Variance estimation after imputation

Retrospective Theses and Dissertations

Iowa State University Capstones, Theses and

Dissertations

2000

Variance estimation after imputation

Jae-Kwang Kim

Iowa State University

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Kim, Jae-Kwang, "Variance estimation after imputation " (2000). Retrospective Theses and Dissertations. 12693. https://lib.dr.iastate.edu/rtd/12693

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Variance estimation after imputation

In-. l a t ' - K w a n g K i m

A d i s s e r t a t i o n s u l ) i n i t t f d t o l l i e g r a d u a t e f a c u l t y in [)artial fullillni<Mit o f t h e re<iuinMn«'nts for t h e d e g r e e o f

D O C T O I i o r I M l l I . O S O l M h ' M a j o r : S t a t i s t i c s M a j o r P r o f e s s o r : W a y n e A. I'uller Iowa S t a t e I ' n i v e r s i t y A m e s . Iowa

2000

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I I ( I r a d u a t r ('oll«'in« Iowa S t a t f r i i i w r s i t y T h i s is t o ctTtify t l i a t t h e Doctoral d i s s e r t a t i o n of J a e - K w a n g K i m h a s m e t t h e d i s s e r t a t i o n r e c | u i r e n i e n t s of Iowa S t a l e I ' l i i v e r s i t y N l a j o r Professor For t h e M a j o r P r o g r a m o r t h e G ' r t e t ' o l l e g e

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TABLE OF CONTENTS

GENERAL INTRODUCTION

I

1 B a s i c Probl«'in I

2 K s t i i n a t i o i i ill I In-pr(>st>iis('uf iiouri's|jons»' 1

;{ l)iss«'rlatioii O r g a n i z a t i o n l i t ' f c n ' i i a ' s 1

LITERATURE REVIEW

(I

1 P f t ' l i i n i i i a r i c s (i •j R cp li cH t i o n V'ariaiHc K.STi n i a t i o n W i t h o u t Noiircspoiisc I I 2.1 I lit-jackkiiiff nu'tliotl 1-J

2 . 2 Halanrc'd ri'p«'at»'(l rcplicalioii I I

•J Hot D f c k I m p u t a t i o n .Mcthoils I')

I Ti ll ' . M u l t i p l e I m p u t a t i o n .Xpiiroacli I S 1.1 l i a y c s i a n .Ju.stiHcation "211 1.2 R a i u l o m i z a t i o n Validity 2 2 !.;{ F a y ' s K.xamplo 2 1 ') Q u a s i - R a n d o m i z a t i o n .Approach 2 7 G O t h o r a p p r o a c h e s 2 9 ti.l K a l t o n a n d K i s h ' s a p p r o a c h 2 9 6 . 2 S a r n d a l ' s a p p r o a c h ^{0 6.;5 Tollefson a n d Fullor's a p p r o a c h .12

I V

(i.-l F a y ' s a p p r o a c h

licft'ieiiccs 3 1

VARIANCE ESIMATION AFTER IMPUTATION

3 7

A b s t r a c t 3 7 I I n t r o d u c t i o n 3 S J A X a i l r t i i i r [•.srnMcilKJii Mi'tiiuil -5'.) 3 Kxtc'iisioiis T o U a i i d o m I i i i p u t a t i o n 11 1 .Jaci\knif<' Mctluxl 11 ('oiu|)U' X S u r v e y Dcsiffiis IN ' ) . l D c t c r i n i i i i s t i c i m p u t a t i o n IN K a n d o m i i n p u t a l i t > n 'y'l (i C o n c l u d i n g K c n u u k s "i l Ackno\vl('il}!,in(Mits K c f c r c n c c s "i") A p p e n d i x A •')() A p p c ' n d i x 11 (il

INFERENCE PROCEDURES FOR HOT DECK IMPUTATION

. . (i3

A l ) s t r a c t (i3 1 I n t r o d u c t i o n (il 2 M o d e l s for I n i p u t a t i o n (i') 2 . 1 N o t a t i o n ()o 2 . 2 P o p u l a t i o n .Model .Approach 6 8 2 . 3 R e s p o n s e .Model . A p p r o a c h (>9 2.-1 Hot dcck imputation 70 3 E s t i m a t i o n .After H o i D e c k I m p u t a t i o n : P o p u l a t i o n M o d e l .Approach . . 72

\ ' 5 \ ' a r i ; u u ( ' K s t i i n a t i o i i 8 8 (i S i i m i l a t i o i i S t u d i e s 5)(i (i.l KxperiiiUMit O n t ' 5)0 G.'J Kx|j«'rinuMil two 10;{ A p p e i u l i x lUN Urferriu'c's ! l-'5

REPLICATION VARIANCE ESTIMATION FOR MULTI-PHASE

STRATIFIED SAMPLING

12:]

A l j s t r a c t 1 2 3

1 l i i t r u d i K t i o i i 1 2 1

•J A s y n i p t o t i c I'lDpt'itics I'Jt)

•J A [{('plication M c l l i o d for t h e Kt'\v«'if»lit«-(1 Kxpaiisioii K s t i i n a t o r I'Ui

1 U t ' p l i i a t i o i i \'ariaiic(' Kstiiiiatioii I'or t h e D o u b l e Mxpansiuu lOstiinator . . I l l

o U e p l i c a l i o n V a r i a n c e l^stiiiiator for M u l t i - I ' l i a s e S a m p l i n g I I')

(i A p p l i c a t i o n t o tli«'JOUO I S ( ' e n s u s 1 1 9 (i.l i n t r o f l u c t i o n 11!) (i.2 P o i n t Kstiinatioii l ' ) 0 G.3 X'ariance Kstiination 1 A p p e n t l i x 1 •')•') A . N o t a t i o n l o o B . P r o p e r t i e s of t h e proposetl v a r i a n c e «' s t i i n at or 1 5 6 C . D e r i v a t i o n o f t w o p h a s e v a r i a n c e w h e n t h e s e c o n d p h a s e is s t r a t i f i e d poisson s a m p l i n g l o 9 R e f e r e n c e s 1()8

GENERAL CONCLUSIONS

170

R e f e r e n c e s 171

V l l Taljlc I ThIJU' 2 T a b l e T a b l e I l a b l e •') T a b l e (i Table 7 T a b l e 8 T a b l e y

LIST OF TABLES

i s t r a l i o n of t h e p s e i u l o <lata s«'t for U a o a n d S l i a o inetlioil 17 I l l u s t r a t i o n o f t h e p r o f j o s e d pseiulo d a t a s e t 18

M»'aii. v a r i a i i e e . a n d s t a i u l a r d i z e d v a r i a n c e of t h e point <'sliina-t o r s u n d e r <'sliina-t h e f o u r clilfer<'sliina-tMi<'sliina-t i m p u <'sliina-t a <'sliina-t i o n s c l u ' i n e s in exp«'riineii<'sliina-t

o n e (lO.UOO s a m p l e s ) IKi

Nb'an. relativt* bia.s. v a r i a n c e , s l a n d a r d i z e t l varianc<' o f l l u ' vari­

a n c e e s t i m a t o r in e x p e r i m e n t o n e ( lU.UOU sampU-s ) 117 S t a n t l a r d i z e t l n K ' a n C . l . w i d t h , s l a n d a r d i z e i l v a r i a n c e o f ( ' . I . w i d t h . a n d c o v e r a g e in e x p e r i m e n t o u e (lU.OOU s a m p l e s ) 118 Pojjiilation p a r a m e t e r s f o r s i m u l a t i o n in e x p e r i m e n t t w o 118 .Mean, v a r i a n c e , a n d s l a i u l a r d i z e d v a r i a n c e o f t l i e p o i n t e s t i m a ­ t o r s u n d e r different i m p u t a t i o n s c h e m e s i n experiiiK'nt t w o ( 1 0 . 0 0 0 s a m p l e s o f s i z e 100) 119 •Mean, r e l a t i v e b i a s , v a r i a n c e , s t a n d a n l i z e d v a r i a n c e o f t h e vari­ a n c e e s t i m a t o r in e x p e r i m e n t t w o (10.000 s a m p l e s ) 120 S t a n d a r d i z e d m e a n C M . w i d t h , s t a n d a r d i z e d v a r i a n c e ( M . w i d t h , a n d c o v e r a g e in e x p e r i m e n t t w o (10.000 s a m | ) l e s ) 121

Vlll

Tahlc lU M e a n a n d variaiici' o f l l i c point o s t i m a t o r of 0.x uiuliT t h e t h n v

difftMcnl i m p u t a t i o n schciiu's in ( > x i ) c > r i i i R'iit t w o wlu'ii Z i s a l w a y s

o b s o r v t ' d 121

Table 11 M e a n a n d v a r i a n c e o f t h e v a r i a n c e e s t i m a t o r , t h e m e a n l e n g t h o f 9.j/c ("I (("I w i d t h ) . I h e c o v e r a R c ' o f *)') Vf C I . ;iiid t h e r e l a t i v e b i a s o f t h e variance''<fi!!!al<>r in <'Nperi!n<'tit t w o wlwn / alwMy;

I

GENERAL INTRODUCTION

1

Basic Problem

III m a n y s a i i i p U ' s u r v e y s , s o i i i r u l t h e u n i t s c u n t a r t i ' d <lu n o t r e s p o n d . O t h e r u n i t s

m a y r e s p o n d t o s o m e h u t n o t a l l < | i u ' s t i o n s heiii}^ a s k e t l . 1 In- p r o h l e m o f inissiinj; d a t a i n

s u r v e y s a i n i ) r m j i ; i s c a l l e t l t h e | ) r o l ) l e m o f i i o i i n .

T h e p r o b l e m s c r e a t e i l by iionr»'spoiise a r e well e x i d a i i u ' d by U u b i n (1!)S7):

r i n ' s e inissiiiff valiH's i n t e n d e d by survey d e s i g n t o l)e o b s e r v e d not o n l y m e a n less e[lit i«Mit e s t i m a t e ' s i i e e a u s e of t h e r«'dut e d siz«' o f d a t a b a s e b u t a l s o t hat s t a n d a r f l c o m i j h ' t e - d a t a m e t h o d s l annot b«' imiiH'diately u s e d t o analyz«' t h e d a t a . .Moreover, p o s s i b l e biases <'xist b e c a u s e t h e r e s p o n d i ' i i t s a r e o f t e n s y s t e m a t i c a l l y d i l f e r e n t f r o m t h e noiirespoiuU'nts: of p a r t i c u l a r conciMii. t h e s e biases a r e difficult t o e l i m i n a t e s i n c e t h e precise rea.sons for iu)nr<'sponse ar«' u s u a l l y n o t k n o w n . ( p . 1 )

It is c o m m t j i i p r a c t i c e t o d i s t i n g u i s h between u n i t . wluMi s o n u ' o f t h e

u n i t s c o n t a c t e d d o n o t r e s p o n d b e c a u s e of iiot-at-liomes. refusals, i n a b i l i t y t o p a r t i c i p a t e , a n d untracecl u n i t s , a n d i l t i n r t o i i n s p o i i M . when s o m e b u t n o t a l l o f t h e responses a r e a v a i l a b l e . I t e m n o n r e s p o n s e a r i s e s b e c a u s e of i t e m r e f u s a l s , " d o i r t k n o w s ", o m i ss i o n a n d a n s w e r s d e l e t e d in e d i t i n g . I h e p r o b l e m of v a r i a n c e e s t i m a t i o n in t h e pr e s en c e of i t e m n o n r e s p o n s e will b e a d d r e s s e d i n t h i s work.

2

2 Estimation in the presense of nonresponse

r i i o l i t e r a t u r e o n t h e e s t i m a t i o n i ) r o b l e m in th<' p r e s e n c e of n o n r e s p o n s e is c o m p a r ­ a t i v e l y r c v e n t ; r e v i e w papers inc l u d e O h a n d Sc h e u r e n ( IDS.'J). K o t t ( I ! ) 9 1 ) . a n d Brick ancl K a l t o n ( 1!)9()). M e t h o d s p r o p o s e d i n t h i s l i t e r a t u r e c a n h e roughly grcjupecl i n t o t h e following c a t e g o r i e s (not nuitiially e x c l u s i v e ) :

( i ) I' v o n d u n s l i a s t d o n ( ' o i n p h ti Itj H t r o r d i d I

W h e n s o m e variables a r e not r e c o r d e d for s o m e cjf t h e u n i t s , a simi>le e x p e d i e n t is t o cliscard i n c o m p l e t e l y recorcled u n i t s ancl t o a n a l \ x e o n l y t h e u n i t s w i t h c o m p l e t e d a t a . D e l e t i o n g o f units is e a s y t o c a r r y cnit a n d m a y h e satisfactcjry w i t h snutll a m c n i n t s o f missing d a t a . B i a s will o c c u r in t h e pcjint e s t i m a t e a s well a s in t h e v a r i a n c e - c o v a r i a n c c e s t i m a t e , u n l e s s t l i e p o p u l a t i o n m e a n for r e s p o n d e n t s is ecpial

t o t h a t cjf nonresponclcnts ( s e e . e . g . K a l t o n |) . 7). Also, t h e fracticju of

clis-carclecl u n i t s will h e iion-negligil>le w h e n t h e n u m b e r of i t e m s in t h e c|uestiomiaire is l a r g e .

( i i ) \ \ ( Kjhling A d j u s t m i nt

In \ \ t i y h t i n y ndju.'itiin iil for t h e n o n r e s p o n s e p r o b l e m , t h e w e i g i i t s o f s p e c i h e d r e s p o n d e n t s a r e increased s o t h a t t h e y r e p r e s e n t t h e n o n r e s p o n c l e n t s . W e i g h t i n g a d j u s t m e n t is primarily used t o c o m | ) e n s a t e f o r u n i t n o n r e s p o n s e . 1 h e m a i n o b ­ j e c t i v e of t h e weighting a d j u s t m e n t i s t o r e d u c e b i a s in s u r v e y e s t i m a t e s b y m a k i n g eacii r e s p o n d e n t represent t h e c o r r e c t f r a c t i o n o f t h e t a r g e t p o p u l a t i o n .

(iii) I m p u l a t i o n P m n d u r r s

I m p u t a t i o n m e a n s inserting v a l u e s f o r m i s s i n g i t e m s . I m p u t a t i o n is useful in deal­

( h ) I ' s e s t l i c s a i n t ' s u r v e y weiglits f o r a l l iltMiis. u n l i k e s e p a r a t e w e i g h t i n g a d j u s t ­ m e n t for e a c h i t e m .

( b ) l i e t a i n s a l l t h e r e p o r t e d d a t a f o r u s e i n m u l t i v a r i a t e a n a l y s i s , u n l i k e t h e coin-[Dlete c a s e a p p r o a c h .

1 h e r e a r e s e v e r a l i m p u t a t i o n m e t h o c l s u s e d i n p r a c t i c e . Hot d e c k im|JUtatioti is t h e i m p u t a t i o n p r o c e d u r e in w h i c h t h e v a l u e a s s i g n e d for a m i s s i n g i t e m is tak<'n from r e s p o n d e n t s in t h e c u r r e n t s a m p l e . .Many o f the- htjl d e c k i m p u t a t i o n p r o c e d u r e s s t a r t w i t h a d i v i s i o n of t h e s a m p l e i n t o c e l l s l)asetl «;n au.xiliary v a r i a b l e s k i u n v n for b o t h t h e n ' s p o n d e n t s a n d n o n n ' s p o n d e n t s . [ h e cell is c a l l e d t h e iinpul(tlii»n n i l . O n e of tli<' c o m m c j i i l y u s e d hcjt d e c k i m p u t a t i o n m e t h o d s is .•iiiiipti rttinloiii h o t dtrfc n n p n t n h o n . w h e r e n o n r e s p o n d e n t s a r e assigne(l v a l u e s f r o m r e s p o n d e n t s in t h e s a n i e i m p u t a t i o n cell w i t h eciual p r o b a b i l i t i e s o f s e l e c t i o n . I n s p i t e o f i t s c o n v e n i e n c e , t r t ' a t i n g t h e i m p u t e d values a s if t h e y a r e t r u e values a n d m a k i n g i n f e r e n c e u s i n g s t a n d a r d f o r m u l a s s h o u l d b e u.secl w i t h c a u t i o n . The s t a n ­ d a r d v a r i a n c e e s t i m a t o r s , in p a r t i c u l a r , l e a d t o u n d e r e s t i m a t i o n b e c a u s e t h e a d d i t i o n a l v a r i a b i l i t y d u e t o m i s s i n g v a l u e s a n d i m p u t a t i o n is n o t t a k e n i n t o a c c o u n t .

W e will r e v i e w e.xisting m e t l u n l s o f v a r i a n c e e s t i m a t i o n for i m p u t e d d a t a a n d suggest a l t e r n a t i v e m o t h o d s .

3 Dissertation Organization

T h e d i s s e r t a t i o n c o n s i s t s o f t h r e e rc'search p a p e r s . T h e d i s s e r t a t i o n i s o r g a n i z e d a s follows: • C h a p t e r 2 E x i s t i n g m e t h o d s o f v a r i a n c e e s t i m a t i o n a f t e r i m p u t a t i o n , i n c l u d i n g t h e a p p r o a c h o f R u b i n ( 1 9 S 7 ) a n d t h e a p p r o a c h o f R a o ( 1 9 9 6 ) a r e r e v i e w e d . T h e p r o s a n d cons

1

of t h e e x i s t i n g niethocls a r e discussed.

• C l i a p t e r

A v a r i a n c e e s t i m a t i o n m e t l i o d based o n s i n g l e i m p u t a t i o n is p r o p o s e d i n t h i s p a p e r . I his is b a s i c a l l y a n e.xtension of t h e a j i p r o a c h o f Kao( UMKi). 1 h e propos«'d m e t h o d is usefid f o r e s t i m a t i n g t h e variance of a s m o o t h f u n c t i o n of l i n e a r e s t i m a t o r s .

• C h a p t e r t

In t h i s p a p e r , a v a r i a n c e e s t i m a t i o n m<'tho«l b a s e d o n t w o o r m o r e i m p u t e d values for i ' a c h m i s s i n g i t e m is p r o p o s e d . In p a r t i c u l a r , a iirocedur** calleil fully eflicient f r a c t i o n a l i m p u t a t i o n is |iro|)osed a n d v a r i a n c e e s t i m a t i o n l o r t h e jirocedure is p r e s e n t e d . • C h a i J t e r j In t h i s p a p e r , a v a r i a n c e e s t i m a t i o n m e t h o d f o r mulli-pha.se s a m p l i n g is preseiitetl a n d a p p l i e d t o t h e 2 0 0 0 I S C e n s u s of I ' o p u l a t i t j i i . • C h a p t e r (i C o n c l u s i o n s a r e m a d e .

References

B r i c k . .J. .\I. a n d K a l t o n . ( I . (199(i) " H a n d l i n g m i s s i n g d a t a in s u r v e y r«'search." S l a t i t

-/ical M f t h o d a i n S h d i c a l R i s c a r c h , 21.>238.

K a l t o n . CI. (198.'{) ('otnpfn.-^atiiiy f o r .•<iirct y d a t a . I n s t i t u t e o f S o c i a l Research.

K o t t . P . S . ( 1 9 9 4 ) . n o t e o n h a n d l i n g n o n r e s p o n s e i n s a m p l e s u r v e y s . " J o u r n a l of t i n

• )

O h . II. L. a i u l S c l u ' u r o i i . K. J . (IlKS;}). "W'cigliling acljiistinciits for unit n o i i - n ' s p o i i s c . "

In liicnnipli If D a Id ill S u m pit S u r r t ijt<. \'oliinii J, Tlit o n j tiinl l i i b l i o y n i p h i r s . W . ( I .

M a d o w . I. O i k i n . a i u l 1). B. Ruhiii ( r d s . ). .\«'\v N'ork: AcaclcMiiic l ^ c s s . 1 1:{-IS1.

Rao. .). N. K . ( 1*)%). " O n v a r i a i u f c s t i i u a t i o i i with i m p u t e d s u r v e y d a t a . " .Journal of

tin Aiiif rican Statislicul Assnciatiuu. 1)1. 191)-.')l)l).

(i

LITERATURE REVIEW

1 Preliminaries

A | ) o | ) u l a l i o n o f .V i d c n t i l i a i j i c clcintMits is (U-noti-d i)y / = { 1 . "J \ }. A s u b s e t ol l l u ' |)o|Julalioii is s«'U'( t«'<l a n d caiU'd a sani|)lr. 1 In- s e l e c t i o n o f sanii)les uses a >^et of p r o h a h i l i t y n d e s calU-d t i i e s t u n p l i n y iiitcliiini.>iii. Let . 1 d e n o t e t h e set of indices in i h e s a m p l e . Deliiu' t h e s a m p l e s e l e c t i o n indicator f u n c t i o n

l j = < 1 if J € . 1

( i . n

I ) i f j ^ . i

a n d

B = ( / ,

I

k ) . ( l . - J )

Let / H B ) d e n o t e a s a m p l i n g iiiechanisin t h a t a s s i g n s proi)al)ilities. s u m m i n g t o o n e . t o t l i e 2'^ p o s s i b l e

B

v e c t o r s .

.Associated w i t h t h e j - l \ \ e l e m e n t of t h e p o p u l a t i o n is a v«'ctor o f c h a r a c t e r i s t i c s d e n o t e d by a n d t h e p o i j u l a t i o n of vectors is d e n o t e d by

Y = ( y ,

y

, v ) .

( l . ; J )

L e t t h e p o p u l a t i o n ( j u a n t i t y o f i n t e r e s t b e f

(yi

y

.v) find let U b e a n e s t i m a t o r o f 0 \ ba.sed o n t h e s a m p l e . T h e t r a d i t i o n a l s u r v e y s a m p l i n g a p p r o a c h t r e a t s

Y

a s fixed

i

A n e s t i m a t o r 0 is CH

II

CC

I

dtsKju itiibiasid for 0 \ if

/• (tf 1 ^ ) = 0 s . ( l . l ) w l i c r c JT - { y ,

y

.v}. /-' ( O \ a n d Y.b ' I f n o l c s t h e s u i n i i u n a t i o n o v e r a l l possil)l«'

B.

A n u l l i c r l u o i l f of i n f o r i ' n r r assiuncs t h a t l l i c p o p u l a t i u n v c r t o r

Y

is a r a n d o m s a m p l e f r o m a n i n l i n i t f s n p c r i ^ p n l a t i o i i . i ll*' inodcl-l)as<'d aj)j)roai li in s u r v e y s a m p l i n g m a k e s i n f e r e n c e s b a s e d o n tin- c o n d i t i o n a l d i s t r i h u l i o n o f

Y

f^iven t h e s a m p l e o u t c o m e

.1.

N o t e t i i a t t h i s c o n d i t i o n a l d i s t r i b u t i o n is d e t e r m i n e d b y t h e s a m p ^ m ^ m e c h a n i s m a s well a s by th«' d i s t r i b u t i o n o f t h e variable

Y.

I h e d e | ) e n d e n c e o n t h e sampling; m e c h a n i s m c a n b e a v o i d e d if t h e s a m i j l i n g m e c h a n i s m is iijiioniblt. W'e f o r m a l i z e t h e coticept in D e i i n i t i o n

1 . 1 .

Definition 1.1

l . t l l l n (lislnbulion of Y bi <li i i o t t d bij C ( Y ) a n d ctilltti t i n s u i n r p o p il­ l a t i o n i n o d t l . [ a ! /»(B) bf Ihf suuiplintj t m r i t a n i s i n . l i n n . /J(B) lynorabli a n d i r t l u s u p i rpopulatioti riiodd if a n d only if

£ ( Y | . \ ) = Z : ( Y ) .

(l..-j)

w h i n

£ ( Y | . 1 ) /.S

lli( conditional d i s t r i b u l i o n o f \ y i v m t l u .sampit o u t r o i m

. 1 .

I.et

X

= ( X i . - - - . x , v ) b e a vector of v a l u e s f o r a secx}nd v a r i a b l e , w h e r e t h e t r u e v e c t o r

X

is k n o w n f o r t h e population. .A sufficient c o n d i t i o n for t h e i g n o r a b i l i t y o f t h e

sampling mechanism is that it can be described by the conditional independence of

Y

a n d

B

g i v e n

X.

In D a w i d ' s (1979) n o t a t i o n .

Y L B | X

( l . ( i )

m e a n s t h a t t h e v a r i a b l e

Y

is i n d e p e n d e n t o f t h e s a m p l e s e l e c t i o n i n d i c a t o r v a r i a b l e

s

s t r a t i l i e i l r a i u l o i n s a m p l i n g a n d t h e a u x i l i a r y v a r i a b l o X is t h e i n d i c a t o r v c c t o r f o r s t r a t a , t h e n t h f c o n t l i t i o n ( l . O ) h o l d s b o c a u s c . i n t h e s a m e s t r a t u m , t h e p r o b a b i l i t y o f s a n i j i l e s e l e c t i o n is t h e s a m e for all e l e m e n t s . H e n c e , t h e s a m p l i n g m e c h a n i s m is i n d e p e n d e n t of t h e v a l u e o f V i n t l i e s t r a t u m . R u b i n ( 1 9 7 0 ) . S c o t t a n d S m i t h ( 1 9 7 3 ) . a i u l S u g d e n a n d S m i t h ( 1 9 8 1 ) d i s c u s s i g n o r a b i l i t y .

I.et u s a s s u m e t h a t t h e l i n i t e p o p u l a t i o n I i s m a d e u|) of (1 i m p u t a t i o n cells. W i t h i n e a c h ci'll </. cy = 1 (1. t h e »'lemeiits ar<' i d e n t i c a l l y a n d i n d e p e n d e n t l y d i s t r i b u t e d w i t h

m e a n it., a n d v a r i a n c e . i.e.

w h e r e I j d e n o t e s t h e s e t o f i n d i c e s f o r t h e i m i) U t a t i u n c e l l . W e c a l l l l u ' m o d e l ( 1 . 7 )

t h e i i n p u t d t i o i i ct II niotit I.

Lemma 1.1

A s s n n i i roinlilioii ( l . ( ) ) with t i n i t t t-rilinrij ra r ni b h X h f i m j t i n i i u l i r a t o r

r t c l n r f o r i i i i p u l a l i o n cflls nn<l (I s s u h k that

Till II till s a m p l i n g iiucliaiiisni i s iynovablt u n d t r snpt rpopnhition m o d i I ( 1 . 7 ) .

Proof. I

,et b e a n y m i ' a s u r a b l e s e t in t h e s i g m a - f i e l d rT

( V ' )

gj-nerated b y t i i e r a n d o m v a r i a b l e V. I h e n . b y t h e d e h n i t i o n o f c o n d i t i o n a l i n d e i K ' u d e n c e . for i <E I j . U < l ' r ( / , = 1) < I . I . - - . . V . ( l . S ) P r (V; € > ' . / . = ! \ i e l ' j ) = P r ( V ; € >• I i € { ' . j ) P v ( l , = \ \ , e l ' j ) . .Mso. Pr( V ; € .s' I

i e f ' . j J , = i )

P r ( y . € S . l , = 1 I < €

r . j )

P r { I , = \ \ i € r . j ) P r ( V ; e S \ i € u n d e r ( 1 . 8 ) . H e n c e . £ ( v ; | / € r , . / . = i ) = r ( y ; | / € r , ) .

9

S i m i l a r l y .

£ ( V ; I / € I ' , . ! , = 0 ) = £ ( V ; I / € T y ) .

S o . t l i c r e s u l t follows. •

I . c m m a I.I iiii[>lii'>< t tial l l i e ' l i s t r i i m t ioii o f I I n - s j i m p l f i i p a r t is llii' s a m i ' a s t h a t of iioii-samplcd p a r t . Tliat is

v ; | . i

~ / € / ' , ( I . ! ) )

for <'a< li ( c l l <j. <j = I

(livvii iioiircsiHJiist'. t h e original s a m p l e .1 is (Iccoiiiposcd i n t o tin- sj't of rt'sponcU'iits. A n - a n d t h e s»'t of n o n r c s p o n d c n t s . . 1 \ / . l)«'lin«' t h e rcs|)onsc i i u l i c a t o r fimrtioii

1 V, r e s p o n d s if sami)U'<l

/ = 1 V ( l . i U )

0 V, dot's n o t r«'spoiul if s a m p l e d

l i , =

a n d t h e assot iatecl vt-ctor

R = ( / ^ , l i s ) . ( l . l l )

I h e d i s t r i b u t i o n o f

R

is called t h e r es j) ons e m e c h a n i s m . .Vote t h a t t h e r«'siJonse m e c h a n i s m is u s u a l l y u n k n o w n a n d i s sp«'cilied by t h e mo<lel. C o n d i t i o n a l i n f e r e n c e for

Y

g i v e n

R

r e q u i r e s t h e specification o f t h e r e s p o n s e m e c h a n i s m . I g n o r a b i l i t y o f t h e r e s p o n s e m e c h a n i s m i s d e f i n e d in D e f i n i t i o n 1.2

Definition 1.2

L d C { Y \ b t t h t c o n d i t i o n a l d i f t t r i b u t i o n

o/Y

y i r t i i t i n r t a l i z t d sanipli . 1 . a n d t i n r t n l i z t d r t s p o m U t t t .s A f i . T l u n , tht rf.'iponst n i t c h a i i i .'ini (.s iynortibli u n d e r t h t i n o d t l if

£ ( Y | . - L . l f i ) = £ ( Y | . - l ) .

( 1 . 1 2 )

10

R u h i n ( lf)7(). p . ' i S ' J ) lU'fiiictl t l i r n ' s p o i i s e i i u ' c l i H i i i s i n t o b o a m i s s i n g a t r a n t l o m

( M A 11) n u ' c l i a i i i s m if

w h e r e t h e n o t a t i o n (1.13) m e a n s t h a t t h e r e s p o n s e i n d i c a t o r v a r i a b l e

R

is i i u h ' p e n d e n t

uf the study variable

Y.

conditiuna! on the au.xiliary variable

X

and tlie "^ainph'

B.

Lemma 1.2

I.it llu auiiliurij v a n a b h

X

bt llit i m p i i t a t i o n n i l r u r i a b h dijiiitd i n l.( iiniKi 1.1. A s s t i i m thai

a n d t h a t I / k M A H condiHon ( I . I J ) h o l d s , fhtii llu n s p o i i x m t r l i u n t s n i i s Kjuortiblt unilt r lilt t i i o d t l ( l . O j .

Proof.

I l u ' proof is ((uite s i m i l a r t o t h a t of L e m m a "J.l. Let >" b e a n y m e a s u r a b l e s e t i n t h e sigma-li«'ld ^ ( V ) gen«'rated b y t h e r a n d o m v a r i a b l e V. I ' h e n . b y t h e delinitiou o f t h e c o n d i t i o n a i i n d e | ) e n d e n e e . f o r t € I

R ± Y | ( X . B )

( i . i ; n 0 < P r ( / / , = 1) < 1. I ^ A ( 1 . 1 1 ) Pr ^ S . It, = I \ I e r. j . I, = I ) = P r l \ ' , ^ S \ i€ r, j . / , = l ) X | > r ( / f , = 1 I / e r , . / . = 1) a n d b y t h e d e l i n i t i o u of /?, P r ( / f , = I 1 / €

r , J ,

= 1) = = 1 I / €

r , ) .

r i i e r e f o r e . P r ( v ; € . s I i € [',J. I, = \ . H, = I ) P r ( > ; e

s . H , =

11

i e t ' j . I , = \ )

P r ( H . = I I / € / ; . / , = I ) P r ( y ; € . v | t € L ' j . l , = 1) u n d e r ( 1 . 1 4 ) . H e n c e , t h e result follows.

11 ( l i v e n l . t ' i i u u a 1.1 a n d L r n i m a l . J . w e a r c abU- t o s a y l l i a t n i o d f l ( 1 . 7 ) . l o g e t l u T w i t h a n i g n o r a b l e s a m p l i n g i n o r l i a n i s i n a n d a n i g n o r a b l e i v s p o n s c n u ' c i i a n i s i n . p r o d u c i ' o b s e r v a t i o n s i n a n i m p u t a t i o n ccll t h a t a r c d i s t r i b u t e d i d e n t i c a l l y a n d itKle|jendently. r i i a t is. V, \ ( . \ . . \ n ) - ( / ' . . t ; ) . ( 1 . 1 5 )

2 Replication Variance Estimation Without Nonresponse

I n t h i s s e i t i o n . w e c o n s i d e r t h « ' ca.st- w h e n t h e r e i s n o n o n r c s | ) i i n s e i n t h e s a m p l e . I n m a n y s t i u T n ' s . d a t a a r e c o l l e c l e d f r o m i n d i v i d u a l s o r u n i t s s a m p l e < l u s i n g c o m p h ' x s a m p l e d e s i g n s t h a t i n c l i u l e v a r y i n g i ^ r o b a b i l i t i c s a n d n o n - i n d e p e n d e n t s « ' l c c t i o n s . O n « ' a p p r o a c h t o e s t i m a t i n g t h e s t a n < l a r d e r r o r t)f t h e e s t i m a t o r i s t o l i n c a r i z * ' t h e e s t i m a t o r u s i n g a T a y l o r s c r i « ' s e x p a n s i o n a n d t h e n us»' s t a i u l a r d s a m p h * s u r v e y v a r i a n c e • • s t i m a t i o n m e t h o d t o e s t i m a t e t h e p r e c i s i o n o f t h e l i n e a r i / . i ' i l s t a t i s t i c . . \ n a d v a i U a g e o f t h e l i n e a r i z a t i o n m e t l i o t l i s t h a t it i s a p p l i c a b l e t o g e n e r a l s a m [ ) l i n g d e s i g n , b u t a d i s a d v a n t a g e i s t h a t it i n v o l v e s t l u ' i l e r i v a t i o n o f a s e p a r a t e v a r i a n c e e s t i m a t i o n f o r m u l a f o r e a c h s t a t i s t i c . .An a l t e r n a t i v e a p p r o a c h is t o u s e a r e p l i c a t i o n m e t h o d . Two p o p u l a r m e t h o d s i n surv<>y s a m p l i n g a r e t h e j a c k k n i f e a n d b a l a n c e d r e p e a t e d r e p l i c a t i o n ( l i l i U ) . W o l t e r (198')) a n d R u s t a n d R a o (199G) p r o v i d e g o o t l r e v i e w s of t h e r e p l i c a t i o n l i t e r a t u r e a s a p p l i e d t o coinple.N s a m p l e s u r v e y s .

Let 0 b e t h e p o p u l a t i o n p a r a m e t e r o f i n t e r e s t a n d let 0 b e t h e e s t i m a t o r of 0 ba.sed o n t h e full s a m p l e . T o e s t i m a t e s a m p l i n g e r r o r s , s u b s a m p l e s f r o m t h e s a m p l e a r e <lrawn a m i 0 is c o m p u t e t l f r o m e a c h s u b s a m p l e . Dilferent w a y s o f s u b s a m p l i n g f r o m t h e full s a m p l e c o r r e s p o n d t o d i f f e r e n t r e p l i c a t i o n m e t h o d s . I h e s u b s a m p l e s a r e c a l l e d r e p l i c a t e s a m p l e s a n d t h e s t a t i s t i c s c a l c u l a t e d f r o m t h e s e r e p l i c a t e s a r e calletl r e p l i c a t e e s t i m a t e s .

12

T h e \ a r i a n a ' o f tin* s a m p l e c s t i m a l o r ^ is e s t i i n a t e c l froni t h e r e p l i c a t e e s t i m a t e s b y

I

' W = Z ' *

k=i

w h e r e Qik) is t h e A'-th e s t i m a t e of 0 b a s e d o n t h e oljservatioiis i i u h u l e c l i n t h e A-th r i ' p l i i a t e . /, is t h e n u m b e r o f r e p l i i a t e s . a n i l c^. is a factor a s s o c i a t e d w i t h r e p l i c a t e A* a n d d e t e r m i n e d by t h e r e p l i c a t i o n m < ' t h o d . W h e n t h e o r i g i n a l e s t i m a t o r

0

is a l i n e a r e s t i m a t o r of t h e f o r m = C-MT)

16.1

w h e r e u \ = tr, ( . 1 ) . t h e A-th r e p l i c a t e ol 0 c a n b e w r i t t e n a s I t . I

w h e r e d e n o t e s t h e r e p l i c a t e weight f o r t h e / - t h unit of t h e A'-th r<'plicat«'.

[ h e following l e m m a p r o v i d e s a n e c e s s a r y c o n d i t i o n for a v a r i a n c e e s t i m a t o r t o Ix' u n b i a s e d .

Lemma 2.1

L d tht o r n j u m l t s l h i m t o r 0 bt a l i i i i u r t.'itinKitor of t i n f o r m i n ( J . 171. If a ri pliraliort variniict i. s l i m a t o r I i n ( J . K J j

/.•»

( Us k j i i n n b i d s i d f o r t i n ( d i s i y i i ) rariaiicf of 0 a n d always takt.s i i o i i n t y a t i r t r a l i K s . tlnii wi h a r t

H A-= 1 . 2 . • • • . / . (2.1!)) i6.> ie.» f o r all

z ,

salisfyiiKj I V/r I = 0 . (2.2U)

Proof. Let

' • W = t - - ' ( E

- E

f c = l \ i 6 . » i € . \ '

B y (2."20) Hiul tlif u i i b i a s o d n c s s o f I w e h a v e

('Ml ^} =

"-S i net' P i - ( \ " - ( « , ) < U | . F ) = 0 . w e iiav»> r { o ^ ) = 0

for a l l sainpU's. Siiicc a r c all p o s i t i \ c h y tin* lUiiiiicgatixciH'ss of I (Jj^ a g a i n . ( 2. 2 0)

follows. •

r i i c r('i)ruat«' f a c t o r c^. is c h o s e n s o t h a t c^. - " ' • ) ^i] estiniat«'s \ d r (»', V, / , ) .

I l u l e r s t r i c t unhia.seclness o f I (^0^ . w e h a v e

I

-Y . n - ( " • ! " - '!•,)* P r ( / € .1) = / r - l ' r ( / < = . » ) [ ! - P r ( / € . \ ) ] . ( 2 . 2 1 ) k = l

Kc|uality ( 2 . 2 1 ) h o l d s b e c a u s e th«' left s i d e of ( 2 . 2 1 ) is t h e e x p e c t e d valiu' o f \ ' for t h e p a r t i c u l a r p o i ) u l a t i o n JF w h o s e // values a r e Z I T O S for a l l u n i t s exce|)t f o r t h e / - t h

e l e m e n t . T h e right s i d e o f ( 2 . 2 1 ) is tin- d e s i g n v a r i a n c e of 0 for t h e p o p u l a t i o n JF.

2.1 The jackknife method

T h e j a c k k n i f e m e l l i o t l . w h i c h o r i g i n a l l y w a s (U-signed t o e s t i m a t e t h e l)ia.s o f a n e s t i ­ m a t o r by d e l e t i n g o n e f l a t u m f r o m t h e o r i g i n a l d a t a set a n d r e c a l c u l a t i n g t h e e s t i m a t o r b a s e d o n t i i e rest of t h e d a t a , h a s b e c o m e a v a l u a b l e t o o l for t h e v a r i a n c e e s t i m a t i o n s i n c e t h e w o r k of l u k e y (1!).')S). h i a n i n f i n i t e p o p u l a t i o n c o n t e x t . Tukey (li).')S) s u g ­ g e s t e d t h a t e a c h r e p l i c a t e e s t i m a t e m i g h t b e r e g a r d e d a.s a n i n d e p e n d e n t a n d i d e n t i c a l l y d i s t r i b u t e d r a n d o m v a r i a b l e , w h i c h in t u r n s u g g e s t s a v e r y s i m p l e v a r i a n c e e s t i m a t o r . I n t h e f i n i t e p o p u l a t i o n s a m p l i n g c o n t e x t , e a c h j a c k k n i f e r e p l i c a t e d e l e t e s o n e u n i t a n d m o d i f i e s t h e w e i g h t s o f o t h e r s .

1-1

E x a m p l e 2 . 1 / . I ndt r timijU n i n d o n i s n t n p i m t j o f s i z t n from a Jinitf p o p u l a t i o n

of s i z t .V. Ihi jnckkiitft r a r i n n c t t s t i m a t o r i s d i j i t u d b y ((juntioti ( J . 1 0 ) with < t =

/ j ~ ' ( n — 1 ) ( 1 — a n d tcf''' = (;i — 1) ' l u r , i f i k and i f ) ' ' = 0 . I'litsr

L'aluts s a t i s f y ( J . 10) with = 1 a n d ( 2 . 2 1 ) .

2 . l o r stratijii d r a n d o m s a m p l i n i j . Itt V/,, b( t h i r a l u t o j tht i - t h t h m t n t i n s t r a t u m h . Lt I ct; — {ill, — I) u ' h u i th( unit { h i ) i s d t l i t t d f o r thi k - t h r t p l i c a t i a n d l i t

ti'i,, if a u i u t i n s t r a t u m (j is d i l t t i d (j ^ h

' (iii^ — 1) ' i j u n i t [ h j ) / . s d t I t t i l l i ^ j U i f u n i t ( h i ) I S i l l I t t i l l .

T h t n f 2 . 1 9 ) h o l d s , win n c/,, = ( : / , , i . • • • . r/.,//) w i t h ^ [ if h = ij. a n d zi,,., = 0 o t h i r u ' i s t .

2.2 Balanced repeated replication

UalaiKi'd rt'peati'cl n ' p l i c a t i o i i ( l i K l t ) w a s first [)ro|)oscHl i)y M c C ar t l i y (lf)()^)) for tlu* c a s e wlu-rc t w o c l u s l i ' r s pi-r s t r a t u m a r c sampU'cl w i t h rcplaaMiiciit in t h e first s t a g e o f s a m p l i n g . I n t h e t w o - c l u s l e r - p < ' r - s t r a t u i n tlesigii w i t h / / s t r a t a , a m i n i m a l s e t o f L b a l a n c e d h a l f - s a m p l e s m a y b«' constriictecl f r o m a n /, x /, l l a d a m a r t l m a t r i x ( s e e . e . g . VVolter. 19 8 5) by c h o s i n g a n y / / r o l n m n s e x c l u d i n g t h e column of all + r s . w h e r e

H < L < H ^ L e t b e t h e e l e m e n t of t h e l l a d a m a r d m a t r i x satisfying = 0 (2.2'J) k = \ for a l l h a n d = M i / -k=l

T l i o A'-tli of t l u ' lincHr e s t i m a t o r of t h e f o r m / / 1 >,=i 1-1 c a n b e w r i t t e n a s ( 2 . 2 1 ) S i n c e t h e B R R uses u n l v lialf of t h e o r i g i n a l s a m p h ' . it m a y p r o f h u ' e \<'r\' u n s t a b l e e s t i m a t e s for s o m e n o n l i n e a r s t a t i s t i c s i n r e l a t i v e l y s m a l l samples. I'o a v o i d a n o m a l i e s .

Kay (H)S1) s u g g e s t e d u s i n g

w h e r e U < <'> < 1.

3 Hot Deck Imputation Methods

T h e r e a r e a vari«>ty o f i m p u t a t i o n m e t h o d s u s e d in practice, a s not«'il b y K a l t o n a n d K a s p r z y k (1!)S()). I h e h o t d e c k i m p u t a t i o n m e t h o d s t a r t s with t h e ilivision o f t l u ' s a m p l e i n t o several i m p u t a t i o n c<*lls. M a n y hot d e c k i m p u t a t i o n metho<ls a s s i g n th«' v a l u e f r o m a recorcl w i t h a response- t o a recorcl w i t h a missing v al ue o n t h a t i t e m . T h e s e r e c o r d s will b e c a l l e i l t h e d o n o r a m i r t c i p i m l . res|)ectively. O f t e n , t h e v a l u e s for a set o f r e l a t e d m i s s i n g i t e m s a r e tak<'n f r o m t h e s a n u ' d o n o r , t o p r e s e r v e s o m e o f t h e m u l t i v a r i a t e r e l a t i o n s h i p s .

H o t d e c k i m p u t a t i o n m e t h o d s c a n b e r o u g h l y classified into t h e following c a t e g o r i e s :

( i ) S e q u e n t i a l Hot D e c k I m p u t a t i o n

S o m e h o t d e c k i m p u t a t i o n p r o c e d u r e s i m p u t e t h e value f r o m t h e r e c o r d i n t h e s a m e cell t h a t w a s l a s t r e a d b y t h e c o m p u t e r . T h i s is p a r t l y b a s e d o n a belief t h a t , if t h e d a t a a r e a r r a n g e d i n s o m e g e o g r a p h i c o r d e r , a d j a c e n t u n i t s i n t h e cell

1 ( )

will ti'iid t o Ix' iiiort' s i m i l a r t h a n r a n d o m l y chosen u n i t s i n t h e cell. O n e problem with t h e setpiential iiot d e c k i m p u t a t i o n is t h a t it m a y e a s i l y mak»' nudti|)le uses of donors, a feature tluit l e a d s t o a loss of [jrecision in s u r v e y e s t i m a t e s .

(ii) Uaiidoin Hot Deck I i n i j u t a t i o n

respondent is c h o s e n a t r a n d o m w i t h i n a n i m p u t a t i o n cell, a n d t h e selectecl

respondent's value is assigiK-d t h e iu)nres|)oiident. l o preserve m u l t i v a r i a t e

relationships, values f r o m t h e s a m e d o n o r a r e used for all m i s s i n g i t e m s of a recor<l. T h e seh'ction »)f d o n t u s c a n Ix perforiiud e i t h e r witlireplacement o r w i t h o u t -rei)lacem»'nt. F i n i h e r i n o r e . o n e may h a v e m o r e than o n e iinput<'il value for each missiii(f i t e m .

(iii) N'earest-.Xeighhor Hot l)e<k I m p u t a t i o n

This hot deck nu-thod assigns a iionrespondent t h e value of t h e "nearest" resi)ijn-deiit. when- " n e a r e s t " is defined in t e r m s of a tlistance f u n c t i o n of t h e auxiliary variables.

Random hot deck i m p u t a t i o n involves r a n d o m selection of d o n o r s . This r a n d o m selection mechanism i n t r o d u c e s w h a t is t<'rm«'tl i m p u t a t i o n v a r i a n c e , a n d this iniputatii>n

variance reduces t h e precision of th«' survey e s t i m a t e s .

. \ s reviewed by Hrick a i u l K a l t o n (l(M)(i), t h e r e a r e two m a i n m e t h o d s for reducing i m p u t a t i o n variance. O n e i s t h r o u g h a s a m p l e design for st'Iecting d o n o r s within each i m p u t a t i o n cell. For i n s t a n c e , selecting d o n o r s by simple r a n d o m samiiling without replacement is |)referable l o simi)le r a n d o m s a m p l i n g of flonors w i t h replacement. By

minimizing the multiple use of tlonors. the without-replacement design leads to a IO\V<T

i m p u t a t i o n variance.

•A second approach is t o u s e f r n c t i o r u i l i i i i p u l a l i o n . which involves dividing nonre-spondents" records i n t o p a r t s a n d i m p u t i n g s e p a r a t e l y t o «'ach p a r t . For e.xainple. each

17

rc'spoiuU'iit might b e d i v i d e d i n t o three p a r t s , e a c h ot which is allocated a weight ol o n e - t h i r d o f t h e n o i n c s p o i u l e n t ' s original w e i g h t . I hen s e p a r a t e donors a r e c hosen for eac h p a r t . If we have only o n e i m p u t e d value for e a c h nonr(>spondent. t h e n we will call t h e p r o c e d u r e sitiijU ( h o t d i c k ) i i i i p u t i i t i o i t .

Example 3.1

Siippo.-^i. i n a siiii[jlt m i i d u i i t .•minptf o f s i z i i i . r u n i t s n s p o n d m i d t n d o n o t r t . s p o n d t o i t t n i t j . [ I n i m p i i t t d v a U n f o r l u i s s i n t j u n i t i i s d i n o t i d Oij i j ' • I h t i n i p u t i d f s t i i i i i i t o r o f tli( p o p n l d t i o n niiiiii ) i s

HI = " ' I E H ' A I •

( i t l/( J

I f till i f i t l i - n plitcniif lit h o t d a k i m p u t a t i o n i s u s ( d . t h i n t i n r a r i n n r i o j i/i i s . c o n ­ d i t i o n a l o n r . HI . . / \ i t r ( ! i i ) = \ <"•((/,.) + — / * . (.s,*) w h I rt !•= t U a n d tjr - H y.-I f t i l t u u t h o u t - r i p l a n nil n t h o t d i c k i n i p u t n t i o n i s u s i d with r > i n . t h i n t i n r a r i a i i c i o f UI c o n d i t i o n a l o n r . \ ' a r ( t i i ) = \ ' i i r ( ! j , . ) + ( s ^ ) . (:{.27) I IJ \ / \ / F o r f r a c t i o n a l i m p u t a t i o n w i t h t h i n u m b t r o f i m p u t a t i o n ( i j u a l t o c . a n d t i n w i t h -r i p l a c t i t i f n t h o t d ( c k i m p u t a t i o n i s u s f d i n d ( p i n d i n t l y c t i i i n s . t i n v a n a n c t o f iji i s . c o n d i t i o n a l o n r , \ ' a r ( y i ) = \ ' a r ( i j r ) + l i i s ; ) . (;j.28) c n -U t n c t , w i t l w u t - r c p l a c t n n n t h o t d e c k i m p u t a t i o n a n d f r a c t i o n a l i m p u t a t i o n n d u c t t h ( i m p u t a t i o n v a r i a n c e r t l a t i i ' f t o w i t h - r t p l u c t m t n l i m p u t a t i o n .

IS

Fractional i i n p u t a t i o i i is cliscussrfl by K a l t o n a n d Kisli (19SI) a n d Kay (1996). A (lifForcnt. but r e l a t i ' d , a p p r o a c h is m u l t i p l e i m p u t a t i o n , which is discussed in t l u ' next section.

4 The Multiple Imputation Approach

.Multiple impntaticjii. proposed by R u b i n (1078). is a jjrocedure for handlinp; inissin^^ d a t a t h a t alUnvs t h e d a t a analyst tcj use s t a n d a r d Iechnic|ues of analysis desip,iu'd for c o m p l e t e d a t a , w h i l e a t t h e s a m e t i m e providing a m e t h o d t o e s t i m a t e t h e uncertainty d u e t o t h e m i s s i n g d a t a .

. \ c o m p r e h e n s i v e description of iiudti|jle i n t p u t a t i o i i is given in R u b i n (19S7). Rubin (1987) devot<'s a gocnl d e a l of c h a p t e r •'} t o specifying recpiin'ments for t h e \ a i i d i l y of m u l t i p l e i m p u t a t i o n inference u n d e r t h e nujch'l ba.s«'d a p p r o a c h . His arginnents in t h a t c h a p t e r a r e for t h e Bayesian a p p r o a c h , w h e r e inferences a r e m a d e using t h e postericjr m e a n a n d t h e p o s t e r i o r variance. R u b i n (1987) d e v o t e s C h a p t e r I t o conditions for t h e validity of multiph* i m p u t a t i o n in t h e r a n d o m i z a t i o n f r a m e w o r k .

•Multiple i m p u t a t i o n c a n b e c h a r a c t e r i z e d by t h e m e t h o d of g e n e r a t i n g tlie i m p u t e d values a n d by t h e variance fornuila. ['he variance f o r m i d a d i r e c t l y uses the c o m p l e t e -s a m p l e variance e -s t i m a t o r -s o t h a t it c a n b e i m p l e m e n t e d ea-sily u-sing t h e exi-sting -soft­ ware.

Let On b e t h e c o m p l e t e s a m p l e e s t i m a t o r of tiie p a r a m e t e r 0 a n d l „ = \ (V,„,„) b e t h e c o m p l e t e s a m p l e variance e s t i m a t o r of 0,^. The full s a m p l e V„,,„ is decomposed a s

Vjum = (>',65. V'„u»). w h e r e is t h e p a r t of w i t h / / , = 1 a n d V„,„ is t h e part of

> j(i ni 11 h /?i — 0 .

M u l t i p l e i m p u t a t i o n involves r e p e a t i n g t h e i m p u t a t i o n process independently M t i m e s . T h e i m p u t e d values a r e g e n e r a t e d from t h e posterior d i s t r i b u t i o n of V,,,,, given Yobs- After m u l t i p l e i m p u t a t i o n , w e h a v e M d a t a s e t s . T h u s w e c a n construct M

sep-19

a r a l e s t a t i s t i c s a n d M e s t i m a t o r s of v a r i a n c e baseil o n t h e a u g m e n t e d s a m p l e . Let t h e

statistics b e a n d I /(i).„ ^ i(M).n f'Ji" t h e e s t i m a t o r a n d e s t i m a t o r of

variance, respectively. 1 hen. t h e m u l t i p l e i m p u t a t i o n e s t i m a t o r of 0 is

M O M . n = . U - ' ( 1 . 2 9 ) (=1 a n d t h e a s s o c i a t e d variance e s t i m a t o r is t \ i. n = V\/.ri + — — — I h i. n - ( l.;{0) where \ l i = i a n d M f l u. n = ( A / - 1 ) " ' (i . ; J - ' ) 1-1

T h e t y p i c a l assum|>lions associated with multiiile i m p u t a t i o n a r e

I m | / • ; 0 ( i.:{;}) a n d j n n n [/•.' ( T ^ ^ , , . ) - T = U. (l.iM) where a n d = lim 0\i,„ A / X T-^.n = liin A/-+ X

In t h e Bayesian a p p r o a c h , t h e d i s t r i b u t i o n u.sed in ( I..};}) a n d ( I.^M) is t h e conditional

d i s t r i b u t i o n of 0 given \[,b, under t h e a.ssumed model. In t h e cla.ssical model-based a p ­

proach. t h e d i s t r i b u t i o n is t h e d i s t r i b u t i o n of V'^6s u n d e r t h e a s s u m e d model. In t h e r a n d o m i z a t i o n a p p r o a c h , t h e d i s t r i b u t i o n used i n (-1.33) a n d (1.34) is t h e joint d i s t r i b u ­ tion of t h e s a m p l i n g m e c h a n i s m a n d t h e response m e c h a n i s m .

20

4.1 Bayesian Justification

C h a p t e r of R u b i n (19S7) cU-als with t h f valichty of multiph' i m p u l a t i o i i in Uu*

Bayesian fraiiu'work. l b review tliat a p p r o a c h , we assume that t h e e s t i m a t o r 0„ based o n tlie c o m p h ' t e sainph* is t h e posterior m e a n of 0 under the a s s u m e d Bayesian inotiel

/ ( i i u h u l i n g b o t h t h e likelihoo<l a n d tlie prior d e n s i t y ) . That is.

()„ = i-:,{01 (1.;{••))

Also, let \ „ lie t h e |)osterior variance of 0 u n d e r t l u ' model / . That is.

i ; . = i / « ; 1 (i.iUi)

According t o M e n g (l!)f) l . i).') l:}). a liayesian iiUKlel / satisfying ( 1.•{•")) a n d ( l.^Ui) is said

t o b e c o i i t j i m i l l t o t h e analysis using I I sing t h e terminology of congeniality, u e

s u m m a r i z e t h e m a i n results in c h a p t e r •'{ of R u b i n (1?)S7).

Result 4 . 1 .1 .-.s 1/;n f

I h i l l

( i ) [''or lilt c o i n p l d t s i i i n p h . t i n l i m j t s i n n m o d i I f i.s v o n i j t m i i l l o llii itiialifsi.-i tisiiiij (iiiil

( i t ) t h i i u i p i i h d r a l u i.'i ( i n d r u u ' i i f r o m llit r o i i d i t i o i i i i l i l i s l r i b u l i o i i | V/fc,) " /

^ m i s ( j i i ' f i i Vi.fc, u n d t r l f i t B i i i j i s i i i n m o d t l f .

T h i l l , n n d t r n o i i n s p o n s t . Ihf l i n y t f i i a i i m o d d f /.s coiujiiuiil t o Iht i i n i i l y s i s usiiiij l l n i m p u i f d p a i r ( ^ A / . n . /.v/.n) c a l c u l u l t d f r o m a n d (.(..W). its . \ / —> : x . T h i i l i s .

= E f ( 0 I (

a n d

7

'X.N

= I V L i J . (-l.^W)

21

Proof,

l o slunv t h e

T'C

|ualii'n's ( 1.37) a n d ( l.iJS). n u l c t h a t

r { 0 \ Yj,.,) = I

r ( o \ ) v ; , . . , ) P { ) | > " 6 . ) ' i o )

I

v;,,,) for / = I . 2 . - - -

. M .

S o . by till' law of l a r g e tiumlicrs.

/ • ; ( 0 | V j , . i = Mm £ ; / • . • ( « I > A , . i ; r ) 1 = I 1 -i — 1 a n d

+\- {i-:{0\

v;,u,) i V

m

}

I -1 = -1 I " «=I a l m o s t surely. •

By Uesult 1.1. we h a v e t h e desired relation ( 1.3i}) a n d ( t..M) under t h e posterior

distribution of 0 given In Uesult l . l . t h e r e ar«' t w o m o d e l s involvt-d. I ' h e first

motlel is called t h e a n a l y s t ' s model, which is u.sed in ( I.•{•')) a n d ( l.-'Ui). I lu- secoiul

m o d e l is called t h e i m p u t e r ' s m o d e l , which is used in calculating C { ) ' , n i s I Uesuh

1.1 re(|uires t h a t t h e t w o models b e t h e s a m e . . \ s is observed in Kay (1991.11)92). .\Ieng (199-1). l{ul)in (n)9()). a n d Schafer (1997). if t h e a n a l y s l ' s model is dilferent f r o m t h a t of i m p u t e r . then t h e m u l t i p l e i m p u t a t i o n e s t i m a t o r m a y b e bia.sed. not only for variance estinuitiou but also for p o i n t e s t i m a t i o n .

22

4.2 Randomization Validity

Multiple i i n p u t a t i o i i . which is based on t h e Bayesiaii p a r a d i g m , c a n h e evaitiatetl u n d e r t h e fre(|uenlist p a r a d i g m , where t l u ' p o p u l a t i o n values a r e t r e a t e d a s fixed a n d inferences are hased o n tin* s a m p l i n g d i s t r i b u t i o n gen«'rated by r e p e t i t i o n s of th»' sami)le seU'ction procedure a n d a motlel for resi)onse probai)ilities.

Hubin(l!)S7. p.118) g a v e t h e delinition of pro|)er i i n p u l a l i o n . w h i c h is a key concept for t h e randoniization validity of multiple i m p u t a t i o n . T h e delinition of a proper i m p u

-tati(jn procedure t r e a t s t h e comi)lele samjjle V a s lixed. a n d t h e res|}onse indication

vector

R

as t h e r a n d o m variable. F o r coinplet(> s a m i i l e statistics a n d \ „ . a im|Mitalion

m e t h o d is called profx r uncler t h e assumeil r e s p o n s e mechanism if

I = 0,,. i c i )

l-H x.M I Vv.m} = K,. (("2)

a n d

I Vvi.m} = I ((";{)

The subscript H is used h e r e t o em|)hasize t h a t t h e reference d i s t r i b u t i o n is with respect t o t h e assumed response m e c h a n i s m on H.

[ h e main conclusion regarcling randomization validity with prop«'r i m p u t a t i o n is well s u m m a r i z e d in i t u b i n (19S7):

Result 1.1; If t h e c o m p h ' t e - d a t a inference is randomization valid a n d t h e imiltiple-imputation p r o c e d u r e is p r o p e r , t h e n t h e infinite-/^ r e p e a t e d im­ putation inference is randomization-valid u n d e r t h e posited r e s p o n s e mech­ anism. ( p . 119)

T h e conditions of p r o p e r i m p u t a t i o n a r e difficult t o verify. O n e i m p u t a t i o n procedure

is t o generate t h e missing p a r t V'„,„ from t h e conditional d i s t r i b u t i o n £ ( V „ , „ | of

2:}

p r o p i r i n i p u l d t i o i i . liayesianly p r o p e r impiilalioii is nol suMic ieiit for p r o p e r imi)iilalioii.

r i u ' following t l i c o r r m a t l e r n p t s l o clarify l l u ' rolationsliips.

Theorem 4.1

I f a m u l t i p U i m p u t a t i o n i j t n t r a t t t l f r o m C { \ „ u 3 | ^uha) U'^iny tlit f i a i j t s i u i i m o d t l f s d t i s j i t s ( C ' l ) , t i n n i t a l s o s a t i s j i t s { ( ' • { } .

Proof. [.

c t tl li<> h!1\' gi\<'n full ^ainpl'' I' s t i n i a t o r of I).

lU'

t l i c law of l a r a e tiumlxMs.

1 . a n d l<:=l = / • / ( O n

I yJ,.,)

\ t ^lini^ T T H " ^

~

= I/(<),. I

Now,

I V„,m) = l-U / (^r. I ^ .6,) I v.,,,

= /•,'/< IV/ — I'. J {0,X I I \ ,.h, I Flirt l i o n n o r e .

A

h(«x.,. I

-Av (<),.! v:,.,) IV

= Vn E f { o „ I I V,„

whore i h e e q u a l i t y ( l.;J9) follows from t h e decomposilion

(l.:5!)) ( 1 . 1 0 )

V

[ q

I v..m] = Vft

[ E f

{

q

i

K b . )

1

+ i

- n

[ i / {

q

I V . 6 , ) I v;,.,

w i t h Q = d „ - E f I VLi,) a n d so E f[ Q \ \ = 0 . The e q u a l l y (1.1 0 ) holds because

by a s s u m p t i o n (C'l).

Sunu- autliurs. for rxampU* Kay (19!)'2). have cpn'stioiKnl t h e validity of t h e m u l t i p l e i m p u t a t i o n u n d e r t h e f r e q u e n t i s t response probability m o d e l . W e will s t u d y t h i s in t h e n e x t subsection.

4 . 3 F a y ' s E x a m p l e

Fay( 19!)1.1(M)2) us«'d a li<'riiouHi model t o i l l u s t r a t e t h e ililliculty of c r e a t i n g pro|)er i m p u t a t i o n s a s a g«'iieral purposi' m<'thodology. W e s u p p r e s s t h e subscript n in t h e e s t i m a t o r s t o simplify t h e n o t a t i o n . S u p p o s e we liav<' a s i m p l e r a n d o m s a m p l e of size ii for variable V t a k i n g o n l y 0 t)r 1. .Vssumi' t h a t , for simplicity, t h e lirst r c-U'inents a r e

observetl a n d t h e | ) a r a m e t e r of int*'ri'st is 0 \ — A " ' 51. = i f^is'iine tin- uniform

r e s p o n s e mechanism a n d u s e th«' H<'rnouHi model t o c n ' a t e lh<> m u l t i p l e i m p u t a t i o n , t h e n w<' use M w i t h ( l . l l ) a s t l u ' e s t i m a t o r of ( ) \ a n d u s e w i t h M (-1.-12) a n d M

•Jo

t o e s t i m a t e t h e variance of 0 \ i . riien, letting n r ^ p ^ ( 0 . I) and . V ' n —> 0. we have

E [ O ^ \ \ )

Var{0^ I Y)

I-(IK i

Y)

O s - 1 _{- O s )}

< r

' O s { \ - O s ) - N ~ ' ) O s Oy

IxTaus*' t l u ' res|)ondents c a n b e regarded a s a s i m p l e r a n d o m s a m p l e from t h e p o p u l a t i o n . Hence.

i : { i \

i Y )

= \ „ r ( ( l ^

| Y ) .

Now. a s s u m e that for e a c h unit i . we have .V, t a k i n g e i t h e r u or b a s possible values. W e want t o e s l i m a t t ' 0 , , = / ' r ( V = 1..V = a ) a n d O t , = l ' r { ) = 1. .V = b ) . Let

II — I I , , -I- III, an<l /• = r , + n , . If we h a v e c o m p l e t e response, t h e n w v will u s e

0., = f / , ; '

^ v;/( . V , = </1

1 = 1

0 , = = M

1=1

a n d t h e variance-covariance e s t i m a t o r for {O.t.Oi,^ is

\ =

( i - y , . ) 0

0 ( i - t f . )

w i t h d„ = tfu + Oh.

If we h a v e missing d a t a a n d i m p u t e using t h e a p p r o x i m a t e Bayesian b o o t s t r a p m e t h o d [jroposed by Rui)in a n d S c h e n k e r (l!)86). tlien we u s e

M { ' ) 0..M = . 1 / : i (=1

4

, u = (=1

2(i

WIKTC

jc) _ { Y . y j i \ , = " ) + T .

v;"7(.v, = </

1=1 i = r+ 1 ijU) "h.i =

=

E

> • ; " ' / ( . V , = b ) 1=1 l = r+l a n d t l u ' varinii(»'-c-u\ariaiu-(' e s t i m a t o r for is

/

l \ , = ; r V\/ + (1 + i (1 + .\/ ' I Ihi.ih + M ) li\i :rr\/ + (1 +

wluTf I \/ is dcliiK'd in ( l. l'i) a n d

m I h l : . = I h l . i h = Ihi.bh — •*' * f = | (=1

riuMi. since tiie resj)onileiits a r e a siinpli- r a n d o m sam|)U' of size r .

/

K { T . . ) = 0 s ( 1 - 0 S

_L -L ^ y i (" i i - ' . i ) () i

"u 'T, V "u / '" V ".i / \ / >•

C "II-'•u ^ f ^ i ±. J . J . f '•l.-' l.y' i \ V / V " b / ' • " f . " i l V " ( . / >• ljUt I '«;•

/

\

V / = t f v ( l 0 , s -£u.+ i ^ "f, r V ri„ y r V M„ m j i ( l _ + i [ i _ ( i t ) -r \ lia " h / "f, r \ " 6 /

Hence. overestimates t h e variances of a n d tf5,x. u n d e r s t i m a t e s t h e c o v a r i a n c e o f

27

Kubin (1990) fxplaiiK'd tluit if t lie auxiliary variabk- .V is not used by t h e i m p u t c r t o croatc t l u ' m u l t i p l e i m p u t a t i o n , b u t is used l)y tlit> u l t i m a t e analyst t o tlefine t'slimands. then tlu* i m p u t a t i o n may not IK- p r o p e r . In t h e above e x a m p l e , tlie . A H U i m p u t a t i o n is

proper for 0 = + Oi,. l)ut is imjiroijer for 0 = 0.^— Oi,. Ruliin (1!)9()) a r g u e s t h a t t h e

nHilti|)le i m p u t a t i o n is still conlidenee-proper in tlie s e n s e that it produ<<'s a variance ('stin)ate t l i a t is t o o large.

5 Quasi-Raiidomizatioii Approach

T h e r a n d o m i z a t i o n aj)proaeh. wliieli tri'ats tlu- |)opulation vector

Y

a s lixed. has

played a d o m i n a n t role in t h e design a n d analysis of s a m p l e surveys, [{andumization inference re<|uires t h a t units Ix' selec t<'d by iinihabilili/ snniplinij. which is characterized by t h e folhjuing two projjerties:

1. I'he s a m p l i n g distribution is d e t e r m i n e d by t h e s a m p l e r b»'fore a n y ij values a r e known.

2. Kvery u n i t h a s a positive ( k n o w n ) probal)ility of s«'lection.

riie key ingredient of t h e r a n d o m i v a t i o n approach, a k n o w n probability of selection, is lost when s o m e of t h e d a t a a r e m i s s i n g .

Ciiven t h e e x i s t e n c e of n o i n e s p o n d e n t s . o n e approach is t o regard t h e respoiid<Mits as t h e second pha.se s a m p l e in a two-pha.se sam|)le design. T h i s is niaiie possible by treating

the //,"s as random variables. It is necessary to specify a probability niod«'l for

R.

T h e r a n d o m i z a t i o n v e r s i o n o f i n f e r e n c e f o r i m p u t a t i o n i s c a l l e d t h e q i K L s i - r a n d o i n i z a t i o i i approach, a t c r u : suggested by O h a n d .Seh«Hireii (l9N.n.

T h e r e a r e t w o main differences b e t w e e n t h e s a m p l i n g distribution a n d t h e response distribution i n t h e q u a s i - r a n d o m i z a t i o n approach. F i r s t , t h e sampling d i s t r i b u t i o n is determined b y t h e sampler before a n y observations ar«' taken. O n t h e o t h e r h a n d .

wo may have (lifrerenl response m e c h a n i s m s for ditrerent i t e m s . Second, t l i e s a m p l i n g dislriljution is k n o w n , under t h e c o n t r o l of tlu' survey s t a t i s t i c i a n . O n t h e o t h e r h a n d , t h e response d i s t r i l j u t i o n is u n k n o w n a m i needs s o m e f o r m of modelling.

If t h e finite population is p a r t i t i o n e d into

CI

i m p u t a t i o n cells, t h e usual

([uasi-raiidoniization a p p r o a c h assunu's t h e following respons*' m e c h a n i s m .

( k . l ) ['"or e a c h cell i ] — I . - - . ( i . all i t e m s { h } , g t o have

tiie s a m e r e s p o n s e prol)al)ility. p j = P r { l i , — 1 | / € ^ j ) . where / , tlenotes t h e s«'t of indices for t h e 7-th i m p u t a t i o n c«>ll.

( R . i ) Kor e v e r y / = 1. • • • . .V. l ^ r { l i , — I ) > 0.

We will call t h i s u n i f o r m n s i w i i s i intcli<itiisin i n l l n i i i m p u l a l i o t i n i l .

Uao a n d Shat) used w f i f j l i t u l h o i i l i c k u n p t i i a t t o n . which selects tlonors with

replacenn'iil witli t h e prohaliility of selection being |)rop(jrtiunal ttj t h e s a m p l i n g weights. This produces a n unbiased e s t i m a t o r u n d e r assinnptions ( H. 1) a n d (H.2). W hile t lu' pro-cedur«' is o f t e n saitl t o be design uidjia.sed. unbia.sedn«'ss r«'(iuires t h e responsj* probability model a s s u m p t i o n s ( l { . l ) a n d ( l t . 2 ) .

riie adjusttnl jackknife v a r i a n c e t ' s l i m a t u r for w e i g h t e d hot deck i m p u t a t i o n , pro­

posed by R a o a n d S h a o is constructi'd by c h a n g i n g I'very i m p u t e t l value for t h e

jackknife replicatc* when a r e s p o n d e n t is deleted. I h e v a r i a n c e e s t i m a t o r c a n b e w r i t t e n a s I. ₂ (••i.i;}) where r; + (I - f i j ) { ' h + - .7, (o.-l-l) J=1 j€.lnr. with 1 J •' (o.-io)