# Disordered Steps?

"I have been analyzing a mathematics examination with the partial credit model. The test consists of 47 items, each of which can have ratings from 0 to 9. On examining the output, I find some step calibrations are out of order. This occurs because a higher proportion of persons get extreme ratings than get intermediate ratings. Thus the rating of a partially correct performance has a more difficult step calibration than the rating of a perfect performance. Of course, as one goes up the scale, the accumulated category count of all those who got each rating or lower monotonically increases. It would appear that such accumulated counts give a clearer picture of the scale. Do you agree with my reasoning?"

Dear Fred:
You observe that "some step calibrations are out of order". The difficulty of reaching a higher step is less than that of a lower step. This shows that not all your categories are modal. Some categories are never most probable at any ability level. David Andrich sees this "disordering" as a symptom of item or rating scale invalidity.

Geoff Masters points out that step difficulty is not the difficulty of performing the category, but only of observing the category relative to its previous category. Imagine a test in which children are asked to count from 1 to 10. The rating scale is:

does not count up to 1 correctly:0
counts up to 1 thru 6 correctly:1
counts up to 7 correctly:2
counts up to 8 thru 10 correctly:3

Each category represents greater success than the previous category. Observing category 2, however, is unlikely when compared to categories 1 or 3. Consequently step difficulties will be disordered. The step from 2 to 3 will be easier than from 1 to 2. This has nothing to do with the difficulty of the tasks. It is determined entirely by our peculiar specification of the rating scale, a specification that David would criticize.

Geoff suggests that instead of thinking of step difficulties, we think of Rasch-Thurstone thresholds (obtained via step difficulties). You have hit on Thurstone's approach with your "accumulated counts." F. Samejima and P. McCullagh also use models like Thurstone's, but it is not a measurement model because it does not allow parameter separability, i.e., it is not "test-free" nor "sample-free" nor has sufficient statistics.

Thurstone's idea, though, is good for interpretation. The Thurstone threshold is that "median" point on the underlying variable at which categories below are as probable as categories above. These thresholds are always in order along the variable. The interval between thresholds can represent that part of the variable corresponding to a category. As Geoff points out, Rasch-Thurstone thresholds can be calculated from step difficulties. These are the thresholds reported by Rasch programs.

Here are the results for your Item 27 (rescaled in WITS). Category 5 is rarely observed, so it is "hard" (58.8) to get into, and "easy" (47.4) to get out of. The category probability curves show only categories 0, 1, 4, 6, 7 as modal. The thresholds are ordered, but crowded together, because each category corresponds to only a narrow interval on the variable.

```----------------------------------
RATING     COUNT   STEP  Rasch-Thurstone THRESHOLD
----------------------------------
0          15    -        -
1          11   52.1     49.3
2           7   53.1     51.3
3           5   53.0*    52.0
4           8   49.8*    52.6
5           2   58.8     53.6
6           7   47.4*    53.9
7           5   55.2*    56.8
----------------------------------
* = disordered step calibrations

CATEGORY PROBABILITIES: MODES
P      ---------------------------------------
R  1.0 +                                     +
O      +                                     +
B      +00                                777+
A      +  00                            77   +
B   .8 +    00                        77     +
I      +      0                     77       +
L      +       00                  7         +
I      +         0               77          +
T   .6 +          0             7            +
Y      +          0            7             +
.5 +           0          7              +
O      +            0         7              +
F   .4 +            0        7               +
+             0      *6666            +
R      +              0    *     66          +
E      +        111111*   67       66        +
S   .2 +      11       *4**4         66      +
P      +   111      22***7  4          666   +
O      +111      222 *3***3  44           666+
N      +     2222 ***66*55****5***           +
S   .0 +***************    0*****************+
E      ---------------------------------------
39    43    47    51    55    59    63

RASCH-THURSTONE THRESHOLDS: MEDIANS
p= .5 +0----------1--23456---7--------------+
---------------------------------------
39    43    47    51    55    59    63
MEASURE
```

Disordered Steps?, F Shaw, B Wright, J Linacre … Rasch Measurement Transactions, 1992, 6:2 p. 225

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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