sta108_handout3

sta108_handout3 - Handout 3 Analysis of variance approach...

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Unformatted text preview: Handout 3 Analysis of variance approach to regression Consider the identity Y i- Y = ( Y i- Y ) + ( Y i- Y i ) (1) Since Y i = Y i , we conclude that Y is the average of the fitted values { Y i } . So Y i- Y is the deviation of Y i from the mean and Y i- Y i is the residual. Square both sides of (1) and sum over i . Then it will turn out that X ( Y i- Y ) 2 = X ( Y i- Y ) 2 + X ( Y i- Y i ) 2 + 2 X ( Y i- Y )( Y i- Y i ) . It can be shown that the cross product term ( Y i- Y )( Y i- Y i ) equals zero. So we have X ( Y i- Y ) 2 = X ( Y i- Y ) 2 + X ( Y i- Y i ) 2 . This is an important identity. Here are some notations. SSTO = ( Y i- Y ) 2 [ SSTO stands for total sum of squares], SSR = ( Y i- Y ) 2 [ SSR stands for regression sum of squares], and SSE = ( Y i- Y i ) 2 . So we have the following important identity SSTO = SSR + SSE. Degrees of freedom : Degrees of freedom for SSTO is n- 1. Degrees of freedom for SSR is equal to # of beta parameters estimated - 1 . Since we have estimated two beta parameters, df ( SSR ) = 1. Degrees of freedom for SSE is equal to n - #of beta parameters estimated . So, df ( SSE ) = n- 2. Note that we have df ( SSTO ) = df ( SSR ) + df ( SSE ) . Mean squares: For any sum of squares, the mean square is defined to be the sum of squares divided by its degrees of freedom. So MSE = SSE df ( SSE ) = SSE n- 2 , MSR = SSR df ( SSR ) = SSR, MSTO = SSTO df ( SSTO ) = SSTO n- 1 ....
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sta108_handout3 - Handout 3 Analysis of variance approach...

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