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.     

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.     

The Joint Multivariate Modeling of Multiple Mixed Response Sources: Relating Student Performances with Feedback Behavior

The present study concerns a Dutch computer-based assessment, which includes an assessment process about information literacy and a feedback process for students. The assessment is concerned with the measurement of skills in information literacy and the feedback process with item-based support to improve student learning. To analyze students’ feedback behavior (i.e. feedback use and attention time), test performance, and speed of working, a multivariate hierarchical latent variable model is proposed. The model can handle multivariate mixed responses from multiple sources related to different processes and comprehends multiple measurement components for responses and response times. A flexible within-subject latent variable structure is defined to explore multiple individual latent characteristics related to students’ test performance and feedback behavior. Main results of the computer-based assessment showed that feedback-information pages were less visited by well-performing students when they relate to easy items. Students’ attention paid to feedback was positively related to working speed but not to the propensity to use feedback.     

计算机化多维测验中作答时间和作答精度数据的联合分析

随着心理与教育测量研究的发展和科技的进步,计算机化(大规模)测验逐渐受到人们的关注。为探究在计算机化多维测验中如何利用作答时间数据来辅助评估多维潜在能力,以及为我国义务教育阶段教育质量监测提供数据分析方法上的理论支持。本研究以2012年和2015年国际学生能力评估(PISA)计算机化数学测验数据为例,提出了一种可同时利用作答时间和作答精度数据的联合作答与时间的多维Rasch模型。根据新模型对PISA数据的分析结果,表明引入作答时间数据,不仅有助于提高模型参数的估计精度,还有助于数据分析者利用被试的作答时间信息来做进一步的决策和干预(e.g., 对异常作答行为或预备知识的诊断)。     

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.     

On the Speed Sensitivity Parameter in the Lognormal Model for Response Times and Implications for High-Stakes Measurement Practice

In high-stakes testing, often multiple test forms are used and a common time limit is enforced. Test fairness requires that ability estimates must not depend on the administration of a specific test form. Such a requirement may be violated if speededness differs between test forms. The impact of not taking speed sensitivity into account on the comparability of test forms regarding speededness and ability estimation was investigated. The lognormal measurement model for response times by van der Linden was compared with its extension by Klein Entink, van der Linden, and Fox, which includes a speed sensitivity parameter. An empirical data example was used to show that the extended model can fit the data better than the model without speed sensitivity parameters. A simulation was conducted, which showed that test forms with different average speed sensitivity yielded substantial different ability estimates for slow test takers, especially for test takers with high ability. Therefore, the use of the extended lognormal model for response times is recommended for the calibration of item pools in high-stakes testing situations. Limitations to the proposed approach and further research questions are discussed.     

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.     

The Graded Response Differential Discrimination Model Accounting for Extreme Response Style

Extreme response style or, more generally, individual differences in response spacing have been shown to be an influential bias when analyzing questionnaire data. Recently a promising model adjusting for this bias — the differential discrimination model — has been proposed. An advantage to other related approaches is that the model can be fitted using standard structural equation modeling software. However, the model is designed for analyzing continuous item responses, whereas graded response formats are certainly more prominent in behavioral sciences. To resolve this limitation, the present article extends the differential discrimination model to analyzing graded responses. Empirical examples as well as a small simulation study are presented.     

Separating response‐execution bias from decision bias: Arguments for an additional parameter in Ratcliff's diffusion model

Diffusion model data analysis permits the disentangling of different processes underlying the effects of experimental manipulations. Estimates can be provided for the speed of information accumulation, for the amount of information used to draw conclusions, and for a decision bias. One parameter describes the duration of non‐decisional processes including the duration of motor‐response execution. In the default diffusion model, it is implicitly assumed that both responses are executed with the same speed. In some applications of the diffusion model, this assumption will be violated. This will lead to biased parameter estimates. Consequently, we suggest accounting explicitly for differences in the speed of response execution for both responses. Results from a simulation study illustrate that parameter estimates from the default model are biased if the speed of response execution differs between responses. A second simulation study shows that large trial numbers (N>1,000) are needed to detect whether differences in response‐execution times are based on different execution times.     

多维对数正态作答时间模型：对潜在加工速度多维性的探究

在心理与教育测量中, 潜在加工速度反映学生运用潜在能力解决问题的效率。为在多维测验中探究潜在加工速度的多维性并实现参数估计, 本研究提出多维对数正态作答时间模型。实证数据分析及模拟研究结果表明：(1)潜在加工速度具有与潜在能力相匹配的多维结构; (2)新模型可精确估计个体水平的多维潜在加工速度及与作答时间有关的题目参数; (3)冗余指定潜在加工速度具有多维性带来的负面影响低于忽略其多维性所带来的。     

项目反应时间的对数偏正态模型

近年来,项目反应时间数据的建模是心理和教育测量领域的热门方向之一。针对反应时间的对数正态模型和Box-Cox正态模型的不足,本文在van der Linden的分层模型框架下基于偏正态分布建立一个反应时间的对数线性模型,并成功给出模型参数估计的马尔科夫链蒙特卡罗（Markov Chain Monte Carlo, MCMC）算法。模拟研究和实例分析的结果均表明,与对数正态模型和Box-Cox正态模型相比,对数偏正态模型表现出更加优良的拟合效果,具有更强的灵活性和适用性。     

A Multivariate Multilevel Approach to the Modeling of Accuracy and Speed of Test Takers

Response times on test items are easily collected in modern computerized testing. When collecting both (binary) responses and (continuous) response times on test items, it is possible to measure the accuracy and speed of test takers. To study the relationships between these two constructs, the model is extended with a multivariate multilevel regression structure which allows the incorporation of covariates to explain the variance in speed and accuracy between individuals and groups of test takers. A Bayesian approach with Markov chain Monte Carlo (MCMC) computation enables straightforward estimation of all model parameters. Model-specific implementations of a Bayes factor (BF) and deviance information criterium (DIC) for model selection are proposed which are easily calculated as byproducts of the MCMC computation. Both results from simulation studies and real-data examples are given to illustrate several novel analyses possible with this modeling framework. The authors thank Steven Wise, James Madison University, and Pere Joan Ferrando, Universitat Rovira i Virgili, for generously making available their data sets for the empirical examples in this paper.     

A hierarchical latent response model for inferences about examinee engagement in terms of guessing and item-level non-response

In low-stakes assessments, test performance has few or no consequences for examinees themselves, so that examinees may not be fully engaged when answering the items. Instead of engaging in solution behaviour, disengaged examinees might randomly guess or generate no response at all. When ignored, examinee disengagement poses a severe threat to the validity of results obtained from low-stakes assessments. Statistical modelling approaches in educational measurement have been proposed that account for non-response or for guessing, but do not consider both types of disengaged behaviour simultaneously. We bring together research on modelling examinee engagement and research on missing values and present a hierarchical latent response model for identifying and modelling the processes associated with examinee disengagement jointly with the processes associated with engaged responses. To that end, we employ a mixture model that identifies disengagement at the item-by-examinee level by assuming different data-generating processes underlying item responses and omissions, respectively, as well as response times associated with engaged and disengaged behaviour. By modelling examinee engagement with a latent response framework, the model allows assessing how examinee engagement relates to ability and speed as well as to identify items that are likely to evoke disengaged test-taking behaviour. An illustration of the model by means of an application to real data is presented.     

Psychometric Modeling of response speed and accuracy with mixed and conditional regression

Human performance in cognitive testing and experimental psychology is expressed in terms of response speed and accuracy. Data analysis is often limited to either speed or accuracy, and/or to crude summary measures like mean response time (RT) or the percentage correct responses. This paper proposes the use of mixed regression for the psychometric modeling of response speed and accuracy in testing and experiments. Mixed logistic regression of response accuracy extends logistic item response theory modeling to multidimensional models with covariates and interactions. Mixed linear regression of response time extends mixed ANOVA to unbalanced designs with covariates and heterogeneity of variance. Related to mixed regression is conditional regression, which requires no normality assumption, but is limited to unidimensional models. Mixed and conditional methods are both applied to an experimental study of mental rotation. Univariate and bivariate analyzes show how within-subject correlation between response and RT can be distinguished from between-subject correlation, and how latent traits can be detected, given careful item design or content analysis. It is concluded that both response and RT must be recorded in cognitive testing, and that mixed regression is a versatile method for analyzing test data.I am grateful to Rogier Donders for putting his data at my disposal.     

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.     

Speed of reasoning and its relation to reasoning ability

The study investigates empirical properties of reasoning speed which is conceived as the fluency of solving reasoning problems. Responses and response times in reasoning tasks are modeled jointly to clarify the covariance structure of reasoning speed and reasoning ability. To determine underlying abilities, the predictive validities of two cognitive covariates, namely perceptual and executive attention, are investigated. A sample of N = 230 test takers completed a reasoning test, Advanced Progressive Matrices (APM), and attention tests indicating perceptual and executive attention. For modeling responses the two-parameter normal ogive model, and for modeling response times the two-parameter lognormal model was applied. Results suggest that reasoning speed is a unidimensional construct representing significant individual differences, and that reasoning speed and ability are negatively correlated but clearly distinguishable constructs. Perceptual and executive attention showed differential effects on reasoning speed and reasoning ability, i.e., reasoning speed is explained by executive attention only, while reasoning ability is explained by both covariates. Implications for the assessment of reasoning 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.     

Measurement of Ability in Adaptive Learning and Assessment Systems when Learners Use On-Demand Hints

Adaptive learning and assessment systems support learners in acquiring knowledge and skills in a particular domain. The learners’ progress is monitored through them solving items matching their level and aiming at specific learning goals. Scaffolding and providing learners with hints are powerful tools in helping the learning process. One way of introducing hints is to make hint use the choice of the student. When the learner is certain of their response, they answer without hints, but if the learner is not certain or does not know how to approach the item they can request a hint. We develop measurement models for applications where such on-demand hints are available. Such models take into account that hint use may be informative of ability, but at the same time may be influenced by other individual characteristics. Two modeling strategies are considered: (1) The measurement model is based on a scoring rule for ability which includes both response accuracy and hint use. (2) The choice to use hints and response accuracy conditional on this choice are modeled jointly using Item Response Tree models. The properties of different models and their implications are discussed. An application to data from Duolingo, an adaptive language learning system, is presented. Here, the best model is the scoring-rule-based model with full credit for correct responses without hints, partial credit for correct responses with hints, and no credit for all incorrect responses. The second dimension in the model accounts for the individual differences in the tendency to use hints.     

Shaping the location of a pigeon's peck: effect of rate and size of shaping steps.

For several pigeons, pecking at particular locations within a ten-inch-wide response area was reinforced by grain presentations. The reinforced locations changed systematically to "shape" response location back and forth across the area. The rate and size of these shifts in reinforced locations were varied in both between-subject and within-subject comparisons to evaluate the influence of these variables on the shaping process. Larger step sizes produced larger shifts in location for all sizes inspected, with all sizes from .5 to 3.0 inches effective in shaping behavior. More rapid steps were approximately as effective as slower steps for all rates of shift inspected from 25 reinforcers to 400 reinforcers per step. These data suggest that shaping peck location proceeds most efficiently with rapid, relatively large shifts in criterion performance.     

Response persistence under ratio and interval reinforcement schedules

In Experiment 1, rats were exposed to progressive-ratio schedules of food reinforcement while other rats were exposed simultaneously to yoked-interval schedules that arranged equivalent interreinforcer intervals but required only a single response at the end of the interval for food delivery. In Experiment 2, a within-subject yoked-control procedure was employed in which pigeons were exposed to alternating sessions (one per day) of progressive-ratio schedules and yoked-interval schedules as described above. In both experiments, responding under the yoked-interval schedule persisted beyond the point at which responding under the progressive-ratio schedule had ceased. The progressive-ratio schedules controlled break-and-run distributions, and the yoked-interval schedules controlled more even distributions of responses in time. Response rates decreased and postreinforcement pauses increased over time within individual sessions under both schedules. The results suggest that responding maintained by interval schedules is more persistent than that maintained by ratio schedules. The limitations and implications of this conclusion are discussed in the context of other investigations of response strength and behavioral momentum.