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

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 Comparison of Response Time Threshold Scoring Procedures in Mitigating Bias From Rapid Guessing Behavior

Rapid guessing (RG) is a form of non-effortful responding that is characterized by short response latencies. This construct-irrelevant behavior has been shown in previous research to bias inferences concerning measurement properties and scores. To mitigate these deleterious effects, a number of response time threshold scoring procedures have been proposed, which recode RG responses (e.g., treat them as incorrect or missing, or impute probable values) and then estimate parameters for the recoded dataset using a unidimensional or multidimensional IRT model. To date, there have been limited attempts to compare these methods under the possibility that RG may be misclassified in practice. To address this shortcoming, the present simulation study compared item and ability parameter recovery for four scoring procedures by manipulating sample size, the linear relationship between RG propensity and ability, the percentage of RG responses, and the type and rate of RG misclassifications. Results demonstrated two general trends. First, across all conditions, treating RG responses as incorrect produced the largest degree of combined systematic and random error (larger than ignoring RG). Second, the remaining scoring approaches generally provided equal accuracy in parameter recovery when RG was perfectly identified; however, the multidimensional IRT approach was susceptible to increased error as misclassification rates grew. Overall, the findings suggest that recoding RG as missing and employing a unidimensional IRT model is a promising approach.

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.

Bayesian hierarchical response time modelling-A tutorial

Response time modelling is developing rapidly in the field of psychometrics, and its use is growing in psychology. In most applications, component models for response times are modelled jointly with component models for responses, thereby stabilizing estimation of item response theory model parameters and enabling research on a variety of novel substantive research questions. Bayesian estimation techniques facilitate estimation of response time models. Implementations of these models in standard statistical software, however, are still sparse. In this accessible tutorial, we discuss one of the most common response time models-the lognormal response time model-embedded in the hierarchical framework by van der Linden (2007). We provide detailed guidance on how to specify and estimate this model in a Bayesian hierarchical context. One of the strengths of the presented model is its flexibility, which makes it possible to adapt and extend the model according to researchers' needs and hypotheses on response behaviour. We illustrate this based on three recent model extensions: (a) application to non-cognitive data incorporating the distance-difficulty hypothesis, (b) modelling conditional dependencies between response times and responses, and (c) identifying differences in response behaviour via mixture modelling. This tutorial aims to provide a better understanding of the use and utility of response time models, showcases how these models can easily be adapted and extended, and contributes to a growing need for these models to answer novel substantive research questions in both non-cognitive and cognitive contexts.

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.

Disentangling Different Aspects of Change in Tests with the D-Diffusion Model

Diffusion-based item response theory models are measurement models that link parameters of the diffusion model (drift rate, boundary separation) to latent traits of test takers. Similar to standard latent trait models, they assume the invariance of the test takers' latent traits during a test. Previous research, however, suggests that traits change as test takers learn or decrease their effort. In this paper, we combine the diffusion-based item response theory model with a latent growth curve model. In the model, the latent traits of each test taker are allowed to change during the test until a stable level is reached. As different change processes are assumed for different traits, different aspects of change can be separated. We discuss different versions of the model that make different assumptions about the form (linear versus quadratic) and rate (fixed versus individual-specific) of change. In order to fit the model to data, we propose a Bayes estimator. Parameter recovery is investigated in a simulation study. The study suggests that parameter recovery is good under certain conditions. We illustrate the application of the model to data measuring visuo-spatial perspective-taking.

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.

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 Rapid Guessing Behaviors in Computer-Based Testlet Items

In traditional test models, test items are independent, and test-takers slowly and thoughtfully respond to each test item. However, some test items have a common stimulus (dependent test items in a testlet), and sometimes test-takers lack motivation, knowledge, or time (speededness), so they perform rapid guessing (RG). Ignoring the dependence in responses to testlet items can negatively bias standard errors of measurement, and ignoring RG by fitting a simpler item response theory (IRT) model can bias the results. Because computer-based testing captures response times on testlet responses, we propose a mixture testlet IRT model with item responses and response time to model RG behaviors in computer-based testlet items. Two simulation studies with Markov chain Monte Carlo estimation using the JAGS program showed (a) good recovery of the item and person parameters in this new model and (b) the harmful consequences of ignoring RG (biased parameter estimates: overestimated item difficulties, underestimated time intensities, underestimated respondent latent speed parameters, and overestimated precision of respondent latent estimates). The application of IRT models with and without RG to data from a computer-based language test showed parameter differences resembling those in the simulations.

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.

Using a Response Time-Based Expected A Posteriori Estimator to Control for Differential Speededness in Computerized Adaptive Test

This study investigates using response times (RTs) with item responses in a computerized adaptive test (CAT) setting to enhance item selection and ability estimation and control for differential speededness. Using van der Linden's hierarchical framework, an extended procedure for joint estimation of ability and speed parameters for use in CAT is developed following van der Linden; this is called the joint expected a posteriori estimator (J-EAP). It is shown that the J-EAP estimate of ability and speededness outperforms the standard maximum likelihood estimator (MLE) of ability and speededness in terms of correlation, root mean square error, and bias. It is further shown that under the maximum information per time unit item selection method (MICT)-a method which uses estimates for ability and speededness directly-using the J-EAP further reduces average examinee time spent and variability in test times between examinees above the resulting gains of this selection algorithm with the MLE while maintaining estimation efficiency. Simulated test results are further corroborated with test parameters derived from a real data example.

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.

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.

Validating Test Score Interpretations Using Time Information

A validity approach is proposed that uses processing times to collect validity evidence for the construct interpretation of test scores. The rationale of the approach is based on current research of processing times and on classical validity approaches, providing validity evidence based on relationships with other variables. Within the new approach, convergent validity evidence is obtained if a component skill, that is expected to underlie the task solution process in the target construct, positively moderates the relationship between effective speed and effective ability in the corresponding target construct. Discriminant validity evidence is provided if a component skill, that is not expected to underlie the task solution process in the target construct, does indeed not moderate the speed-ability relation in this target construct. Using data from a study that follows up the German PIAAC sample, this approach was applied to reading competence, assessed with PIAAC literacy items, and to quantitative reasoning, assessed with Number Series. As expected from theory, the effect of speed on ability in the target construct was only moderated by the respective underlying component skill, that is, word meaning activation skill as an underlying component skill of reading competence, and perceptual speed as an underlying component skill of reasoning. Accordingly, no positive interactions were found for the component skill that should not underlie the task solution process, that is, word meaning activation for reasoning and perceptual speed for reading. Furthermore, the study shows the suitability of the proposed validation approach. The use of time information in association with task results brings construct validation closer to the actual response process than widely used correlations of test scores.

How Do Test Takers Interact With Simulation-Based Tasks? A Response-Time Perspective

Many traditional educational assessments use multiple-choice items and constructed-response items to measure fundamental skills. Virtual performance assessments, such as game- or simulation-based assessments, are designed recently in the field of educational measurement to measure more integrated skills through the test takers' interactive behaviors within an assessment in a virtual environment. This paper presents a systematic timing study based on data collected from a simulation-based task designed recently at Educational Testing Service. The study is intended to understand the response times in complex simulation-based tasks so as to shed light on possible ways of leveraging response time information in designing, assembling, and scoring of simulation-based tasks. To achieve this objective, a series of five analyses were conducted to first understand the statistical properties of the timing data, and then investigate the relationship between the timing patterns and the test takers' performance on the items/task, demographics, motivation level, personality, and test-taking behaviors through use of different statistical approaches. We found that the five analyses complemented each other and revealed different useful timing aspects of this test-taker sample's behavioral features in the simulation-based task. The findings were also compared with notable existing results in the literature related to timing data.

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.

An IRT-Multiple Indicators Multiple Causes (MIMIC) Approach as a Method of Examining Item Response Latency

The analysis of response time has received increasing attention during the last decades, since evidence from several studies supported the argument that there is a direct relationship between item response time and test performance. The aim of this study was to investigate whether item response latency affects person's ability parameters, in that it represents an adaptive or maladaptive practice. To examine the above research question data from 8,475 individuals completing the computerized version of the Postgraduate General Aptitude Test (PAGAT) were analyzed. To determine the extent to which response latency affects person's ability, we used a Multiple Indicators Multiple Causes (MIMIC) model, in which every item in a scale was linked to its corresponding covariate (i.e., item response latency). We ran the MIMIC model within the Item Response Theory (IRT) framework (2-PL model). The results supported the hypothesis that item response latency could provide valuable information for getting more accurate estimations for persons' ability levels. Results indicated that for individuals who invest more time on easy items, their likelihood of success does not improve, most likely because slow and fast responders have significantly different levels of ability (fast responders are of higher ability compared to slow responders). Consequently, investing more time for low ability individuals does not prove to be adaptive. The opposite was found for difficult items: individuals spending more time on difficult items increase their likelihood of success, more likely because they are high achievers (in difficult items individuals who spent more time were of significantly higher ability compared to fast responders). Thus, it appears that there is an interaction between the difficulty of the item and person abilities that explain the effects of response time on likelihood of success. We concluded that accommodating item response latency in a computerized assessment model, can inform test quality and test takers' behavior, and in that way, enhance score measurement accuracy.

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 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.