Bootstrap Confidence Intervals for Regression Coefficients under Censored Data

2 002 , V ol. 13, N o.2 p p 355～363

B o ot s t rap Con fide n c e In t e rv al s f or R e g re s s ion Co e f fic ie n t s u n de r Ce n s ore d

D at a

K il H o Ch o^{1 )} . S e o n g H w a Je on g^{2 )}

A b s trac t

U sin g t h e Bu ckley - J am es m et h od, w e con st ru ct b oot st r ap confiden ce in t erv als for t h e r eg r es sion coefficien t s un der t h e cen sor ed dat a . A n d w e com p ar e th ese confiden ce in t erv als in t erm s of t h e cov er ag e pr ob abilit ie s an d th e ex p ect ed confiden ce in t erv al len g th s th r ou gh M ont e Carlo sim u lation .

1 . In tro du c ti on

In in du st rial life t est in g an d m edical follow - up stu dies , cen sorin g is com m on b ecau se of t im e lim it s an d oth er re st rict ion s on dat a collection .

W e con sider th e lin ear r eg r es sion m odel

Yi = 0 + 1X i + i ( i = 1 , , n )

w h er e t h e r espon s es Yi ar e right cen s or ed , an d th e err or s i ar e in dep en den tly an d iden t ically dist ribu t ed (i.i.d .) r an dom v ariable s w ith m ean zer o an d finit e v arian ce.

1. Pr ofes s or , Departm ent of St at istics , Kyungpook National Univ er sit y , T aegu , 702- 701, Kor ea

E - m ail : khcho@knu .ac.kr

2. Depar tm ent of St atist ics , Kyungpook Nation al Univ er sity , T aegu , 702- 701, Kor ea

If Y_i is cen s or ed at t im e C_i, t h en th e cen sor ed r esp on se Z_i can b e w rit t en by

Z_i = Y_{i i} + C_i( 1 - _i) = m in ( Y_i, C_i) ( i = 1 , , n ) , w h er e C_i , , C_n ar e cen sorin g v alu es an d _i ar e t h e in dicat or v ariab les

i =

{

M iller (1976) con sider ed th e m eth od s of e st im at in g t h e r eg r es sion coefficient s of a lin ear r eg r es sion m odel u n der t y pe I cen sor ed dat a . Bu ckley an d J am es (1979 ) su g g est ed t h e m eth od t o a ccom m odat e cen s or ed ob serv at ion s t o estim at e r egr es sion coefficient s .

Un der un cen s or ed dat a , th e lea st squ ar es e st im at or s of ₀ an d ₁ ar e t h e v alu es a an d b w h ich s at isfy t h e equ ation s

n

i = 1( Y_i - a - b X _i ) = 0

an d

n

i = 1( X _i - X )( Y_i - b X _i ) = 0 .

T o accom m odat e cen s or ed ob s erv at ion s , let

Y^*_i = Y_{i i} + E ( Y_i Y_i > C_i) ( 1 - _i) i = 1 , , n .

T h en t h e est im at or of ₁ by Bu ckley an d J am es m et h od is t h e v alu e b w h ich s at isfies t h e equ ation

n

i = 1( X _i - X ) ( Y^*_i - b X _i) = 0 . S in ce E ( Y_i Y_i > C_i) is

u nk n ow n , w e adopt a self- con sist en cy appr oach in Bu ckley an d J am es (1979 ).

Rit ov (1990) an d Currie (1996 ) inv estig at ed t h e a sy m pt ot ic pr opert ies of t h e Bu ck ley - J am es estim at or .

T h e b oot st r ap m eth od h a s b een in tr odu ced by Efron (1979 ) w h o g av e a s eries of ex am ples t o illu st r at e t h e v alidit y of t h e appr oach for a w ide cla s s of st atist ics . T h e accu r acy of t h e b oot st rap appr ox im ation t o t h e sam plin g dist rib ut ion of v ariou s st atist ics h a s b een in v est ig at ed by Bick el an d F r eedm an (1981). Efr on (1981 an d 1987 ) h a s dev elop ed m et h ods for con st ru ct in g appr ox im at e confiden ce in t erv als for a par am et er . T h es e int erv als ar e con stru ct ed from t h e b oot st r ap dist ribu tion of an est im at or .

In t his paper , w e con sider t h e con stru ction of b oot st rap con fiden ce in t erv als for t h e regr es sion coefficient s by t h e Bu ckley - J am es m eth od . A n d w e com par e t h e cov er ag e pr ob abilities an d th e ex pect ed confiden ce in t erv al len gt h s for t h e se

con fiden ce int erv als t hr ou gh M ont e Carlo sim u lation .

2 . B o ot s trap Con fide n c e In t erv al s

T h e b oot st rap m eth od is a u sefu l t ool for con st ru ct in g con fiden ce int erv al or b ein g applicable in situ at ion s w h er e ex plicit ly an aly tical solu tion can n ot b e fou n d or t h e e st im at or is m ath em atically com plicat ed . A ls o, t h e confiden ce in t erv als b y b oot str ap m et h od do n ot r equir e th eor et ical calcu lat ion .

Let X = ( X _{1 ,}X _{2 ,} , X _n) b e th e r an dom s am ple fr om an u nkn ow n dist ribu t ion F , an d ( X _{1 ,}X _{2 ,} , X _n) b e an estim at or of . T h e b oot str ap m eth od is ex t r em ely sim ple in prin ciple.

S t ep 1. Con st ru ct t h e sam ple prob abilit y dist ribu tion F .

S t ep 2. W it h F fix ed, dra w a r an dom sam ple of size n fr om F, say X ^*_i F , i = 1 , 2 , , n .

Call t his "t h e b oot st r ap s am ple ",

X ^* = ( X ^*1, X ^*2, , X ^*n) .

S t ep 3. A ppr ox im at e th e sam plin g dist ribu tion of by t h e b oot str ap dist ribu tion of

* ( X ^*₁ , X ^*₂, , X ^*_n) .

W e con sider Bu ck ley - J am es estim at or for a lin ear r eg r es sion m odel un der cen s or ed dat a by b oot st r ap r esam plin g m eth od .

T h e r esidu al, , is

= Y - X ( ₁ , ₂ , , _n ).

T h en th e cent er ed r esidu als , _i , ar e g iv en by

i = _i - 1

n

n

j = 1 j , i = 1 , 2 , , n.

Let F_n b e t h e em pirical distribut ion of _i. T h at is , F_n put s m a s s 1/ n at _i an d x d F_n = 0 . Giv en Y , let ^*₁, , ^*_n b e con dit ion ally in depen dent r an dom v ariable s w it h com m on dist ribu t ion F_n. A n d let ^* b e t h e n - dim en sion al v ect or w h ose i- th com pon ent is ^*_i. W e put

Y^* = X + ^*.

In form ally , ^* is obt ain ed by re sam plin g t h e cen t er ed r esidu als . N ow , w e con st ru ct

b oot str ap s am ple ( Z^*i , ^*i , X i) , w h er e

Z^*_i = m in ( Y^*_i , C^*_i) an d

*

i =

{

T h en t h e est im at or s of r eg r es sion coefficien t s b y Bu ckley - J am es m et h od ar e g iv en by

1

* = { ^{u Y*}i ( X _i- X ) + ^c Y_i^*( ₁^*) ( X _i- X )}^/ {i = 1ⁿ ( X _i- X )² }

an d

0

* = n^{- 1}{ ^{u Y*}i + ^c Y_i^*( ₁^*)}^- 1

* X ,

w h er e ^u an d ^c den ot e su m m at ion s ov er t h e un cen sor ed an d t h e cen s or ed v alu es only , r espect iv ely .

W e con st ru ct t h e b oot st r ap con fiden ce in t erv als of t h e r egr es sion coefficien t s u n der cen sored dat a a s t h e follow in g s ;

S t ep 1. Gen er at e b oot st rap sam ples ( Z^{* b}i , ^{* b}i , X ^bi ) , i = 1 , , n , b = 1 , , B . S t ep 2. Calculat e th e b oot st rap est im at or s ^{* b} ( 0

* b, 1

* b) corr esp on din g t o each b oot str ap sam ples , ( Z^{* b}_i , ^{* b}_i , X ^b_i ) , i = 1 , , n ,

b = 1 , , B .

S t ep 3 . Con st ru ct t h e con fid en ce in t er v als of r eg r e s sion coefficien t s b y t h r ee b oot st r ap m et h od s t h at ar e p er cen t ile , b ia s corr ect ed (B C ) p er cen t ile , an d b ia s cor r ect ed an d a cceler at ed (B Ca ) p er cen t ile .

P er cent ile con fiden ce int erv al is con st ru ct ed by u sin g th e em pirical distribut ion fu n ct ion of b oot st r appin g estim ation s . Let G b e th e em pirical dist ribu tion fu n ction of t h e b oot str ap dist ribu tion of ^*. If t h e b oot str ap dist ribu tion can b e obt ain ed by M ont e Carlo m et h od , th en G is est im at ed from t h e b oot st r ap replicat ion s

* 1, , ^{* B} a s

G ( t) = 1 B

B

b = 1I ( ^{* b} t )

w h er e t is a r eal v alu e an d B is a r eplication nu m b er of b oot str ap . Let G^{- 1} ( ) b e t h e 100 per cen tile of ^*,

G^{- 1}( ) = {t : G ( t ) }.

T h er efor e, G^{- 1}( ) is th e B t h v alu e in t h e or dered list of ^{* b} an d G^{- 1}( 1 - ) is t h e B ( 1 - ) t h v alu e in t h e or der ed list of ^{* b}. T h en , 100 ( 1 - ) % p er cent ile con fiden ce int erv al of regr es sion coefficient is

( ^{G- 1(} 2 ) , G^{- 1}( 1 -

2 ))^.

Bia s corr ect ed confiden ce int erv al is t h e m et h od t o con stru ct int erv al by corr ect in g t h e bia s of t h e b oot st r ap distribut ion . Bia s corr ect ion is ob t ain ed by cent erin g t h e st an dar d n orm al cu m ulat iv e distribut ion fu n ct ion (cdf). T h e bia s corr ect ion z0 is giv en by

z0 = ^{- 1}(G ( )) = ^{- 1}[B¹

B

b = 1I( ^{* b} )]

w h er e ^{- 1} is t h e inv er s e fu n ction of th e st an dar d n orm al cdf. T h en 100 ( 1 - ) % b ia s corr ect ed p er cen tile confiden ce int erv al is

( ^{G- 1(} 1) , G^{- 1}( ₂))

w h er e ₁ = (^{2 z}0 + z _{/ 2}) ^{an d} 2 = (^{2 z}0 + z_{1 - / 2})^.

If G ( ) = 0 . 5 , t h at is , if h alf of th e b oot st r ap distribut ion of ^* is les s t h an th e ob serv ed v alu e , t h en z0= 0 an d t h e p er cen tile m et h od an d t h e B C m et h od a gr ee.

Bia s corr ect ed an d a cceler at ed (BCa ) m et h od h a s m ore sat isfa ct ory cov er ag e pr ob ab ilit y th an percent ile m et h od . W e com pu t e t h e bia s corr ect ion z₀ in BC

m et h od an d acceler at ion a giv en b y

a =

n

i = 1( ( ) - ( i ))³

6 {^{i = 1n ( ( ) - ( i ))2}}^{2 / 3}

w h er e

( ) = 1

n

n

i = 1 ( i ) ,

an d ( i) is t h e est im at or defin ded on th e sub s am ple w h ich arises w h en t h e i

t h ob serv ation h a s b een delet ed . T h en 100 ( 1 - ) % B Ca con fiden ce in t erv al of r eg r es sion coefficien t is

( ^{G- 1( 3) , G- 1( 4)} )

w h er e

3=

(

)

an d

4=

(

)

If z0 an d a ar e zer o t h en 3 = z / 2 an d 4 = z1 - / 2. T hu s t h e percent ile m et h od an d th e BCa m eth od agr ee .

3 . E x am ple an d S im u l atio n S tu dy

A s t h e dat a for th e ex am ple, w e u s e S t anford h eart t ran splan t dat a w hich w a s g iv en by M iller (1976). T h e dat e for t h e st art of t h e tr an splan t at ion pr ogr am at S t an for d w a s on Oct ob er 1 in 1967, an d t h e cut - off dat e for th is an aly sis w a s on A pril 1 in 1974. Durin g t his t im e in t erv al, 69 pat ien t s r eceiv ed h eart t r an splant s , an d t h e len g t h s of th eir surv iv al, in day s , aft er tr an splan t at ion ar e con sider ed a s r espon se v ariable s . A n d ag es of patient s at t ran splan t ar e con sider ed a s an ex plan at ory v ariable. In accor dan ce w it h Bu ckley - J am es m et h od w e t ak e t h e r espon se v ariables t o log surv iv al tim es t o t h e b a s e 10.

N ow , w e com pu t e th e b oot str ap regr es sion coefficient s for h eart tr an splan t dat a

t hr ou g h t h e 1,000 b oot st r ap r eplication s , an d con st ru ct confiden ce in t erv als of th em b y t h e p er cen tile, BC an d B Ca m et h od s . T h e re sult s ar e list ed in T able 3.1.

T ab l e 3 .1 Reg re s sion coefficien t s an d con fiden ce int erv als

Regr es sion coefficient s ₀ = 3.5962 ₁ = - 0.0278

P er cen tile int erv al ( 1.1575 , 4.9726 ) ( - 0.0761 , - 0.0009 ) BC in t erv al ( 2.1690 , 5.6167 ) ( - 0.0635 , - 0.0025 ) B Ca int erv al ( 2.1032 , 5.2167 ) ( - 0.0635 , - 0.0025 )

A lso, w e inv e st ig at e t h e b oot str ap e st im at es an d t h e b oot st r ap confiden ce int erv als th rou g h th e M ont e Carlo sim u lat ion . W e a s su m e t h at a s a lifetim e dist ribu tion , W eibull dist ribu t ion (W eib ) w it h p ar am et er s = 1 . 0 an d = 0 . 8 (decr ea sin g failu r e r at e ), = 1 . 0 (con st an t failur e r at e ) an d = 1 . 5 (in cr ea sin g failu r e r at e ) ar e con sider ed . A n d a s cen sorin g dist rib ut ion , ex pon en t ial (Ex p ) dist ribu tion w ith par am et er s w hich t h e cen sorin g r at es ar e appr ox im at ely 10 % an d 30 % , r esp ect iv ely , ar e con sidered. T h e M on t e Carlo sim ulat ion s ar e p erform ed for all com bin at ion s of lifet im e distribut ion s an d cen sorin g dist ribu tion s w it h sam ple sizes 40 an d 60. T h e sim ulat ion ex perim ent is p erform ed 1,000 r eplicat ion s an d 1,000 b oot st r ap r eplicat ion s for each ca se. T h e crit eria u sed t o com par e th e con fiden ce int erv als ar e cov er ag e pr ob ability an d th e ex pect ed len g th of con fiden ce int erv al. W e u se 0.90 a s a n om in al pr ob abilit y . T h e r esu lt s of th ese sim u lation s ar e g iv en in T able 3.2. W e do n ot list th e r esu lt s for s am ple size 60 b ecau s e of sim ilarity a s s am ple size 40. F r om t his t able , w e can ob serv e t h e follow in g fact s ; (1) T h e cov er ag e pr ob ab ilit ies of b oot str ap con fiden ce int erv als ar e n early n om in al cov er ag e 0.90.

(2) T h e cov er ag e pr ob ab ilit ies of all th e con fiden ce int erv als decr ea s e a s cen sorin g r at e in crea ses .

(3 ) T h e cov er ag e pr ob abilities of all th e con fiden ce in t erv als in cr ea se a s n in crea ses .

(4 ) BCa m et h od am on g t h e b oot str ap con fiden ce in t erv als h a s t h e sh ort est ex pect ed con fiden ce len gt h .

T h er efor e, w e can kn ow t h at th e b oot str ap m et h od is v ery effect iv e for confiden ce int erv als of r eg re s sion coefficien t s u n der t h e cen sored dat a .

T able 3 .2 Boot strap confiden ce int erv als

M et h od s

Cen s orin g

r at e 10% 30%

0 1 0 1

W eib (1.0, 0.8 )

P ercent ile Len g th 16.941 0.358 9.414 0.393

Cov er ag e 0.883 0.887 0.869 0.883

BC Len g th 13.081 0.355 10.180 0.362

Cov er ag e 0.891 0.897 0.880 0.894

BCa Len g th 11.197 0.337 9.341 0.351

Cov er ag e 0.900 0.901 0.900 0.906

M et h ods

Cen s orin g

r at e 10% 30%

0 1 0 1

W eib (1.0, 1.0)

P er cent ile Len g th 15.974 0.351 9.638 0.217

Cov er ag e 0.879 0.888 0.873 0.887

B C Len g th 15.537 0.395 12.417 0.228

Cov er ag e 0.899 0.897 0.897 0.895

BCa Len g th 10.192 0.365 9.340 0.161

Cov er ag e 0.901 0.901 0.899 0.902

M et h ods

Cen s orin g

r at e 10% 30%

0 1 0 1

W eib (1.0, 1.5 )

P er cent ile Len g th 15.968 0.351 11.794 0.335

Cov er ag e 0.891 0.894 0.880 0.890

B C Len g th 13.614 0.338 14.301 0.341

Cov er ag e 0.894 0.895 0.892 0.897

BCa Len g th 13.964 0.337 10.162 0.306

Cov er ag e 0.895 0.901 0.896 0.902

R e f ere n c e s

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2. Bu ckley , J . an d J am es , I. R . (1979). Lin ear r egr es sion w it h cen sor ed dat a . B iom e tr ica, 66, 429 - 436.

3. Currie, I. D . (1996 ). A n ot e on Bu ckley - J am es estim at or s for cen sor ed dat a . B iom e tr ick a, 83, 912- 915

4. Efron , B . (1979 ). Boot st r ap m et h ods ; A n oth er look at th e j ackk nife. A nnals of S ta tis tics , 7, 1- 26.

5. Efron , B . (1981). N onp ar am etric st an dar d error s an d confiden ce in t erv al.

Canad ian J ournal of S ta t is t ics, 9, 139 - 172.

6. Efron , B . (1987 ). Bet t er b oot str ap con fiden ce int erv al, J ournal of th e A m er ican S ta t is tical A s s ocia t ion, 82, 171- 185.

7. F r eedm an , D . (1981). Boot st r appin g r egr es sion m odel. A nnals of S ta t is tics , 9, 1218 - 1228.

8. M iller , R . G. (1976 ), Lea st s qu ar es r eg r es sion w it h cen sor ed dat a , B iom e tr ika, 63, 449 - 464.

9. Rit ov , Y . (1990), E st im at ion in a lin ear r egr es sion m odel w it h cen sor ed dat a . A nnals of S ta t is tics , 18, 303- 328.

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