# Item Response Tree Models

Item response tree (IRTree) models are latent variable models for fully ordered and partially ordered multi-categorical data. The categories are item response options, for example, for multiple-choice items and Likert scale items. Missing responses and “not applicable” types of responses can be included in the model if they are treated as response options, even when not offered in an explicit way.

The response options are fully or partially ordered in a tree with nodes. A different model can be specified per node to differentiate between the probabilities of the branches leaving the node.

The probability of a response option is the product of the probabilities of the branches to reach an option. For example, if one can fully order the four response options of an item (e.g., A>B>C>D) there would be three nodes, each with two branches: Node 1 differentiates between A&B&C (branch 1) and D (branch 0). Node 2 differentiates between A&B (branch 1) and C (branch 0). Node 3 differentiates between A (branch 1) and B (branch 0). The probability to respond with a C is the probability of the A&B&C branch from node 1 multiplied with the probability of the C branch from node 2. To facilitate estimation, the responses are recoded as a 1-1-1 (A), 1-1-0 (B), 1-0-NA (C) and 0-NA-NA (D). A possible model is one with the same latent variable in each node but with node specific item parameters (discrimination, difficulty). An alternative model would be a cumulative latent variable model, with just ability 1 for node 1, abilities 1&2 for node 2, and abilities 1&2&3 for node 3.

Response omission models of the IRTree type are not full order models but partial order models instead. For example, the first node is response vs. omission, and the second node is conditional on whether a response is given and contrasts a correct with an incorrect response. The recoded responses are 1-1 (correct response), 1-0 (incorrect response), and 0-NA (omission). Both nodes share the ability latent variable, but the first node also has an extra latent variable for responding vs. not responding. In this way one can test whether response omission is informative for ability estimation.

Yet another application is for rating scale items and the identification of response styles. Response styles are modeled as latent variables for the nodes that differentiate between more extreme and less extreme points on the Likert scale.

IRTree models have two major advantages. First, they are very flexible, more flexible than other families of models for categorical data. Second, they do not need special purpose software because they can be estimated with most existing item response model software.

**Paul De Boeck** has a Ph.D. degree from the KU Leuven in Belgium (Flanders), where he also has spent most of his career. From 2009 to 2012, he was affiliated with the University of Amsterdam. He is a former president of the Psychometric Society (2008) and was the first editor of the Applied Research and Case Studies section of Psychometrika.

He is interested in individual differences in various domains, and in quantitative approaches in general. His early quantitative work concerns primarily classification models based on disjunctive and conjunctive rules (HICLAS), while his more recent work concerns model development and applications in the domain of item response theory (IRT) and logistic mixed models. Within IRT he focuses on explanatory models and explanatory measurement, and recently also on IRTree models for response scales (e.g., Likert scales), decision making and cognitive processes.

Recently he also started research on the low replication rate in psychological studies. Based on an analysis of the results of a large-scale replication study published in Science, the probability of null hypotheses seems very low. This may be due to a rather large variation of effect size that is difficult if not impossible to capture following existing methodologies while it generates statistically significant results and misleadingly hopeful confidence intervals and effect size estimates when using the present ways of designing studies and analyzing the data from those studies.

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irtree R script | 2.38 KB |