notes_10_8 - November 19, 2003 Before we begin, I want to...

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1 November 19, 2003 Before we begin, I want to address the question of comparing means. If we have 4 means that we want to compare, we have: 0 1234 : H µµµµ === If we want to test this hypothesis, we reconceptualize it as: 01 2 23 34 :0 0 0 H µµ −= However, what about 14 ? Clearly, this is addressed using the above hypothesis since: 14 122334 µµ µµµµµµ −=−+−+− However, when we test this hypothesis, we use estimates, and perhaps what results is: 1 2 3 4 4 5 6 7 x x x x = = = = It is conceivable that a difference of 1 (which is the difference between adjacent categories) is NOT statistically significant, whereas a difference of 3 is (7-4). So, how do we reconcile this? This is understood if you remember your ANOVA course. When doing ANOVA, we compare means, and then we can do multiple comparisons to find out which means are different. There are several techniques for multiple comparisons, including Tukey, Scheffe, and Bonferroni. The differences between these methods lies in the method for controlling Type I error. The Tukey method controls Type I error for all possible pairwise comparisons, Bonferroni controls Type I error for the number of contrasts decided upon, and Scheffe controls the type I error rate for all possible linear combinations of the means. Clearly, Scheffe is the most conservative test. So, what is the point? The F test that we conduct on the means, or the regression coefficients, is like the Scheffe test. It actually tests not only the specified linear combinations, but also any linear combination of those specified. Thus, since
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2 14 122334 µµ µµµµµµ −=−+−+− It will be included in the test. Perhaps an example will help. (Example done in class, not provided here.) In the case where we are testing the hypothesis: 01 i : ... 0 : At least one of the 0 p a H H ββ β == = Then we are really asking the question “Is the regression useful?” Since we are testing all the regression coefficients. Another way to look at this problem is to consider the multiple correlation. Recall that we have a model, 0 1 2 345 yN S N S N A S S e =+ + + + + + (This is the PPVT model from last class). Note that: is an observed value ˆ let be the predicted value i i y y Thus, 0 1 2 ˆ i ybb Nb Sb N N Ab S S =+ + + + + TO be concrete, for person 1, note that they have a predicted value of: 2 3 4 5 ˆ *0 *10 *8 *21 *22 i b b b b + + + + Note that this was obtained by plugging in the values of N, S, NS, NA, SS We can get the predicted value by now plugging in the estimated regression coefficients: ˆ 39.697 .063*0 .370*10 .374*8 1.523*21 .410*22 i y =++ −+ + =81.40 We will note that the observed value for person one was 68. Therefore, the error is given by the difference between the two values: 68 – 81.4 = -13.4. So we want to know something about the quality of the regression. We can test the coefficients, or we can consider the multiple correlation between the dependent variable, y, and the predictors (in this case, N, S, NS, NA, SS).
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This note was uploaded on 04/16/2008 for the course EDUCATION 771 taught by Professor Keller during the Fall '08 term at UMass (Amherst).

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notes_10_8 - November 19, 2003 Before we begin, I want to...

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