notes6 - Chapter 2 Timothy Hanson Department of Statistics,...

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Chapter 2 Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I 1 / 21
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2.7 Analysis of variance approach to regression (pp. 63–72) If x is useless, i.e. β 1 = 0, then E ( Y i ) = β 0 . In this case β 0 is estimated by ¯ Y . The i th deviation about this grand mean can be written: deviation about grand mean z }| { Y i - ¯ Y = explained by model z }| { ˆ Y i - ¯ Y + slop left over z }| { Y i - ˆ Y i Our regression uses line explains how Y varies with x . We are interested in how much variability in the Y 1 , . . . , Y n is soaked up by the regression model. 2 / 21
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Partitioning the SSTO partitioning the total sum of squares (SSTO): SSTO = n X i =1 ( Y i - ¯ Y ) 2 = ( ( n - 1) S 2 Y ) . SSTO is a measure of the total (sample) variation of Y ignoring x . The sum of squares explained by the regression line is given by SSR = n X i =1 ( ˆ Y i - ¯ Y ) 2 . The sum of squared errors measures how much Y varies around the regression line SSE = n X i =1 ( Y i - ˆ Y i ) 2 . It happily turns out that SSR + SSE = SSTO . 3 / 21
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Analysis of variance (ANOVA) table Restated : The variation in the data (SSTO) can be divided into two parts: the part explained by the model (SSR), and the slop that’s left over, i.e. unexplained variability (SSE). Associated with each sum of squares are their degrees of freedom (df) and mean squares, forming a nice table: Source SS df MS E ( MS ) Regression SSR= n i =1 ( ˆ Y i - ¯ Y ) 2 1 SSR 1 σ 2 + β 2 1 n i =1 ( x i - ¯ x ) 2 Error SSE= n i =1 ( Y i - ˆ Y ) 2 n - 2 SSE n - 2 σ 2 Total SSTO= n i =1 ( Y i - ¯ Y ) 2 n - 1 4 / 21
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Another test of H 0 : β 1 = 0 Note : E ( MSR ) > E ( MSE ) β 1 6 = 0. Loosely, we expect MSR to be larger than MSE when β 1 6 = 0. So testing whether the simple linear regression model explains a significant amount of the variation in Y is equivalent to testing H 0 : β 1 = 0 versus H a : β 1 6 = 0. Consider the ratio MSR / MSE . If H 0 : β 1 = 0 is true, then this should be near one. In fact F * = MSR MSE F 1 , n - 2 when H 0 : β 1 = 0 is true. So E ( F * ) = ( n - 2) / ( n - 4) which goes to one as n → ∞ (when β 1 = 0). 5 / 21
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F-test in ANOVA table This leads to an F-test of H 0 : β 1 = 0 versus H a : β 1 6 = 0 using F * = MSR / MSE : If F * > F 1 , n - 2 (1 - α ) then reject H 0 : β 1 = 0 at significance level α . Note : F * = ( t * ) 2 and so the F-test is completely equivalent to the Wald t-test based on t * = b 1 / se ( b 1 ) for H 0 : β 1 . Toluca data:
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This note was uploaded on 12/14/2011 for the course STAT 704 taught by Professor Staff during the Fall '11 term at South Carolina.

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notes6 - Chapter 2 Timothy Hanson Department of Statistics,...

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