Conceptually, the Poisson Counts model has an infinite number of categories, 0, 1, 2, onwards. In practice, due to time limits and other constraints, there is a finite maximum possible. If that maximum is much higher than the highest observed category, then the Poisson model continues to apply from a practical perspective. If the observed highest is close to the absolute maximum, then use a constrained rating scale. In Winsteps, do this with
ISRANGE=
to set the 0 to absolute maximum range of categories, and
SFUNCTION=
to set the function controlling the Andrich thresholds
SFUNCTION=2 is a good starting value.
The Winsteps program can analyze Poisson count data, with a little work. Poisson counts are a very long rating (or partial credit) scale with pre-set structure. The Andrich Thresholds are loge(n), n=1 upwards. You can define a structure anchor file in this way:
XWIDE=2
STKEEP=YES
CODES = 00010203040506070809101112131415161718192021222324252627282930313233343536373839+
+40414243444546474849505152535455565758596061626364656667686970717273747576777879+
+8081828384858687888990919293949596979899
SAFILE=*
0 0 ; placeholder for the bottom count
1 0 ; the value corresponding to log(1) - the pivot point for the item measure
2 .693 ; the value corresponding to loge(2)
3 1.099 ; the value corresponding to loge(3)
|
4 1.386
5 1.609
6 1.792
7 1.946
8 2.079
9 2.197
10 2.303
11 2.398
12 2.485
13 2.565
14 2.639
15 2.708
16 2.773
17 2.833
18 2.890
19 2.944
20 2.996
21 3.045
22 3.091
23 3.135
24 3.178
25 3.219
26 3.258
27 3.296
|
28 3.332
29 3.367
30 3.401
31 3.434
32 3.466
33 3.497
34 3.526
35 3.555
36 3.584
37 3.611
38 3.638
39 3.664
40 3.689
41 3.714
42 3.738
43 3.761
44 3.784
45 3.807
46 3.829
47 3.850
48 3.871
49 3.892
50 3.912
51 3.932
|
52 3.951
53 3.970
54 3.989
55 4.007
56 4.025
57 4.043
58 4.060
59 4.078
60 4.094
61 4.111
62 4.127
63 4.143
64 4.159
65 4.174
66 4.190
67 4.205
68 4.220
69 4.234
70 4.248
71 4.263
72 4.277
73 4.290
74 4.304
75 4.317
|
76 4.331
77 4.344
78 4.357
79 4.369
80 4.382
81 4.394
82 4.407
83 4.419
84 4.431
85 4.443
86 4.454
87 4.466
88 4.477
89 4.489
90 4.500
91 4.511
92 4.522
93 4.533
94 4.543
95 4.554
96 4.564
97 4.575
98 4.585
99 4.595
*
|
Arrange that the observations have an upper limit much less than 99, or extend the range of CODES= and SAFILE= to be considerably wider than the observations. Winsteps can go up to category 32767.
Use UASCALE= to multiply all Poisson Andrich Thresholds by a constant to adjust the "natural" form of the Poisson counts to the actual discrimination of your empirical Poisson process, or do the multiplication yourself, which can be different for different items. You need to adjust the constant so that the average overall mean-square of the analysis is about 1.0. See RMT 14:2 about using mean-squares to adjust logit user-scaling. (The Facets program does this automatically, if so instructed.)
But my experience with running Poisson counts in the Facets program (which supports them directly) is that most "Poisson count" data do not match the Poisson process well, and are more usefully parameterized as a rating (or partial credit) scale. There is nearly always some other aspect of the situation that perturbs the pure Poisson process, especially by placing a ceiling value on the observations.
Theory: See Wikipedia . The Poisson model is P(k events in an interval) = e-λ λk /k! so that P(k)/P(k-1) = λ/k, which is implemented in Winsteps as ln(P(k)/P(k-1)) = B - D - ln(k)/UASCALE where B is the person ability, D is the item difficulty, and 1/UASCALE is the Poisson-scale discrimination parameter.
Binomial trials: same as above, with loge(n(m-n+1)) where m is a fixed number of trials and n is the number of successes for n=0 to m, so that there a m thresholds. We can use UASCALE= to adjust for scale discrimination as above.
Negative (inverse) binomial trials: same as above where m is the number of trials and n is a fixed number of successes. For convenience, enter the data as x = (m-n) = number of failures, which will go from 0 to a large number (similar to Poisson counts above). Again we can pre-compute the Andrich thresholds for every observation = log (probability of observing x failures / probability of observing x-1 failures) when we are targeting n successes. If the number of successes is constant for each item, then we can use Winsteps. If it varies across observations within an item, we must use Facets. We can use UASCALE= to adjust for scale discrimination as above.
