二階段分層叢集抽樣的設計效應估計：以TIMSS 2007調查研究為例 An Estimation of the Design Effect for the Two-Stage Stratified Cluster Sampling Desig

Abstract

大型調查研究常採用多階段分層叢集抽樣，檢視臺灣學生歷年來參與TIMSS 研究的結果顯示，其平均表現之標準誤常較他國稍大。應TIMSS 專責抽樣之單位要求，本研究旨在推導設計二階段分層叢集抽樣時即可預估標準誤的公式，以利選擇較佳分層架構，達減少標準誤的目標，並進行三個分析檢查其有效性。分析一將30 個參與TIMSS 2007 的國家資料，用該公式與《TIMSS 技術手冊》建議之刀切重複取樣法，分別估算各國學生平均科學成績之標準誤，發現兩者之線性相關達 .98。分析二以29 個連續參加TIMSS 2003和2007 的國家資料，論述利用該公式與現有輔助變項預估將要進行調查的誤差之實用性。分析三探討當叢集的分層輔助變項為連續量時，不同分層數與二階段分層叢集抽樣誤差間的關係，以預估臺灣學生在TIMSS 2011 平均科學成績之標準誤。文後尚提出四階段的評估流程，供相關研究預估主要依變項平均值之標準誤時做參考。 Most large-scale educational surveys utilize a multi-stage stratified cluster sampling design. Past findings revealed that the standard errors of Taiwan students’ mean performances were slightly larger than other countries’. In response to the request by the institute in charge of TIMSS sampling, this study was launched to derive a formula that could estimate the standard error of population mean prior to conducting a two-stage stratified cluster sampling design. This formula could then be used to select an optimal stratification framework that could reduce the size of standard error to an acceptable level. Its validity was investigated in three subsequent studies. In the first study, standard errors for 30 TIMSS 2007 participating countries were estimated according to the newly derived formula as well as by the jackknife replication. The correlation between the two sets of standard errors amounted to 0.98. The second study investigated the practicality of using the new formula in addition to auxiliary variables for predicting standard errors on the data of 29 countries that participated in both TIMSS 2003 and 2007. The third study explored the relationship between the number of stratum and the standard errors under a two-stage stratified cluster sampling design when the auxiliary variables for stratification were continuous. This paper closed by suggesting a four-step procedure to facilitate researchers in estimating standard errors of means during the planning stage of sampling design.

Keywords

• 大型評量
• 抽樣架構計畫
• 降低抽樣誤差
• 複雜抽樣設計
• 變異量估計
• large-scale assessment
• planning sampling framework
• sampling error reduction
• complex survey design
• variance estimation