多點記分無參數試題反應理論與順序理論整合模式程式設計與應用

Program design and application of the combined model of nonparameter IRT and OT Theory

吳世能(Shih-Neng Wu)

指導教授 ： 劉湘川

亞洲大學/資訊學院/資訊工程學系碩士班/碩士(2008年)

摘要

二點記分於傳統認知測驗之各種試題上常採用，但常發生受試者同分之情況，無法真正區分出受試者的能力差異。某些試卷若使用多點記分，也常限於分析方法而只採取同質試題，例如只使用選擇題。劉湘川(2007a)考慮所有可能異質加權系統，兼顧記分之標準化與規格化，並與順序理論整合，提出改進之「標準規格化多點記分順序理論」。
參數型試題反應理論必須滿足局部獨立之限制，只宜分析無順序關係之試題，無參數試題反應理論則無局部獨立之限制，可與試題順序理論結合。Ramsay(1991)首先將各領域廣泛使用之核平滑化無參數回歸估計法轉化應用於二點記分之無參數試題反應理論，稱之為「核平滑化無參數試題反應理論」。
劉湘川(2007b)進一步以多點記分之積差相關加權替代二點記分之點二系列相關加權，提出擴張改進之「多點記分核平滑化無參數IRT模式」，結合上述的「標準規格化多點記分順序理論」，發表「多點記分核平滑化無參數IRT之順序理論』。
本研究運用「多點記分核平滑化無參數IRT之順序理論」，開發相關程式並以國小四、五年級數學領域「分數的加減」單元為範圍自編紙筆測驗進行實證研究。主要得到以下結論：
一、 本研究有效樣本122人中，透過多點記分及標準規格化之得分，只有16人有同分情形，再運用積差相關加權後之得分，則已無同分情形，可有效區隔受試者之得分，增加試券鑑別功能。
二、 試題解答機率隨能力提升而遞增，以本研究所舉的11題跟分數減法相關的試題而言，從低能力之.0468~.6750到中能力的.6080~.9495到高能力的.8845~1，可做為教學者評量時之參考。
三、 能力值不同，試題順序結構隨之不同，顯示不同能力值分數減法之概念結構不同，對此單元之學習有不同的路徑，可供教學者規劃教學順序及個別化補救教學之參考。

關鍵字

多點記分 ； 標準規格化得分 ； 核平化無參數IRT模式 ； 試題順序理論

並列摘要

Dichotomous Response Testing is widely used in the items of traditional cognitive tests, but examinees often receive the same grades in such items and their abilities cannot be distinguished. Some other tests using Polytomous Response Testing are constrained by the analytical methods and only homogeneous items are used, e.g. multiple choice questions. Hsiang-Chuan Liu (2007a) considered all possible heterogeneous weighted averages systems as well as the standardization and normalization of scoring, integrated the systems with Item Ordering Theory, and proposed the improved theory - Ordering Theory for Standardized and Normalized Polytomous Response Testing
Parametric Item Response Theory has the constraint of local independency and can only be used to analyze items without ordering relationship, while Nonparametric Item Response Theory is not constrained and can be combined with the Item Ordering Theory. Ramsay (1991) first applied Kernel Smoothing Nonparametric Regression Model to Nonparametric IRT Models for dichotomous response testing, and introduced “Kernel Smoothing Nonparametric IRT Models”.
Hsiang-Chuan Liu (2007b) further used the weighted averages of the product-moment correlation in Polytomous Response Testing to substitute that of the point-biserial correlation in ichotomous Response Testing, and proposed a expanded, improved model - Kernel Smoothing Nonparametric IRT Models for Polytomous Response Testing. Combining with “Ordering Theory for Standardized and Normalized Polytomous Response Testing”, he published “Kernel Smoothing Nonparametric IRT Models for Polytomous Response Testing”.
This study applies “Kernel Smoothing Nonparametric IRT Models for Polytomous Response Testing” to develop programs. An empirical study was also conducted by using the unit “The Addition and Subtraction of Fractions” in fourth/fifth-grade math to design written tests.
This study concludes that:
I. Out of the 122 effective samples, only 16 examinees receive the same grades through Polytomous Response Testing based on standardization-normalization scoring. None of them receive the same grades if the weighted averages of the product-moment correlation are used. This model can distinguish the examinees’ grades and increase the degree of differentiation of the tests.
II. The possibility for a question correctly answered is higher with the increase of the examinees’ abilities. For examples, the 11 questions relative to the subtraction of fractions in this study concludes that .0468 - .6750 stands for low ability, .6080 - .9495 for average ability, and .8845 - 1 for high ability. These can be useful reference when teachers are examining the examinees’ performance.
III. The item ordering structures differ when the abilities are different, showing that examinees with different levels of abilities differ in their conceptual structures for the subtraction of fractions and in their path of learning in this unit. Teachers can thus plan the order of teaching and provide individual remedial teaching.

並列關鍵字

Polytomous Response Testing ； Normalization-standardization Scoring ； Kernel Smoothing Nonparametric IRT Models ； Item Ordering Theories

參考文獻

劉湘川（2000），點二系列相關試題鑑別指數之值譜數分析及其在IRT上之應用。測驗統計年刊第八輯，1-22頁。台中市：國立台中師範學院。

劉湘川（2001b），核平滑化試題選項特徵曲線與選項關聯結構整合擴充模式。測驗統計年刊第九輯，1-18頁。台中市。國立台中師範學院。

劉湘川（2002a），高階相關比累進加權核平滑化試題選項綜合模式。測驗統計年刊第十輯，197-218頁。台中市。國立台中師範學院。

劉湘川（2002b），多重加權核平滑化試題選項綜合模式及其轉化應用。師範學院教育學術論文發表。嘉義市。國立嘉義大學。

劉湘川（2003a），混合型語義結構分析之研究。測驗統計年刊第十一輯，1-16頁。台中市。國立台中師範學院。