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# class03_23 - 1.017/1.010 Class 23 Analyzing Regression...

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1 1.017/1.010 Class 23 Analyzing Regression Results Analyzing and Interpreting Regression Results Least-squares estimation methods provide a way to fit linear regression models (e.g. polynomial curves) to data. Once a model is obtained it is useful to be able to quantify: 1. The significance of the regression 2. The accuracy of the parameter estimates and predictions The significance of the regression can be analyzed with an ANOVA approach. Estimation and prediction accuracy are related to the means and variances of the regression parameters. Regression ANOVA The regression term is not significant (it does not explain any of the y variability) if the following hypothesis is true: H0: E [ y ( x )] = h ( x ) A = a 1 That is, the mean of y is a constant that does not depend on the independent variable x . This hypothesis can be tested with a statistic based on the following sums- of-squares: = = = = n i i i n i n 1 2 1 1 ; ) ( y m m y SST y y ( ) [] = + + = = n i i i i x x H H 1 2 2 3 2 1 ˆ ˆ ˆ ] ˆ [ ] ˆ [ a a a y A Y A Y SSE SSE - SST SSR = SST measures the y variability if the regression model is not used. SSE measures the y variability if the regression model is used. SSR measures the y variability explained by the regression model.

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2 The statistic used to test significance of the regression is the ratio of the mean sums of squares for regression and error: 1 m- SSR MSR = m n = SSE MSE ) , ( MSE MSR MSE MSR = F R E [ MSR ] depends on the magnitudes of the regression coefficients a 2 , . .. a m while E [ MSE ] does not.
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class03_23 - 1.017/1.010 Class 23 Analyzing Regression...

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