Unformatted text preview: Chapter 8: Regression Diagnostics
Linear Model: 0 After we fit the above linear model to the data, we wonder whether the assumptions (1) Linear relationship between X and Y 0 and (2) (3) are right has normal distribution, we need to check whether the normal Sometimes, we also assume that assumption is right for the data We will use residuals ê= Y – Ŷ as the statistic for such diagnostics. Review: Ŷ = X = X(XTX)-1XTY= HY ̂ ̂ H is called hat matrix = Properties of estimated residual 0 , where is the ith diagonal element in H ̂ 1 Properties of hat matrix: , , 0 ∑ 1 ∑ Leverage ∑ 1 As Thus 1 0 is called the leverage of the ith observation (case). In weighted least squares, some attention needs to paid on how to define residual 0 Residual: Deficiencies: 1) ∑ because the weights are ignored. 2) ̂ depends We thus define residual as ) Pearson residuals or weighted residuals The pattern of residuals when the model is correct Plot vs 1) Scatter around zero ̂ 0 ̂ 1 is not quite constant but variability is small. 2) 3) Residuals are correlated but correlation is not visible. 1. Plot: Check to see if the linear mean function is adequate ˆ ˆ ei versus y i Remedy: add x 2 terms into the model 2. Check to see if the constant variance assumption is valid ˆ ˆ Plot: ei versus y i Remedy: variance stabilizing transformations 3. Sometimes, we assume that the residuals have normal distribution ˆ Plot: QQ plot of ei to check the normality assumption. Diagnostic for Nonconstant Variance Suppose now that In general, we assume
:λ | depends on λ and Z
2 A score test: 0 |, |, ̃ exp λ ̃ 1) Regress Y on X , compute 1 2) Compute the regression of
SSreg(z) be the sum of squares for Regression df=q 3) ~ Remedy: Variance Stabilizing Transformation √: 1 : | | | 0 | | | 1 log Y: √ ...
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