10-regression-part4-handout

10-regression-part4-handout - STA 3024: Regression Douglas...

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STA 3024: Regression Douglas Whitaker Department of Statistics 19 March 2012 Douglas Whitaker (Department of Statistics) STA 3024: Regression 19 March 2012 1 / 34
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Multiple Linear Regression In simple linear regression, we explained one response variable ( Y ) with one explanatory variable ( X ). In multiple linear regression, we explain one response variable ( Y ) with more than one explanatory variable ( X 1 ,X 2 ,...,X p ). Douglas Whitaker (Department of Statistics) STA 3024: Regression 19 March 2012 2 / 34
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Theoretical Models Simple Linear Regression Y i = α + βX i + e i Multiple Linear Regression Y i = α + β 1 X 1 i + β 2 X 2 i + ··· + β p X pi + e i Y i = β 0 + β 1 X 1 i + β 2 X 2 i + ··· + β p X pi + e i Douglas Whitaker (Department of Statistics) STA 3024: Regression 19 March 2012 3 / 34
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SLR Assumptions Random sample Independent observations Linear relationship between response and explanatory variable Error terms are normally distributed e i ∼ N (0 ) Constant variance (same σ for each e i ) Douglas Whitaker (Department of Statistics) STA 3024: Regression 19 March 2012 4 / 34
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MLR Assumptions Random sample Independent observations Linear relationship between response and each explanatory variable (that is, slope for an explanatory variable is the same for all fixed values of other explanatory variables) Error terms are normally distributed e i ∼ N (0 ) Constant variance (same σ for each e i ) Douglas Whitaker (Department of Statistics) STA 3024: Regression 19 March 2012 5 / 34
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Hypotheses for ANOVA H 0 : the model is useless vs. H A : the model is not useless Said another way H 0 : β = 0 vs. H A : β 6 = 0 Douglas Whitaker (Department of Statistics) STA 3024: Regression 19 March 2012 6 / 34
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ANOVA (Simple) Source SS df MS F Regression SSR 1 SSR 1 SSR/ 1 SSE/ ( N - 2) Error SSE N - 2 SSE N - 2 Total SST N - 1 Douglas Whitaker (Department of Statistics) STA 3024: Regression 19 March 2012 7 / 34
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Hypotheses for ANOVA H 0 : the model is useless vs. H A : the model is not useless Said another way H 0 : β 1 = β 2 = ··· = β p = 0 vs. H A : β i 6 = 0 for some i Douglas Whitaker (Department of Statistics) STA 3024: Regression 19 March 2012 8 / 34
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ANOVA (Multiple) Source SS df MS F Regression SSR p SSR p SSR/p SSE/ ( N - p - 1) Error SSE N - p - 1 SSE N - p - 1 Total SST N - 1 Douglas Whitaker (Department of Statistics) STA 3024: Regression 19 March 2012 9 / 34
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A few things to note For the same data, SST will stay the same irrespective of the number of predictors As the number of predictors goes up, SSR goes up As SSR goes up, SSE goes down As SSR goes up, SSR SST = R
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This note was uploaded on 03/20/2012 for the course STA 3024 taught by Professor Ta during the Spring '08 term at University of Florida.

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10-regression-part4-handout - STA 3024: Regression Douglas...

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