(See Table 1, Table 3.2, Table 12, Table 21, Graphs)
There are several ways of conceptualizing a rating scale item. They all contain exactly the same measurement information, but communicated in different ways. Usually, one of these alternatives will be most meaningful for your audience.
The plots corresponding to these approaches are shown in Table 21, and also on the Graphs screen.
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|CATEGORY OBSERVED|OBSVD SAMPLE|INFIT OUTFIT|| ANDRICH |CATEGORY|
|LABEL SCORE COUNT %|AVRGE EXPECT| MNSQ MNSQ||THRESHOLD| MEASURE|
|-------------------+------------+------------++---------+-------- |
| 0 0 378 20| -.87 -1.03| 1.08 1.19|| NONE |( -2.07)| 0 Dislike
| 1 1 620 34| .13 .33| .85 .69|| -.86 | .00 | 1 Neutral
| 2 2 852 46| 2.24 2.16| 1.00 1.47|| .86 |( 2.07)| 2 Like
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0. Distribution of frequencies of observation in each category. For a rating-scale for which inferences will be made at the category level, we like to see a uniform process at work. When applied to a sample, this would produce a smooth distribution of category frequencies. Unimodal without sharp peaks or troughs. No statistical test is intended, but merely Berkson's "inter-ocular traumatic test" (= what hits you between the eyes).
1. If you conceptualize the rating scale in terms of the probability of individual categories (Andrich's approach), then the Andrich Thresholds are of interest. The Andrich thresholds are the points at which adjacent categories are equally probable.
CATEGORY PROBABILITIES MODES - Andrich Thresholds at intersections
P -------------------------------------------------------------
R 1.0 00000000 22222222
O 0000000 2222222
B .8 000 222
A 000 222
B .6 00 22
I .5 00*111111111*22
L .4 111|00 22|111
I 111 | 00 22 | 111
T .2 1111 | 22*00 | 1111
Y 1111111 22222 00000 1111111
.0 ********222222222222222 | | 000000000000000********
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-5 -4 -3 -2 -1 0 1 2 3 4 5
PERSON [MINUS] ITEM MEASURE
2. If you conceptualize the rating scale in terms of average ratings on the model (predicted) item characteristic curve (ICC), then "Category measures" are of interest. The "Category Measures" are the points on the latent variable at which the expected score on the item equals the category number. The Rasch-half-point thresholds define the ends of each category interval. These are shown in Table 12.5.
EXPECTED SCORE OGIVE MEANS
E -------------------------------------------------------------
X 2 222222222222
P 2222222
E 1.5----------------------------------------2222*
C 111 *
T 111 *
E 1-------------------------------111 *
D 111 * *
111 * *
S .5--------------------00000 * *
C 000000* * *
O 0 000000000000 * * *
R ------------------------------------------------------------
E -5 -4 -3 -2 -1 0 1 2 3 4 5
PERSON [MINUS] ITEM MEASURE
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|CATEGORY STRUCTURE | SCORE-TO-MEASURE | 50% CUM.| COHERENCE|ESTIM|
| LABEL MEASURE S.E. | AT CAT. ----ZONE----|PROBABLTY| M->C C->M|DISCR|
|------------------------+---------------------+---------+----------+-----|
| 0 NONE |( -2.07) -INF -1.19| | 62% 42%| | 0 Dislike
| 1 -.86 .07 | .00 -1.19 1.19| -1.00 | 54% 71%| .73| 1 Neutral
| 2 .86 .06 |( 2.07) 1.19 +INF | 1.00 | 85% 78%| 1.19| 2 Like
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3. If you conceptualize the rating scale in terms of the probability of accumulated categories (Thurstone's approach), then Rasch-Thurstonian thresholds = "50% Cumulative Probabilities" are of interest. 50% Cum Probability is the point at which the probability of being observed in the categories below = the probability of being observed in this category or above. These are shown in Table 12.6.
Thurstonian thresholds are category boundaries on the latent variable, where we define "boundary" to mean "if someone has an ability measure exactly on a category boundary, then that person has a 50% chance of being observed in a category below the boundary (including categories down to the bottom of the rating scale) and a 50% chance of being observed in a category above the boundary (including categories up to the top of the rating scale)". This is the same definition that we apply to the only category boundary for a dichotomous item, between 0 and 1. Someone exactly at the 0-1 boundary (the item difficulty) has this same 50-50 chance.
The Rasch-Thurstonian thresholds also approximate the item difficulties when the rating scales are dichotomized between the categories below the target category and those at and above the target category. This is useful when the probability of "success" on a polytomous item must be computed. www.rasch.org/rmt/rmt233e.htm
MEDIANS - Cumulative probabilities
P -------------------------------------------------------------
R 1.0 ********222222222222222
O 0 1111111 22222
B .8 0 111 222
A 0 111 22
B .6 0 11 22
I .5 0----------------------111---------222-----------------------
L .4 0 | 11 | 22
I 0 | 11 | 222
T .2 0 | 111 | 222
Y 0 | 11111 2222222
.0 0 | | 111111111111111********
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-5 -4 -3 -2 -1 0 1 2 3 4 5
PERSON [MINUS] ITEM MEASURE
