1 観察研究のための統計推測 - general misspecification model approach - S. Eguchi, ISM & GUAS This talk is a part of co-work with J. Copas, University of Warwick ISM Seminar, 15 Jan,

2 Observational study Experimental Study (Interventional Study) Observational Study case-control cohort Randomization cross-sectional Meta analysis

3 Hidden Bias Selection bias sampling bias self-selection bias losses to follow up length bias lead-time bias membership bias Berkson's bias Measurement bias information bias observer bias ascertainment bias bias due to digit preference recall bias interviewer bias

4 Incomplete Data Missing data Censoring,Truncation Group allocation Cf. MCAR, MAR Ignorable or Informative Random or Nonrandom

5 Review Cornfield (1951,55)Heckman (1976) Rosenbaum, Rubin (1983) Little (1985) Rubin (1976) Little, Rubin (2002) Rosenbaum (2002) Copas, Li (1997) Copas, Eguchi (2001) Robins (1993)

6 Strong model If Z has Z f (z,  Let Y = h(Z) be a many-to-one mapping. then Y has Cf. EM algorithm (Dempster, Laird, Rubin, 1977)

7 Standard inference

8 Tubular Neighborhood M Strong model Copas, Eguchi (2001)

9 Mis-specification

10 Weak model where

11 Strong and weak models Strong model Weak model

12 Example (MAR) Z = (X, R)Y,Y, h h, 

13 Sensitivity method Get the interpretable bound over { M } 

14 How small  is “small” ? from a counterfactual The most important situation is Any selectivity parameter cannot be estimated.

15 Key idea  Keep small qualitatively! Find an ancillary statistic U on  Build the confidence interval by U  Don’t fix  explicitly

16 Two MLEs Unobserved MLE Observed MLE

17 Under strong model

18 Under weak o(  )-model

19 Joint asymptotics

20 U = S | T The conditional distribution If T were observed by T = t,

21 Acceptability T has Hypothesis H : 

22 Envelope region

23 The worst case

24

25

26  1 2

27   1 2

28  1 2

29 Conclusion Double the confidence interval! The worst case occurs when

30 Today’s claim k-times Inflated region

31 Example 95% Confidence Interval レ レ × 0 0 0

32         おわりに ２倍の信頼区間の使い方 Bayesian model for  Acceptability