- 學位論文
以加權最小平方法分析重複有序之資料
Weighted Least Squares Analysis for Repeated Ordinal Data
摘要
本研究分成四部份，第一部分說明研究背景、研究目的，以及論文整體架構；第二部份為研究方法，說明本研究的理論推導；第三部份為實例說明，將上部份的理論運用於實例中，觀察其分析結果；最後一部分為本研究的結論。 主要的數學理論有多變數中央極限定理、Delta method、加權最小平方法，當應變數屬於類別型時,加權最小平方法是第一個被應用來分析重複試驗資料的方法。雖然加權最小平方法大多用於分析類別型的重複試驗資料，但本研究發現當資料型態是次序型態時，也能用加權最小平方法來做分析，而且類別型的反應變數試驗結果不再只會分少數幾類，只要是分有限類都可用本文的方法進行分析，而如何用加權最小平方法來分析重複有序之資料即是本研究的主要目的。 此用的方法是將每位試驗者所得的觀測值轉成排序值，再用多變數中央極限定理、Delta method、加權最小平方法和本文使用的方法來檢定樣本間是否有統計上的差異。此方法並不需要假設重複試驗的每個時間點為獨立，只需假設樣本中的資料為多項的取樣。在實例的部分，將會用本文探討的方法來檢定時間點的排序值是否有線性、二次的統計差異。
並列摘要
A new approach to analyze the repeated outcomes is proposed. By transforming each of subjects to a rank component vector and then applying the multivariate central limit theory and the delta method, the proposed method can be used to test the difference within group and between groups. This methodology makes no assumptions concerning the time dependence among the repeated measurements. It is based only on the multinomial distribution for count data. The practical examples testing the linear and quadratic components of the time effect illustrate the use of the proposed method. The underlying model for the weighted least squares approach is the multinomial distribution. Although the distribution assumptions are much weaker, one still must make some basic assumptions concerning the marginal distributions at each time point. In addition, the assumptions of specific ordinal data methods such as the proportional odds model may be inappropriate. In all of these situations, nonparametric methods for analyzing repeated measurements may be of use. The proposed method is to assign ranks to repeated measurements from the smallest value to the largest value for each subject. The vector of rank means can be computed by the linear transformation of these ranks. Then the multivariate central limit theory and the delta method are applied to obtain the test statistics. The methods make no assumptions concerning the distribution of the response variable. Two practical examples will be illustrated the use of the proposed method.