ISSN 0439-755X
CN 11-1911/B

Acta Psychologica Sinica ›› 2016, Vol. 48 ›› Issue (12): 1625-1630.doi: 10.3724/SP.J.1041.2016.01625

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The weighted-score logistic model and Monte Carlo simulation study

JIAN Xiaozhu1,2; DAI Buyun2; DAI Haiqi2   

  1. (1 School of Education, Jinggangshan University, Ji’an 343009, China) (2 School of Psychology, Jiangxi Normal University; Jiangxi Key Laboratory of Psychology and Cognitive Science, Nanchang 330022, China)
  • Received:2014-11-18 Published:2016-12-24 Online:2016-12-24
  • Contact: DAI Buyun, E-mail:


Item difficulty and item emphases are the two fundamental properties for the polytomously scored item. Thus, it is necessary to use a special parameter, the weighted-score parameter or the item full mark, to express the emphases of the polytomously scored item. In the previous studies, the researchers had proposed eight polytomous models, e.g., Graded-Response Model, Partial Credit Model, etc. In all the polytomous models, several item difficult parameters are used to represent the item difficulty based on the dichotomous Logistic model. Thus, the polytomous models may not effectively give expression to the item emphases of the polytomous item. A new polytomous model, the weighted-score logistic model (WSLM), is proposed in this study. On the basis of the item emphases of the polytomously scored item, the WSLM model adds the weighted-score parameters into the dichotomous logistic model. The WSLM includes only one difficulty parameter (i.e., the average difficulty parameter) to represent the overall item difficulty, which obviously differs from the other polytomous models. Moreover, in the WSLM, the probability of an examinee responding in category , is of certain functional relation with the average difficulty parameter, discrimination parameter, and the score that the examinee have obtained on this item. Thus, the probability of responding in category under the WSLM can be expressed as . According, the probabilities that an individual will receive the category scores of 0, 1, 2, …, under the WSLM are expressed by: respectively. And all the above probabilities add up to 1. Then, the probability that an examinee will receive a category score of or higher on a polytomously scored item is . It should be noted that, the WSLM reduces to the dichotomous logistic model if . Similarly, the probabilities of the WSLM can also be graphically represented via the category response curves and operating characteristic curves. What’s more, the shapes of the category response curves and operating characteristic curves of the WSLM are very similar to those of GRM. The item full mark is determined when designing the test, which can be considered as the indirect reflection of the common understanding of the weight of the item score. Thus, the item full mark of the polytomously scored item can be treated as the item weighted-score parameter, which does not need to be estimated. Just like the dichotomous logistic model, the discrimination parameter and average difficulty parameter of the WSLM can be estimated by using the classical MMLE/EM algorithm. For a mixed test containing both the dichotomously scored and polytomously scored items, the MMLE/EM algorithm can also be used to estimate all the item parameters. Following the basic procedure of MMLE/EM algorithm, we have written the item parameter estimation programs using the Visual Basic Program, and have successfully estimated the discrimination parameters and the average difficulty parameters of both the dichotomous and polytomous items under the WSLM. A Monte Carlo simulation study was conducted to investigate the performance of WSLM. The results of the simulation tests demonstrated that the ABS and RMSE of item parameters were relatively small. The numerical values of the item full marks can’t almost affect the ABS and RMSE of item parameters. When the scores on the items were not consecutive, the ABS and RMSE of item parameters were relatively small. Moreover, the ABS and RMSE of item parameters under the WSLM was as small as those under the dichotomous Logistic model. In summary, the item-parameter recovery in the simulation tests under the WSLM was effective and acceptable.

Key words: IRT, Logistic model, weighted-score Logistic model, polytomous models.