Challenges on Imprecise Inference for the measure of association in - PowerPoint PPT Presentation

Challenges on Imprecise Inference for the measure of association in 2x2 tables Chel Hee Lee 1 , elis Bickis 2 Mik 1 Clinical Research Support Unit Community Health and Epidemiology College of Medicine University of Saskatchewan 2 Department of Mathematics and Statistics University of Saskatchewan 9th WPMSIIP, Durham, England 6 September 2016 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 1 / 14

Beta-Binomial Model (Walley, 1991) n=2,m=1 n=5,m=1 1.0 1.0 0.8 0.8 0.6 0.6 CDF CDF 0.4 0.4 (1,3) (1,6) (1.4,2.6) (1.4,5.6) 0.2 0.2 (1.8,2.2) (1.8,5.2) (2.2,1.8) (2.2,4.8) (2.6,1.4) (2.6,4.4) 0.0 (3,1) 0.0 (3,4) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 p p n=10,m=4 n=20,m=8 1.0 1.0 0.8 0.8 0.6 0.6 CDF CDF 0.4 0.4 (4,8) (8,14) (4.4,7.6) (8.4,13.6) 0.2 0.2 (4.8,7.2) (8.8,13.2) (5.2,6.8) (9.2,12.8) (5.6,6.4) (9.6,12.4) 0.0 (6,6) 0.0 (10,12) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 p p Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 2 / 14

Beta-Binomial Model (Chel) E [ θ ] E [ Y ] 0 0 20 0 20 0 −0.5 0.5 3 −3 −1.5 5 0.1 0.2 0.3 8 5 6 . . . 1 0 . 0 0 0 0 1 1 1 1 0 0 15 1 15 1 0 0 1 1 0 0 0 0 ξ 0 ξ 0 10 1 10 1 0 0 1 1 1 1 0 0 0 0 5 5 0 0 0 0 1 1 5 . 0 0 0 0 5 10 15 20 0 5 10 15 20 ξ 1 ξ 1 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 3 / 14

Beta-Binomial Model (Chel) 1.0 0.8 0.6 CDF 0.4 0.2 0.0 −40 −20 0 20 40 odds ratio Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 4 / 14

Uniform-Binomial Model 0.9 0.8 7 0.6 0.5 1 0 . 10 0.4 8 0.3 6 0.2 4 0.1 2 0 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 1 0 5 10 15 20 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 5 / 14

Persistent Pulmonary Hypertension (Walley, 1996, Sec. 5) Survived Yes No Total TRT CT 6 10 ECMO 9 9 CT means a conventional therapy and ECMO means extracorporeal membrane oxygenation. Assumptions A constant chance of survival under each treatment. Outcomes are independent for different babies. Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 6 / 14

Persistent Pulmonary Hypertension Imprecise Beta Model θ 6 c ( 1 − θ c ) 4 θ 9 L ( θ c , θ e | n ) ∝ e θ st c − 1 ( 1 − θ c ) s ( 1 − t c ) − 1 θ st e − 1 ( 1 − θ e ) s ( 1 − t e ) − 1 π ( θ c , θ e ) ∝ c e θ st c + 5 ( 1 − θ c ) s ( 1 − t c )+ 3 θ st e + 8 ( 1 − θ e ) s ( 1 − t e ) − 1 π ( θ c , θ e | n ) ∝ e Inferences about θ e − θ c : θ e ≤ θ c H 0 H 1 : θ e > θ c This can be answered by calculating P ( H 1 | n ) and P ( H 1 | n ) Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 7 / 14

Can we work with log-odds? General form Y 2 (levels) (fixed) Yes No Total Y 1 Yes n 11 n 10 n 1 (type) No n 01 n 00 n 0 L ( n 11 , n 01 | p 11 , p 01 ) p n 11 11 ( 1 − p 11 ) n 1 − n 11 p n 01 01 ( 1 − p 01 ) n 0 − n 01 ∝ � � n 11 θ 1 + n 01 θ 2 − n 1 log ( 1 + e θ 1 ) − n 2 log ( 1 + e θ 2 ) = exp where θ = log ( p 11 / ( 1 − p 11 )) and θ 2 = log ( p 01 / ( 1 − p 01 )) . Subsuquently, log-odds ratio (LOR) is found by θ 1 − θ 2 . P ( L > 0 | n 11 , n 01 ) can be also found numerically. Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 8 / 14

Consider All Margins Are Fixed General form Y 2 (fixed) Yes No Total Y 1 Yes n 11 n 10 n 1 + No n 01 n 00 n 0 + (fixed) n + 1 n + 0 n L ( p 11 , p 1 + , p + 1 ) � n � p n 11 11 ( p 1 + − p 11 ) n 10 ( p + 1 − p 11 ) n 01 ( 1 − p 1 + − p + 1 + p 11 ) n 00 = n ij � n � �� where θ = p 11 p 00 = n 11 log ( θ ) − A ( θ ) − log , exp n ij p 10 p 01 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 9 / 14

Can we work with percent agreement? Three outcomes are considered P ( Y 1 = 1 , Y 2 = 0 ) = p 1 P ( Y 1 = 0 , Y 2 = 1 ) = p 2 P [( Y 1 = 1 , Y 2 = 1 ) or ( Y 1 = 0 , Y 2 = 0 )] = p a where p 1 + p 2 + p 3 = 1. p n 1 1 p n 2 2 p n 3 L ( Y | p ) ∝ where n 3 = n − n 1 − n 2 3 e θ 1 e θ 2 � � � � = exp { n 1 + n 2 1 + e θ 1 + e θ 2 1 + e θ 1 + e θ 2 � � � � 1 n − n log + log } 1 + e θ 1 + e θ 2 n 1 , n 2 , n 3 Subsequently, we are interested in the inference P ( p 1 − p 2 > 0 ) = P ( log ( p 1 / p 2 )) = P ( p 1 > p 2 ) . Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 10 / 14

Cohen’s Kappa Definition ( p 11 + p 00 ) − ( p 1 + p + 1 + p 0 + p + 0 ) κ ( p ) = 1 − ( p 1 + p + 1 + p 0 + p + 0 ) 2 ( p 11 − p 1 + p + 1 ) = p 1 + p + 1 − 2 p 1 + p + 1 p 00 p 11 − p 01 p 10 = p 11 p 00 − p 01 p 10 + ( p 01 + p 10 ) / 2 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 11 / 14

Can We Work With Four Cells? Mik’s Talk log p ij = ξ 1 θ 1 + ξ 2 θ 2 + ξ 3 θ 3 − A ( θ ) , i , j = 1 , 2 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 12 / 14

Matter of Visualization 1.2 1.0 ● 0.8 0.6 0.4 ● ● ● ● 0.2 ● ● ● 0.0 ● ● −0.2 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 13 / 14

Matter of Visualization Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 13 / 14

Matter of Visualization Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 13 / 14

Matter of Visualization Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 13 / 14

References References I Diaconis, P. and Ylvisaker, D. (1979). Conjugate Priors for Exponential Families. Ann. Statist., 7(2):269–281. Walley, P. (1991). Statistical reasoning with imprecise probabilities. Chapman and Hall, London;. Walley, P. (1996). Inferences from Multinomial Data: Learning about a Bag of Marbles. Journal of the Royal Statistical Society. Series B (Methodological), 58(1):pp. 3–57. Chel Hee Lee, Mik , elis Bickis (USASK) Imprecise Inference 2016-SEP-06 14 / 14