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University of Southern California Dissertations and Theses
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Comparing dependent groups with missing values: an approach based on a robust method
(USC Thesis Other)
Comparing dependent groups with missing values: an approach based on a robust method
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C O M P A R I N G D E P E N D E N T G R O U P S 1
C om pa r i ng D e pe nde nt G r oups w i t h M i s s i ng V a l ue s : A n A ppr oa c h B a s e d on a R obus t M e t hod
J i nxi a M a
D e pa r t m e nt of P s y c hol ogy
U ni ve r s i t y of S out he r n C a l i f or ni a
C O M P A R I N G D E P E N D E N T G R O U P S 2
T ab l e T ab l e T ab l e T ab l e of of of of C on t e n t s C on t e n t s C on t e n t s C on t e n t s
A b s t r ac t A b s t r ac t A b s t r ac t A b s t r ac t ..................................................................................................................................3 ..................................................................................................................................3 ..................................................................................................................................3 ..................................................................................................................................3
I n t r od u c t i on ............................................................................................................................4 I n t r od u c t i on ............................................................................................................................4 I n t r od u c t i on ............................................................................................................................4 I n t r od u c t i on ............................................................................................................................4
M e t h od .....................................................................................................................................9 M e t h od .....................................................................................................................................9 M e t h od .....................................................................................................................................9 M e t h od .....................................................................................................................................9
S i m u l at i on S i m u l at i on S i m u l at i on S i m u l at i on D e s i gn .................................................................................................................11 D e s i gn .................................................................................................................11 D e s i gn .................................................................................................................11 D e s i gn .................................................................................................................11
R e s u l t s ....................................................................................................................................13 R e s u l t s ....................................................................................................................................13 R e s u l t s ....................................................................................................................................13 R e s u l t s ....................................................................................................................................13
C on c l u s i on s ............................................................................................................................13 C on c l u s i on s ............................................................................................................................13 C on c l u s i on s ............................................................................................................................13 C on c l u s i on s ............................................................................................................................13
R e f e r e n c e s ..............................................................................................................................15 R e f e r e n c e s ..............................................................................................................................15 R e f e r e n c e s ..............................................................................................................................15 R e f e r e n c e s ..............................................................................................................................15
A p p e n d i x................................................................................................................................19 A p p e n d i x................................................................................................................................19 A p p e n d i x................................................................................................................................19 A p p e n d i x................................................................................................................................19
T ab l e s ................................................................................................................................21- 23 T ab l e s ................................................................................................................................21- 23 T ab l e s ................................................................................................................................21- 23 T ab l e s ................................................................................................................................21- 23
F i gu r e s ...............................................................................................................................24- 25 F i gu r e s ...............................................................................................................................24- 25 F i gu r e s ...............................................................................................................................24- 25 F i gu r e s ...............................................................................................................................24- 25
C O M P A R I N G D E P E N D E N T G R O U P S 3
A b s t r ac t A b s t r ac t A b s t r ac t A b s t r ac t
I n t hi s s t udy , w e i nve s t i g a t e t he r obus t m e t hods f or c om pa r i ng t w o or m or e g r oups w he n t he r e
a r e m i s s i ng va l ue s . W e us e t he 20% - t r i m m e d m e a n a s t he m e a s ur e of l oc a t i on a nd c onduc t a
pe r c e nt i l e boot s t r a p m e t hod t o t e s t t he r obus t ne s s w he n c om pa r i ng g r oups w i t h m i s s i ng va l ue s .
B ot h s ke w e d a nd he a vy - t a i l e d di s t r i but i ons a r e c ons i de r e d. O ur r e s ul t s i ndi c a t e t ha t t he m e t hod
t e s t e d r e m a i ns s a t i s f a c t or i l y r obus t w he n c om pa r i ng t w o or f our g r oups w i t h e qua l s a m pl e s i z e s
of 30 a nd 5 m i s s i ng va l ue s pe r g r oup.
K e y w or ds : boot s t r a p, m i s s i ng va l ue , r obus t , t r i m m e d m e a n
C O M P A R I N G D E P E N D E N T G R O U P S 4
C om pa r i ng D e pe nde nt G r oups w i t h M i s s i ng V a l ue s : A n A ppr oa c h B a s e d on a R obus t M e t hod
I n t r od u c t i on I n t r od u c t i on I n t r od u c t i on I n t r od u c t i on
M i s s i ng da t a i s a pe r va s i ve pr obl e m i n m a ny r e s e a r c h a r e a s a nd t he r e a s ons a r e m ul t i f ol d.
M i s s i ng va l ue s oc c ur w he n pa r t i c i pa nt s dr op out , w he n m e a s ur e m e nt s f a i l , w he n t he r e s ponde nt s
l e a ve bl a nks t o que s t i ons i n s ur ve y s , w he n c ol l e c t e d da t a g e t l os t , or w he n m e a s ur e m e nt s a r e not
i n a c c or da nc e w i t h s om e pr i or know l e dge of t he da t a , i .e ., i m pl a us i bl e . S o how t o de a l w i t h
m i s s i ng va l ue s w he n c om pa r i ng t w o or m or e g r oups i s bot h a ve r y i nt e r e s t i ng a nd pr a c t i c a l t opi c
t o r e s e a r c he r s .
O ne m e t hod f or de a l i ng w i t h m i s s i ng va l ue s i s s i m pl y e x c l udi ng s uc h c a s e s , know n a s
c om pl e t e c a s e a na l y s i s . B ut t hi s a ppr oa c h m i g ht r e s ul t i n i ne f f i c i e nt e s t i m a t i on, w hi c h i n t ur n
m i g ht r e s ul t s i n a s ubs t a nt i a l r e duc t i on i n pow e r w he n t e s t i ng hy p ot he s e s ( e .g., L i a ng, W a ng ,
R obi ns , & C a r r ol l , 2004) . A l t e r na t i ve m e t hods f or ha ndl i ng m i s s i ng obs e r va t i ons a r e
s um m a r i z e d i n s e ve r a l books ( e .g., A l l i s on, 2001; L i t t l e & R ubi n, 2002; M c K ni ght , S i da ni &
F i gue r e do, 2007; M ol e nbe r ghs & K e nw a r d, 2007; D a ni e l s & H oga n, 2008; S c ha f e r , 1997) .
W he n de a l i ng w i t h m e a ns , c om m on a ppr oa c he s i nc l ude m a x i m um l i ke l i hood e s t i m a t i on
( I b r a hi m , C he n & L i ps i t z , 1999; I b r a hi m , L i ps i t z & H or t on, 2004) , t he w e i ght e d a dj us t m e nt
m e t ho d ( C oc hr a n, 1977 ) , s i ngl e i m put a t i o n ( R a o & S i t t e r , 1995 ) , a nd m ul t i pl e i m put a t i o n ( R ubi n,
1987; L i t t l e & R ubi n, 2002) . M a xi m um l i ke l i hood us e s t he E M a l gor i t hm or ot he r a l g or i t hm s t o
obt a i n m a xi m um l i ke l i hood e s t i m a t e s of t he m e a ns a nd t he c ova r i a nc e m a t r i x unde r s om e
di s t r i but i ona l a s s um pt i ons , i n m a ny c a s e s unde r t he a s s um pt i on of m ul t i va r i a t e nor m a l i t y . W he n
t hi s a s s um pt i on i s m e t , M L pr oduc e s pa r a m e t e r e s t i m a t e s t ha t ha ve t he a dva nt a ge s of ons i s t e nc y ,
a s y m pt ot i c e f f i c i e nc y , a nd a s y m pt ot i c nor m a l i t y , w hi c h m e a ns t ha t a s s a m pl e s i z e i nc r e a s e s , t he
C O M P A R I N G D E P E N D E N T G R O U P S 5
e s t i m a t e s c onve r ge t o t he t r ue pa r a m e t e r va l ue s , t he e f f i c i e nc y i m pr ove s , a nd t he a ppr ox i m a t i on
of t he s a m pl i ng di s t r i but i on t o nor m a l i t y i m pr ove s ( A l l i s on, 2003) . H ow e ve r , a t t e nt i on s houl d be
pa i d t o t he r obus t ne s s of t he m e t hod, be c a us e t he M L m e t hod r e qui r e s t he s pe c i f i c a t i on of a
pa r a m e t r i c di s t r i but i on f or t he c ova r i a t e s . T hus i t i s pos s i bl e t o s pe c i f y t he i nc or r e c t c ova r i a t e
di s t r i but i on. ( I b r a hi m , C he n & L i ps i t z , 1999) . T he w e i ght e d a dj us t m e nt m e t hod i s a m odi f i c a t i on
of c om pl e t e c a s e a na l y s i s , w hi c h i nvol ve s di f f e r e nt i a l l y w e i ght i ng t he c om pl e t e c a s e s t o a dj us t
f or bi a s . T he w e i ght e d e s t i m a t or s a r e of t e n r e l a t i ve l y s i m pl e t o c om put e , but t he c om put a t i on of
a ppr opr i a t e s t a nda r d e r r or i s c ha l l e ngi ng. S t a t i s t i c a l pa c ka ge s a r e a va i l a bl e f or c om put i ng t he
a s y m pt ot i c s t a nda r d e r r or s , how e ve r , t he s e pr ogr a m s t y p i c a l l y t r e a t t he w e i ght s a s f i xe d a nd
know n, w he r e a s nonr e s pons e w e i g ht s a r e c om put e d f r om t he obs e r ve d da t a a nd a r e s ubj e c t t o
s a m pl i ng unc e r t a i nt y ( L i t t l e & R ubi n, 2002) . B ot h s i ng l e a nd m ul t i pl e i m put a t i on m e t hods ha ve
t he pr a c t i c a l a dva nt a ge of a l l ow i ng s t a nda r d c om pl e t e - da t a m e t hods of a na l y s i s t o be us e d. T he
di s a dva nt a ge of s i ng l e i m put a t i on i s t ha t i t c a nnot r e f l e c t s a m pl i ng va r i a bi l i t y or unc e r t a i nt y
a bout t he c or r e c t m ode l f or nonr e s pons e . M ul t i pl e i m put a t i on r e c t i f i e s t he di s a dva nt a ge of s i ng l e
i m put a t i on by r e pl a c i ng e a c h m i s s i ng va l ue w i t h a ve c t or of 2 ≥ D i m put e d va l ue s . T he r e s ul t i ng
D c om pl e t e da t a a na l y s i s c a n be e a s i l y c om bi ne d t o c r e a t e a n i nf e r e nc e t ha t va l i dl y r e f l e c t s
s a m pl i ng va r i a bi l i t y be c a us e of t he m i s s i ng va l ue s ( L i t t l e & R ubi n, 2002) .
T he r e a r e t w o g e ne r a l s t r a t e gi e s f or c om pa r i ng de pe nde nt g r oups . T he f i r s t i s t o t e s t
t he hy p ot he s i s t ha t t he m a r gi na l di s t r i but i ons a r e i de nt i c a l , a nd t he ot he r i s t o t e s t t he hy p ot he s i s
t ha t t he m a r gi na l di s t r i but i ons ha ve e qua l m e a s ur e s of l oc a t i on. L e t t he J de pe nde nt g r oups ha ve
unknow n m a r gi na l di s t r i but i on f unc t i ons ) ( ) , ( ) , (
2 1
x F x F x F
J
⋯ . T o c om pa r e t he di s t r i but i ons of
t he s e J g r oups , t he nul l hy p ot he s i s i s
) ( ) ( ) ( :
2 1 0
x F x F x F H
J
= = = ⋯ , ( 1)
C O M P A R I N G D E P E N D E N T G R O U P S 6
i .e ., t he m a r gi na l di s t r i but i on f unc t i ons of t he J g r oups a r e i de nt i c a l .
R a t he r t ha n t e s t i ng ( 1) a c om m on g oa l i s t o t e s t
J
H θ θ θ = = = ⋯
2 1 0
: , ( 2)
w he r e
j
θ i s s om e m e a s ur e of l oc a t i on a s s oc i a t e d w i t h t he j t h m a r gi na l di s t r i but i on ) , , 1 ( J j ⋯ = .
L u dbr ook ( 2008) s ug ge s t e d us i ng a pe r m ut a t i on t e s t w he n c om pa r i ng g r oups ba s e d on
m e a ns . T he pe r m ut a t i on t e s t i s a t y p e of s t a t i s t i c a l s i g ni f i c a nc e t e s t i n w hi c h t he di s t r i but i on of
t he t e s t s t a t i s t i c unde r t he nul l hy p ot he s i s i s obt a i ne d by c a l c ul a t i ng a l l pos s i bl e va l ue s of t he t e s t
s t a t i s t i c unde r r e a r r a nge m e nt s of t he l a be l s on t he obs e r ve d da t a poi nt s . F or t e s t i ng t he a r i t hm e t i c
di f f e r e nc e be t w e e n t w o m e a ns , L udbr ook s ug ge s t e d c a l c ul a t i ng t he P va l ue w i t h t he f or m ul a :
T w o- s i de d P = {A l l pe r m ut a t i ons i n w hi c h t he m e a n di f f e r e nc e ≥ t ha t obs e r ve d, i n e i t he r
di r e c t i on} / {A l l pos s i bl e pe r m ut a t i ons , r e t a i ni ng t he s a m e g r oup s i z e s }. ( 3)
A pos i t i ve f e a t ur e of L udbr ook's pe r m ut a t i on t e s t i s t ha t w he n t e s t i ng t he hy p ot he s i s of i de nt i c a l
di s t r i but i ons ( 1) , t he e xa c t pr oba bi l i t y of a T y p e I e r r or c a n be de t e r m i ne d. B ut a s a m e t hod f or
t e s t i ng ( 2) , i t c a n be uns a t i s f a c t or y e ve n w he n t he r e a r e no m i s s i ng va l ue s ( e .g., B oi k, 1987;
R om a no, 1990) .
I n t e r m s of m i s s i ng da t a i m put a t i on m e t hods , a s i m pl e s t r a t e gy f or c om pa r i ng t he
m a r gi na l m e a ns i s t o c om put e a c onf i de nc e i nt e r va l us i ng a nor m a l or t a ppr ox i m a t i on i n t he
us ua l m a nne r . I t i s know n, how e ve r , t ha t t hi s a ppr oa c h c a n be uns a t i s f a c t or y , a s not e d, f or
e x a m pl e , by L i a ng, S u a nd Z ou ( 2008) a s w e l l a s W a ng a nd R a o ( 2002) . M or e ove r , e ve n w he n
t he r e a r e no m i s s i ng va l ue s , c onc e r ns a bout r e l a t i ve l y l ow pow e r a nd i na c c ur a t e c onf i de nc e
i nt e r va l s a r i s e w he n s a m pl i ng f r om a he a vy - t a i l e d di s t r i but i on ( e .g., W i l c ox , 2005) . W he n t he
g oa l i s t o c om pa r e r obus t m e a s ur e s of l oc a t i on, i t a ppe a r s t ha t no i m put a t i on s t r a t e gy ha s be e n
pr opos e d a nd s t udi e d.
C O M P A R I N G D E P E N D E N T G R O U P S 7
A n a ppr oa c h t o m i s s i ng va l ue s us i ng a n e m pi r i c a l l i ke l i hood m e t hod, ba s e d on s i ng l e
i m put a t i on, w a s de r i ve d by L i a ng e t a l . ( 2008) , w hi c h i s r e a di l y a da pt e d t o t he pr obl e m of
c om put i ng a c onf i de nc e i nt e r va l f or
D
µ t he popul a t i on m e a n a s s oc i a t e d w i t h Y X D − = ,
w he r e X a nd Y a r e t he de pe nde nt r a ndom va r i a bl e s be i ng c om pa r e d. A c onc e r n, how e ve r ,
i s t ha t w he n de a l i ng w i t h he a vy - t a i l e d di s t r i but i ons , l a r ge s a m pl e s i z e s m i g ht be ne e de d t o
g e t a r e a s ona bl y a c c ur a t e c onf i de nc e i nt e r va l f or
D
µ e ve n w he n t he r e a r e no m i s s i ng
va l ue s . F or e x a m pl e , s uppos e D ha s t he c ont a m i na t e d nor m a l di s t r i but i on
) 10 / ( ) ( ) 1 ( ) ( x x x H Φ + Φ − = ε ε , ( 4)
w he r e ) ( x Φ i s t he c um ul a t i ve di s t r i but i on f unc t i on of t he s t a nda r d nor m a l di s t r i but i on. B a s e d on
a s i m ul a t i on w i t h 5000 r e pl i c a t i ons , i f t he s a m pl e s i z e i s 1 0 0 = n , 1 . 0 = ε , a nd t he g oa l i s t o
c om put e a c onf i de nc e i nt e r va l f or t he m e a n, t he a c t ua l T y p e I e r r or i s e s t i m a t e d t o be 0.084.
U s i ng t he B a r t l e t t - c or r e c t i on s t udi e d by D i C i c c i o, H a l l a nd R om a no ( 1991) , t he a c t ua l
pr oba bi l i t y c ove r a ge i s 0.078. W he n s a m pl i ng f r om a s ke w e d he a vy - t a i l e d di s t r i but i on, pr a c t i c a l
c onc e r ns a r e e ve n e x a c e r ba t e d. E ve n i f a c c ur a t e pr oba bi l i t y c ove r a ge c oul d be a t t a i ne d, out l i e r s
a nd he a vy - t a i l e d di s t r i but i ons c a n i nf l a t e t he s t a nda r d e r r or of t he s a m pl e m e a n, s o pow e r c a n be
r e l a t i ve l y l ow w he n c om pa r i ng g r oups . T a ke t he e x a m pl e of t he c ont a m i na t e d nor m a l
di s t r i but i on ( E qua t i on 4) . F i gur e 1 s how s t he s t a nda r d nor m a l a nd t he c ont a m i na t e d nor m a l
pr oba bi l i t y de ns i t y f unc t i on c or r e s pondi ng t o E qua t i on ( 4) w i t h 1 . 0 = ε . T he c ont a m i na t e d
nor m a l c ur ve i n F i g ur e 1 i s i n da s he d l i ne s . I t m i g ht s e e m t ha t t he nor m a l di s t r i but i on pr ovi de s a
g ood a ppr ox i m a t i on of t he c ont a m i na t e d nor m a l , but t he s t a nda r d nor m a l ha s va r i a nc e 1, a nd
t he c ont a m i na t e d nor m a l ha s va r i a nc e 10.9. W he n s a m pl i ng f r om t hi s c ont a m i na t e d nor m a l
di s t r i but i on, bot h W e l c h' s a nd S t ude nt ' s m e t hod f or c om pa r i ng t he m e a ns of t w o g r oups ha ve
pow e r a ppr ox i m a t e l y 0.278 w he n t e s t i ng a t t he 0.05 l e ve l w i t h e qua l s a m pl e s i z e of 25 a nd w he n
C O M P A R I N G D E P E N D E N T G R O U P S 8
t he di f f e r e nc e be t w e e n t he m e a ns i s 1 ( W i l c ox , 2005) .
“ M os t da t a s e t s a r e de m ons t r a bl y nonnor m a l ” ( A l l i s on, 2003) . M os t da t a a r e be l l - s ha pe d,
but how w e l l w e c a n a ppr ox i m a t e t he be l l - s ha pe d da t a t o t he nor m a l di s t r i but i on i s i n que s t i on
be c a us e of t he he a vy t a i l s , out l i e r s , a nd s ke w ne s s of t he da t a i n m a ny c a s e s . H e a vy - t a i l e d
di s t r i but i ons , r oughl y m e a ni ng di s t r i but i ons f or w hi c h out l i e r s a r e l i ke l y t o oc c ur , a ppe a r t o be
qui t e c om m on ba s e d on m ode r n out l i e r de t e c t i on t e c hni que s , a s pr e di c t e d by T uke y ( 1960) . T hi s
c oul d l e a d t o poor c ont r ol ove r t he pr oba bi l i t y of a T y p e I e r r or a nd i na c c ur a t e pr oba bi l i t y
c ove r a ge , e s pe c i a l l y w he n c om pa r i ng m e a ns , or r e l a t i ve l y l ow pow e r . M a ny m e t hods ha ve be e n
de ve l ope d i n a n a t t e m pt t o de a l w i t h t hi s pr obl e m , w hi c h i nc l ude m e t hods ba s e d on r obus t
m e a s ur e s of l oc a t i on ( e .g. W i l c ox , 2005) , but no r e s ul t s a r e a va i l a bl e f or t e s t i ng ( 2) t o c om pa r e
m or e t ha n t w o g r oups w i t h m i s s i ng va l ue s ba s e d on r obus t m e a s ur e s of l oc a t i on.
T he f oc us he r e i s on t he 20% t r i m m e d m e a n. W he n s a m pl i ng f r om a nor m a l di s t r i but i on,
l i t t l e e f f i c i e nc y i s l os t w he n us i ng t he 20% t r i m m e d m e a n, ve r s us t he s a m pl e m e a n. H ow e ve r , f or
a he a vy t a i l e d di s t r i but i on, t he e f f i c i e nc y of t he 20% t r i m m e d m e a n c a n be m uc h hi g he r t ha n t he
m e a n. B ut t oo m uc h t r i m m i ng ( t he m a x i m um a m ount of t r i m m i ng i s 50% t r i m m i ng , i .e ., t he
m e di a n) c a n l e a d t o poor e f f i c i e nc y w he n s a m pl i ng f r om a nor m a l di s t r i but i on. T he c hoi c e of
20% t r i m m i ng i s a c om pr om i s e be t w e e n no t r i m m i ng a nd t he m e di a n ( R os e nbe r ge r & G a s ko,
1983; W i l c ox , 1997; W i l c ox , 2001) . A not he r c ons i de r a t i on i n c hoos i ng how m uc h t r i m m i ng i s i n
t e r m s of hy p ot he s i s t e s t i ng . T oo l i t t l e t r i m m i ng r e s ul t s i n poor c ont r ol ove r t he pr oba bi l i t y of a
T y p e I e r r or , poor pr oba bi l i t y c ove r a ge of t he c onf i de nc e i nt e r va l , unde s i r a bl e pow e r pr ope r t i e s ,
a nd r e l a t i ve l y l ow pow e r unde r de pa r t ur e s f r om nor m a l i t y . H ow e ve r , w he n s a m pl i ng f r om a
nor m a l di s t r i but i on, t oo m uc h t r i m m i ng r e s ul t s i n l ow pow e r . A ga i n, t he 20% t r i m m i ng i s a
r e a s ona bl y g ood c om pr om i s e be t w e e n t he m e a n a nd t he m e di a n ( S t a udt e & S he a t he r , 1990;
C O M P A R I N G D E P E N D E N T G R O U P S 9
W i l c ox , 2001) .
M e t h od M e t h od M e t h od M e t h od
F i r s t l y l e t us ha ve a br i e f r e vi e w of t he t r i m m e d m e a n. B a s e d on a r a ndom s a m pl e :
n
X X ⋯ ,
1
, a γ - t r i m m e d m e a n i s c om put e d a s f ol l ow s . L e t
) ( ) 1 ( n
X X ≤ ≤ ⋯ be t he va l ue s w r i t t e n
i n a s c e ndi ng or de r a nd l e t ] [ γ n g = , 5 . 0 0 < ≤ γ , w he r e ] [ γ n i s t he g r e a t e s t i nt e g e r l e s s t ha n or
e qua l t o γ n . T he n t he γ - t r i m m e d m e a n i s
∑
−
+ =
−
=
g n
g i
i
t X
g n
X
1
) (
2
1
. ( 5)
W e a r e f oc us i ng on 20% t r i m m e d m e a n, s o he r e 2 . 0 = γ .
T o t e s t ( 2) , w he r e now θ i s t a ke n t o be t he popul a t i on 20% t r i m m e d m e a n, t he t e s t
s t a t i s t i c i s
∑
=
− =
J
j
t j t
X X Q
1
2
)
ˆ
( , ( 6)
w he r e
j t
X
ˆ
i s t he 20% t r i m m e d m e a n of t he j t h g r oup a nd
∑
=
=
J
j
j t t
J X X
1
/
ˆ
. T he s t r a t e gy i s t o
a ppr ox i m a t e t he nul l di s t r i but i on of Q vi a a boot s t r a p m e t hod. I n m or e de t a i l , s uppos e t he J
g r oups of da t a a r e s t or e d i n t hi s J n × m a t r i x
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
nJ n
J
X X
X X
⋯
⋮
⋯
1
1 11
. ( 7)
F i r s t , s e t
t j i j i j
X X C
ˆ
− = , w he r e
t j
X
ˆ
i s t he 20% t r i m m e d m e a n of t he j t h c ol um n of t he
da t a . T ha t i s , s hi f t t he e m pi r i c a l di s t r i but i ons s o t ha t t he nul l hy p ot he s i s i s t r ue . N e x t a boot s t r a p
s a m pl e i s obt a i ne d by r e s a m pl i ng, w i t h r e pl a c e m e nt , n r ow s of da t a f r om t he m a t r i x
C O M P A R I N G D E P E N D E N T G R O U P S 10
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
nJ n
J
C C
C C
⋯
⋮
⋯
1
1 11
( 8)
y i e l di ng
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
* *
1
*
1
*
11
nJ n
J
C C
C C
⋯
⋮
⋯
( 9)
F or t he j t h c ol um n of t he boot s t r a p da t a j us t g e ne r a t e d, c om put e t he 20% t r i m m e d m e a n
ba s e d on t hi s boot s t r a p s a m pl e . L a be l i t
*
ˆ
t j
X . T he n c om put e
∑
=
− =
J
j
t t j
X X Q
1
2 * * *
)
ˆ
( , ( 10)
w he r e
∑
=
=
J
j
t j t
J X X
1
* *
/
ˆ
. R e pe a t t hi s pr oc e s s B t i m e s y i e l di ng
* *
1
, ,
B
Q Q ⋯ . P ut t he s e B va l ue s
i n a s c e ndi ng or de r :
*
) (
*
) 1 ( B
Q Q ≤ ≤ ⋯ . L e t α be t he nom i na l T y p e I e r r or pr oba bi l i t y , t he n r e j e c t
t he hy p ot he s i s of e qua l 20% t r i m m e d m e a n i f
*
) ( c
Q Q > , w he r e ] ) 1 [ ( B c α − = .
T he i ni t i a l s t r a t e gy w a s t o us e 1 0 0 = B i n or de r t o r e duc e e xe c ut i on t i m e , but t hi s w a s f ound t o
be uns a t i s f a c t or y i n t e r m s of c ont r ol l i ng t he pr oba bi l i t y of a T y p e I e r r or . I n c r e a s i ng B t o 300
l e d t o g ood c ont r ol ove r t he T y p e I e r r or f or a r a nge of s i t ua t i ons , but e x c e pt i ons oc c ur r e d. F or
t he s e l a t t e r s i t ua t i ons , s i m ul a t i ons w e r e r e pe a t e d w i t h 2 0 0 0 = B . T he e x e c ut i on t i m e i s a r ound 2
hour s f or e a c h boot s t r a p l oop. A l a r ge r va l ue f or B w a s not us e d t o a voi d l ong e r e x e c ut i on t i m e ,
t houg h t hi s m i g ht s a c r i f i c e s om e pow e r . L o s s of pow e r i s due t o t he f i ni t e ne s s of B i n pr a c t i c e
a nd t he r a ndom ne s s of boot s t r a p s a m pl i ng . I n c r e a s i ng t he num be r of boot s t r a p s a m pl e s B w i l l
a l w a y s i nc r e a s e t he pow e r of t he t e s t ( R a c i ne a nd M a c K i nnon,2004; D a vi ds on a nd M a c K i nnon,
2000) . J oc ke l ( 1986) pr opos e d a m e t hod t o a ppr ox i m a t e t he l ow e r bound of pow e r l os s , de not e d
C O M P A R I N G D E P E N D E N T G R O U P S 11
by
D
B e ) , ( α , w hi c h i s c a l l e d D w a s s e f f i c i e nc y . T o a c qui r e a D w a s s e f f i c i e nc y of 94.5% , t he
va l ue of B s houl d be 999 w he n t e s t i ng a t 0 5 . 0 = α l e ve l . W he n 2 0 0 0 = B , t he D w a s s e f f i c i e nc y
i s 96.1% . B ne e ds t o be 3360 t o a c qui r e a D w a s s e f f i c i e nc y of 97% a nd 30239 t o a c qui r e 99% .
O t h e r s t ud i e s a b out t he pow e r l os s a n d t he c ho i c e of
B
i nc l ude D a vi d s o n a nd M a c K i nn on ( 20 00)
a nd R a c i ne a nd M a c K i nnon ( 2004) . T he y e i t he r pr opos e d a pr e t e s t pr oc e dur e f or c hoos i ng B , or
c a m e up w i t h a m odi f i e d P va l ue t o r e duc e t he i m pa c t of f i ni t e ne s s a nd r a ndom ne s s of boot s t r a p
s a m pl i ng on t he pow e r of t he t e s t . I n e i t he r s e ns e , 2 0 0 0 = B i s a pr e t t y l a r ge va l ue .
A n a l t e r na t i ve a ppr oa c h w a s c ons i de r e d but a ba ndone d be c a us e i t t ur ne d out t o pe r f or m
uns a t i s f a c t or i l y w he n c om pa r i ng m or e t ha n t w o g r oups w i t h m i s s i ng va l ue s . B r i e f l y , t he da t a i n
t hi s m e t hod a r e not c e nt e r e d, a nd boot s t r a p s a m pl e s a r e obt a i ne d by r e s a m pl i ng r ow s of da t a t ha t
a r e a va i l a bl e . I n t he pr oc e dur e , a w e i ght e d e s t i m a t e of t he g r a nd m e a n i s us e d i ns t e a d of t he
us ua l e s t i m a t e d g r a nd m e a n i n or de r t o he l p de a l w i t h t he he t e r os c e da s t i c i t y a m ong t he m a r gi na l
di s t r i but i ons ( s e e A ppe ndi x ) .
S i m u l at i on S i m u l at i on S i m u l at i on S i m u l at i on D e s i gn D e s i gn D e s i gn D e s i gn
S i m ul a t i ons w e r e us e d t o s t udy t he f i ni t e s a m pl e pr ope r t i e s of t he m e t hod i n S e c t i on
2. D a t a w e r e g e ne r a t e d f r om one of f our di s t r i but i ons : nor m a l , s ke w e d a nd l i g ht - t a i l e d,
non- s ke w e d a nd he a vy - t a i l e d, a nd s ke w e d a nd he a vy - t a i l e d. M or e pr e c i s e l y , a h and g − −
di s t r i but i on w a s us e d t o g e ne r a t e t he da t a .
T o e l a bor a t e , l e t Z be a r a ndom va r i a bl e g e ne r a t e d f r om a s t a nda r d nor m a l di s t r i but i on.
T he n
C O M P A R I N G D E P E N D E N T G R O U P S 12
⎪
⎪
⎩
⎪
⎪
⎨
⎧
=
>
−
=
0 ) ,
2
e x p(
0 ) ,
2
e x p(
1 ) e x p(
2
2
g i f
hZ
Z
g i f
hZ
g
gZ
W ( 11)
ha s a h and g − − di s t r i but i on. H e r e g a nd h a r e t w o pa r a m e t e r s t ha t de t e r m i ne t he s ke w ne s s
a nd he a vy - t a i l e dne s s of t he di s t r i but i on. I n s hor t , w he n 0 = = h g , t he di s t r i but i on of W i s
s t a nda r d nor m a l . W he n 0 > g , t he W di s t r i but i on i s s ke w e d. T he bi g ge r t he va l ue of g , t he
m or e s ke w e d. W he n 0 > h , t he W di s t r i but i on i s he a vy - t a i l e d. T he bi g ge r t he va l ue of h , t he
m or e he a vy - t a i l e d.
T he f our di s t r i but i ons c ons i de r e d he r e a r e : s t a nda r d nor m a l ) 0 , 0 ( = = h g , a s y m m e t r i c
l i g ht - t a i l e d ) 0 , 2 . 0 ( = = h g , s y m m e t r i c he a vy - t a i l e d ) 2 . 0 , 0 ( = = h g , a nd a s y m m e t r i c he a vy -
t a i l e d ) 2 . 0 , 2 . 0 ( = = h g . T a bl e 1 pr ovi de s s om e pr ope r t i e s of t he g - a nd- h di s t r i but i on, w he r e
1
κ
i ndi c a t e s s ke w ne s s a nd
2
κ i ndi c a t e s kur t os i s . G r a ph s of t he s e di s t r i b ut i ons a r e s how n i n F i gur e 2.
W e us e d R t o g e ne r a t e t he da t a f r om t he h and g − − di s t r i but i on, a nd t he n pl ot t e d a n e s t i m a t e of
t he di s t r i but i on w i t h t he R f unc t i on a ke r d, w hi c h i s a n a da pt i ve ke r ne l de ns i t y e s t i m a t or de r i ve d
by W i l c ox ( 2012) .
W e f oc us e d on s m a l l s a m pl e s w i t h m e di um a m ount of m i s s i ng va l ue s i n t hi s s t udy .
T he s a m pl e s i z e us e d w a s 3 0 = n w i t h 5 m i s s i ng va l ue s ( i .e ., 6 / 1 m i s s i ng ) pe r g r oup. W he t he r
our r e s ul t s a l s o a ppl y f or ot he r s a m pl e s i z e s w i t h ot he r a m ount s of m i s s i ng va l ue s r e m a i ns
unknow n. L e t
i
n be t he num be r of m i s s i ng va l ue s i n t he i t h g r oup. W e c om pa r e d 2, 4, a nd 6
g r oups . W he n t w o g r oups w e r e c om pa r e d, ) 5 , 5 ( ) , (
2 1
= n n . W he n f our g r oups w e r e c om pa r e d,
) 5 , 5 , 5 , 5 ( ) , , , (
4 3 2 1
= n n n n . W he n s i x g r oups w e r e c om pa r e d, ) 5 , 5 , 5 , 5 , 5 , 5 ( ) , , , , , (
6 5 4 3 2 1
= n n n n n n .
C O M P A R I N G D E P E N D E N T G R O U P S 13
R e s u l t s R e s u l t s R e s u l t s R e s u l t s
T a bl e 2 r e por t s t he e s t i m a t e d T y p e I e r r or s ba s e d on 2000 r e pl i c a t i ons . T he num be r of
r e pl i c a t i ons w a s c hos e n t o bot h ke e p e x e c ut i on t i m e a t a r e a s ona bl e l e ve l a nd t he T y p e I e r r or
e s t i m a t e d r e a s ona bl y a c c ur a t e .
G e ne r a l l y s pe a ki ng, t hi s m e t hod pe r f or m s s a t i s f a c t or i l y w he n c om pa r i ng 2 a nd 4 g r oups ,
a nd pa r t i a l l y s a t i s f a c t or i l y w he n c om pa r i ng 6 g r oups , w i t h s a m pl e s i z e 3 0 = n a nd m i s s i ng
pr opor t i on 6 / 1 = pe r g r oup. A l l of t he t y p e I e r r or s e s t i m a t e d a r e be l ow 0.07 w he n c om pa r i ng 2
g r oups . W he n c om pa r i ng 4 or 6 g r oups , a l l of t he t y p e I e r r or s e s t i m a t e d a r e be l ow 0.05 ( s om e
a r e e ve n be l ow 0.04 a nd 0.03, w hi c h i s t oo c ons e r va t i ve ) . R e f e r r i ng t o B r a dl e y 's c r i t e r i on ( 1978) ,
w hi c h s t a t e s t ha t t he e s t i m a t e d T y p e I e r r or s houl d be be t w e e n 0.025 a nd 0.075, our r e s ul t s s how
t ha t a l l e s t i m a t e d T y p e I e r r or s w he n c om pa r i ng 2 a nd 4 g r oups a r e i n t hi s r a nge , but t he r e s ul t s
f or 6 g r oups g o be y o nd t hi s r a nge unde r t he di s t r i but i on of s y m m e t r i c he a vy - t a i l e d a nd t he
di s t r i but i on of a s y m m e t r i c he a vy - t a i l e d w i t h e s t i m a t e d T y p e I e r r or s be i ng 0.022 a nd 0.020
r e s pe c t i ve l y .
T he a l t e r na t i ve a ppr oa c h m e nt i one d i n s e c t i on 2 ha d e s t i m a t e d t y p e I e r r or s a l l be l ow
0.03 w he n c om pa r i ng 2 g r oups , a nd w e nt a s hi g h a s 0.092 f or 4 = J a nd 0.29 f or 6 = J . ( S e e
A ppe ndi x )
C on c l u s i on s C on c l u s i on s C on c l u s i on s C on c l u s i on s
I n s um m a r y , t he m e t hod i n S e c t i on 2 w a s f ound t o be r obus t w he n c om pa r i ng 2 a nd 4
de pe nde nt g r oups w i t h s m a l l s a m pl e s i z e s ( 3 0 = n ) a nd a m ode r a t e a m ount of m i s s i ng va l ue s
( 6 / 1 m i s s i ng va l ue s pe r g r oup) a c r os s t he f our s i m ul a t e d di s t r i but i ons : s t a nda r d nor m a l ,
a s y m m e t r i c l i g ht - t a i l e d, s y m m e t r i c he a vy - t a i l e d, a nd a s y m m e t r i c he a vy - t a i l e d. T he m e a s ur e of
C O M P A R I N G D E P E N D E N T G R O U P S 14
l oc a t i on be i ng c om pa r e d w a s 20% t r i m m e d m e a n, a nd t he r e s ul t s w e r e r e ve a l e d i n e s t i m a t e d
T y p e I e r r or s . W he n c om pa r i ng 6 g r oups , t he m e t hod r e m a i ne d r obus t unde r t he di s t r i but i ons of
s t a nda r d nor m a l a nd a s y m m e t r i c l i g ht - t a i l e d. A s t he di s t r i but i on be c a m e m or e he a vy - t a i l e d, t he
r e s ul t s be c a m e t oo c ons e r va t i ve a nd w e nt be l ow 0.025. W he t he r t hi s m e t hod s t i l l r e m a i ns r obus t
w he n t he s a m pl e s i z e i s s m a l l e r or t he pr opor t i on of m i s s i ng va l ue s i s l a r ge r a nd une qua l ne e ds
t o be e x pl or e d f ur t he r i n t he f ut ur e .
C O M P A R I N G D E P E N D E N T G R O U P S 15
R e f e r e n c e s R e f e r e n c e s R e f e r e n c e s R e f e r e n c e s
A l l i s on, P . D . ( 2001) . M i s s i ng dat a. T hous a nd O a ks , C A : S a g e .
A l l i s on, P . D . ( 2003) . M i s s i ng da t a t e c hni que s f or s t r uc t ur a l e qua t i on m ode l i ng . J our nal of
A bnor m al P s y c hol ogy , 112, 4, 545- 557.
B oi k, R . J . ( 1987) . T he F i s he r - P i t m a n pe r m ut a t i on t e s t : A non- r obus t a l t e r na t i ve t o t he nor m a l
t he or y F t e s t w he n va r i a nc e s a r e he t e r oge ne ous . B r i t i s h J our nal of M at he m at i c al and
St at i s t i c al P s y c hol ogy , 40, 26- 42.
B r a dl e y , J . V . ( 1978) R obus t ne s s ? B r i t i s h J our nal of M at he m at i c al and St at i s t i c al P s y c hol ogy ,
31, 144- 152.
C oc hr a n, W . G . ( 1977) . Sam pl i ng t e c hni que s ( 3r d E d.) . N e w Y or k: W i l e y .
D a ni e l s , M . J . a nd H og a n, J . W . ( 2008) . M i s s i ng dat a i n l ongi t udi nal s t udi e s : s t r at e gi e s f or
B ay e s i an m ode l i ng and s e ns i t i v i t y anal y s i s . B oc a R a t on, F L : C ha pm a n & H a l l / C R C .
D a vi ds on, R . a nd M a c K i nnon, J . G . ( 2000) . B oot s t r a p t e s t s : H ow m a ny boot s t r a ps ? E c onom e t r i c
R e v i e w s , 19, 55- 68.
D om hof , S ., B r unne r , E . a nd O s g ood, D . ( 2002) . R a nk pr oc e dur e s f or r e pe a t e d m e a s ur e s w i t h
m i s s i ng va l ue s . Soc i ol ogi c al M e t hods & R e s e ar c h , 30, 367- 393.
E f r on, B . a nd T i bs hi r a ni , R . J . ( 1993) A n i nt r oduc t i on t o t he boot s t r ap. N e w Y or k: C ha pm a n &
H a l l .
H a l l , P . ( 1988a ) . O n s y m m e t r i c boot s t r a p c on f i de nc e i nt e r va l s . J our nal of t he R oy al St at i s t i c al
Soc i e t y , S e r i e s B , 50, 35- 45.
H a l l , P . ( 1988b) . T he or e t i c a l c om pa r i s on of boot s t r a p c onf i de nc e i nt e r va l s . A nnal s of St at i s t i c s ,
16, 927- 953.
H a l l , P . a nd W i l s on, S . R . ( 1991) . T w o g ui de l i ne s f or boot s t r a p hy p ot he s i s t e s t i ng . B i om e t r i c s ,
C O M P A R I N G D E P E N D E N T G R O U P S 16
47, 757- 762.
H oa gl i n, D . C . ( 1985) S um m a r i z i ng s ha pe num e r i c a l l y : T he g - a nd- h di s t r i but i ons . I n D . H oa gl i n,
F . M os t e l l e r a nd J . T uke y ( E ds .) , E x pl or i ng dat a t abl e s , t r e nds , and s hape s ( pp. 461{515) .
N e w Y or k: W i l e y .
I b r a hi m , J . G ., C he n, M . H . a nd L i ps i t z , S . R . ( 1999) . M ont e C a r l o E M f or m i s s i ng c ova r i a t e s i n
pa r a m e t r i c r e gr e s s i on m ode l s . B i om e t r i c s , 55, 591- 596.
I b r a hi m , J . G ., L i ps i t z , S . R . a nd H or t on, N . ( 2001) . U s i ng a ux i l i a r y da t a f or pa r a m e t e r
e s t i m a t i on w i t h noni g nor a bl e m i s s i ng out c om e s . A ppl i e d St at i s t i c s , 50, 361- 373.
J oc ke l , K .- H . ( 1986) . F i ni t e s a m pl e pr ope r t i e s a nd a s y m pt ot i c e f f i c i e nc y of M ont e C a r l o t e s t s .
A nnal s of St at i s t i c s , 14, 336- 347.
K e s e l m a n, H . J ., L i x , L . M . a nd K ow a l c huk, R . K . ( 1998) . M ul t i pl e c om pa r i s on pr oc e dur e s f or
t r i m m e d m e a ns . P s y c hol ogi c al M e t hods , 3, 123- 141.
L i a ng, H ., S u, H . a nd Z ou, G . ( 2008) . C on f i de nc e i nt e r va l s f or a c om m on m e a n w i t h m i s s i ng
da t a w i t h a ppl i c a t i ons i n a n A I D S s t udy . C om put at i onal St at i s t i c s & D at a A nal y s i s , 53,
546- 553.
L i t t l e , R . J . A . a nd R ubi n, D . ( 2002) . St at i s t i c al anal y s i s w i t h m i s s i ng dat a, 2nd E d. N e w Y or k:
W i l e y .
L i u, R . G . a nd S i ng h, K . ( 1997) . N ot i ons of l i m i t i ng P va l ue s ba s e d on da t a de pt h a nd boot s t r a p.
J our nal of t he A m e r i c an St at i s t i c al A s s oc i at i on , 92, 266- 277.
L u dbr ook, J . ( 2008) . O ut l y i ng obs e r va t i ons a nd m i s s i ng va l ue s : H ow s houl d t he y be ha ndl e d?
C l i ni c al and E x pe r i m e nt al P har m ac ol ogy & P hy s i ol ogy , 35, 670- 678.
M c K ni ght , P . E ., M c K ni g ht , K . M ., S i da ni , S . a nd F i gue r e do, A . J . ( 2007) . M i s s i ng dat a: a
ge nt l e i nt r oduc t i on. N e w Y or k: G ui l f or d P r e s s .
C O M P A R I N G D E P E N D E N T G R O U P S 17
R a c i ne , J . a nd M a c K i nnon, J . G . ( 2007) . S i m ul a t i on- ba s e d t e s t s t ha n c a n us e a ny num be r of
s i m ul a t i ons . C om m uni c at i ons i n St at i s t i c s - Si m ul at i on and C om put at i on, 36, 357- 365.
N g, M . a nd W i l c ox , R . R . ( 2010) . C om pa r i ng t he R e g r e s s i on S l ope s of I n de pe nde nt G r oups .
B r i t i s h J our nal of M at he m at i c al and St at i s t i c al P s y c hol ogy , 63, 319- 340.
R og a n, J . C ., K e s e l m a n, H . J . a nd M e ndoz a , J . L . ( 1979) . A na l y s i s of r e pe a t e d m e a s ur e m e nt s .
B r i t i s h J our nal of M at he m at i c al and St at i s t i c al P s y c hol ogy , 32, 269- 286.
R om a no, J . P . ( 1990) . O n t he be ha vi or of r a ndom i z a t i on t e s t s w i t hout a g r oup i nva r i a nc e
a s s um pt i on. J our nal of t he A m e r i c an St at i s t i c al A s s oc i at i on 85, 686- 692.
R ubi n, D . B . ( 1976) . I n f e r e nc e a nd m i s s i ng da t a . B i om e t r i k a, 63, 581- 592.
R ubi n, D . B . ( 1987) . M ul t i pl e i m put at i on f or nonr e s pons e i n s ur v e y s . N e w Y or k: W i l e y .
S t a udt e , R . G . a nd S he a t he r , S . J . ( 1990) . R obus t e s t i m at i on and t e s t i ng . N e w Y or k: W i l e y .
T e m pl , M . a nd F i l z m os e r , P . ( 2008) . V i s ual i z at i on of m i s s i ng v al ue s us i ng t he R - pac k age V I M .
U npubl i s he d doc t or a l di s s e r t a t i on, D e pt . of S t a t i s t i c s a nd P r oba bi l i t y T he or y , V i e nna
U ni ve r s i t y of T e c hnol ogy .
T uke y , J . W . ( 1960) . A s ur ve y of s a m pl i ng f r om c ont a m i na t e d nor m a l di s t r i but i ons . I n I . O l ki n
e t a l . ( E ds .) C ont r i but i ons t o P r obabi l i t y and St at i s t i c s . S t a nf or d, C A : S t a nf or d
U ni ve r s i t y P r e s s .
V a l l e j o, G ., F e r nnde z , M . P ., L i va c i c - R oj a s , P . E . a nd T ue r o- H e r r e r o, E . ( 2011) . C om pa r i s on of
M ode r n M e t hods f or A na l y z i ng R e pe a t e d M e a s ur e s D a t a W i t h M i s s i ng V a l ue s .
M ul t i v ar i at e B e hav i or al R e s e ar c h, 46, 900- 937.
W a ng , Q .H . a nd R a o, J . N . K . ( 2002) . E m pi r i c a l l i ke l i hood- ba s e d i nf e r e nc e i n l i ne a r m ode l s w i t h
m i s s i ng da t a . Sc andanav i an J our nal of St at i s t i c s , 29, 563- 576.
C O M P A R I N G D E P E N D E N T G R O U P S 18
W e s t f a l l , P . H . a nd Y oung, S . S . ( 1993) . R e s am pl i ng bas e d m ul t i pl e t e s t i ng . N e w Y or k: W i l e y .
W i l c ox , R . R . ( 2005) . I nt r oduc t i on t o r obus t e s t i m at i on and hy pot he s i s t e s t i ng ( 2nd e d.) . S a n
D i e go, C A : A c a de m i c P r e s s .
W i l c ox , R . R . ( 2006) . A not e on i nf e r e nc e s a bout t he m e di a n of di f f e r e nc e s c or e s . E duc at i onal
and P s y c hol ogi c al M e as ur e m e nt , 66, 624- 630.
W u, P . C . ( 2002) . C e nt r al l i m i t t he or e m and c om par i ng m e ans , t r i m m e d m e ans one - s t e p M -
e s t i m at or s and m odi f i e d one - s t e p M e s t i m at or s unde r non- nor m al i t y . U npubl i s he d
doc t or a l di s s e r t a t i on, D e pt . of E duc a t i on, U ni ve r s i t y of S out he r n C a l i f or ni a .
C O M P A R I N G D E P E N D E N T G R O U P S 19
A p p e n d i x A p p e n d i x A p p e n d i x A p p e n d i x
A s m e nt i one d i n S e c t i on 2, t he a l t e r na t i ve m e t hod t ur ne d out t o pe r f or m uns a t i s f a c t or i l y
w he n c om pa r i ng 2 a nd m or e t ha n 2 g r oups w i t h m i s s i ng va l ue s . B r i e f l y , t hi s m e t hod e s t i m a t e s
t he c om m on m e a s ur e of l oc a t i on a nd t he n c he c ks t o s e e how de e pl y i t i s ne s t e d w i t hi n t he
boot s t r a p va l ue s obt a i ne d w he n r e s a m pl i ng f r om t he or i gi na l va l ue s . T ha t i s , t he da t a a r e not
c e nt e r e d, a nd boot s t r a p s a m pl e s a r e obt a i ne d by r e s a m pl i ng r ow s of da t a t ha t a r e a va i l a bl e f r om
t he or i gi na l da t a s e t
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1
1 11
F ur t he r m or e , i n t he pr oc e dur e , a w e i ght e d e s t i m a t e of t he g r a nd t r i m m e d m e a n i s us e d
i ns t e a d of t he us ua l e s t i m a t e d g r a nd t r i m m e d m e a n i n or de r t o he l p de a l w i t h t he
he t e r os c e da s t i c i t y a m ong t he m a r gi na l di s t r i but i ons . T he r e s ul t s of t hi s m e t hod a r e s how n i n
T a bl e 3.
W he n c om pa r i ng 2 g r oups , a l l of t he e s t i m a t e d T y p e I e r r or s a r e be l ow 0.03. O nl y w he n
t he di s t r i but i on i s s t a nda r d nor m a l i s t he T y p e I e r r or a bove 0.025 ( r e f e r r i ng t o B r a dl e y 's
c r i t e r i on) . W he n c om pa r i ng 4 g r oups , t he r e s ul t s s how t ha t t he e s t i m a t e d T y p e I e r r or s r e m a i n i n
B r a dl e y 's r a nge ( 0.055 a nd 0.051 r e s pe c t i ve l y ) unde r t he di s t r i but i ons of a s y m m e t r i c a nd
s y m m e t r i c he a vy - t a i l e d, but t he r e s ul t s g o be y o nd t he r a nge unde r t he di s t r i but i ons of s t a nda r d
nor m a l a nd l i g ht - t a i l e d. T hi s i s not nor m a l be c a us e us ua l l y t he r e s ul t s unde r nor m a l di s t r i but i on
a r e be t t e r t ha n unde r ot he r di s t r i but i ons . W he n t he num be r of c om pa r e d g r oups r e a c he s 6, t he
e s t i m a t e d T y p e I e r r or s s e t a s l a r ge a s 0.295, w hi c h i s a c om pl e t e c ol l a ps e .
O ve r a l l , t hi s m e t hod di d not pe r f or m w e l l w he n c om pa r i ng 2, 4, or 6 g r oups . F or 2
g r oups , t he r e s ul t s a r e w a y t oo c ons e r va t i ve . F or 4 g r oups , t he r e s ul t s a r e pa r t i a l l y i n t he i de a l
C O M P A R I N G D E P E N D E N T G R O U P S 20
r a nge , but onl y w i t h he a vy - t a i l e d di s t r i but i ons . W he n c om pa r i ng 6 g r oups , t hi s m e t hod c ol l a ps e s .
C O M P A R I N G D E P E N D E N T G R O U P S 21
T a bl e 1
Som e pr ope r t i e s of t he g- and- h di s t r i but i on.
C O M P A R I N G D E P E N D E N T G R O U P S 22
T a bl e 2
E s t i m at e d t y pe I e r r or s w i t h 0 5 . 0 = α and 3 0 = n .
C O M P A R I N G D E P E N D E N T G R O U P S 23
T a bl e 3
E s t i m at e d t y pe I e r r or s w i t h 0 5 . 0 = α and 3 0 = n ( al t e r nat i v e m e t hod) .
C O M P A R I N G D E P E N D E N T G R O U P S 24
F i gur e 1. S t a nda r d nor m a l a nd c ont a m i na t e d nor m a l di s t r i but i ons . .
C O M P A R I N G D E P E N D E N T G R O U P S 25
F i gur e 2. F our di s t r i but i ons i n da t a s i m ul a t i on.
Abstract (if available)
Abstract
In this study, we investigate the robust methods for comparing two or more groups when there are missing values. We use the 20%-trimmed mean as the measure of location and conduct a percentile bootstrap method to test the robustness when comparing groups with missing values. Both skewed and heavy-tailed distributions are considered. Our results indicate that the method tested remains satisfactorily robust when comparing two or four groups with equal sample sizes of 30 and 5 missing values per group.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Ma, Jinxia
(author)
Core Title
Comparing dependent groups with missing values: an approach based on a robust method
School
College of Letters, Arts and Sciences
Degree
Master of Arts
Degree Program
Psychology
Publication Date
11/19/2012
Defense Date
11/19/2012
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
bootstrap,missing value,OAI-PMH Harvest,robust,trimmed mean
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Wilcox, Rand R. (
committee chair
), John, Richard S. (
committee member
), McArdle, John J. (
committee member
)
Creator Email
jinxia.ma@gmail.com,jinxiama@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-116024
Unique identifier
UC11289446
Identifier
usctheses-c3-116024 (legacy record id)
Legacy Identifier
etd-MaJinxia-1311.pdf
Dmrecord
116024
Document Type
Thesis
Rights
Ma, Jinxia
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
bootstrap
missing value
robust
trimmed mean