階層線性模型與一般線性迴歸模型之比較與樣本數之選取

The Comparison of Hierarchical Linear Model and General Linear Regression Modal with Sample Sizes Selection

嚴雯婷(Wen-Ting Yan)

指導教授 ： 施葦

國立臺北大學/商學院/統計學系/碩士(2011年)

摘要

階層線性模型 (Hierarchical Linear Model, HLM) 是一種將迴歸模式擴展到階層資料結構的統計分析技術。由於階層線性模型涉及到不同層級的樣本數，因此在樣本規模的選擇上更加複雜。本研究透過模擬方法，考慮不同組內相關性下，針對階層線性模型和一般線性迴歸模型的解釋能力以及固定效果和隨機效果變異數的型一誤機率與檢定力進行比較。並且透過模擬方法找出不同條件下較佳模型與所需最少的樣本數組合，做為實務上考量抽取樣本數的限制或是成本問題下的一個樣本數組合參考建議。

　　透過模擬結果發現，不論資料相關性的高低，一般線性迴歸模型的模型解釋能力皆優於階層線性模型。但並不建議使用一般線性迴歸模型檢測固定效果。階層線性模型在低相關資料中，欲使對固定效果的檢測準確，所建議樣本數至少為5組，每組30個樣本數，且隨著組數的增加可將組內樣本數減少。在具有中、高相關資料下，階層線性模型在組數為20 以上時，對固定效果的檢測有良好表現，且隨著組數越大，可選擇越小的組內樣本數進行固定效果的檢測。低相關資料下，僅階層線性模型於100組，每組50個樣本數下，才能較準確檢測截距和斜率的組間變異。在中、高相關資料下，建議一般線性迴歸模型在小組數大組內樣本數下，適合檢測截距和斜率的組間變異。階層線性模型則建議於大組數小組內樣本數進行截距組間變異的檢測；而斜率組間變異的檢測則需要大組數大組內樣本數才有良好表現。

關鍵字

階層線性模型；一般線性迴歸模型；組內相關係數；樣本數大小

並列摘要

The hierarchical linear model is a technique of the statistics analysis that expands the regression model to the data with hierarchical structure. Because the hierarchical linear model involves different levels of data, the choice of sample sizes gets complicated. Through simulation, this research compares the hierarchical linear model with the general linear regression model by the predictability, typeⅠerror probability and power of testing fixed effect and random effect. Furthermore, appropriate model and sample sizes are suggested.

The research shows that, regardless of the strength of intra-class correlation coefficient, the general linear regression models are better than the hierarchical linear models in predictability. For testing fixed effect, the general linear regression model is not recommended. Under weak intra-class correlation, the number of groups of at least 5 with 30 observations in each group is recommended for HLM. As the number of groups increase, the number of observation within each group can be reduced. Under median and strong intra-class correlation, the number of groups of at least 20 is recommended and the number of observation in each group can be reduced as the number of groups increase.

For testing random effect, under weak intra-class correlation, the GLM is not recommended and large number of groups with large sample size in each group is recommended for HLM. Under median and strong intra-class correlation, small number of group with large sample size in each group are recommended for GLM; large number of group with small sample size in each group are recommended in testing the variation of intercepts for HLM; large number of groups with large sample size in each group are recommended in testing the variation of slopes for HLM.

並列關鍵字

Hierarchical linear model； Regression model； Intra-class Correlation Coefficient； Sample sizes

參考文獻

2、 簡嘉惠，2010，「階層線性模式與一般線性迴歸模式檢定力的比較」。

3、 Bassiri, D. (1988). “Large and small sample properties of maximum likelihood estimates for the hierarchical linear model.” Unpublished doctoral dissertation, Department of Counseling, Educational Psychology and Special Education, Michigan State University.

5、 Browne, W. & Draper, D. (2000). “Implementation and performance issues in the Bayesian and likelihood fitting of multilevel models.” Computational Statistics, 15:391-420

7、 Cohen, J. (1988) Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ:Eribaum.

9、 Hofmann, D.A. (1997). “An overview of the logic and rationale of hierarchical linear models.” Journal of Management, 23(6): 723-744.