, The Netherlands: CITO

Kutylowski A 1991 Sequential Models for Rating Scales, Rasch Measurement Transactions, 5:3 p. 161

Linacre JM 1991. Structured rating scales. Sixth International Objective Measurement Workshop. April. Chicago. ERIC TM 016615

Verhelst N.D., Glas C.A.W. & De Vries H.H. (1997) A Steps model to analyze partial credit. In W.J. van der Linden & R.K. Hambleton (Eds.), Handbook of modern item response theory (pp. 123 - 138) New York: Springer.

-----------------------------------------------------
Observed       Implied            Response
Score          Dichotomies        Interpretation
-----------------------------------------------------
      Glas-Verhelst  "Success" ("Steps" model)
----------------------------------------------------
Rating    Step  Step  Step     Steps  Successes
level      1      2    3       Tried  Scored
----------------------------------------------------
3-Hard     1      1    1         3      3
 2         1      1    0         3      2
 1         1      0    M         2      1
0-Easy     0      M    M         1      0
----------------------------------------------------

      Linacre "Failure"	model
----------------------------------------------------
Rating    Step  Step  Step     Steps  Successes
level      1      2    3       Tried  Scored
----------------------------------------------------
3-Hard     M      M    1         1      1
 2         M      1    0         2      1
 1         1      0    0         3      1
0-Easy     0      0    0         3      0
----------------------------------------------------

       Andrich "Rating Scale" model
       Masters "Partial Credit" model
----------------------------------------------------
Rating    Step  Step  Step     Steps  Successes
level      1      2    3       Tried  Scored
----------------------------------------------------
3-Hard     1      1    1         3      3
 2         1      1    0         3      2
 1         1      0    0         3      1
0-Easy     0      0    0         3      0
----------------------------------------------------

Key: 1=Passed, 0=Failed, M=Not administered

Beyond Partial Credit: Rasch Success and Failure Models, J Linacre … Rasch Measurement Transactions, 1991, 5:2 p. 155

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
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