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# note6a - STAT5044 Regression and Anova Inyoung Kim 1 49...

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STAT5044: Regression and Anova Inyoung Kim 1 / 49

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Outline 1 How to check assumptions 2 / 49
Assumption Linearity: scatter plot, residual plot Randomness: Run test, Durbin-Watson test when the data can be arranged in time order. Constant variance: scatter plot, residual plot (ABS-residual plot); Brown-Forsythe test, Breusch-Pagan Test Normality of error: Box-plot, histogram, normal probability plot; Shapiro-Wilks test, Kolmogorov-Smirnov, Anderson-Darling Remark: Normality probability plot provides no information if the assumption of linearity and/or constant variance are violated 3 / 49

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Influential point Combination of large absolute residual and high leverage ( h ii ) Leverage: diagonal value of Hat matrix ( H ) H = h 11 h 12 ··· h 1 n h 21 h 22 ··· h 1 n . . . . . . . . . . . . h n 1 h n 2 ··· h nn High leverage large h ii 4 / 49
Residual Three types: Ordinary r : r i = y - ˆ y , where E ( r i ) = 0 and var ( r i ) = ( 1 - h ii ) σ 2 Standardized: r i ˆ σ 1 - h ii Studendized (or Jackknife): r i ˆ σ ( i ) 1 - h ii t n - 2 where, ˆ σ 2 ( i ) = ( j r 2 j ( i ) ) / ( n - p - 1 ) and (p+1) is the number of parameter h ii is the leverage which is the diagonal value of Hat matrix. r j ( i ) = y j - ˆ y j ( i ) = y j - ( ˆ β 0 ( i ) + ˆ β 1 ( i ) x j ) 5 / 49

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Properties of residuals Sum to zero: r i = 0 Are not independent 6 / 49
Residual Jackknife r i ( i ) = y i - ˆ y i ( i ) N ( 0 , σ 2 1 - h ii ) where the subindex (i) indicate that estimate without point i. residual for y i computed using regression without y i then scaling. Studendized residual: r i ( i ) q ˆ var ( r i ( i ) ) r i ( i ) = y i - ˆ y i ( i ) = y i - [ ˆ β 0 ( i ) + ˆ β 1 ( i ) x i ] 7 / 49

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Studendized residual r i ( i ) q ˆ var ( r i ( i ) ) = r i ˆ σ ( i ) 1 - h ii by Fact 1 and 2. Fact 1: r i ( i ) = r i 1 - h ii Fact 2: r 2 i ( i ) = ( n - p σ 2 - r 2 i 1 - h ii ˆ σ ( i ) = ( n - p σ 2 - r 2 i 1 - h ii n - p - 1 8 / 49
Residual Using Fact1 r i ( i ) = r i 1 - h ii , we have Var ( r i ( i ) ) = Var ( r i ) ( 1 - h ii ) 2 = σ 2 1 - h ii . But σ 2 is unknown. We use ˆ σ 2 ( i ) r i ( i ) = Y i - ˆ Y i ( i ) r i ( i ) = r i 1 - h ii 9 / 49

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Residual Studendized residual r i 1 - h ii r ˆ σ 2 ( i ) 1 - h ii = r i q ˆ σ 2 ( i ) ( 1 - h ii ) ˆ σ 2 ( i ) = j r 2 j ( i ) n - p - 1 where j r 2 j ( i ) = ( n - p σ 2 - r 2 i 1 - h ii NOTE: “large” residual if | r j ( i ) | > 3 An expression for the distribution of the standardized residuals was obtained (Weisberg, 1985). 10 / 49
Studendized residual r i ( i ) q ˆ var ( r i ( i ) ) = r i ˆ σ ( i ) 1 - h ii t n - p - 1 You don’t need to know how to prove this in our class! (beyond our class scope) 11 / 49

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Comparison with standardized residual Standardized residual: r i - 0 p var ( r i ) = r i - 0 p σ 2 ( 1 - h ii ) r i p ( 1 - h ii σ 2 If one has outliers with large absolute “residual”, then ˆ σ 2 may not be a good measurement. Residuals are not independent and have different variances. The distribution of the standardized residual is not a t distribution. People usually ignore these problems.
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note6a - STAT5044 Regression and Anova Inyoung Kim 1 49...

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