Information geometry of Statistical inference with selective sample S. Eguchi, ISM & GUAS This talk is a part of co-work with J. Copas, University of Warwick

Local Sensitivity Approximation for Selectivity Bias. J. Copas and S. Eguchi J. Royal Statist. Soc. B, 63 (2001), (

Summary

Statistical model Probability model

Near parametric exact parametric non-parametric near-parametric

Def. (X, Y, Z) is ignorable Ignorability

Observational status

Binary response with missing

Non-ignorable cases

Small degree of non-ignorability

Distributional expression

Tangent space and neighborhood

Exponential map

Tubular Neighborhood M

Decomposition

tangent and normal

Conditional Distribution

Calibration

Rosenbaum’s log odd ratio

Counterfactual

Guide line

Non-ignorable missing

Selectivity region

Unstable or Misspecifying

Regression formulation

Heckman model

Likelihood

Likelihood analysis where

Profile likelihood of 

N = 1435, n = 1323 

Skin cancer data

Various pattern of bias

Group comparison

Non-random allocation

Selection bias

Effect of sentence Z = 1 prison Z = 2 community service Z = 3 probation Y = ratio of reconviction Logistic model

Selectivity regions Probation effect Community service effect 01 ‐1‐1 ‐1‐1 1 0     C.I.

two-group comparison

Likelihood

Analysis

UK National Hearing Survey The effect of occupational noise Case (high level noise) Control Response Y is threshold of 3kHz sound

Case mean Control mean Pooled s. d. t-statistic Standard analysis supports high significance Conventional result

Non-random allocation

Future problem

Arnold,B.C. and Strauss, D.J. (1991) Bivariate distributions with conditionals in prescribed exponential families. J.Roy.Statist.Soc., B, 53, Begg,C.B., Satagopan, J.M. and Berwick, M.(1998) A new strategy for evaluating the impact of epidemiologic risk factors for cancer with application to melanoma. J. Am. Statist. Assoc., 93, Bowater, R.J.,Copas, J.B., Machado, O.A. and Davis, A.C. (1996) Hearing impairment and the log-normal distribution. Applied Statistics, 45, Chambers, R.L.and Welsh, A.H. (1993) Log-linear models for survey data with non- ignorable non-response. J.Roy.Statist.Soc., B, 55, Copas, J. B.and Li, H. G. (1997) Inference for non-random samples (with discussion). J. Roy. Statist. Soc.,B, 59, Copas, J.B. and Marshall, P. (1998) The offender group reconviction scale:a statistical reconviction score for use by probation offers. Applie Statistics, 47, References

Cornfeld,J.,Haenszel,W.,Hammond,E.C.,Lilien eld,A.M.,Shimkin,M.B.and Wyn- der,E.L.(1959) Smoking and lung cancer:recent evidence and a discussion of some questions. J.Nat.Cancer Institute, 22, Davis,A.C.(1995) Hearing in Adults. London:Whurr. Foster, J.J.and Smith,P.W.F.(1998) Model based inference for categorical survey data subject to nonignorable nonresponse. J. Roy. Statist. Soc, B, 60, Heckman, J.J.(1976) The common structure of statistical models of truncation,sample selection and limited dependent variables,and a simple estimator for such models. Ann. Economic and Social Measurement, 5, Heckman, J.J. (1979) Sample selection bias as a specifcation error. Econometrica, 47, Kershaw, C. (1999) Reconvictions of offenders sentenced or discharged from prison in 1994, England and Wales. Home Office Statistical Bulletin, 5/99. London: HMSO.

Lin, D.Y., Pasty, B.M.and Kronmal, R.A.(1998) Assessing the sensitivity of regression results to unmeasured confounders in observational studies. Biometrics, 54, Little, R. J. A. (1985) A note about models for selectivity bias. Econometrica, 53, Little,R.J.A. (1995) Modelling the dropout mechanism in repeated-measures studies J. Am. Statist. Assoc., 90, Little,R.J.A. and Rubin, D.A.(1987) Statistical Analysis with Missing Data. New York: Wiley. McCullagh, P. and Nelder, J.A. (1989) Generalize Linear Models. 2nd ed. London: Chapman and Hall. Rosenbaum, P.R. (1987) Sensitivity analysis for certain permutation inferences in matched observational studies. Biometrika, 74, Rosenbaum, P.R. (1995) Observational Studies. New York: Springer

Rosenbaum, P.R. and Krieger,A.M.(1990) Sensitivity of two-sample permutation inferences in observational studies.J.Am.Statist.Assoc., 85, Rosenbaum, P.R. and Rubin,D.B.(1983)Assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome. J. Roy. Statist. Soc., B, 45, Scharfstein, D,O., Rotnitzy, A. and Robins, J. M. (1999) Adjusting for non- ignorable drop-out using semiparametric nonresponse models (with discussion). J. Amer. Statist.Assoc.,94, Schlesselman,J.J.(1978)Assessing effects of confounding variables. Am. J. Epidemiology, 108, 3-8. White,H.(1982)Maximum likelihood estimation of misspecified models. Econometrica, 50, 1-26.