A flexible approach to modelling over‐, under‐ and equidispersed count data in IRT: the Two‐Parameter Conway–Maxwell–Poisson model

Several psychometric tests and self-reports generate count data (e.g., divergent thinking tasks). The most prominent count data item response theory model, the Rasch Poisson Counts Model (RPCM), is limited in applicability by two restrictive assumptions: equal item discriminations and equidispersion (conditional mean equal to conditional variance). Violations of these assumptions lead to impaired reliability and standard error estimates. Previous work generalized the RPCM but maintained some limitations. The two-parameter Poisson counts model allows for varying discriminations but retains the equidispersion assumption. The Conway–Maxwell–Poisson Counts Model allows for modelling over- and underdispersion (conditional mean less than and greater than conditional variance, respectively) but still assumes constant discriminations. The present work introduces the Two-Parameter Conway–Maxwell–Poisson (2PCMP) model which generalizes these three models to allow for varying discriminations and dispersions within one model, helping to better accommodate data from count data tests and self-reports. A marginal maximum likelihood method based on the EM algorithm is derived. An implementation of the 2PCMP model in R and C++ is provided. Two simulation studies examine the model's statistical properties and compare the 2PCMP model to established models. Data from divergent thinking tasks are reanalysed with the 2PCMP model to illustrate the model's flexibility and ability to test assumptions of special cases.Correction for this article: https://doi.org/10.1111/bmsp.1231

Comparison of random‐effects meta‐analysis models for the relative risk in the case of rare events - a simulation study

Pooling the relative risk (RR) across studies investigating rare events, for example, adverse events, via meta‐analytical methods still presents a challenge to researchers. The main reason for this is the high probability of observing no events in treatment or control group or both, resulting in an undefined log RR (the basis of standard meta‐analysis). Other technical challenges ensue, for example, the violation of normality assumptions, or bias due to exclusion of studies and application of continuity corrections, leading to poor performance of standard approaches. In the present simulation study, we compared three recently proposed alternative models (random‐effects [RE] Poisson regression, RE zero‐inflated Poisson [ZIP] regression, binomial regression) to the standard methods in conjunction with different continuity corrections and to different versions of beta‐binomial regression. Based on our investigation of the models' performance in 162 different simulation settings informed by meta‐analyses from the Cochrane database and distinguished by different underlying true effects, degrees of between‐study heterogeneity, numbers of primary studies, group size ratios, and baseline risks, we recommend the use of the RE Poisson regression model. The beta‐binomial model recommended by Kuss (2015) also performed well. Decent performance was also exhibited by the ZIP models, but they also had considerable convergence issues. We stress that these recommendations are only valid for meta‐analyses with larger numbers of primary studies. All models are applied to data from two Cochrane reviews to illustrate differences between and issues of the models. Limitations as well as practical implications and recommendations are discussed; a flowchart summarizing recommendations is provided

Simulation study: Meta-analysis for rare events

In this simulation study, we examined the performance of different alternative random-effects meta-analysis models which pool the RR in settings where the events of interests are rare. This is challenging to standard models of meta-analysis

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

Several psychometric tests and self-reports generate count data, e.g., divergent thinking tasks. The most prominent count data IRT model, the Rasch Poisson Counts Model (RPCM), is limited in applicability by two restrictive assumptions: equal item discriminations and equidispersion (conditional mean = conditional variance). Violations of these lead to impaired reliability and standard error estimates. Previous work generalized the RPCM but maintained some limitations. The Two-Parameter Poisson Counts Model (2PPCM) allows varying discriminations but retains the equidispersion assumption. The Conway-Maxwell-Poisson Counts Model (CMPCM) allows for modeling over- and underdispersion (conditional mean < resp. > conditional variance) but still assumes constant discriminations. The present work introduces the Two-Parameter Conway-Maxwell-Poisson (2PCMP) model which generalizes these three models to allow for varying discriminations and dispersions within one model, helping to better accommodate data from count data tests and self-reports. A marginal maximum likelihood method based on the Expectation-Maximization algorithm is derived. An implementation of the 2PCMP model in R and C++ is provided. Two simulation studies examined the model’s statistical properties and compared the 2PCMP model to established models. Data from divergent thinking tasks were re-analyzed with the 2PCMP model to illustrate the model’s flexibility and ability to test assumptions of special cases

The Dark Triad is Dead, Long Live the Dark Triad: An Item-Response Theoretical Examination of the Short Dark Tetrad

Research on the Dark Tetrad (Machiavellianism, narcissism, psychopathy, and sadism) is burgeoning. Therefore, psychometrically sound measures are required to ensure that predictions based on these constructs are reliable and valid. A recently devised scale to assess all traits simultaneously is the Short Dark Tetrad (SD4; Paulhus et al., 2021). Previous studies demonstrated the general suitability of the SD4 in terms of classical test theory, but item response theoretical (IRT) analyses are still missing, whereby IRT approaches extract more information on the items and scales and tests assumptions untested in classical test theory. Thus, we evaluated the subscales of the SD4 using IRT modeling (N = 594). Unlike the sadism subscale, the Machiavellianism, narcissism, and psychopathy subscales had satisfactory fit of the two-parameter polytomous IRT models and provided ample information across the respective trait continua. The sadism subscale, however, exhibited problems related to item discrimination (i.e., in differentiating between individuals with similar levels) and item (category transition) difficulties (i.e., quantifying how “difficult” it is to endorse higher compared to next lower response categories). The results thus emphasize the need for a thorough revision of the items of the sadism scale

Understanding Ability and Reliability Differences Measured with Count Items: The Distributional Regression Test Model and the Count Latent Regression Model

In psychology and education, tests (e.g., reading tests) and self-reports (e.g., clinical questionnaires) generate counts, but corresponding Item Response Theory (IRT) methods are underdeveloped compared to binary data. Recent advances include the Two-Parameter Conway-Maxwell-Poisson model (2PCMPM), generalizing Rasch’s Poisson Counts Model, with item-specific difficulty, discrimination, and dispersion parameters. Explaining differences in model parameters informs item construction and selection, but has received little attention. We introduce two 2PCMPM based explanatory count IRT models: The Distributional Regression Test Model for item covariates, and the Count Latent Regression Model for person covariates. Estimation methods are provided and satisfactory statistical properties observed in simulations. Two examples illustrate how the models help understanding tests and underlying constructs

Every Trait Counts: Marginal Maximum Likelihood Estimation for Novel Multidimensional Count Data Item Response Models with Rotation or l1-Regularization for Simple Structure

This is the repository for a manuscript of the same title which is currently under review. For the sake of blinded reviews, the information on this repository is somewhat reduced at the moment. Upon completing the review process, we are going to add more information here. All files pertaining to the manuscript are course already available here

Every Trait Counts: Marginal Maximum Likelihood Estimation for Novel Multidimensional Count Data Item Response Models with Rotation or l1-Regularization for Simple Structure

The framework of multidimensional item response theory (MIRT) offers psychometric models for various data settings, most popularly for dichotomous and polytomous data. Less attention has been devoted to count responses. A recent growth in interest in count item response models (CIRM)---perhaps sparked by increased occurrence of psychometric count data, e.g., in the form of process data, clinical symptom frequency, number of ideas or errors in cognitive ability assessment---has focused on unidimensional models. A few recently proposed unidimensional CIRMs rely on the Conway-Maxwell-Poisson distribution as the conditional response distribution which allows to model conditionally over-, under-, and equidispersed responses. In this article, we generalize one of those CIRMs to the multidimensional case, introducing the Multidimensional Two-Parameter Conway-Maxwell-Poisson Model (M2PCMPM) class. Using the Expectation-Maximization (EM) algorithm, we develop marginal maximum likelihood estimation methods, primarily for exploratory M2PCMPMs. The resulting discrimination matrices are rotationally indeterminate. We pursue the goal of obtaining a simple structure for them by (1) rotating and (2) regularizing the discrimination matrix. Recent IRT research has successfully used regularization of the discrimination matrix to obtain a simple structure (i.e., a sparse solution) for dichotomous and polytomous data. We develop an EM algorithm with lasso ($\ell_1$) regularization for the M2PCMPM and compare (1) and (2) in a simulation study. We illustrate the proposed model with an empirical example using intelligence test data