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 = ( ˆ Y i- ¯ Y ) + ( Y i- ˆ Y i ) (1) Since ∑ Y i = ∑ ˆ Y i , we conclude that ¯ Y is the average of the fitted values { ˆ Y i } . So ˆ Y i- ¯ Y is the deviation of ˆ Y i from the mean and Y i- ˆ 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 ( ˆ Y i- ¯ Y ) 2 + X ( Y i- ˆ Y i ) 2 + 2 X ( ˆ Y i- ¯ Y )( Y i- ˆ Y i ) . It can be shown that the cross product term ∑ ( ˆ Y i- ¯ Y )( Y i- ˆ Y i ) equals zero. So we have X ( Y i- ¯ Y ) 2 = X ( ˆ Y i- ¯ Y ) 2 + X ( Y i- ˆ 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 = ∑ ( ˆ Y i- ¯ Y ) 2 [ SSR stands for ”regression sum of squares”], and SSE = ∑ ( Y i- ˆ 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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## This note was uploaded on 09/28/2011 for the course STA STA 108 taught by Professor Jiang during the Summer '09 term at UC Davis.

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sta108_handout3 - Handout 3 Analysis of variance approach...

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