Regression Analysis, AVONA, T-test, and Coefficient of Determination-ECO6416

# Regression Analysis, AVONA, T-test, and Coefficient of Determination-ECO6416

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Regression Analysis, ANOVA, T-test, and Coefficient of Determination There are very direct relationships among linear regression, analysis of variance, t-test and the coefficient of determination. The following small data set is for illustrating the connections among the above statistical procedures, and therefore relationships among statistical tables: X1 4 5 4 6 7 7 8 9 9 11 X2 8 6 8 10 10 11 13 14 14 16 Suppose we apply the t-test . The statistic is t = 3.207, with d.f. = 18. The p-value is 0.003 indicating a very strong evidence against the null hypothesis. Now, by introducing a dummy variable x with two values, say 0 and 1, representing the two data sets, respectively, we are able to apply regression analysis : x 0 0 0 0 0 0 0 0 0 0 y 4 5 4 6 7 7 8 9 9 11 x 1 1 1 1 1 1 1 1 1 1 y 8 6 8 10 10 11 13 14 14 16 Among other statistics, we obtain a large slope = m = 4 0, indicating the rejection of the null hypothesis. Notice that, the t-statistic for the slope is: t-statistic = slope/(its standard error) = 4/
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Unformatted text preview: 1.2472191 = 3.207, which is the t-statistic we obtained from the t-test. In general, the square of t-statistic of the slope is the F-statistic in the ANOVA table; i.e., t m 2 = F-statistic Moreover, the coefficient of determination r 2 = 0.36, which is always obtainable from the t-test, as follows: r 2 = t 2 / (t 2 + d.f.). For our numerical example, the r 2 is (3.207) 2 / [(3.207) 2 + 18] = 0.36, as expected. Now, applying ANOVA on the two sets of data, we obtain the F-statistic = 10.285, with d.f. 1 = 1, and d.f. 2 = 18. The F-statistic is not large enough; therefore, one must reject the null hypothesis. Note that, in general, F α , (1, n) = t 2 α/2 , n . For our numerical example, F = t 2 = (3.207) 2 = 10.285, as expected. As expected, by just looking at the data, all three tests indicate strongly that the means of the two sets are quite different....
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## This note was uploaded on 10/07/2011 for the course ECO 6416 taught by Professor Staff during the Spring '08 term at University of Central Florida.

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