The purposes of the study were to compare the relative performances of three fixed item parameter calibration methods (FIPC) in item and ability parameter estimation and to examine how the ability estimates obtained from these different methods affect interpretations using reported scales of different lengths. Through a simulation design, the study was divided into two stages. The first stage was the calibration stage, where the parameters of pretest items were estimated. This stage investigated the accuracy of item parameter estimates and the recovery of the underlying ability distributions for different sample sizes, different numbers of pretest items, and different types of ability distributions under the three-parameter logistic model (3PL). The second stage was the operational stage, where the estimated parameters of the pretest items were put on operational forms and were used to score examinees. The second stage investigated the effect of item parameter estimation had on the ability estimation and reported scores for the new test forms. It was found that the item parameters estimated from the three FIPC methods showed subtle differences, but the results of the DeMars method were closer to those of the separate calibration with linking method than to the FIPC with simple-prior update and FIPC with iterative prior update methods, while the FIPC with simple-prior update and FIPC with iterative prior update methods performed similarly. Regarding the experimental factors that were manipulated in the simulation, the study found that the sample size influenced the estimation of item parameters. The effect of the number of pretest items on estimation of item parameters was strong but ambiguous, likely because the effect was confounded by changes of both the number of the pretest items and the characteristics of the pretest items among the item sets. The effect of ability distributions on estimation of item parameters was not as evident as the effect of the other two factors. After the pretest items were calibrated, the parameter estimates of these items were put into operational use. The abilities of the examinees were then estimated based on the examinees’ response to the existing operational items and the new items (previously called pretest items), of which the item parameters were estimated under different conditions. This study found that there were high correlations between the ability estimates and the true abilities of the examinees when forms containing pretest items calibrated using any of the three FIPC methods. The results suggested that all three FIPC methods were similarly competent in estimating parameters of the items, leading to satisfying determination of the examinees’ abilities. When considering the scale scores, because the estimated abilities were very similar, there were small differences among the scaled scores on the same scale; the relative frequency of examinees classified into performance categories and the classification consistency index also showed the interpretation of reported scores across scales were similar. The study provided a comprehensive comparison on the use of FIPC methods in parameter estimation. It was hoped that this study would help the practitioners choose among the methods according to the needs of the testing programs. When ability estimates were linearly transformed into scale scores, the lengths of scales did not affect the statistical properties of scores, however, they may impact how the scores are subjectively perceived by stakeholders and therefore should be carefully selected.

本論文探討當自變數存在測量誤差時，羅吉斯迴歸模型的估計問題，並將此結果應用在電腦化適性測驗中的線上校準問題。在變動長度電腦化測驗的假設下，我們證明了估計量的強收斂性。試題反應理論被廣泛地使用在電腦化適性測驗上，其假設受試者在試題的表現情形與本身的能力，可以透過試題特徵曲線加以詮釋，羅吉斯迴歸模式是最常見的試題反應模式。藉由適性測驗的施行，考題的選取可以依據不同受試者，選擇最適合的題目。因此，相較於傳統測驗而言，在適性測驗中，題目的消耗量更為快速。在題庫的維護與管理上，新試題的補充與試題校準便為非常重要的工作。線上試題校準意指在線上測驗進行中，同時進行試題校準。因此，受試者的能力估計會存在測量誤差。從統計的觀點，線上校準面臨的困難，可以解釋為在非線性模型下，當自變數有測量誤差時的實驗設計問題。我們利用序貫設計降低測量誤差，得到更精確的估計，相較於傳統的試題校準，可以節省更多的時間及成本。我們利用處理測量誤差的技巧，進一步應用序貫設計的方法，處理在線上校準中，受試者能力存在測量誤差的問題。 / In this dissertation, we focus on the estimate in logistic regression models when the independent variables are subject to some measurement errors. The problem of this dissertation is motivated by online calibration in Computerized Adaptive Testing (CAT). We apply the measurement error model techniques and adaptive sequential design methodology to the online calibration problem of CAT. We prove that the estimates of item parameters are strongly consistent under the variable length CAT setup. In an adaptive testing scheme, examinees are presented with different sets of items chosen from a pre-calibrated item pool. Thus the speed of attrition in items will be very fast, and replenishing of item pool is essential for CAT. The online calibration scheme in CAT refers to estimating the item parameters of new, un-calibrated items by presenting them to examinees during the course of their ability testing together with previously calibrated items. Therefore, the estimated latent trait levels of examinees are used as the design points for estimating the parameter of the new items, and naturally these designs, the estimated latent trait levels, are subject to some estimating errors. Thus the problem of the online calibration under CAT setup can be formulated as a sequential estimation problem with measurement errors in the independent variables, which are also chosen sequentially. Item Response Theory (IRT) is the most commonly used psychometric model in CAT, and the logistic type models are the most popular models used in IRT based tests. That's why the nonlinear design problem and the nonlinear measurement error models are involved. Sequential design procedures proposed here can provide more accurate estimates of parameters, and are more efficient in terms of sample size (number of examinees used in calibration). In traditional calibration process in paper-and-pencil tests, we usually have to pay for the examinees joining the pre-test calibration process. In online calibration, there will be less cost, since we are able to assign new items to the examinees during the operational test. Therefore, the proposed procedures will be cost-effective as well as time-effective.