Can fast and slow intelligence be differentiated?

Killing Two Birds with One Stone: Accounting for Unfolding Item Response Process and Response Styles Using Unfolding Item Response Tree Models

Two research streams on responses to Likert-type items have been developing in parallel: (a) unfolding models and (b) individual response styles (RSs). To accurately understand Likert-type item responding, it is vital to parse unfolding responses from RSs. Therefore, we propose the Unfolding Item Response Tree (UIRTree) model. First, we conducted a Monte Carlo simulation study to examine the performance of the UIRTree model compared to three other models - Samejima's Graded Response Model, Generalized Graded Unfolding Model, and Dominance Item Response Tree model, for Likert-type responses. Results showed that when data followed an unfolding response process and contained RSs, AIC was able to select the UIRTree model, while BIC was biased toward the DIRTree model in many conditions. In addition, model parameters in the UIRTree model could be accurately recovered under realistic conditions, and mis-specifying item response process or wrongly ignoring RSs was detrimental to the estimation of key parameters. Then, we used datasets from empirical studies to show that the UIRTree model could fit personality datasets well and produced more reasonable parameter estimates compared to competing models. A strong presence of RS(s) was also revealed by the UIRTree model. Finally, we provided examples with R code for UIRTree model estimation to facilitate the modeling of responses to Likert-type items in future studies.

A Psychometric Perspective on the Associations between Response Accuracy and Response Time Residuals

We provide an alternative psychometric perspective on the empirical statistical dependencies observed between response accuracy residuals (RARs) and response time residuals (RTRs) in the context of the van der Linden model. This perspective emphasizes the RAR (or parts of the RAR) as being exogenous and having a directional influence on response time. Our simple and theoretically justifiable perspective adds to previous joint response time/accuracy models and comports with recent generalizations of the D-diffusion IRT model incorporating person-by-item interactions, and thus similarly reproduces many of the recently highlighted empirical findings concerning the associations between RARs and RTRs. Using both empirical and simulation-based results, we show how our psychometric perspective has both applied and interpretational implications. Specifically, it would suggest that (1) studies of item parameter estimate heterogeneity in relation to response times may reflect more of a psychometric artifact (due to the exogenous effects of the RARs) as opposed to providing insights about the response process (e.g., the application of different response strategies) and that (2) efforts to use RTRs as indicators of latent proficiency should attend to the anticipated interactions between the latent proficiency and RAR on response times. The validity of our psychometric perspective against alternatives likely relies on appeals to theory; the best perspective to take may vary depending on the test setting.

A Recent Development of a Network Approach to Assessment Data: Latent Space Item Response Modeling for Intelligence Studies

This article aims to provide an overview of the potential advantages and utilities of the recently proposed Latent Space Item Response Model (LSIRM) in the context of intelligence studies. The LSIRM integrates the traditional Rasch IRT model for psychometric data with the latent space model for network data. The model has person-wise latent abilities and item difficulty parameters, capturing the main person and item effects, akin to the Rasch model. However, it additionally assumes that persons and items can be mapped onto the same metric space called a latent space and distances between persons and items represent further decreases in response accuracy uncaptured by the main model parameters. In this way, the model can account for conditional dependence or interactions between persons and items unexplained by the Rasch model. With two empirical datasets, we illustrate that (1) the latent space can provide information on respondents and items that cannot be captured by the Rasch model, (2) the LSIRM can quantify and visualize potential between-person variations in item difficulty, (3) latent dimensions/clusters of persons and items can be detected or extracted based on their latent positions on the map, and (4) personalized feedback can be generated from person-item distances. We conclude with discussions related to the latent space modeling integrated with other psychometric models and potential future directions.

Conditional Dependence across Slow and Fast Item Responses: With a Latent Space Item Response Modeling Approach

There recently have been many studies examining conditional dependence between response accuracy and response times in cognitive tests. While most previous research has focused on revealing a general pattern of conditional dependence for all respondents and items, it is plausible that the pattern may vary across respondents and items. In this paper, we attend to its potential heterogeneity and examine the item and person specificities involved in the conditional dependence between item responses and response times. To this end, we use a latent space item response theory (LSIRT) approach with an interaction map that visualizes conditional dependence in response data in the form of item-respondent interactions. We incorporate response time information into the interaction map by applying LSIRT models to slow and fast item responses. Through empirical illustrations with three cognitive test datasets, we confirm the presence and patterns of conditional dependence between item responses and response times, a result consistent with previous studies. Our results further illustrate the heterogeneity in the conditional dependence across respondents, which provides insights into understanding individuals' underlying item-solving processes in cognitive tests. Some practical implications of the results and the use of interaction maps in cognitive tests are discussed.

A Latent Space Diffusion Item Response Theory Model to Explore Conditional Dependence between Responses and Response Times

Traditional measurement models assume that all item responses correlate with each other only through their underlying latent variables. This conditional independence assumption has been extended in joint models of responses and response times (RTs), implying that an item has the same item characteristics fors all respondents regardless of levels of latent ability/trait and speed. However, previous studies have shown that this assumption is violated in various types of tests and questionnaires and there are substantial interactions between respondents and items that cannot be captured by person- and item-effect parameters in psychometric models with the conditional independence assumption. To study the existence and potential cognitive sources of conditional dependence and utilize it to extract diagnostic information for respondents and items, we propose a diffusion item response theory model integrated with the latent space of variations in information processing rate of within-individual measurement processes. Respondents and items are mapped onto the latent space, and their distances represent conditional dependence and unexplained interactions. We provide three empirical applications to illustrate (1) how to use an estimated latent space to inform conditional dependence and its relation to person and item measures, (2) how to derive diagnostic feedback personalized for respondents, and (3) how to validate estimated results with an external measure. We also provide a simulation study to support that the proposed approach can accurately recover its parameters and detect conditional dependence underlying data.

A Modeling Framework to Examine Psychological Processes Underlying Ordinal Responses and Response Times of Psychometric Data

This article presents a joint modeling framework of ordinal responses and response times (RTs) for the measurement of latent traits. We integrate cognitive theories of decision-making and confidence judgments with psychometric theories to model individual-level measurement processes. The model development starts with the sequential sampling framework which assumes that when an item is presented, a respondent accumulates noisy evidence over time to respond to the item. Several cognitive and psychometric theories are reviewed and integrated, leading us to three psychometric process models with different representations of the cognitive processes underlying the measurement. We provide simulation studies that examine parameter recovery and show the relationships between latent variables and data distributions. We further test the proposed models with empirical data measuring three traits related to motivation. The results show that all three models provide reasonably good descriptions of observed response proportions and RT distributions. Also, different traits favor different process models, which implies that psychological measurement processes may have heterogeneous structures across traits. Our process of model building and examination illustrates how cognitive theories can be incorporated into psychometric model development to shed light on the measurement process, which has had little attention in traditional psychometric models.

A method for quantifying individual decision thresholds of latent print examiners

In recent years, 'black box' studies in forensic science have emerged as the preferred way to provide information about the overall validity of forensic disciplines in practice. These studies provide aggregated error rates over many examiners and comparisons, but errors are not equally likely on all comparisons. Furthermore, inconclusive responses are common and vary across examiners and comparisons, but do not fit neatly into the error rate framework. This work introduces Item Response Theory (IRT) and variants for the forensic setting to account for these two issues. In the IRT framework, participant proficiency and item difficulty are estimated directly from the responses, which accounts for the different subsets of items that participants often answer. By incorporating a decision-tree framework into the model, inconclusive responses are treated as a distinct cognitive process, which allows inter-examiner differences to be estimated directly. The IRT-based model achieves superior predictive performance over standard logistic regression techniques, produces item effects that are consistent with common sense and prior work, and demonstrates that most of the variability among fingerprint examiner decisions occurs at the latent print evaluation stage and as a result of differing tendencies to make inconclusive decisions.

Should Intelligence Tests Be Speeded or Unspeeded? A Brief Review of the Effects of Time Pressure on Response Processes and an Experimental Study with Raven's Matrices

Intelligence tests are often performed under time constraints for practical reasons, but the effects of time pressure on reasoning performance are poorly understood. The first part of this work provides a brief review of major expected effects of time pressure, which includes forcing participants to skip items, convoking a mental speed factor, constraining response times, qualitatively altering cognitive processing, affecting anxiety and motivation, and interacting with individual differences. The second part presents data collected with Raven's matrices under three conditions of speededness to provide further insight into the complex effects of time pressure, with three major findings. First, even mild time pressure (with enough time available for all participants to complete the task at a leisurely pace) induced speeding throughout the whole task, starting with the very first item, and participants sped up more than was actually required. Second, time pressure came with lower confidence and poorer strategy use and a substantial decrease of accuracy (d = 0.35), even when controlling for response time at the item level-indicating a detrimental effect on cognitive processing beyond speeding. Third, time pressure disproportionately reduced response times for difficult items and participants with high ability, working memory capacity, or need for cognition, although this did not differentially affect ability estimates. Overall, both the review and empirical sections show that the effects of time pressure go well beyond forcing participants to speed or skip the last few items and make even mild time constraints inadvisable when attempting to measure maximal performance, especially for high-performing samples.

Testing Replicability and Generalizability of the Time on Task Effect

The time on task (ToT) effect describes the relationship of the time spent on a cognitive task and the probability of successful task completion. The effect has been shown to vary in size and direction across tests and even within tests, depending on the test taker and item characteristics. Specifically, investing more time has a positive effect on response accuracy for difficult items and low ability test-takers, but a negative effect for easy items and high ability test-takers. The present study sought to test the replicability of this result pattern of the ToT effect across samples independently drawn from the same populations of persons and items. Furthermore, its generalizability was tested in terms of differential correlations across ability tests. To this end, ToT effects were estimated for three different reasoning tests and one test measuring natural sciences knowledge in 10 comparable subsamples with a total N = 2640. Results for the subsamples were highly similar, demonstrating that ToT effects are estimated with sufficient reliability. Generally, faster answers tended to be more accurate, suggesting a relatively effortless processing style. However, with increasing item difficulty and decreasing person ability, the effect flipped to the opposite direction, i.e., higher accuracy with longer processing times. The within-task moderation of the ToT effect can be reconciled with an account on effortful processing or cognitive load. By contrast, the generalizability of the ToT effect across different tests was only moderate. Cross-test relations were stronger in relative terms if performance in the respective tasks was more strongly related. This suggests that individual differences in the ToT effect depend on test characteristics such as their reliabilities but also similarities and differences of their processing requirements.

Modeling Conditional Dependence of Response Accuracy and Response Time with the Diffusion Item Response Theory Model

In this paper, we propose a model-based method to study conditional dependence between response accuracy and response time (RT) with the diffusion IRT model (Tuerlinckx and De Boeck in Psychometrika 70(4):629-650, 2005, https://doi.org/10.1007/s11336-000-0810-3 ; van der Maas et al. in Psychol Rev 118(2):339-356, 2011, https://doi.org/10.1080/20445911.2011.454498 ). We extend the earlier diffusion IRT model by introducing variability across persons and items in cognitive capacity (drift rate in the evidence accumulation process) and variability in the starting point of the decision processes. We show that the extended model can explain the behavioral patterns of conditional dependency found in the previous studies in psychometrics. Variability in cognitive capacity can predict positive and negative conditional dependency and their interaction with the item difficulty. Variability in starting point can account for the early changes in the response accuracy as a function of RT given the person and item effects. By the combination of the two variability components, the extended model can produce the curvilinear conditional accuracy functions that have been observed in psychometric data. We also provide a simulation study to validate the parameter recovery of the proposed model and present two empirical applications to show how to implement the model to study conditional dependency underlying data response accuracy and RTs.

An item response tree model with not-all-distinct end nodes for non-response modelling

The non-response model in Knott et al. (1991, Statistician, 40, 217) can be represented as a tree model with one branch for response/non-response and another branch for correct/incorrect response, and each branch probability is characterized by an item response theory model. In the model, it is assumed that there is only one source of non-responses. However, in questionnaires or educational tests, non-responses might come from different sources, such as test speededness, inability to answer, lack of motivation, and sensitive questions. To better accommodate such more realistic underlying mechanisms, we propose a a tree model with four end nodes, not all distinct, for non-response modelling. The Laplace-approximated maximum likelihood estimation for the proposed model is suggested. The validation of the proposed estimation procedure and the advantage of the proposed model over traditional methods are demonstrated in simulations. For illustration, the methodologies are applied to data from the 2012 Programme for International Student Assessment (PISA). The analysis shows that the proposed tree model has a better fit to PISA data than other existing models, providing a useful tool to distinguish the sources of non-responses.

Accounting for individual differences in speed in the discretized signed residual time model

With advances in computerized tests, it has become commonplace to register not just the accuracy of the responses provided to the items, but also the response time. The idea that for each response both response accuracy and response time are indicative of ability has explicitly been incorporated in the signed residual time (SRT) model (Maris & van der Maas, 2012, Psychometrika, 77, 615-633), which assumes that fast correct responses are indicative of a higher level of ability than slow correct responses. While the SRT model allows one to gain more information about ability than is possible based on considering only response accuracy, measurement may be confounded if persons show differences in their response speed that cannot be explained by ability, for example due to differences in response caution. In this paper we propose an adapted version of the SRT model that makes it possible to model person differences in overall speed, while maintaining the idea of the SRT model that the speed at which individual responses are given may be indicative of ability. We propose a two-dimensional SRT model that considers dichotomized response time, which allows one to model differences between fast and slow responses. The model includes both an ability and a speed parameter, and allows one to correct the estimates of ability for possible differences in overall speed. The performance of the model is evaluated through simulation, and the relevance of including the speed parameter is studied in the context of an empirical example from formative educational assessment.

Faking Detection Improved: Adopting a Likert Item Response Process Tree Model

With the increasing popularity of noncognitive inventories in personnel selection, organizations typically wish to be able to tell when a job applicant purposefully manufactures a favorable impression. Past faking research has primarily focused on how to reduce faking via instrument design, warnings, and statistical corrections for faking. This article took a new approach by examining the effects of faking (experimentally manipulated and contextually driven) on response processes. We modified a recently introduced item response theory tree modeling procedure, the three-process model, to identify faking in two studies. Study 1 examined self-reported vocational interest assessment responses using an induced faking experimental design. Study 2 examined self-reported personality assessment responses when some people were in a high-stakes situation (i.e., selection). Across the two studies, individuals instructed or expected to fake were found to engage in more extreme responding. By identifying the underlying differences between fakers and honest respondents, the new approach improves our understanding of faking. Percentage cutoffs based on extreme responding produced a faker classification precision of 85% on average.

Combining Clickstream Analyses and Graph-Modeled Data Clustering for Identifying Common Response Processes

Complex interactive test items are becoming more widely used in assessments. Being computer-administered, assessments using interactive items allow logging time-stamped action sequences. These sequences pose a rich source of information that may facilitate investigating how examinees approach an item and arrive at their given response. There is a rich body of research leveraging action sequence data for investigating examinees' behavior. However, the associated timing data have been considered mainly on the item-level, if at all. Considering timing data on the action-level in addition to action sequences, however, has vast potential to support a more fine-grained assessment of examinees' behavior. We provide an approach that jointly considers action sequences and action-level times for identifying common response processes. In doing so, we integrate tools from clickstream analyses and graph-modeled data clustering with psychometrics. In our approach, we (a) provide similarity measures that are based on both actions and the associated action-level timing data and (b) subsequently employ cluster edge deletion for identifying homogeneous, interpretable, well-separated groups of action patterns, each describing a common response process. Guidelines on how to apply the approach are provided. The approach and its utility are illustrated on a complex problem-solving item from PIAAC 2012.

Using Response Times and Response Accuracy to Measure Fluency Within Cognitive Diagnosis Models

The recent "Every Student Succeed Act" encourages schools to use an innovative assessment to provide feedback about students' mastery level of grade-level content standards. Mastery of a skill requires the ability to complete the task with not only accuracy but also fluency. This paper offers a new sight on using both response times and response accuracy to measure fluency with cognitive diagnosis model framework. Defining fluency as the highest level of a categorical latent attribute, a polytomous response accuracy model and two forms of response time models are proposed to infer fluency jointly. A Bayesian estimation approach is developed to calibrate the newly proposed models. These models were applied to analyze data collected from a spatial rotation test. Results demonstrate that compared with the traditional CDM that using response accuracy only, the proposed joint models were able to reveal more information regarding test takers' spatial skills. A set of simulation studies were conducted to evaluate the accuracy of model estimation algorithm and illustrate the various degrees of model complexities.

Neurocognitive Psychometrics of Intelligence: How Measurement Advancements Unveiled the Role of Mental Speed in Intelligence Differences

More intelligent individuals typically show faster reaction times. However, individual differences in reaction times do not represent individual differences in a single cognitive process but in multiple cognitive processes. Thus, it is unclear whether the association between mental speed and intelligence reflects advantages in a specific cognitive process or in general processing speed. In this article, we present a neurocognitive-psychometrics account of mental speed that decomposes the relationship between mental speed and intelligence. We summarize research employing mathematical models of cognition and chronometric analyses of neural processing to identify distinct stages of information processing strongly related to intelligence differences. Evidence from both approaches suggests that the speed of higher-order processing is greater in smarter individuals, which may reflect advantages in the structural and functional organization of brain networks. Adopting a similar neurocognitive-psychometrics approach for other cognitive processes associated with intelligence (e.g., working memory or executive control) may refine our understanding of the basic cognitive processes of intelligence.

Modeling Differences Between Response Times of Correct and Incorrect Responses

While standard joint models for response time and accuracy commonly assume the relationship between response time and accuracy to be fully explained by the latent variables of the model, this assumption of conditional independence is often violated in practice. If such violations are present, taking these residual dependencies between response time and accuracy into account may both improve the fit of the model to the data and improve our understanding of the response processes that led to the observed responses. In this paper, we propose a framework for the joint modeling of response time and accuracy data that allows for differences in the processes leading to correct and incorrect responses. Extensions of the standard hierarchical model (van der Linden in Psychometrika 72:287-308, 2007. https://doi.org/10.1007/s11336-006-1478-z ) are considered that allow some or all item parameters in the measurement model of speed to differ depending on whether a correct or an incorrect response was obtained. The framework also allows one to consider models that include two speed latent variables, which explain the patterns observed in the responses times of correct and of incorrect responses, respectively. Model selection procedures are proposed and evaluated based on a simulation study, and a simulation study investigating parameter recovery is presented. An application of the modeling framework to empirical data from international large-scale assessment is considered to illustrate the relevance of modeling possible differences between the processes leading to correct and incorrect responses.

Formal Modeling of the Resistance to Peer Influence Questionnaire: A Comparison of Adolescent Boys and Girls With and Without Mild-to-Borderline Intellectual Disability

Items of the Resistance to Peer Influence Questionnaire (RPIQ) have a tree-based structure. On each item, individuals first choose whether a less versus more peer-resistant group best describes them; they then indicate whether it is "Really true" versus "Sort of true" that they belong to the chosen group. Using tree-based item response theory, we show that RPIQ items tap three dimensions: A Resistance to Peer Influence (RPI) dimension and two Response Polarization dimensions. We then reveal subgroup differences on these dimensions. That is, adolescents with mild-to-borderline intellectual disability, compared with typically developing adolescents, are less RPI and more polarized in their responses. Also, girls, compared with boys, are more RPI, and, when high RPI, more polarized in their responses. Together, these results indicate that a tree-based modeling approach yields a more sensitive measure of individuals' RPI as well as their tendency to respond more or less extremely.

A heteroscedastic hidden Markov mixture model for responses and categorized response times

Various mixture modeling approaches have been proposed to identify within-subjects differences in the psychological processes underlying responses to psychometric tests. Although valuable, the existing mixture models are associated with at least one of the following three challenges: (1) A parametric distribution is assumed for the response times that—if violated—may bias the results; (2) the response processes are assumed to result in equal variances (homoscedasticity) in the response times, whereas some processes may produce more variability than others (heteroscedasticity); and (3) the different response processes are modeled as independent latent variables, whereas they may be related. Although each of these challenges has been addressed separately, in practice they may occur simultaneously. Therefore, we propose a heteroscedastic hidden Markov mixture model for responses and categorized response times that addresses all the challenges above in a single model. In a simulation study, we demonstrated that the model is associated with acceptable parameter recovery and acceptable resolution to distinguish between various special cases. In addition, the model was applied to the responses and response times of the WAIS-IV block design subtest, to demonstrate its use in practice.

A New Perspective on the Multidimensionality of Divergent Thinking Tasks

In the presented work, a shift of perspective with respect to the dimensionality of divergent thinking (DT) tasks is introduced moving from the question of multidimensionality across DT scores (i.e., fluency, flexibility, or originality) to the question of multidimensionality within one holistic score of DT performance (i.e., snapshot ratings of creative quality). We apply IRTree models to test whether unidimensionality assumptions hold in different task instructions for snapshot scoring of DT tests across Likert-scale points and varying levels of fluency. It was found that evidence for unidimensionality across scale points was stronger with be-creative instructions as compared to be-fluent instructions which suggests better psychometric quality of ratings when be-creative instructions are used. In addition, creative quality latent variables pertaining to low-fluency and high-fluency ideational pools shared around 50% of variance which suggests both strong overlap, and evidence for differentiation. The presented approach allows to further examine the psychometric quality of subjective ratings and to examine new questions with respect to within-item multidimensionality in DT.

An Overview of Models for Response Times and Processes in Cognitive Tests

Response times (RTs) are a natural kind of data to investigate cognitive processes underlying cognitive test performance. We give an overview of modeling approaches and of findings obtained with these approaches. Four types of models are discussed: response time models (RT as the sole dependent variable), joint models (RT together with other variables as dependent variable), local dependency models (with remaining dependencies between RT and accuracy), and response time as covariate models (RT as independent variable). The evidence from these approaches is often not very informative about the specific kind of processes (other than problem solving, information accumulation, and rapid guessing), but the findings do suggest dual processing: automated processing (e.g., knowledge retrieval) vs. controlled processing (e.g., sequential reasoning steps), and alternative explanations for the same results exist. While it seems well-possible to differentiate rapid guessing from normal problem solving (which can be based on automated or controlled processing), further decompositions of response times are rarely made, although possible based on some of model approaches.

Nonlinear Indicator-Level Moderation in Latent Variable Models

Linear, nonlinear, and nonparametric moderated latent variable models have been developed to investigate possible interaction effects between a latent variable and an external continuous moderator on the observed indicators in the latent variable model. Most moderation models have focused on moderators that vary across persons but not across the indicators (e.g., moderators like age and socioeconomic status). However, in many applications, the values of the moderator may vary both across persons and across indicators (e.g., moderators like response times and confidence ratings). Indicator-level moderation models are available for categorical moderators and linear interaction effects. However, these approaches require respectively categorization of the continuous moderator and the assumption of linearity of the interaction effect. In this article, parametric nonlinear and nonparametric indicator-level moderation methods are developed. In a simulation study, we demonstrate the viability of these methods. In addition, the methods are applied to a real data set pertaining to arithmetic ability.

Modeling Nonlinear Conditional Dependence Between Response Time and Accuracy

The most common process variable available for analysis due to tests presented in a computerized form is response time. Psychometric models have been developed for joint modeling of response accuracy and response time in which response time is an additional source of information about ability and about the underlying response processes. While traditional models assume conditional independence between response time and accuracy given ability and speed latent variables (van der Linden, 2007), recently multiple studies (De Boeck and Partchev, 2012; Meng et al., 2015; Bolsinova et al., 2017a,b) have shown that violations of conditional independence are not rare and that there is more to learn from the conditional dependence between response time and accuracy. When it comes to conditional dependence between time and accuracy, authors typically focus on positive conditional dependence (i.e., relatively slow responses are more often correct) and negative conditional dependence (i.e., relatively fast responses are more often correct), which implies monotone conditional dependence. Moreover, most existing models specify the relationship to be linear. However, this assumption of monotone and linear conditional dependence does not necessarily hold in practice, and assuming linearity might distort the conclusions about the relationship between time and accuracy. In this paper we develop methods for exploring nonlinear conditional dependence between response time and accuracy. Three different approaches are proposed: (1) A joint model for quadratic conditional dependence is developed as an extension of the response moderation models for time and accuracy (Bolsinova et al., 2017b); (2) A joint model for multiple-category conditional dependence is developed as an extension of the fast-slow model of Partchev and De Boeck (2012); (3) An indicator-level nonparametric moderation method (Bolsinova and Molenaar, in press) is used with residual log-response time as a predictor for the item intercept and item slope. Furthermore, we propose using nonparametric moderation to evaluate the viability of the assumption of linearity of conditional dependence by performing posterior predictive checks for the linear conditional dependence model. The developed methods are illustrated using data from an educational test in which, for the majority of the items, conditional dependence is shown to be nonlinear.

Response Mixture Modeling: Accounting for Heterogeneity in Item Characteristics across Response Times

In item response theory modeling of responses and response times, it is commonly assumed that the item responses have the same characteristics across the response times. However, heterogeneity might arise in the data if subjects resort to different response processes when solving the test items. These differences may be within-subject effects, that is, a subject might use a certain process on some of the items and a different process with different item characteristics on the other items. If the probability of using one process over the other process depends on the subject's response time, within-subject heterogeneity of the item characteristics across the response times arises. In this paper, the method of response mixture modeling is presented to account for such heterogeneity. Contrary to traditional mixture modeling where the full response vectors are classified, response mixture modeling involves classification of the individual elements in the response vector. In a simulation study, the response mixture model is shown to be viable in terms of parameter recovery. In addition, the response mixture model is applied to a real dataset to illustrate its use in investigating within-subject heterogeneity in the item characteristics across response times.

A semi-parametric within-subject mixture approach to the analyses of responses and response times

In item response theory, modelling the item response times in addition to the item responses may improve the detection of possible between- and within-subject differences in the process that resulted in the responses. For instance, if respondents rely on rapid guessing on some items but not on all, the joint distribution of the responses and response times will be a multivariate within-subject mixture distribution. Suitable parametric methods to detect these within-subject differences have been proposed. In these approaches, a distribution needs to be assumed for the within-class response times. In this paper, it is demonstrated that these parametric within-subject approaches may produce false positives and biased parameter estimates if the assumption concerning the response time distribution is violated. A semi-parametric approach is proposed which resorts to categorized response times. This approach is shown to hardly produce false positives and parameter bias. In addition, the semi-parametric approach results in approximately the same power as the parametric approach.

Modelling Conditional Dependence Between Response Time and Accuracy

The assumption of conditional independence between response time and accuracy given speed and ability is commonly made in response time modelling. However, this assumption might be violated in some cases, meaning that the relationship between the response time and the response accuracy of the same item cannot be fully explained by the correlation between the overall speed and ability. We propose to explicitly model the residual dependence between time and accuracy by incorporating the effects of the residual response time on the intercept and the slope parameter of the IRT model for response accuracy. We present an empirical example of a violation of conditional independence from a low-stakes educational test and show that our new model reveals interesting phenomena about the dependence of the item properties on whether the response is relatively fast or slow. For more difficult items responding slowly is associated with a higher probability of a correct response, whereas for the easier items responding slower is associated with a lower probability of a correct response. Moreover, for many of the items slower responses were less informative for the ability because their discrimination parameters decrease with residual response time.

Response moderation models for conditional dependence between response time and response accuracy

It is becoming more feasible and common to register response times in the application of psychometric tests. Researchers thus have the opportunity to jointly model response accuracy and response time, which provides users with more relevant information. The most common choice is to use the hierarchical model (van der Linden, 2007, Psychometrika, 72, 287), which assumes conditional independence between response time and accuracy, given a person's speed and ability. However, this assumption may be violated in practice if, for example, persons vary their speed or differ in their response strategies, leading to conditional dependence between response time and accuracy and confounding measurement. We propose six nested hierarchical models for response time and accuracy that allow for conditional dependence, and discuss their relationship to existing models. Unlike existing approaches, the proposed hierarchical models allow for various forms of conditional dependence in the model and allow the effect of continuous residual response time on response accuracy to be item-specific, person-specific, or both. Estimation procedures for the models are proposed, as well as two information criteria that can be used for model selection. Parameter recovery and usefulness of the information criteria are investigated using simulation, indicating that the procedure works well and is likely to select the appropriate model. Two empirical applications are discussed to illustrate the different types of conditional dependence that may occur in practice and how these can be captured using the proposed hierarchical models.

A comparison of item response models for accuracy and speed of item responses with applications to adaptive testing

We compare three modelling frameworks for accuracy and speed of item responses in the context of adaptive testing. The first framework is based on modelling scores that result from a scoring rule that incorporates both accuracy and speed. The second framework is the hierarchical modelling approach developed by van der Linden (2007, Psychometrika, 72, 287) in which a regular item response model is specified for accuracy and a log-normal model for speed. The third framework is the diffusion framework in which the response is assumed to be the result of a Wiener process. Although the three frameworks differ in the relation between accuracy and speed, one commonality is that the marginal model for accuracy can be simplified to the two-parameter logistic model. We discuss both conditional and marginal estimation of model parameters. Models from all three frameworks were fitted to data from a mathematics and spelling test. Furthermore, we applied a linear and adaptive testing mode to the data off-line in order to determine differences between modelling frameworks. It was found that a model from the scoring rule framework outperformed a hierarchical model in terms of model-based reliability, but the results were mixed with respect to correlations with external measures.

Conditional Dependence between Response Time and Accuracy: An Overview of its Possible Sources and Directions for Distinguishing between Them

With the widespread use of computerized tests in educational measurement and cognitive psychology, registration of response times has become feasible in many applications. Considering these response times helps provide a more complete picture of the performance and characteristics of persons beyond what is available based on response accuracy alone. Statistical models such as the hierarchical model (van der Linden, 2007) have been proposed that jointly model response time and accuracy. However, these models make restrictive assumptions about the response processes (RPs) that may not be realistic in practice, such as the assumption that the association between response time and accuracy is fully explained by taking speed and ability into account (conditional independence). Assuming conditional independence forces one to ignore that many relevant individual differences may play a role in the RPs beyond overall speed and ability. In this paper, we critically consider the assumption of conditional independence and the important ways in which it may be violated in practice from a substantive perspective. We consider both conditional dependences that may arise when all persons attempt to solve the items in similar ways (homogeneous RPs) and those that may be due to persons differing in fundamental ways in how they deal with the items (heterogeneous processes). The paper provides an overview of what we can learn from observed conditional dependences. We argue that explaining and modeling these differences in the RPs is crucial to increase both the validity of measurement and our understanding of the relevant RPs.

Hidden Markov Item Response Theory Models for Responses and Response Times

Current approaches to model responses and response times to psychometric tests solely focus on between-subject differences in speed and ability. Within subjects, speed and ability are assumed to be constants. Violations of this assumption are generally absorbed in the residual of the model. As a result, within-subject departures from the between-subject speed and ability level remain undetected. These departures may be of interest to the researcher as they reflect differences in the response processes adopted on the items of a test. In this article, we propose a dynamic approach for responses and response times based on hidden Markov modeling to account for within-subject differences in responses and response times. A simulation study is conducted to demonstrate acceptable parameter recovery and acceptable performance of various fit indices in distinguishing between different models. In addition, both a confirmatory and an exploratory application are presented to demonstrate the practical value of the modeling approach.

Joint Modeling of Ability and Differential Speed Using Responses and Response Times

With computerized testing, it is possible to record both the responses of test takers to test questions (i.e., items) and the amount of time spent by a test taker in responding to each question. Various models have been proposed that take into account both test-taker ability and working speed, with the many models assuming a constant working speed throughout the test. The constant working speed assumption may be inappropriate for various reasons. For example, a test taker may need to adjust the pace due to time mismanagement, or a test taker who started out working too fast may reduce the working speed to improve accuracy. A model is proposed here that allows for variable working speed. An illustration of the model using the Amsterdam Chess Test data is provided.

Distinguishing Fast and Slow Processes in Accuracy - Response Time Data

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.

A test for conditional independence between response time and accuracy

An important distinction between different models for response time and accuracy is whether conditional independence (CI) between response time and accuracy is assumed. In the present study, a test for CI given an exponential family model for accuracy (for example, the Rasch model or the one-parameter logistic model) is proposed and evaluated in a simulation study. The procedure is based on the non-parametric Kolmogorov-Smirnov tests. As an illustrative example, the CI test was applied to data from an arithmetics test for secondary education.

Measuring Ability, Speed, or Both? Challenges, Psychometric Solutions, and What Can Be Gained From Experimental Control

The main challenge of ability tests relates to the difficulty of items, whereas speed tests demand that test takers complete very easy items quickly. This article proposes a conceptual framework to represent how performance depends on both between-person differences in speed and ability and the speed-ability compromise within persons. Related measurement challenges and psychometric models that have been proposed to deal with the challenges are discussed. It is argued that addressing individual differences in the speed-ability trade-off requires the control of item response times. In this way, response behavior can be captured exclusively with the response variable remedying problems in traditional measurement approaches.

A Bivariate Generalized Linear Item Response Theory Modeling Framework to the Analysis of Responses and Response Times

A generalized linear modeling framework to the analysis of responses and response times is outlined. In this framework, referred to as bivariate generalized linear item response theory (B-GLIRT), separate generalized linear measurement models are specified for the responses and the response times that are subsequently linked by cross-relations. The cross-relations can take various forms. Here, we focus on cross-relations with a linear or interaction term for ability tests, and cross-relations with a curvilinear term for personality tests. In addition, we discuss how popular existing models from the psychometric literature are special cases in the B-GLIRT framework depending on restrictions in the cross-relation. This allows us to compare existing models conceptually and empirically. We discuss various extensions of the traditional models motivated by practical problems. We also illustrate the applicability of our approach using various real data examples, including data on personality and cognitive ability.