Global Goodness-of-Fit Tests in Logistic Regression with Sparse Data

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4. Solutions4.1 Modify limiting distribution- X 2 , D are asymptotically normal under n, m i →∝(Osius/Rojek, 1992; McCullagh, 1986)4.2 Grouping observations- Hosmer-Lemeshow test (Hosmer/Lemeshow, 1980)Maybe the standard test with sparse data nowadays,but it has some deficiencies (Hosmer et al, 1997,Bertolini et al., 2000)4.3 Use other tests statistics- X 2 F (Farrington, 1996)X 2 F = X 2 +Ni = 1Approximate moments:ˆtwith Q X ( X ) t WX XTest:−m πi( 1 − 2πî)( ) ( y − )im îπiˆ 1 − πˆN( X | ˆ β ) = N − p −1+ˆ π ( 1−ˆ π )i2EFi iQîii=1N2 p + 1mi−1Var( XF| ˆ β ) = 21− N i=1 mi−1= ˆ , Wˆdiag( m iˆ π (1 î−πi))ZF= .2FVar2E( X ˆF| β )2( X | ˆ β ) 1/ 2is standard normal under the hypothesisProblem: For m i ≡,1 we have X 2 F =N=X−FiO.Kuss, Global Goodness-of-Fit Tests in Logistic Regression with Sparse Data, 2.11.02
- IM-Test (White, 1982; Orme, 1988)Information matrix equation:2 ∂ L − E = ∂β∂β′ ∂L∂LE ∂β∂β′ Estimate both matrices, summation of the maindiagnonal elements yields the ((p+1)×1) vectordˆM= i = 1( y i− πˆ)( 1 − 2 ˆ π i) z iiwithz = (1, x , x )i2i1,2iptTest:IM =1 t −1Mdîs χ 2 -distributed with (p+1) dfandVˆ* t * * t * −1* t *[ Z ( I − X ( X X ) X ) Z ]ˆ 1V = ,M*X = ˆ πi( 1−ˆ πi)X ,Zˆ( 1−ˆ π )(1− 2 ˆ π Z*= πi ii) ,dˆZ as the matrix with the zi as rows.O.Kuss, Global Goodness-of-Fit Tests in Logistic Regression with Sparse Data, 2.11.02
Page 3: Data:Response1 01 Y 1 m 1 -Y 1 m 1C
Page 7 and 8: 3. The Problem of Sparse DataThe χ
Page 9: ... but what is the solution???- In
Page 13 and 14: Example:Occupational hand eczema in
Page 15 and 16: 6. Simulation Results6.1. Null hypo
Page 17 and 18: Back to the example:p-Wert p*-WertX
Page 19: 8. Literature- Agresti A. Categoric

Global Goodness-of-Fit Tests in Logistic Regression with Sparse Data

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