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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

SLIDE 1

Challenges on Imprecise Inference

for the measure of association in 2x2 tables Chel Hee Lee1 Mik

, elis Bickis2

1Clinical Research Support Unit

Community Health and Epidemiology College of Medicine University of Saskatchewan

2Department 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

SLIDE 2

Beta-Binomial Model (Walley, 1991)

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

n=2,m=1

p CDF (1,3) (1.4,2.6) (1.8,2.2) (2.2,1.8) (2.6,1.4) (3,1) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

n=5,m=1

p CDF (1,6) (1.4,5.6) (1.8,5.2) (2.2,4.8) (2.6,4.4) (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

n=10,m=4

p CDF (4,8) (4.4,7.6) (4.8,7.2) (5.2,6.8) (5.6,6.4) (6,6) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

n=20,m=8

p CDF (8,14) (8.4,13.6) (8.8,13.2) (9.2,12.8) (9.6,12.4) (10,12)

Chel Hee Lee, Mik

, elis Bickis (USASK)

Imprecise Inference 2016-SEP-06 2 / 14

SLIDE 3

Beta-Binomial Model (Chel)

E [ θ ]

ξ1 ξ0 −3 −1.5 −0.5 0.5 1 . 5 3

5 10 15 20 5 10 15 20

1 1 1 1 1 1 1 1

E [ Y ]

ξ1 ξ0 0.1 0.2 0.3 . 5 . 5 . 6 . 8

5 10 15 20 5 10 15 20

1 1 1 1 1 1 1 1 Chel Hee Lee, Mik

, elis Bickis (USASK)

Imprecise Inference 2016-SEP-06 3 / 14

SLIDE 4

Beta-Binomial Model (Chel)

−40 −20 20 40 0.0 0.2 0.4 0.6 0.8 1.0

dds ratio

CDF

Chel Hee Lee, Mik

, elis Bickis (USASK)

Imprecise Inference 2016-SEP-06 4 / 14

SLIDE 5

Uniform-Binomial Model

0.1 0.2 0.3 0.4 0.5 0.6 . 7 0.8 0.9 1

5 10 15 20 2 4 6 8 10

0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 1 Chel Hee Lee, Mik

, elis Bickis (USASK)

Imprecise Inference 2016-SEP-06 5 / 14

SLIDE 6

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

SLIDE 7

Persistent Pulmonary Hypertension

Imprecise Beta Model L(θc, θe|n) ∝ θ6

c(1 − θc)4θ9 e

π(θc, θe) ∝ θstc−1

c

(1 − θc)s(1−tc)−1θste−1

e

(1 − θe)s(1−te)−1 π(θc, θe|n) ∝ θstc+5(1 − θc)s(1−tc)+3θste+8

e

(1 − θe)s(1−te)−1 Inferences about θe − θc H0 : θe ≤ θc H1 : θe > θc This can be answered by calculating P(H1|n) and P(H1|n)

Chel Hee Lee, Mik

, elis Bickis (USASK)

Imprecise Inference 2016-SEP-06 7 / 14

SLIDE 8

Can we work with log-odds?

General form Y2 (levels) (fixed) Yes No Total Y1 Yes n11 n10 n1 (type) No n01 n00 n0 L(n11, n01|p11, p01) ∝ pn11

11 (1 − p11)n1−n11pn01 01 (1 − p01)n0−n01

= exp

n11θ1 + n01θ2 − n1 log(1 + eθ1) − n2 log(1 + eθ2)
where θ = log(p11/(1 − p11)) and θ2 = log(p01/(1 − p01)). Subsuquently,

log-odds ratio (LOR) is found by θ1 − θ2. P(L > 0|n11, n01) can be also found numerically.

Chel Hee Lee, Mik

, elis Bickis (USASK)

Imprecise Inference 2016-SEP-06 8 / 14

SLIDE 9

Consider All Margins Are Fixed

General form Y2 (fixed) Yes No Total Y1 Yes n11 n10 n1+ No n01 n00 n0+ (fixed) n+1 n+0 n L(p11, p1+, p+1) = n nij

pn11

11 (p1+ − p11)n10(p+1 − p11)n01(1 − p1+ − p+1 + p11)n00

= exp

n11 log(θ) − A(θ) − log

n nij

where θ = p11p00 p10p01

Chel Hee Lee, Mik

, elis Bickis (USASK)

Imprecise Inference 2016-SEP-06 9 / 14

SLIDE 10

Can we work with percent agreement?

Three outcomes are considered P(Y1 = 1, Y2 = 0) = p1 P(Y1 = 0, Y2 = 1) = p2 P[(Y1 = 1, Y2 = 1)or(Y1 = 0, Y2 = 0)] = pa where p1 + p2 + p3 = 1. L(Y|p) ∝ pn1

1 pn2 2 pn3 3

where n3 = n − n1 − n2 = exp{n1

eθ1

1 + eθ1 + eθ2

+ n2
eθ2

1 + eθ1 + eθ2

−n log
1

1 + eθ1 + eθ2

+ log
n

n1, n2, n3

}

Subsequently, we are interested in the inference P(p1 − p2 > 0) = P(log(p1/p2)) = P(p1 > p2).

Chel Hee Lee, Mik

, elis Bickis (USASK)

Imprecise Inference 2016-SEP-06 10 / 14

SLIDE 11

Cohen’s Kappa

Definition κ(p) = (p11 + p00) − (p1+p+1 + p0+p+0) 1 − (p1+p+1 + p0+p+0) = 2(p11 − p1+p+1) p1+p+1 − 2p1+p+1 = p00p11 − p01p10 p11p00 − p01p10 + (p01 + p10)/2

Chel Hee Lee, Mik

, elis Bickis (USASK)

Imprecise Inference 2016-SEP-06 11 / 14

SLIDE 12

Can We Work With Four Cells?

Mik’s Talk log pij = ξ1θ1 + ξ2θ2 + ξ3θ3 − A(θ), i, j = 1, 2

Chel Hee Lee, Mik

, elis Bickis (USASK)

Imprecise Inference 2016-SEP-06 12 / 14

SLIDE 13

Matter of Visualization

−0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 −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

SLIDE 14

Matter of Visualization

Chel Hee Lee, Mik

, elis Bickis (USASK)

Imprecise Inference 2016-SEP-06 13 / 14

SLIDE 15

Matter of Visualization

Chel Hee Lee, Mik

, elis Bickis (USASK)

Imprecise Inference 2016-SEP-06 13 / 14

SLIDE 16

Matter of Visualization

Chel Hee Lee, Mik

, elis Bickis (USASK)

Imprecise Inference 2016-SEP-06 13 / 14

SLIDE 17

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

f 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