複雜抽樣設計下多元迴歸參數估計式之實證比較

An Empirical Comparison of Parameter Estimators for Multiple Regression under Complex Sampling Design

李秋慧(Chiu-Hui Lee)

指導教授 ： 許玉雪

國立臺北大學/商學院/統計學系/碩士(2010年)

摘要

抽樣調查設計隨著社會型態的不同愈來愈複雜，而由抽樣而得之調查資料，若沒有使用適合的參數估計方法，將使統計分析的精確度降低。本研究彙整過去文獻配合複雜抽樣調查設計之迴歸分析方法，利用(1)最小平方法、(2)加權最小平方法、(3)Quasi-Aitken加權最小平方法等三種方法進行多元迴歸分析。過去研究發現：(1)在不等機率抽樣下，一般使用的最小平方法所得到之迴歸參數估計式會有偏誤產生；(2)使用加權最小平方法可能可以產生一致的估計式，但是其變異卻會比最小平方法大許多；(3)當迴歸模式具有異質性且層與層之間誤差項變異相等下，Quasi-Aitken加權最小平方法的參數估計變異程度會小於最小平方法及加權最小平方法。本研究乃採用中央研究院社會學研究所2007年「臺灣社會變遷基本調查第五期第三次休閒組」的資料進行各種迴歸參數估計式的實證分析，旨在瞭解受訪者的年齡及父母的教育程度對受訪者的教育程度影響如何，並比較在此複雜抽樣原則下，不同迴歸參數估計式的迴歸係數、估計變異程度及模型的解釋力為何。研究結果顯示受訪者的年齡與其受學校教育的年數成反比，又父母親受學校教育年數愈高，受訪者受學校教育年數也愈高，尤其是父親的教育程度對子女的影響程度又較母親為大，且由此實證資料之迴歸參數估計分析結果可知，未來在進行分析時，應先確認資料屬性，若迴歸模式具異質性且層與層之間誤差項變異相等，則採用Quasi-Aitken加權最小平方法進行模型建構，可使迴歸參數估計式具不偏性，且其參數估計的變異程度會比最小平方法及加權最小平方法小。

關鍵字

分層抽樣 ； 迴歸分析 ； 複雜抽樣 ； 最小平方法 ； 加權最小平方法 ； Quasi-Aitken加權最小平方法

並列摘要

The sampling design is getting more complex to comply with a variety of social environment and to increase the precision of sampling survey as well. The traditional estimators used with complex survey may lower the accuracy of the statistical analysis. This study explores the methods of regression analysis on survey data obtained under a complex sampling. Three methods of multiple regression analysis proposed by previous studies, namely, ordinary least squares, weighted least squares, and Quasi-Aitken probability weighted least squares are used in this study for comparison analysis. Previous studies show that the ordinary least squares estimator is biased under the data collected under the unequal probability design; while under the equal probability design the weighted least squares estimator is better than ordinary least squares, but under the unequal probability design weighted least squares estimator may have a larger variance.
This study uses the data of "Taiwan Social Change Survey 2007, Phase 5, Wave 3," collected under a stratified unequal probability sampling by the Institute of Sociology Academia Sinica for empirical comparison of those three methods via comparing the estimates of regression coefficients, RMSE, and . The empirical results consist with previous studies. The results show that there is no big difference among the estimated parameters of those three methods. The results also show that the education year of respondents has significant negative relationship with their age but has positive relationship with their parents’ education year.

並列關鍵字

Stratified sampling ； Regression analysis ； Complex sampling ； Least squares ； Weighted least squares ； Quasi-Aitken weighted least squares

參考文獻

陳建銘(2008)，「分層不等機率抽樣之迴歸參數估計的比較分析」，國立臺北大學統計研究所碩士論文。

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