Rasch’s Poisson counts model is a latent trait model
for the situation in which K tests are administered to
N examinees and the test score is a count [e.g., the
repeated occurrence of some event, such as the number
of items completed or the number of items answered
(in)correctly]. The Rasch Poisson counts model assumes
that the test scores are Poisson distributed random
variables. In the approach presented here, the Poisson
parameter is assumed to be a product of a fixed test
difficulty and a gamma-distributed random examinee
latent trait parameter. From these assumptions, marginal
maximum likelihood estimators can be derived for the
test difficulties and the parameters of the prior gamma
distribution. For the examinee parameters, there are a
number of options. The model can be applied in a situation
in which observations result from an incomplete
design. When examinees are assigned to different subsets
of tests using background information, this information
must be taken into account when using marginal
maximum likelihood estimation. If the focus is on test
calibration and there is no interest in the characteristics
of the latent traits in relation to the background information,
conditional maximum likelihood methods may be
preferred because they are easier to implement and are
justified for incomplete data for test parameter estimation.
Index terms: EM algorithm, incomplete designs,
latent trait models, marginal maximum likelihood estimation,
Rasch Poisson counts model.
Jansen, Margo G. H. (1995). The Rasch Poisson counts model for incomplete data: An application of the EM algorithm. Applied Psychological Measurement, 19, 291-302. doi:10.1177/014662169501900307
Jansen, Margo G. H..
The Rasch Poisson counts model for incomplete data: An application of the EM algorithm.
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