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.
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
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
The URL of this page is www.rasch.org/rmt/rmt52j.htm
Website: www.rasch.org/rmt/contents.htm