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Deviance for Binary Data Notes

# Unm r computes the null deviance for all i g i 0

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Unformatted text preview: ; Y ) − l (b; Y )] N =2 yi log i =1 yi ˆ yi + (ni − yi )log ni − yi ˆ ni − yi D s approximate sampling distribution is chi-square. UNM R computes the null deviance. For all i ˆ g (πi ) = β0 → π = g −1 (β0 ) ˆ ˆ Fitted values yi = ni π ˆ Null deviance: N −2 yi log (ˆ ) + (ni − yi )log (1 − π ) + π ˆ i =1 ni yi with degrees of freedom N − 1. Residual deviance: g (πi ) = β0 + β1 Xi 1 + . . . + βp Xpi ˆ πi are estimated with covariates and yi = ni πi ˆ ˆ UNM Residual deviance: N D = −2 yi log (ˆi ) + (ni − yi )log (1 − πi ) + π ˆ i =1 ni yi with degrees of freedom N − (p + 1). Hypothesis testing: H0 : β1 = β2 = . . . = βp = 0 vs. H1 : at least one βi = 0. D ∗ = Null deviance − Residual deviance dof = (N − 1) − (N − (p + 1)) = p. D ∗ approximately follows a χ2p) . ( UNM Alternatively if we just focus on the Residual deviance (grouped Binomial case), dof = (n1 + n2 + . . . + nN ) − (p + 1) or dof = no....
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