C532 Nov 16 2011

C532 Nov 16 2011 - C532 ANOVA Regression Modeling Y Cao...

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C532: ANOVA & Regression Modeling Nov 16, 2011 Y. Cao 1 Topics: learn more about ANCOVA model : a general case with more than three treatment groups in a clinical trial. Datasets : np423aug2007.sav In the previous tutorial, we studied ANCOVA model with only two treatment groups: sham or chiro group. Today, we’d like to generalize the ANCOVA to a more complicated case, that is, no matter how many (of course, infinity is not our concern.) groups in a clinical trial, we still have tools to deal with it. The ANOVA model with j (j >=3) groups is The ANCOVA model looks like Compare the ANOVA model and the ANCOVA model (also see the lecture note, page 1), we find that the only difference is that in the later model, we add a covariate (continuous) to the former model. That is, in the ANCOVA model, there is one more term (the covariate) in the equation it is . In the above equations, represent new dummy variables to label the group levels ( 1,2,…j ), so there are j groups in all. is the intercept for the multiple linear regression model while are corresponding coefficients to the dummy variables so once SPSS output these numbers we then can write down the above two equations. Table 1 of ANOVA in today’s lecture: the degrees of freedom for the “between groups” is number of groups minus 1 , that is k 1, while in Table 2 of ANOVA, this becomes the “covariate” term with degree of freedom =1 (because it’s a continuous variable!)
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Nov 18, 2009 Y. Cao 2 The total degree of freedom is always =n 1, that is, number of patients (or sample size) minus 1 for all three situations. Why n -1? Because from the whole body of patients, we can obtain a grand mean (or overall mean ) for the outcome variable (Y) therefore reduces (lose) 1 degree of freedom. Table 3 (ANOVA Table for ANCOVA) truly stands for the merits of Table 1 and Table 2 since in Table 3 it is like we have the 2 models that appear in the Tables 1 and 2. Since our main interest is to compare the group means to see if there is any significant difference between some groups or pair of groups, therefore the statistic is defined as MS (mean square) between the groups divided by the MS within the groups (error). If the variance within the groups (error term) is much less than those from the between groups, then the statistic value will be large then it indicates a big difference (or differences) between the groups, then we would prepare to conclude that H o should be rejected (remember that our H o : mean in groups 1= mean in group 2= = mean in group k). Isn’t this intuitively reasonable? At present, since we are using dummy variable to realize the ANCOVA model, the hypothesis naught will shift to a new form: while the alternative hypothesis is Please recognize that in the lecture note we used letter j instead of k (does not matter!) for the ANOVA/ANCOVA models, but now we use k in the 3 ANOVA tables. Also, the number of dummy variables we need is now becomes k -1.
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C532 Nov 16 2011 - C532 ANOVA Regression Modeling Y Cao...

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