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The linear transformation model with frailties for the analysis of item response times.
Br J Math Stat Psychol. 2013 Feb; 66(1):144-68.BJ

Abstract

The item response times (RTs) collected from computerized testing represent an underutilized source of information about items and examinees. In addition to knowing the examinees' responses to each item, we can investigate the amount of time examinees spend on each item. In this paper, we propose a semi-parametric model for RTs, the linear transformation model with a latent speed covariate, which combines the flexibility of non-parametric modelling and the brevity as well as interpretability of parametric modelling. In this new model, the RTs, after some non-parametric monotone transformation, become a linear model with latent speed as covariate plus an error term. The distribution of the error term implicitly defines the relationship between the RT and examinees' latent speeds; whereas the non-parametric transformation is able to describe various shapes of RT distributions. The linear transformation model represents a rich family of models that includes the Cox proportional hazards model, the Box-Cox normal model, and many other models as special cases. This new model is embedded in a hierarchical framework so that both RTs and responses are modelled simultaneously. A two-stage estimation method is proposed. In the first stage, the Markov chain Monte Carlo method is employed to estimate the parametric part of the model. In the second stage, an estimating equation method with a recursive algorithm is adopted to estimate the non-parametric transformation. Applicability of the new model is demonstrated with a simulation study and a real data application. Finally, methods to evaluate the model fit are suggested.

Authors+Show Affiliations

University of Illinois at Urbana-Champaign. cwang49@illinois.eduNo affiliation info availableNo affiliation info available

Pub Type(s)

Journal Article
Research Support, U.S. Gov't, Non-P.H.S.

Language

eng

PubMed ID

22506914

Citation

Wang, Chun, et al. "The Linear Transformation Model With Frailties for the Analysis of Item Response Times." The British Journal of Mathematical and Statistical Psychology, vol. 66, no. 1, 2013, pp. 144-68.
Wang C, Chang HH, Douglas JA. The linear transformation model with frailties for the analysis of item response times. Br J Math Stat Psychol. 2013;66(1):144-68.
Wang, C., Chang, H. H., & Douglas, J. A. (2013). The linear transformation model with frailties for the analysis of item response times. The British Journal of Mathematical and Statistical Psychology, 66(1), 144-68. https://doi.org/10.1111/j.2044-8317.2012.02045.x
Wang C, Chang HH, Douglas JA. The Linear Transformation Model With Frailties for the Analysis of Item Response Times. Br J Math Stat Psychol. 2013;66(1):144-68. PubMed PMID: 22506914.
* Article titles in AMA citation format should be in sentence-case
TY - JOUR T1 - The linear transformation model with frailties for the analysis of item response times. AU - Wang,Chun, AU - Chang,Hua-Hua, AU - Douglas,Jeffrey A, Y1 - 2012/04/17/ PY - 2012/4/18/entrez PY - 2012/4/18/pubmed PY - 2014/1/15/medline SP - 144 EP - 68 JF - The British journal of mathematical and statistical psychology JO - Br J Math Stat Psychol VL - 66 IS - 1 N2 - The item response times (RTs) collected from computerized testing represent an underutilized source of information about items and examinees. In addition to knowing the examinees' responses to each item, we can investigate the amount of time examinees spend on each item. In this paper, we propose a semi-parametric model for RTs, the linear transformation model with a latent speed covariate, which combines the flexibility of non-parametric modelling and the brevity as well as interpretability of parametric modelling. In this new model, the RTs, after some non-parametric monotone transformation, become a linear model with latent speed as covariate plus an error term. The distribution of the error term implicitly defines the relationship between the RT and examinees' latent speeds; whereas the non-parametric transformation is able to describe various shapes of RT distributions. The linear transformation model represents a rich family of models that includes the Cox proportional hazards model, the Box-Cox normal model, and many other models as special cases. This new model is embedded in a hierarchical framework so that both RTs and responses are modelled simultaneously. A two-stage estimation method is proposed. In the first stage, the Markov chain Monte Carlo method is employed to estimate the parametric part of the model. In the second stage, an estimating equation method with a recursive algorithm is adopted to estimate the non-parametric transformation. Applicability of the new model is demonstrated with a simulation study and a real data application. Finally, methods to evaluate the model fit are suggested. SN - 2044-8317 UR - https://www.unboundmedicine.com/medline/citation/22506914/The_linear_transformation_model_with_frailties_for_the_analysis_of_item_response_times_ L2 - https://doi.org/10.1111/j.2044-8317.2012.02045.x DB - PRIME DP - Unbound Medicine ER -