information geometry of statistical inference with selective sample s. eguchi, ism & guas this...

　　 Information geometry of Statistical inference with selective sample

S. Eguchi, ISM & GUAS

This talk is a part of co-work withJ. Copas, University of Warwick

  Local Sensitivity Approximation

for Selectivity Bias.

J. Copas and S. Eguchi

J. Royal Statist. Soc. B, 63 (2001), 871-895. (http://www.ism.ac.jp/~eguchi/recent_preprint.html)

Summary

bounds bleinterpretaGet

for with of modelnear for the

inference theCompare

modelof neiborhoodTubular

analysis ySensitivit

bias Selection

tyIgnorabili

MM

M

Statistical model

),( )....,,(iid

1 xpXX n ～

Statistical model

Probability model

B

xxpBXP )d()()(

dpx RRX ,

Near parametric

),...,( )....,,( 11 nn xxgXX ～

)()),(,KL(min

nOfg

0

exact parametric

non-parametric

near-parametric

status nalobservatio:

variablecovariate:

variableresponse:

Z

X

Y

XZY

Def. (X, Y, Z) is ignorable

Ignorability

Observational status

0 if )(censoring missing

1 if observed is

indicator. nsering)missing(ce is .1Exm.

Z

ZY

Z

GzyfY

Z

z ,,1for , Z|

process. allocationfor label group is .2 Exm

～

1 offrequency

observed of nb: , size sample

1)pr(

index Z missing with responsebinary a be Y

Yf

nN

Y

Binary response with missing

ns

n

f )ˆ1(ˆ variance,

ˆ MLE 2

01

1

)1(

)1|1pr(

pp

pZY

1

)1)( (1 )1( 0

) (1 1

0 1

model

01

01

ppz

ppz

yy

? ?

? ?

0 1

data

N

N

nfnf

yy

))|1(Z( yPpy

Non-ignorable cases

yZ

yZ

|0pr

|1prlogVar2

?" small" is small How ! ~

withˆ Compare

)( ˆ~,

~ 2

10

0

ON

nNsn

pfnfp

fp

Small degree of non-ignorability

zzyyYZ

ZYYZ

f

qp

zfyfxzyf

XZY

11 )1()1( 2'Exm

dim , dim where

),(),()|,(

| 2 Def

Distributional expression

1)},({Var

0)},({)},({ :),(

),KL( min:

3 Def.

,

zy

ZyzYzy

fgg

YZ

ZY

f

ff

YZYZYZ

Tangent space and neighborhood

)( 2

, KL

)(),()exp(:),(

32

0

Ofg

g

zyy,zfzyg

YZYZ

YZ

YZYZ

Exponential map

Tubular Neighborhood

M

M

) 1 (

)( )(

),(),,(

),(),( where

),( ),( ),(

2

1

,1

,

2

1

2

1

ij

jkkjfijjif

ZT

q

YT

p

jijiij

vvuu

xzSIvv

ySIuu

vuzvyuzy

zY

Decomposition

q

j qjjjjj

p

i piiiii

vuvu

vuvuzy

1 1

1 1

....

....,

tangent and normal

Conditional Distribution

0 , argsolve ~

).. ,.,. (

).. ,.,. ( when

).. 1)( ,()|(

).. 1)( ,()|(

*2

1

1*

1

1

*2

1

|

1

2

1

|

|*

ySEI

uu

vv

vuIzfyzg

vuIyfzyg

ZYgz

Tqy

Tpz

qjjjyZYZ

piiizYZY

,..1

22

22

|,2

2

1

|2

0""

)(

)KL(

))|((logVar 4. Def.

pvizz

Tz

zf

ZΥΥｚ

YZfz

i

Z

Υ

gf

yzg

Calibration

Rosenbaum’s log odd ratio

)|0(

)0(

)0(

)0( log

yr

r

r

y|r

const

.2|| ifReject

)1 ,0( )(

testScore

0 ::hypothesis a Testing

0

,1

2

1

0

TH

NzyNT

H

kk

N

k

～

Counterfactual

Guide line

)(2

2

22power local

0 ::hypothesis a Testing

2

1

2

1

2

1

2

1

0

nNn

N

N

NN

H

)(1 log log

,1 ,

, log,

likelihood-log

1 ,

1,

2

11 1

1,2

1

11

1

22

1

OnN

vyuySI

yfL

zzv

n

k pikiiK

n

k

k

n

k

Non-ignorable missing

2

1

2

1

2

1

ˆ

,..1

2111

2ˆ

21ˆ

ˆ~ , 1If

0"" )ˆ~

()ˆ~

(

)( ˆ~

, argmax~

MLE |

N

nNnp

O

L

pii

Selectivity region

)(Var,)ˆ,1( )ˆ,(1

where

)(22

)0,ˆ( ),( max

*2*

111

*

32*

2*2

2*

*2*

|

uvyun

u

Onuun

LL

Yfpi

kii

? estimate weCan

statistics skewness includes 0

poly Hermitte )(

),( ,),N( ),(

Otherwise

0 if unstable is ˆ

ˆ

*31

3

22

1

212*

*

u

yu

yf

u

ii

～

Unstable or Misspecifying

)())(1( log)( log

))( ( )(

))(( log ),(

) ,1( ,)( ),(

observedfully ,,1;

2

11

,1

1

1 ,

2

1

Oxx

xySI

xyfL

fpxx

Nkx

n

kk

Tn

kk

T

kT

k

n

kkkk

n

kk

TkY

kzkkT

kkT

k

k

Regression formulation

Heckman model

1eXY T

2eXR T

1

1

0

0

2

1

,Ne

e～

0 0

1 0

ZR

ZR

missing is 0

observed is 0

YR

YR

Likelihood

)( )/11

(

1

)0,|(

T

22

T

)(

xxyx

xyrxyf

T

T

xyx

yxrPTT

22 11 ), | 0(

)( )0 ,|E( TT xxrxy

Likelihood analysis

n

ii

Ti xyn,,,L

1

22

)(2

1 log ) (

,)( log)( log1

T

1

N

nii

n

ii xu

i

Ti

ii

xyxu

1

1

12

2

2

where

),,,( max)(|,,

*

LL

n

s

iT

i

n

i

iT

i

ˆxˆy

K*L

ˆxˆy

K*L

*L*L

1

42

1

3

1

3)((0)

)((0)

0(0) , 0(0)

Profile likelihood of

dataaudit work Coventry

,income y ) age ,age ,sex 1, ( 2xN = 1435, n = 1323

Skin cancer data

control case

3231 f 259 2 f

130 3 f 288 4 f

Data Melanoma :2 Table

10 nevi

4321 pppp average bound

.2 .1 .2 .1

.1 .2 .2 .1

.1 .15 .15 .1

.2 .2 .1 .1

.5 .533 .533 .5

.011 0 0

1.580 1.719 .391

.812 .840 .222

.002 1.761 .391

.698 1.02 .063

results Simulation :3 Table

Various pattern of bias

)(),(),()(

model dependence

)(),(),(

modeleffect random

),1, ( ),( |

|

ztezftfg

dttftyfyf

GzyfZY

ZTTZ

TzTYzY

zY ～

Group comparison

)(

),( ,|

),(),( where

)},( 1 ){,(

),()|(

|*

*Y

|||

kT

zzzk

kT

zT

zk

zkYk

YT

z

zTYZΤzY

xd

xd

yfxzY

ztzy

zyyf

tyfzyg

～

Non-random allocation

)(

2

1

)(ˆ ~ ˆ~

2

1

2

1

*

1 1

*1

1

1

zkz

kzkz

kzkz

zkk

z

N

k

G

zzkkzk

kzG

z

G

z I

p

pp

daI

Selection bias

),...,(:~

1ˆPlot GGS R

Effect of sentence

Z = 1 prisonZ = 2 community serviceZ = 3 probationY = ratio of reconviction

Logistic model

433221)},|({logit XddZXYE zz

Selectivity regions

Pro

bati

on e

ffec

t

Community service effect0 1‐1

‐11

0

C.I.

two-group comparison

1 )( sgn ery 2er

)0( 1 1 1 rzy,...,y n

)0( 2 11 rzy,...,y Nn

2 0 1 0 zrzr

n

i

in

i

i yyN

12

12

2

)1

(log2

)(log

n

ni

in

ni

i yy

12

12

2

)1

(log2

)(

) ( ˆ -1

2ˆ )( ˆ 3

2

Likelihood

Analysis

)(

2

121 yyˆ

2

)ˆ(

)(2

1 )ˆ( var 222

NO

N

UK National Hearing Survey

The effect of occupational noise

Case (high level noise)

Control 　

670 n

1441 n

Response Y is threshold of 3kHz sound

710.31 y

893.30 y

351.0s

52.3t f.) d.209(

Case mean

Control mean

Pooled s. d.

t-statistic

Standard analysis supports high significance

Conventional result

) ( 39.5.523 )~

( 3 t

29.0if96.1 )~

( 05.0 zt

Non-random allocation

05.0210.05 Nnnzt

30.023.005.0

Future problem

bounds bleinterpretaGet

for with of modelnear for the

inference theCompare

modelof neiborhoodTubular

analysis ySensitivit

bias Selection

tyIgnorabili

MM

M

Arnold,B.C. and Strauss, D.J. (1991) Bivariate distributions with conditionals in prescribed exponential families. J.Roy.Statist.Soc., B, 53, 365-376.

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, 415-426.

Bowater, R.J.,Copas, J.B., Machado, O.A. and Davis, A.C. (1996) Hearing impairmentand the log-normal distribution. Applied Statistics, 45, 203-217.

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, 157-170.

Copas, J. B.and Li, H. G. (1997) Inference for non-random samples (with discussion). J. Roy. Statist. Soc.,B, 59 ,55-95.

Copas, J.B. and Marshall, P. (1998) The offender group reconviction scale:a statisticalreconviction score for use by probation offers. Applie Statistics, 47, 159-171.

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, 173-203.

Davis,A.C.(1995) Hearing in Adults. London:Whurr.

Foster, J.J.and Smith,P.W.F.(1998) Model based inference for categorical survey datasubject to nonignorable nonresponse. J. Roy. Statist. Soc, B, 60, 57-70.

Heckman, J.J.(1976) The common structure of statistical models of truncation,sampleselection and limited dependent variables,and a simple estimator for such models. Ann. Economic and Social Measurement, 5, 475-492.

Heckman, J.J. (1979) Sample selection bias as a specifcation error. Econometrica, 47, 153-161.

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 ,948-963.

Little, R. J. A. (1985) A note about models for selectivity bias. Econometrica, 53, 1469-1474.

Little,R.J.A. (1995) Modelling the dropout mechanism in repeated-measures studies J. Am. Statist. Assoc., 90, 1112-1121.

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 ,13-26.

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, 493-498.

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, 212-218.

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, 1096-1146.

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.