A Flexible Approach to Modeling Over-, Under- and Equidispersed Count Data in IRT: The Two-Parameter Conway-Maxwell-Poisson Model

Several psychometric tests generate count data, e.g. the number of ideas in divergent thinkingtasks. The most prominent count data IRT model, the Rasch Poisson Counts Model (RPCM)assumes constant discriminations across items as well as the equidispersion assumption of thePoisson distribution (i.e., E(X) = Var(X)), considerably limiting modeling flexibility. Violationsof these assumptions are associated with impaired ability, reliability, and standard error estimates.Models have been proposed to loose the one or the other assumption. The Two-Parameter PoissonCounts Model (2PPCM) allows varying discriminations but retains the equidispersion assumption.The Conway-Maxwell-Poisson Counts Model (CMPCM) that allows for modeling equi- but alsoover- and underdispersion (more or less variance than implied by the mean under the Poisson distribution)but assumes constant discriminations. The present work introduces the Two-ParameterConway-Maxwell-Poisson (2PCMP) model which generalizes the RPCM, the 2PPCM, and the CMPCM(all contained as special cases) to allow for varying discriminations and dispersions withinone model. A marginal maximum likelihood method based on a fixed quadrature Expectation-Maximization (EM) algorithm is derived. Standard errors as well as two methods for latent abilityestimation are provided. An implementation of the 2PCMP model in R and C++ is provided. Twosimulation studies examine the model’s statistical properties and compare the 2PCMP model toestablished methods. Data from divergent thinking tasks are re-analyzed with the 2PCMP modelto illustrate the model’s flexibility and ability to test assumptions of special cases.

Poisson Diagnostic Classification Models: A Framework and an Exploratory Example

Assessments with a large amount of small, similar, or often repetitive tasks are being used in educational, neurocognitive, and psychological contexts. For example, respondents are asked to recognize numbers or letters from a large pool of those and the number of correct answers is a count variable. In 1960, George Rasch developed the Rasch Poisson counts model (RPCM) to handle that type of assessment. This article extends the RPCM into the world of diagnostic classification models (DCMs) where a Poisson distribution is applied to traditional DCMs. A framework of Poisson DCMs is proposed and demonstrated through an operational dataset. This study aims to be exploratory with recommendations for future research given in the end.

RESIDUAL ANALYSIS IN RASCH POISSON COUNTS MODELS

A Rasch Poisson counts (RPC) model is described to identify individual latent traits and facilities of the items of tests that model the error (or success) count in several tasks over time, instead of modeling the correct responses to items in a test as in the dichotomous item response theory (IRT) model. These types of tests can be more informative than traditional tests. To estimate the model parameters, we consider a Bayesian approach using the integrated nested Laplace approximation (INLA). We develop residual analysis to assess model t by introducing randomized quantile residuals for items. The data used to illustrate the method comes from 228 people who took a selective attention test. The test has 20 blocks (items), with a time limit of 15 seconds for each block. The results of the residual analysis of the RPC were promising and indicated that the studied attention data are not well tted by the RPC model.

Exponentially Weighted Moving Averages of Counting Processes When the Time between Events Is Weibull Distributed

There are control charts for Poisson counts, zero-inflated Poisson counts, and over dispersed Poisson counts (negative binomial counts) but nothing on counting processes when the time between events (TBEs) is Weibull distributed. In our experience the in-control distribution for time between events is often Weibull distributed in applications. Counting processes are not Poisson distributed or negative binomial distributed when the time between events is Weibull distributed. This is a gap in the literature meaning that there is no help for practitioners when this is the case. This book chapter is designed to close this gap and provide an approach that could be helpful to those applying control charts in such cases.

Analysis of the Ruff 2 & 7 Test of Attention with the Rasch Poisson Counts Model

Background:Attention is a basic neurocognitive function which is a prerequisite for performance on more complex cognitive tasks. The Ruff 2 & 7 test is a well-known measure of attention with a well-supported theoretical and empirical underpinnings.Objective:The Ruff 2 & 7 test, has not been subjected to rigorous item response theory analysis yet. The purpose of this research was to examine the fit of the Ruff 2 & 7 test to the Rasch Poisson Counts Model (RPCM).Methods:Responses of 138 nonclinical subjects to the Ruff 2 & 7 test were analyzed with the RPCM measurement model using ‘lme4’ package in R. The fit of the individual items (blocks) and the overall test to the model were examined.Results:Findings showed that three out of seven scoring techniques fit the Rasch model. The scoring techniques which fitted the model were total number of characters cancelled, total number of characters correctly cancelled, and total number of characters correctly cancelled minus errors of commission.Conclusion:Three of the scoring techniques fit the RPCM which support the internal validity of the test when these scoring procedures were employed. Therefore, the Ruff 2 & 7 test is psychometrically uni-dimensional when these three scores are computed.