# note20 - STAT5044 Regression and Anova Inyoung Kim 1 13...

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Unformatted text preview: STAT5044: Regression and Anova Inyoung Kim 1 / 13 Outline 1 Inference for GLMs 2 / 13 Deviance and Goodness of Fit The saturated GLM has a separate parameter for each observation. It gives a perfect fit. This sound good, but it is not helpful model It does not smooth the data or have the advantage that a simpler model has. Nonetheless, it serves as a baseline for other models, such as for checking model fit. 3 / 13 Deviance and Goodness of Fit A saturated model explains all variation by the systematic component of the model Let ˜ θ denote the estimate of θ for the saturated model, corresponding to estimated means ˜ μ i = y i for all i . For a particular unsaturated model, denote the corresponding ML estimates by ˆ θ and ˆ μ i For maximized log likelihood L (ˆ μ ; y ) for that model and maximized log likelihood L ( y ; y ) in the saturated case,- 2 [ L (ˆ μ ; y )- L ( y ; y )] describes lack of fit. It is the likelihood-ratio statistic for testing the null hypothesis that the model holds against the alternative that a more general model holds- 2 [ L (ˆ μ )- L ( y ; y )] = 2 ∑ i [ y i ˜ θ i- b ( ˜ θ i )] / a ( φ )- 2 ∑ i [ y i ˜ θ i- b ( ˆ θ i )] / a ( φ ) Usually, a ( φ ) has the form a ( φ ) = φ / ω i , and this statistic equals 2 ∑ i ω i [ y i ( ˜ θ i- ˆ θ i )- b ( ˜ θ i )+ b ( ˆ θ i )] / φ = D ( y ; ˆ μ ) / φ This is called the scaled deviance and...
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note20 - STAT5044 Regression and Anova Inyoung Kim 1 13...

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