不同難度與能力分布下條件測量標準誤的差異問題之研究

The Study of Standard Errors of Measurement on Variant Difficulty and Ability Distributions

一般而言，條件測量標準誤是在大樣本性質的條件下，利用測驗訊息函數所推導出來的，稱為「漸近條件測量標準誤」，為一種近似估計值。但是在實務上，試題題數大多僅數十題，而且其試題難度分布經常是兩極端的題數會較少，因此在估計兩極端的試題難度時，經常會出現不符合大樣本性質的狀況，此時估計的條件測量標準誤與漸近條件測量標準誤是會產生差異的。另外，在大樣本的情況下，漸近條件測量標準誤與能力分布無關，但若題數僅數十題，測量標準誤則可能會受能力分布的影響。本研究擬以在較少題數的情況下，實際模擬估算條件測量標準誤，研究方法以模擬方式產生負偏態、正偏態、常態及雙峰等四種常見的難度與能力分布型態，依據IRT單參數對數模式模擬答題反應資料，計算在各種能力值下估計的條件測量標準誤，之後再與漸近條件測量標準誤進行差異的比較；此外，也採用本研究所提出的「分段平均標準誤」統計量，比較各能力區段估計的測量標準誤之差異。研究結果顯示，在某些能力區段中，估計的條件測量標準誤會異常高於漸近條件測量標準誤。此外，不同的受試者能力分布與試題難度分布的組合，也會對估計的條件測量標準誤造成不同程度的影響。

In general， conditional standard errors of measurement (CSEM) are approximate estimation derived from test information function under large sample properties and are called “asymptotic conditional standard errors of measurement” (asymptotic CSEM).However， large sample properties are usually not satisfied in practice， especially at the tail of difficulty distribution. It makes the parameter estimation of more difficult or easier items discrepant and doubtful. In addition， asymptotic CSEM is irrelevant to the distribution of ability under the condition of large sample size and may be influencedby ability when the number of item is small. This study attempts to compute estimated CSEM under small item size by simulation. The research method was to simulate 4 common distribution types of item difficulty and ability: positive skewness， negative skewness， normal， and two-modal， and thus constituting the 4 difficulty types×4abi-lity types， totally 16 combinations. Dichotomous one-parameter logistic model was used to simulate the responses of examinee and to compute CSEM from the 16 combinations of difficulty and ability. Then the results of estimated CSEM were compared with the results of asymptotic CSEM. Furthermore， this study also presents a statistic about di-vided average standard errors of measurement to assess the estimated CSEM in each ability level. The results reveal that estimated CSEM is higher than asymptotic CSEM at some specific ability levels. In addition， the estimated CSEM changes with different types of ability distribution when the difficulty distribution is fixed. Finally， some con-clusions and suggestions are proposed for future researches.