Disordered Andrich thresholds indicate that a category occupies a narrow interval on the latent variable (usually because the category is observed relatively rarely). This is not a problem unless you need advancing abilities to probably increment smoothly up the categories of the rating scale without skipping narrow categories.
If this is a problem for you, then please collapse categories. This usually requires some experimenting to find the best solution:
1) Look at the average measures for each category. Combine categories with disordered average measures
2) Look at the category frequency. Combine categories with low frequencies
Disordered Andrich thresholds indicate that some categories on the latent variable are narrow. Disordered Andrich thresholds do not violate Rasch models, but they may impact our interpretation of how the rating scale functions.
Example: Imagine a location on the latent variable that is the boundary between two categories. If there are exactly 1,000 people with measures at that boundary. We would expect to observe 500 of them in categories below the boundary and 500 of them in categories above the boundary. Dichotomous items function exactly this way.
dichotomous item 0-1
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category 0 on latent variable
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category 1 on latent variable
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1000 persons here →
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500 persons observed here as 0
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500 persons observed here as 1
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dichotomous item difficulty →
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Polytomous items (RSM, PCM) are move complex. RSM and PCM predict that some of the 1000 will be observed in categories next to the boundary, and some in categories further away, so that there will be 500 in total above the boundary and 500 below the boundary. OK so far?
polytomous item 0-1-2 (ordered categories, ordered Andrich thresholds)
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category 0
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category 1 (wide)
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category 2
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1000 persons here →
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←
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500 persons observed here as 0
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490 persons observed here as 1
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10 persons observed here as 2
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|← Andrich threshold → | (ordered)
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For 3 rating-scale categories 0,1,2 our first boundary is between 0 and 1. If category 1 is very wide, almost all the 500, say 490, will be observed in category 1 and 10 in category 2. The Andrich threshold corresponding to each boundary is basically ln(frequency of category below/frequency of category above), so the Andrich threshold for our first boundary 0-1 is something like ln(500/490) = 0.0 and for the next boundary, 1_2, ln(490/10) = 3.9. The Andrich thresholds are in ascending order.
polytomous item 0-1-2 (ordered categories, disordered Andrich thresholds)
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category 0
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category 1 (narrow)
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category 2
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1000 persons here →
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←
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500 persons observed here as 0
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10 persons as 1
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490 persons observed here as 2
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→ | Andrich threshold |← (disordered)
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But if category 1 is very narrow, only 10 of the 500 may be observed in category 1, and 490 in category 2 above the narrow category 1. The Andrich threshold for our first boundary 0-1 is something like ln(500/10) = 3.9 and for the next boundary, 1_2, ln(10/490) = -3.9. So the categories are disordered, even though the data fit the Rasch model perfectly!
Disordered Andrich thresholds indicate narrow categories on the latent variable. Statistically, these are no problem, but for practical purposes we may want all the categories to be wide enough that the Andrich thresholds are ordered.
The category boundaries in this example are reported as the "Rasch-Thurstone thresholds".
