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Chapter 2 Notes - Example from Table 2.1 data Yjk number of...

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Example from Table 2.1 data Y jk number of chronic medical conditions for women that use similar practitioner services. j = 1 ’town’; j = 2 ’country’ k = 1 , 2 , . . . , k j where k 1 = 26 and k 2 = 23. For ’town’ , ¯ Y 1 · = 1 . 423 and SD = 1 . 17. For ’country’, ¯ Y 2 · = 0 . 913 and SD = 0 . 9. We can try to model Y 0 jk s as independent Poisson ( θ j ) . θ j rate for group j . Recall that if Y i Poisson ( θ ) , the log-likelihood function is l ( θ ; Y ) = X i Y i log ( θ ) - θ X i Y i + X i log ( y i !) Do women have similar levels in the two groups? UNM
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Boxplots of the data by group 0 1 2 3 4 Town 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Country UNM
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Hypothesis testing problem, H 0 : θ 1 = θ 2 = θ vs H 1 : θ 1 6 = θ 2 Two models : Under H 0 , Y jk Poisson ( θ ) . Second model, Y jk Poisson ( θ j ) (nested models). Under H 0 , l 0 = l ( θ ; Y ) = 2 X j = 1 k j X k = 1 y jk log ( θ ) - θ - log ( y jk !) The MLE is ˆ θ = 2 X j = 1 k j X k = 1 y jk / N ; N = 26 + 23 = 49 UNM
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Notice the MLE is ”pooled estimate” of the town and country means.
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