In this paper we elaborate on the potential of the lmer function from the lme4 package in R for item response (IRT) modeling. In line with the package, an IRT framework is described based on generalized linear mixed modeling. The aspects of the framework refer to (a) the kind of covariates -their mode (person, item, person-by-item), and their being external vs. internal to responses, and (b) the kind of effects the covariates have -fixed vs. random, and if random, the mode across which the effects are random (persons, items). Based on this framework, three broad categories of models are described: Item covariate models, person covariate models, and person-by-item covariate models, and within each category three types of more specific models are discussed. The models in question are explained and the associated lmer code is given. Examples of models are the linear logistic test model with an error term, differential item functioning models, and local item dependency models. Because the lme4 package is for univariate generalized linear mixed models, neither the two-parameter, and three-parameter models, nor the item response models for polytomous response data, can be estimated with the lmer function.

Recent work suggests that the positive manifold of individual differences may arise, or be amplified, by a mechanism called mutualism. Kievit et al. (2017) showed that a latent change score implementation of the mutualism model outperformed alternative models, demonstrating positive reciprocal interactions between vocabulary and reasoning during development. Here, we replicated these findings in a cohort of children ( N = 227, 6–8 years old) and expanded the findings in three directions. First, a third wave of data was included, and the findings were robust to alternative model specifications. Second, a simulation demonstrated that data sets of similar magnitude and distributional properties could have, in principle, favored alternative models with close to 100% power. Third, we found support for the hypothesis that mutualistic-coupling effects are stronger and self-feedback parameters weaker in younger children. Together, these findings replicated the work of Kievit et al. (2017) and further support the hypothesis that mutualism supports cognitive development.

We investigate the relation between speed and accuracy within problem solving in its simplest non-trivial form. We consider tests with only two items and code the item responses in two binary variables: one indicating the response accuracy, and one indicating the response speed. Despite being a very basic setup, it enables us to study item pairs stemming from a broad range of domains such as basic arithmetic, first language learning, intelligence-related problems, and chess, with large numbers of observations for every pair of problems under consideration. We carry out a survey over a large number of such item pairs and compare three types of psychometric accuracy-response time models present in the literature: two ‘one-process’ models, the first of which models accuracy and response time as conditionally independent and the second of which models accuracy and response time as conditionally dependent, and a ‘two-process’ model which models accuracy contingent on response time. We find that the data clearly violates the restrictions imposed by both one-process models and requires additional complexity which is parsimoniously provided by the two-process model. We supplement our survey with an analysis of the erroneous responses for an example item pair and demonstrate that there are very significant differences between the types of errors in fast and slow responses.

This study investigates why children who are good at one math skill also perform well on other math tasks and seeks a solution by comparing two influential theories of general intelligence. The g-factor theory (g for 'general intelligence') implies a general math ability that steers all math-related development. In contrast, mutualism theory states that different math skills co-develop, positively influencing each others growth. We examined this with a large longitudinal data set (N ≈ 12.000) that tracked the development of basic (counting ↔ addition) and more advanced (multiplication ↔ division) math skills for a full school year. We used bivariate latent change score models to investigate whether g-factor or mutualism theory provides a better explanation of their co-development. We found that basic and more advanced math skills become more strongly related over time and that co-development is mutually beneficial. Both results support mutualism theory, a dynamic network perspective of cognitive development, where, in this case, growth in a particular math domain positively influences that of other math skills. Our results perhaps reveal the tip of the iceberg when it comes to the intricacies of co-developing math abilities. We discuss the implications of mutualism theory for understanding the dynamics of learning mathematics.