Category boundaries and thresholds

Conceptualizing rating scales and partial-credit response structures for communication can be challenging. Rasch measurement provides several approaches. Choose the one that is most meaningful for you.

 

Look at this edited excerpt of Table 3.2:

 

--------------------------------------------------------------------------------------

|      DATA            |  QUALITY CONTROL  |   Andrich   |  EXPECTATION  |50% Cumulat|

| Category Counts  Cum.| Avge  Exp.  OUTFIT| Thresholds  |  Measure at   |Probability|

|Score   Used   %    % | Meas  Meas   MnSq |Measure  S.E.|Category  Zone |      at   |

--------------------------------------------------------------------------------------

|  0      891   6%   6%|  -.04   -.07  1.3 |             |( -2.27)       |    low    |

|  1      383   3%   9%|   .47    .51  1.3 |  -.91    .04|   -.68   -1.53|   -1.23   |

|  2     1017   7%  15%|  1.07   1.17   .8 | -.15    .04|    .63   -.03|  -.07  |

|  3    12683  85% 100%|  2.15   2.14  1.0 |  1.06    .03|(  2.34)   1.55|    1.29   |

------------------------------------------------------------(Mean)---------(Median)---

 

Here are three ways of conceptualizing and communicating the transition, threshold, boundary between category 1 and category 2:

 

(1) Rasch-half-point thresholds (Zone). Someone at the boundary between "1" and "2" would have an expected rating of 1.5, or 1000 persons at the boundary between "1" and "2" would have an average rating of 1.5. This boundary is the "zone" = "Expected Measure at 1+0.5 or 2 -0.5" which is -.03 logits, the Rasch-half-point threshold. To illustrate this, use the model item characteristic curve. The expected score ogive / model ICC (Table 21.2 - second on list in Graphs menu). The CAT+.25, CAT-0.5, AT CAT, and CAT-.25 columns in the ISFILE= plot points on this ogive. The expected score ogive relates most directly to the estimation of the Rasch parameters. Since it is only one line, it is also convenient for summarizing performance at any point on the latent variable by one number. Crucial points are the points on the variable corresponding to the lower category value + 0.5, i..e, more than the higher adjacent category value - 0.5. These Rasch-half-point thresholds are "average score thresholds" or "Rasch-ICC thresholds".

 

(2) Rasch-Thurstone thresholds (50% Cumulative Probability). Someone at the boundary between "1" and "2" would have a 50% chance of being rated 1 or below, and a 50% chance of being rated 2 or above. This is the Rasch-Thurstone threshold of -.07 logits. To illustrate this, use the cumulative probability curves. The cumulative probability curves (Table 21.3 - and third on list in Graphs menu). The 50%PRB columns in the ISFILE= are the crucial points on these curves. and are the Rasch-Thurstone thresholds, useful for identifying whether a person is most likely to respond below, at or above a certain category.

 

(3) Rasch-Andrich thresholds. Someone at the boundary between "1" and "2" would have an equal chance of being rated 1 or 2. This is the Rasch--Andrich Threshold of -.15 logits. To illustrate this, use the category probability curves. The probability curves (Table 21.1 - and top of list in Graphs menu). The Andrich Threshold in the ISFILE= gives the point of equal probability between adjacent categories. The points of highest probability of intermediate categories are given by the AT CAT values. These probability curves relate most directly to the Rasch parameter values, also called Rasch-Andrich thresholds. They are at the intersection of adjacent probability curves, and indicate when the probability of being observed in the higher category starts to exceed that of being observed in the adjacent lower one. This considers the categories two at a time, but can lead to misinference if there is Rasch-Andrich threshold disordering.

 

d) Empirical average measures. For any particular sample, there is the average ability of the people who scored in any particular category of any particular item. This is the "Average Measure" reported in Table 3.2. This is entirely sample-dependent. It is not reported in ISFILE=. In the Table, the empirical average measure of the persons responding in category 1 is .47 logits, and in category 2 is 1.07 logits. Half-way between is .77 logits.


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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 with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments, George Engelhard, Jr. & Stefanie Wind 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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