NEWTest2Solutions 3

2. (a) In a regression context, what is an outlier? ( 1 mark ) (b) In general, are all outliers influential points? Justify your answer. ( 1 mark ) (c) Fill in the blanks: (i) measures the influence of an observation on all fitted values. (ii) measures the influence of an observation on its own fitted value. (iii) measures the influence of an observation on a par- ticular regression coe ffi cient. ( 3 marks: 1 each ) 3 An outlier is a point that deviates from the model . No , not all outliers are influential points . Consider : line with A * B ✗ ↓ B 1 : without A B not influential Cook 's Distance DFFITS DFBETAS

5. Consider a data set consisting of 24 observations on the variables X =Age and Y =Plasma Level. A Box-Cox transformation was considered and from SAS, we have the following plot: (a) Based on the plot, for practical purposes, what transformation would we use? State the new model and be sure to define any new variables you introduce. ( 3 marks ) (b) What type of model violations is the Box-Cox transformation aimed at correcting? Are there any guidelines to its use and/or particular cases where it works best? ( 2 marks ) 6 Since top + = -0.5 , we would consider Y' = g- 422 Yry . The new model would be : YÉpotBiXi The Box . Cot transformation aims at correcting non - normality of errors and / Or non - constant variance . It is mostly useful for strictly positive quantities and works best for > c- C- 42 ) . + 2

STATS 3A03: Fall 2022 Formula Sheet 1. Preliminaries • S xx = P n i =1 ( x i - x ) 2 = P n i =1 x 2 i - n x 2 • S xy = P n i =1 ( x i - x )( y i - y ) = P n i =1 x i y i - n x y • Cov( X, Y ) = E [( X - μ X )( Y - μ Y )] = E ( XY ) - μ X μ Y • ⇢ = Cov( X, Y ) p Var( X )Var( Y ) r = S xy p S xx S yy r p n - 2 p 1 - r 2 ⇠ t n - 2 2. For the simple linear regression model: Y i = β 0 + β 1 X i + " i , i = 1 , . . . , n • ˆ β 0 = y - ˆ β 1 x ˆ β 1 = S xy S xx Var ⇣ ˆ β 0 ⌘ = σ 2 ✓ 1 n + x 2 S xx ◆ Var ⇣ ˆ β 1 ⌘ = σ 2 S xx • ˆ σ 2 = SSE n - 2 = P ( y i - ˆ y i ) 2 n - 2 • Var(ˆ μ 0 ) = ⇣ 1 n + ( x 0 - ¯ x ) 2 S xx ⌘ σ 2 Var(ˆ y 0 ) = ⇣ 1 + 1 n + ( x 0 - ¯ x ) 2 S xx ⌘ σ 2 • F = MSReg MSE ⇠ F df reg ,df E R 2 = 1 - SSE / SST 3. For the multiple linear regression model: Y = X β + " • H = X ( X 0 X ) - 1 X 0 ˆ β = ( X 0 X ) - 1 X 0 y • Var( ˆ β ) = σ 2 ( X 0 X ) - 1 ˆ σ 2 = SSE / ( n - p - 1). • Var(ˆ μ 0 ) = σ 2 ( x 0 0 ( X 0 X ) - 1 x 0 ) Var(ˆ y 0 ) = σ 2 (1 + x 0 0 ( X 0 X ) - 1 x 0 ) • ˆ β j - β j se ( ˆ β j ) ⇠ t n - p - 1 j = 0 , 1 , . . . p, F = MSReg MSE ⇠ F df reg ,df E • R 2 = 1 - SSE / SST R 2 adj = 1 - SSE / ( n - p - 1) SST / ( n - 1) • F = (SSE red - SSE full ) / ( df red - df full ) SSE full /df full ⇠ F df red - df full ,df full • D i = ( r ⇤ i ) 2 p +1 ⇥ h ii 1 - h ii DFBETAS j,i = ˆ β j - ˆ β j ( i ) ˆ σ ( i ) q ( x 0 x ) - 1 jj DFFITS i = r ⇤ i q h ii 1 - h ii • y ( λ ) i = ⇢ y λ i - 1 λ if λ 6 = 0 , log y i if λ = 0 . • ˆ β WLS = ( X 0 WX ) - 1 X 0 Wy 9