2010-01-24

如何比较一个自变量对三个因变量的影响大小?

ZYS @ 2010-01-23:

Dear Dr. Zhu,

I read your blog post on how to test the differences of two correlation coefficients posted on http://zjz06.spaces.live.com/blog/cns!3F49BBFB6C5A1D86!954.entry. Is it appropriate to use your method for two regression models based on one sample population?

My question is whether there is a appropriate way to test significant difference between regression coefficients of two different models from same sample population? For example in the below table, how would we statistically compare the difference between betas for political interest as predictors of DV in three conditions? This is a repeated measure experiment. the same group participants participated in three conditions in three months. In other words, the research question is whether the impact of political interest on opinion expression is moderated by condition... The reviewer wants a statistical test --but i didn't find a good way to test since they are not independent samples...

 
Opinion expression in Condition 1
Opinion expression in Condition 2
Opinion expression in Condition 3
 

ß

ß

ß

Education

.13*

-.07

-.06

Male

.10*

-.01

.05

Age

-.04

.28***

.31***

White

.04

.03

-.03

Political Interest

.33***

.10*

.08

R-square

.04

.09

.11

F

3.60**

7.48***

9.61***

Is there a better way this problem could be tackled?

庄主 @ 2010-01-24:

Your research question differs from the one in that post, where the issue is to compare the correlation coefficients between two independent variables (IVs) with the same dependent variable (DV) whereas your task is to compare the correlations between the same IV with three different DVs.

One approach you can consider is repeated measures regression in general linear modeling (GLM), in which you form a within-subjects (WS) factor to account for the three DVs, and then regress the WS factor on your IV and control variables.  To test the significance of the differences among the relevant regression coefficients, you use the procedure described in 如何检验两个回归系数的差别?.

Another approach, which I think simpler and easier to be understood, is to test a path model with  equality constraints in SEM (see the diagram below). Aside from your regular hypotheses, you fit a series of nested models to test the null hypothesis that β1 = β2 = β3. For example, you compare the fully unconstrained model in which β1, β2, and β3 are free to be estimated and the fully constrained model in which β1, β2, and β3 are fixed to be the same.  The difference in the resulting Chi-squares between the two models, tested with Chi-square distribution with 2 degrees of freedom, tests the above null hypothesis.  In addition, you can test the fully constrained model with three partially constrained models, each with a pairwise constraint such as β1 = β2, β1 = B3, or β2 = β3, to entertain the possibility that not all three coefficients are exactly the same but some pair(s) of them may be. 

image

With the data in the table you provided, I guess that Condition 1 would be significantly different from Conditions 2 and 3 whereas the latter two might not. 

Good luck with your publication.  If it’s published, please come here to share your experience in getting published in journals.  Many readers of this blog would be very interested in learning from your experience.

如何解读回归与ANOVA的不同结果?

Anonymous @ 2010-01-22:

祝老师好!我是心理专业的学生,想请教一个具体的问题。对一个三类别变量作ANOVA结果显示类别1显著小于类别2和类别3,但将这三类转换为dummy variable,类别1为00,类别2为01,类别3为10,用同样的因变量来做回归时,结果是第一个dummy variable没有显著差异,而第二个dummy variable有显著差异,这是不是意味着类别2显著区别于类别1和类别3呢?如果是,为什么会出现这种结果呢?可能需要加上的是我在做回归时还引入了一些其它的变量,其中有一个会与类别变量有相关。

庄主 @ 2010-01-24:

将一个含3个类别的名目变量转换成两个dummy variables(分别记为D1和D2)后做回归分析,D1反映的是类别1和类别2的差别、D2反映的是类别1和类别3的差别。那么为什么类别1和类别2的差别在ANOVA中显著而在回归中不显示?因为你在回归中加入了其它自变量而成为多元分析而ANOVA是只有一个自变量的单元分析。其实,你可以将那些新加的自变量作为协方差 (covariates) 放入ANOVA的,这时就成了ANCOVA (Analysis of Covariates),其结果应该与你做的回归分析完全一样。

你再问:D2为显著,是否说明了类别2显著区别于类别1和类别3呢?没有。D2只说明类别1和类别3的差别,与类别2无关。D1检验类别1与类别2的差别。如果希望知道类别2与类别3的差别,将类别2设为基准组(即类别2取00值,而类别1为10值),那么D2就是反映类别2与3的差别了。