﻿ Poisson counts - Binomial trials - Negative (inverse) binomial

# Poisson counts, Binomial trials, Negative (inverse) binomial trials

The Winsteps program can analyze Poisson count data, with a little work.

Poisson counts are a 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)

....

99 4.595 ; the value corresponding to loge(99)

*

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.

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. 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.

Binomial trials:

Same as above, with n thresholds loge(n(m-n+1)) where m is fixed number of trials and n is number of successes for n=0 to m.

Negative (inverse) binomial trials:

Same as above, with n thresholds where m is number of trials and n is 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.

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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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