許多社會科學及醫學上的長期追蹤研究上，常會根據研究之需要，而針對某一群人在一段時間內重覆地收集其有關變項（包括類別型反應變項及解釋變項）的資料。這種重覆觀察的資料在統計的文獻上稱為長期追蹤研究資料。在這些長期追蹤研究上，研究者常利用迴歸模型建構的技巧來探討反應變項及解釋變項之間的關係。 一般常用的模型，著重於評估解釋變項對反應變項的當時及短期效應，當解釋變項比反應變項更頻繁地被觀測時，這些模型則不適用。當反應變項可在不同類別間變動時，我們通常有興趣去探討解釋變項如何去影響反應變項的演變或未來走向的趨勢，這種研究可稱之為類別型長期追蹤研究資料的未來趨勢分析。本論文提出了以馬可夫離散時間過程來建立類別型長期追蹤研究資料的模型。此模型不但可以捕捉到解釋變項對反應變項的未來趨勢效應；而且當解釋變項較反應變項更頻繁地被觀測時，本模型也可以利用解釋變項的完整訊息來做出更正確的統計推論。 / Many longitudinal studies in social science and medical science take repeated observations of an categorical outcome, along with several covariates, from follow-up subjects over a certain period of time. Such repeated observations are called longitudinal or panel data in the statistical literature. It is often of interest in these studies to investigate the relationship between the outcome and the covariates through regression modeling techniques. Commonly used models often focus on assessing the contemporary or short term effect of the covariate on the outcome, and can't incorporate time-varying covariates that are observed more or less frequently than the rate we observe the outcome. When the outcome fluctuates among different categories, it is often of interest to assess how covariates effect the evolution or trend of the underlying outcome process. Such assessment can be termed trend analysis of categorical panel data. In this thesis, we propose a Markov chain based regression model for analyzing nominal categorical panel data that are generated by a discrete time outcome process. The proposed model focuses on assessing the trend effect of the covariate on the categorical outcome, and is able to utilize the complete information of the covariates that are observed more or less frequently than the outcome.

實驗設計中，我們假設在區塊中存在一趨勢效應(trend effect)。此趨勢效應影響觀察值，也影響我們對區塊效應(block effect)和處理效應(treatment effect)的估計。此種設計模式不同於一般的區塊設計模式，因此須將趨勢效應加人設計模式中。 　　Bradley and Yeh (1980)研究和討論此種趨勢效應在區塊設計模式中之影響，並定義出無趨勢設計(trend-free design)。所謂無趨勢設計，乃是在區塊設計模式中，趨勢效應被抵消不影響處理效應之分析。Bradley and Yeh (1983)推導了一個線性無趨勢設計存在的必要條件是r(k+1)≡0(mod 2)其中k為區塊大小，r為處理出現的次數。 　　Bradley and Yeh進一步預測任一滿足r(k+1)≡0(mod2)的區塊設計，經過在區塊中處理位置調整後，可變為一個線性無趨勢設計。本篇論文的主要目的乃是在探討給定一GD設計(group-divisibledesigns)，檢驗和推導此預測是否為真。 / Yeh and Bradley conjectured that every binary connected block design with blocks of size k and a constant replication numberr for each treatment can be converted to a linear trend-free design by permuting the positions of theatments within blocks if and only if r(k+1)≡0 (mod 2). Chai and Majumdar (1993) proved that any BIB design which satisfies r(k+1)≡0 (mod 2) is even can be converted to a linear trend-free design. In this thesis, we want to examine this conjecture is true or not for group-divisible designs (GD designs).