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Notes 6 - $ ST3241 Categorical Data Analysis I Generalized...

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& $ % ST3241 Categorical Data Analysis I Generalized Linear Models Some More Discussions 1
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& $ % Deviance A saturated model has a separate parameter for each observation giving a perfect fit. Let ˜ θ denote the estimate of θ for the saturated model, corresponding to estimated means ˜ μ i = y i for all i . Let ˆ θ denote the MLE of θ for the model under consideration. The deviance of the fitted model is defined as D ( y ; ˆ μ ) = - 2[ L μ ; y ) - L ( y ; y )] = 2 N X i =1 [ y i ˜ θ i - b ( ˜ θ i )] /a ( φ ) - 2 N X i =1 [ y i ˆ θ i - b ( ˆ θ i )] /a ( φ ) 2
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& $ % Deviance Usually, a ( φ ) has the form a ( φ ) = φ/w i , and this statistic equals 2 N X i =1 w i [ y i ( ˜ θ i - ˆ θ i ) - b ( ˜ θ i ) + b ( ˆ θ i )] This is called scaled deviance . The greater the scaled deviance, the poorer the fit. For some GLMs, the scaled deviance has an approximate chi-squared distribution. 3
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& $ % Deviance For Poisson Model For Poisson GLMs, ˆ θ i = log ˆ μ i and b ( ˆ θ i ) = exp( ˆ θ i ) = ˆ μ i Similarly, for saturated model ˜ θ i = log y i
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