Now consider the deviation of the observed responses

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Chapter 9 / Exercise 17
Probability and Statistics for Engineering and the Sciences
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Now consider the deviation of the observed responses from their fitted values based on the regression model: i i i i i e X b b Y Y Y = + - = - ) ( 1 0 ^ . When these terms are squared and summed up, this is referred to as the error sum of squares (SSE) . We’ve already encounterd this quantity and used it to estimate the error variance. = - = n i i i Y Y SSE 1 2 ^ ) ( When the observed responses fall close to the regression line, SSE will be small. When the data are not near the line, SSE will be large. Finally, there is a third quantity, representing the deviations of the predicted values from the mean. Then these deviations are squared and summed up, this is referred to as the regression sum of squares ( SSR) . = - = n i i Y Y SSR 1 2 ^ ) ( The error and regression sums of squares sum to the total sum of squares: SSE SSR SSTO + = which can be seen as follows:
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Chapter 9 / Exercise 17
Probability and Statistics for Engineering and the Sciences
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SSR SSE Y Y Y Y Y Y Y Y e Y X e b e b Y Y Y Y Y X b b e Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y SSTO Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y n i i n i i i n i i n i i i n i n i i i i n i i n i i n i i i n i i i n i i n i i i n i i i i n i i n i i i n i i i i i i i n i i i i i i i i i i i i i i i i i i i + = - + - = + - + - = - + + - + - = - + + - + - = - - + - + - = - - + - + - = - = - - + - + - = - + - = - - + - = - + - = - = = = = = = = = = = = = = = = = = 1 2 ^ 1 2 ^ 1 2 ^ 1 2 ^ 1 1 1 1 0 1 2 ^ 1 2 ^ 1 1 0 1 2 ^ 1 2 ^ 1 ^ ^ 1 2 ^ 1 2 ^ 1 ^ ^ 2 ^ 2 ^ 1 2 ^ ^ 2 ^ 2 ^ 2 ^ ^ 2 ^ ^ ^ ^ ) ( ) ( ) 0 ( 2 ) ( ) ( 2 ) ( ) ( ) ( 2 ) ( ) ( ) )( ( 2 ) ( ) ( ) )( ( 2 ) ( ) ( ) ( ) )( ( 2 ) ( ) ( )] ( ) [( ) ( ) ( ) ( The last term was 0 since = = 0 i i i X e e , Each sum of squares has associated with degrees of freedom . The total degrees of freedom is df T = n -1. The error degrees of freedom is df E = n -2. The regression degrees of freedom is df R = 1. Note that the error and regression degrees of freedom sum to the total degrees of freedom: ) 2 ( 1 1 - + = - n n . Mean squares are the sums of squares divided by their degrees of freedom: 2 1 - = = n SSE MSE SSR MSR Note that MSE was our estimate of the error variance, and that we don’t compute a total mean square. It can be shown that the expected values of the mean squares are: = - + = = n i i X X MSR E MSE E 1 2 2 1 2 2 ) ( } { } { β σ σ Note that these expected mean squares are the same if and only if β 1 =0. The Analysis of Variance is reported in tabular form: Source df SS MS F Regression 1 SSR MSR=SSR/1 F=MSR/MSE Error n -2 SSE MSE=SSE/( n -2) C Total n -1 SSTO
F Test of β 1 = 0 versus β 1 0 As a result of Cochran’s Theorem (stated on page 76 of text book), we have a test of whether the dependent variable Y is linearly related to the predictor variable X . This is a very specific case of the t -test described previously. Its full utility will be seen when we consider multiple predictors. The test proceeds as follows: Null hypothesis: 0 : 1 0 = β H Alternative (Research) Hypothesis: 0 : 1 β A H Test Statistic: MSE MSR F TS = * : Rejection Region: ) 2 , 1 ; 1 ( * : - - n F F RR α P -value: *} ) 2 , 1 ( { F n F P - Critical values of the F -distribution (indexed by numerator and denominator degrees’ of freedom) are given in Table B.4, pages 1340-1345.