臺灣博碩士論文加值系統

許多實證研究資料的研究，所處理的資料常常具有階層性的巢狀結構關係(nested structure)，例如，在教育學欲探討影響學生學習效果的關係，收集了學生所屬的班級、老師上課的方式、學生的性別、學生的家庭背景和學生所屬的學校，學生內屬於班級，班級內屬於學校，這就是一具有三階層的資料，具有上述關係的資料，我們就簡稱為階層性資料。傳統對於這類研究的處理，常會以迴歸方法，利用「合計」(aggregation)或「散計」(disaggregation)的技巧，將具有階層性的資料轉變成不具有階層性的資料，再去加以分析，但階層性資料組內常存在著相關的關係，也因此迴歸方法會忽略了組內相關的變異，使得分析結果不夠精確。為了解決這類的問題，近年來發展出另一套新興的統計方法，階層線性模型(Hierarchical Linear Models；HLM)，去加以處理。簡而言之，階層線性模型就是對於各個階層分別建立迴歸模型，再將組內與組間的變異因素考慮進去，作整合性的分析，因此可以避免掉忽略階層性資料組內相關的問題。
参數估計的好壞是個很重要的議題，在HLM領域上也不例外。本篇論文的主桿在於探討在不同實驗設計情況下，參數估計準確度的好壞。所要探討的影響因子有五個，分別為組內個數、組別個數、斜率隨機效果變異數、截距隨機效果變異數、斜率與截距之間的相關係數，所使用的模型為階層一之截距及斜率為階層二結果變項模式(a intercept-and slopes-as-outcomes model) ，依變數為二元反應變數，估計參數共有七個，分別為 (平均截距項)、 (平均斜率項)、 (組別迴歸係數)、 (組間迴歸係數)、 (斜率隨機效果變異數)、 (截距隨機效果變異數)、 (斜率與截距之間的變異數)，我們將利用五個影響因子，交互組合作出96組的資料集，利用SAS軟體模擬分析資料，求出估計參數值，將估計的參數值與原本設定的參數值做比較，利用Bias與MSE的值去判斷其準確性，再將結果做整合性的探討，進而探討影響參數估計的精確度的因子為何。
研究結果，對於樣本數的探討而言，樣本數的大小不會對參數估計的準確度有所影響，因此當樣本數設定的類型與本研究類似時，可以不必考慮到樣本數對於參數估計準確度的影響，因此對於大小樣本而言，用小樣本估計參數即可獲得良好的估計值。對於變異數探討而言，如果斜率隨機效果變異數、截距隨機效果變異數的值太大或斜率與截距之間的相關係數太小時，所估計出來的參數值會有不準確的情形出現。

Nested data structures often exist in many empirical researches. It is traditionally handled with regression techniques to aggregate or dis-aggregate the data so that the data are treated as if they are uncorrelated. However, since the data are actually correlated within the hierarchies, as a results, inaccurate estimates and results are obtained. The recent developed Hierarchical Linear Models (HLM) handles hierarchical correlated data. Hierarchical Linear Models builds separate regression models for each hierarchy, considers variations from both within and among groups, and improves the accuracy of parameter estimates.
The objective of this research is to explore the accuracy of parameter estimate under different situations. The studied mode is “a intercept-and slopes-as-outcomes model”, and we investigate the effects of sample size, cluster size, slope random effects, intercept random effects, and correlation between slopes and intercepts. We consider binary responses. We simulated data based on the combination of the five factors above calculate bias and MSE. Finally we conducted an regression analysis to study the impact of the factors on accuracy of the parameter estimates.
The result showed variances are the most influential factors to the accuracy of parameter estimation, which are random effects of slopes and intercepts, and correlation between slopes and intercepts The bigger the variance of the intercepts and slopes or the smaller of the correlation, the less accurate of the parameter estimation. Among the variances, the random slopes effect is the most significant. However, is an exception. The accuracy does not decrease with the increase of the correlation between slopes and intercepts. Second, for the sample sizes (number of clusters and cluster size) do not have a substantial effect. In other words, the accuracy of the smaller sizes does not differ much from the accuracy of the larger sizes. The possible explanation could be due to the design of sample sizes are not too distinguishable, so the difference of accuracy of parameter estimation is not significant.

目錄

第一章 緒論 1
第一節 研究動機與背景 1
第二節 研究目的 3
第三節 研究範圍與重要性 4
第二章 文獻探討 5
第一節 HLM 簡介 5
第二節 HGLM 簡介 12
第三節 估計方法 14
第四節 過去模擬研究 17
第三章 研究方法 21
第一節 研究設計 21
第二節 分析計畫 27
第四章 研究結果 28
第一節 實驗組合簡介 28
第二節 參數精確度評斷 28
第三節 影響因子對於參數估計精準度之討論 37
第五章 結論與建議 42
第一節 研究結論 42
第二節 研究建議與方向 43
文獻參考 45
附錄 48



圖次

圖 2-1 1 二階層結構資料 6
圖 3 1 1 研究流程圖 26
圖 4 2 1 不同組別個數下 參數估計之Bias圖 29
圖 4 2 2 不同組別個數下 參數估計之Bias圖 30
圖 4 2 3 不同組別個數下 參數估計之Bias圖 30
圖 4 2 4 不同組別個數下 參數估計之Bias圖 31
圖 4 2 5 不同組別個數下 參數估計之Bias圖 31
圖 4 2 6 不同組別個數下 參數估計之Bias圖 32
圖 4 2 7 不同組別個數下 參數估計之Bias圖 32
圖 4 2 8 不同變異數下 參數估計之bias圖 33
圖 4 2 9 不同變異數下 參數估計之bias圖 34
圖 4 2 10 不同變異數下 參數估計之bias圖 34
圖 4 2 11 不同變異數下 參數估計之bias圖 35
圖 4 2 12 不同變異數下 參數估計之bias圖 35
圖 4 2 13 不同變異數下 參數估計之bias圖 36
圖 4 2 14 不同變異數下 參數估計之bias圖 36




表次

表 2-1 1 二階層的常見次模型表 8
表 3-1 1 影響因子的數值表 23
表 3-1 2 不同設定的資料集 24

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