Rank-Based Approach to OptimalScore via Dimension Reduction

Shao-Hsuan WangNational Taiwan University, Taiwan

Nov 2015

1

Rank-based measures

Kendall’s

Concordance Index

Rank correlation

Widely used in medical statistics, epidemiology,economics, and sociology, etc.

2

Rank-based measures

Regression Model

Y : a univariate response

Z (Z , , Z ): multiple covariates1 p

3

Rank-based measures

YRResponse

Y

TZ

TZRComposite score

4

Rank-based measures

(Y T T, Z) (Y, Z)For pair of observations

concordant :T

1 1 and 2 2

T

,

YY andTZ Z Y Y andTZ Z1 2

discordant :TY Y and

1 2 1

TZ Z Y

2 1 2

TY andTZ Z1 2 1 2 1 2 1 2

5

Rank-based measures

Kendall’s P(Y TY, Z T Z ) P(YY T, Z T Z )1 2

Rank correlation1 2 1

T T

2 1 2

rc P(YY , Z Z )1 2 1 2

Concordance IndexT TCI P( Z Z | YY)1 2 1 2

6

Rank-based measures

YRResponse

Y

TZ

TZRComposite score

7

Rank-based measures

There could not exist amonotonic association !!

8

Motivation

Composite score

TZ

g(Z)measurable functions

10

C-max

YRResponse

Concordance-index function :

C(g) P(g(Z

g(Z)RComposite score

)g(Z)|YY)

C C1 2 1 2

(g) C-max : max supgF c

Optimal score : m(Z) such that msupC(g)gF 11c

Intrinsic modelbehind Rank-based measures

M1 Distributional assumption

: Generalized Regression Model (Han 1987)

M2 Structural assumption

: Dimension Reduction (Li 1991, Cook 1991)

12

Intrinsic modelbehind Rank-based measures

M1

a non-degeneratemonotonic function on R

Y G(md (Z),)0

13

Intrinsic modelbehind Rank-based measures

M1

a non-degeneratemonotonic function on R

Y G(md (Z),)0

an unspecifed bivariate function strictlyincreasing at each component for the other

one being fixed14

Intrinsic modelbehind Rank-based measures

M2

Y D G(md (Z),)0

a multivariate polynomial of the unknown degreed 0

15

Intrinsic modelbehind Rank-based measures

M2Dimension Reduction

m (Z) Tm (B Z)d d k 00 0 0

(1)d 0 be the smallest degree such that YZ | md (Z)0

B(2) 0 { 01 , , 0k0 } is a basis of the central subspace (CS)

16

Model Flexibility

Linear regression model Y T0 Z

T Binary Choice model

Accelerated Failure time model

Y I(log(Y)

0 ZT0

0)

Z Generalized linear regression model (GLM)

Non-monotonic regression modelY ( T0

2Z)

17

Types of covariates

all discrete but continuous covariates

Covariates which moments could not exist

18

Theories

Propositions:

(1) Existence m (Z arg maxC(g)d0 g

(2) Uniqueness f (Z)arg maxC(g) f (Z) cm (Z) c

(3) Optimality

d0g

for a ploynomial fd0

d0 1

(z) of the degree

d0 2

d0

g(Z)arg maxC(g) g(Z)T(m (Z))dg

for some monotonic function T0

19

Summary

TZ could not be the best composite score

Model flexibility

Various types of covariates

Optimal score : existence, uniqueness, andoptimality

20

How to estimatedk

0

0

: structural degree: structural dimension

S(B ) : the central subspace0

m (BZ) : the optimal scored k 0

C0 0

max : the C-max

Estimation Procedure

Estimation Procedure

Derivem (Z) by maximizing the concordance index function viaStep1 d

the generalized single-index form of the polynomial

Tips: (1)d p

m(Z) c Z rj T Zd r r 1 pr0r1 rpr j1

n n

I( T Z T Z ,Y Y )(2)

C (m (Z)) C () i j i j

i1 j1n d 0n n n

i1 j1

I(Y Y )i j

Estimation Procedure

Step 2Apply the outer grandient approach to obtain B

Tips : (1)T

k

m (u) m (B u)d d k 00 0 0

(2)

col(S(B)) col( m (u)(m T(u))dW(u))0 puR

d0 d0

Estimation Procedure

Step 3 Derive the estimator of

Tips : (1)

mdkT(Bk Z)

Z BTZkn n

I( T Z T Z,Y Y )(2)ˆ arg max

(3) T

i1 j1

T

i

n n

i1 j1

j i j

I(Y Y )i j

m (B Z) ˆ Zdk k

Estimation Procedure

Step 4 Adopt the concordance-based generalized BIC to estimateTd, k, S(B),m (BZ), and C0 0 0 d0k0

Tips : (1)IC(d,k)

0 max

TnC (m (Blogn kdZ)) (C 1)n dk k

withIC(0,k)1/2(2)

(d,k) arg maxIC(d,k)0d,1 p1

2 k

Asymptotic results

Consistent model selection--- parsimonious model among the class of

Correct models(d ,k )0 0

n-consistency of estimators of TS(B)andm (B Z)0

Asymptotic normality of estimators of C

d0k0 0

max

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Wine Data

• Vinho verde wine : red wine and white wine(from the Minho Region of Northern Portugal)

• Collected from May/2004 -February/2007

• Red wine : sample size (n)=1599White wine : n=4898

•Physicochemical and sensory tests

Wine data

Response (Y):Preferences 0 (bad) -10 (excellent)

11 Covariates (Z) :fixed acidity, volatile acidity,citric acid, residual sugar,chlorides, free sulfur dioxide,total sulfur dioxide, density,PH, sulphates, and alcohol

Wine data

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Wine data

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Thank You !