metaanalysis in clinical trials

8/7/2019 Metaanalysis in Clinical Trials

http://slidepdf.com/reader/full/metaanalysis-in-clinical-trials 1/12

Meta-Analys i s in Cl in ica l Tr ia l s*

R e b e c c a D e r S i m o n i a n a n d N a n L a ir d

AB S T RACT : T h i s p a p e r e x a m i n e s e i g h t p u b l i s h e d r e v i e w s e a c h re p o r t i n g r e s u l t s f r o m s e v e r alr e l a t e d t r ia l s. E a c h r e v i e w p o o l s t h e r e s u l t s f r o m t h e r e l e v a n t t ri a ls in o r d e r t o e v a l u a t et h e e f f i c ac y o f a c e rt a i n t r e a t m e n t f o r a s p e c i f ie d me d i c a l c o n d i t i o n . T h e s e r e v i e w sl a ck c o n s i s t e n t a s s e s s m e n t o f h o m o g e n e i t y o f t r e a t m e n t e f fe c t b e f o r e p o o li n g . W ed i s c u s s a r a n d o m e f fe c ts a p p r o a c h t o c o m b i n i n g e v i d e n c e f r o m a s e r ie s o f e x p e r i m e n t sc o m p a r i n g t w o t r e a t m e n t s . T h i s a p p r o a c h i n c o r p o r a t e s t h e h e t e r o g e n e i t y o f e f fe c ts i nt h e a n a l y s i s o f th e o v e r a l l t r e a t m e n t e f fi ca c y. T h e m o d e l c a n b e e x t e n d e d t o i n c lu d er e l e v a n t c o v a r ia t e s w h i c h w o u l d r e d u c e t h e h e t e r o g e n e i t y a n d a l l o w f o r m o r e s p e c if ict h e r a p e u t i c r e c o m m e n d a t i o n s . W e s u g g e s t a s i m p l e n o n i t e r a t i v e p r o c e d u r e f o r c h a r -a c t e r i z i n g t h e d i s t r i b u t i o n o f t r e a t m e n t e f f e c ts i n a s e ri e s o f s t u d i e s .

K E Y W O R D S : rand om effects model, heter ogen eity of trea tm ent effects, distribution of trea tme nt effects ,covariate information

I N T R O D U C T I O N

M e t a - a n a l y s i s i s d e f i n e d h e r e a s t h e s t at i st ic a l a n a l y s i s o f a c o l l e c ti o n o f

a n a l y t ic r es u l t s f o r t h e p u r p o s e o f i n t e g r a t i n g t h e f i n d i n g s . S u c h a n a l y s e s a r e

b e c o m i n g i n c re a s in g l y p o p u l a r i n m e d i c a l r es e a rc h w h e r e i n f o r m a t i o n o n

e f f ic a c y o f a t r e a t m e n t i s a v a i l a b l e f r o m a n u m b e r o f cl in i ca l s t u d i e s w i t h

s im i l a r t r e a t m e n t p r o to c o l s . I f c o n s i d e r e d s e p a ra t e ly , a n y o n e s t u d y m a y b e

e i t h e r t o o s m a l l o r t o o l im i t e d i n s c o p e t o c o m e t o u n e q u i v o c a b l e o r g e n e r -

a l iz a b le c o n c l u s i o n s a b o u t t h e e f fe c t o f t r e a t m e n t . C o m b i n i n g t h e f i n d i n g s

a c r o s s s u c h s t u d i e s r e p r e s e n t s a n a t t r ac t iv e a l t e r n a t i v e to s t r e n g t h e n t h e e v i-

d e n c e a b o u t t h e t r e a t m e n t e f f i c a c y .

T h e m a i n d i f fi c u lt y i n in t e g r a t i n g t h e r e s u l ts f r o m v a r i o u s s t u d i e s s t e m s

f r o m t h e s o m e t i m e s d i v e r s e n a t u r e o f t h e s tu d i e s, b o t h i n t e r m s o f d e s ig n

a n d m e t h o d s e m p l o y e d . S o m e a r e c a re f u l ly c o n t ro l l e d r a n d o m i z e d e x p e r i -

Yale University, S choo l of Medicine, N ew Hav en, C onnecticut (R.D.); Harv ard University,School of Public Health, Boston, Massachusetts (N.L.)

*This research was sup po rted by gra nt CA09424-03 from the National Cancer Institute andgra nt GM -29745 from th e National Institute of Health. W e are grateful to Frederick M osteller,Tom Lou is, an d Katherine H alvorsen for critical readings o f various drafts, encouragem ent, andadvice.

Address reprint request to: Rebecca DerSimo nian, Yale University School of Medicine, P.O. Box 3333,New Haven, CT 06510.

Received Ma rch 25, 1986; accepted April 7, 1986.

Con trolled C linical Trials 7:177-188 (1986) 177© ElsevierScience Pub lishing Co., In c. 1986 0197-2456/86/$3.5052 Vanderbi|t Av e., New York, New York 10017



178 R. DerSimonian and N. Laird

m e n t s w h i l e o t h e r s a r e l e s s w e l l c o n t ro l l ed . B e c a u s e o f d i f fe r in g s a m p l e s iz e s

a n d p a t i e n t p o p u l a t i o n s , e a c h s t u d y h a s a d i f f e re n t le v e l o f s a m p l i n g e r r o r

a s w e l l. T h u s o n e p r o b l e m i n c o m b i n i n g s t u d i e s f o r i n t e g ra t i v e p u r p o s e s i s

t h e a s s i g n m e n t o f w e i g h t s t h a t r e f le c t t h e r e l a ti v e " v a l u e " o f t h e i n f o r m a t i o np r o v i d e d i n a s t u d y . A m o r e d i f f ic u l t i s s u e i n c o m b i n i n g e v i d e n c e i s t h a t o n e

m a y b e u s i n g i n c o m m e n s u r a b l e s t u d ie s t o a n s w e r t h e s a m e q u e s t i o n . A r -

m i t a g e [ 1] e m p h a s i z e s t h e n e e d f o r c a r e fu l c o n s i d e r a t i o n o f m e t h o d s i n d r a w -

i n g i n f e r e n c e s f r o m h e t e r o g e n e o u s b u t l o g ic a ll y r e l a te d s t u d i e s . I n t h is s e t t in g ,

t h e u s e o f a r e g r e s s i o n a n a l y s i s t o c h a r a c te r i ze d i f f e r e n c e s i n s t u d y o u t c o m e s

m a y b e m o r e a p p r o p r i a t e [2].

T h i s p a p e r d i s c u s s e s a n a p p r o a c h t o m e t a - a n a l y s i s w h i c h a d d r e s s e s t h e s e

t w o p r o b l e m s . I n t h is a p p r o a c h , w e a s s u m e t h a t t h e r e is a d i s t r ib u t i o n o f

t r e a t m e n t e f f ec t s a n d u t il iz e t h e o b s e r v e d e f f e c t s f r o m i n d i v i d u a l s t u d i e s t o

e s t i m a t e t h i s d is t r ib u t i o n . T h e a p p r o a c h a l l o w s f o r t r e a t m e n t e f f e c ts t o v a r y

a c ro s s s t u d i e s a n d p r o v i d e s a n o b j e c t iv e m e t h o d f or w e i g h t i n g t h a t c a n b e

m a d e p r o g r e s s i v e l y m o r e g e n e r a l b y i n c o r p o r a t i n g s t u d y c h a ra c t er i st ic s i n to

t h e a n a l y s is . W e i ll u s tr a te t h e u s e o f t h is m o d e l i n s e v e r a l e x a m p l e s , a n d

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

t e s t i n g a n d e s t i m a t i o n .

DATABA S E

I n a s y s t e m a t i c s e a r c h o f t h e f ir st t e n i s s u e s p u b l i s h e d i n 1 9 82 o f e a c h o f

f o u r w e e k l y j o u r n a ls (NEJM, JAMA, BMJ, a n d Lancet), H a l v o r s e n [ 3 ] f o u n d

o n l y o n e a r ti cl e ( o u t o f 5 89 ) t h a t c o n s i d e r e d c o m b i n i n g r e s u l ts u s i n g f o r m a l

s t a t i s t i c a l m e t h o d s . O u r d a t a c o n s i s t o f a n a d h o c c o l l e c t i o n o f s u c h a r t i c l e s

f r o m t h e m e d i c a l l it e ra t u re f o u n d t h r o u g h r e f er e n c e s p r o v i d e d b y c o l le a g u e s

a n d t h r o u g h b i b l i o g r a p h i c r e f e r e n c e s i n a r t i c l e s a l r e a d y l o c a t e d [ 4 - 1 1 ] . T h e

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

o u r a t t e n t i o n s i n c e o u r o r i g i n a l a n a l y s e s [ 1 2 - 1 4 ] .

W e e x a m i n e i n d e t a i l e i g h t r e v i e w a r t i c l e s e a c h r e p o r t i n g r e s u l t s f r o m

s e v e r a l r e l a t e d t r i a l s . E a c h r e v i e w p o o l s t h e r e s u l t s f r o m t h e r e l e v a n t t r i a l s i n

o r d e r t o e v a l u a t e t h e e f f i c a c y o f a c e r ta i n t r e a t m e n t f o r a s p e c i f i e d m e d i c a l

c o n d i t i o n . I n m o s t o f th e s e r e v i e w s t h e o r ig i n al i n v e s t i g a t o r s p o o l t h e r e s u l t s

f r o m t h e r e l e v a n t t r ia ls a n d e s t i m a t e a n o v e r a l l t r e a t m e n t e f fe c t w i t h o u t f i rs tc h e c k i n g w h e t h e r t h e t r e a t m e n t e f f e ct a c r o s s th e t ri al s i s c o n s t a n t . O t h e r s

e x c l u d e s o m e t r i a l s a n d c o m b i n e t h e r e s u l t s o n l y f r o m t r i a l s t h a t a r e s i m i l a r

i n d e s i g n a n d i m p l e m e n t a t i o n . T h e i n v e s ti g a to r s w h o d o c h e c k fo r h o m o -

g e n e i t y o f t r e a t m e n t e f f e ct b e f o r e p o o l i n g u s e d i f f e re n t c r it er ia t o a s s e s s t h is

h o m o g e n e i t y . W i th t w o e x c e p t i o n s [ 6,1 1], t h e r e v i e w s c o n s id e r r a n d o m i z e d

t ri al s on l y . T h e t w o r e v i e w s t h a t in c l u d e n o n r a n d o m i z e d s t u d i e s a n a ly z e t h e

d a t a fr o m th e tw o g r o u p s o f s t u d ie s ( r a n d o m i z e d a n d n o n r a n d o m i z e d ) s e p -

a r a te l y . I n t h is s t u d y , w e r e s t ri c t o u r a t t e n t i o n t o t h e r e s u l t s o f r a n d o m i z e d

t r i a l s o n l y . W e f i r s t d e s c r i b e t h e e i g h t r e v i e w s i d e n t i f y i n g e a c h b y i t s f i r s t

a u t h o r , a n d i n T a b le 1 s u m m a r i z e t h e m e t h o d s u s e d i n e a c h r e v ie w :

W i n s h i p : A r e v i e w o f e i g h t tr ia ls t h a t c o m p a r e t h e h e a l i n g r a te s i n d u o d e n a l

u l c e r p a t i e n t s t r e a t e d w i t h c i m e t i d i n e o r p l a c e b o t h e r a p y [4 ].



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1 8 0 R. DerSimonian an d N. Laird

C o n n : A r e v i e w o f n i n e t ri al s t h a t c o m p a r e t h e s u r v i v a l r a t e s in a l c o h o li c

h e p a t i t is p a t i e n t s w i t h s t e r o i d s o r c o n t r o l t h e r a p y [5 ].

M i a o : A r e v i e w o f s ix tr ia ls t h a t c o m p a r e g a s tr ic w i t h s h a m f r e e z in g i n t h e

t r e a t m e n t o f d u o d e n a l u l c e r [ 6]. In a d d i t i o n , t h is r e v i e w c o n s i d e r s 1 4 o b -s e r v at io n a l a n d t w o c o n t r o l le d b u t n o n r a n d o m i z e d s t u d ie s .

D e S i l v a : A r e v i e w o f si x t ri a ls t h a t e v a l u a t e t h e e f f e c t o f l i g n o c a i n e o n t h e

i n c i d e n c e o f v e n t r i c u l a r f i b ri ll a ti o n i n a c u t e m y o c a r d i a l i n f a r c t i o n [7 ]. T h i s

s t u d y o r i g i n a l l y c o n s i d e r e d 1 5 t r i a l s b u t b e c a u s e t h e t r i a l s v a r y w i d e l y i n

t r e a t m e n t s c h e d u l e s a n d d o s e s , s o m e c ri te ri a fo r a d e q u a c y o f t r e a t m e n t

a r e e s t a b l i s h e d a n d o n l y s i x t r i a l s t h a t f u l f i l l t h e s e r e q u i r e m e n t s a r e a n a -

l y z e d .

S t j e r n s w a r d : A r e v i e w o f f iv e t ri al s t h a t c o m p a r e t h e 5 - y e a r s u r v i v a l r a t e s o f

p a t i e n t s w i t h c a n c e r o f t h e b r e a s t t r e a t e d w i t h s u r g e r y p l u s r a d i o t h e r a p y

o r s u r g e r y a l o n e [ 8 ] .

B a u m : A r e v i e w o f 26 tr ia l s t h a t e v a l u a t e t h e e f f ic a c y o f a n t i b i o t ic s i n t h e

p r e v e n t i o n o f w o u n d i n f e c t i o n f o ll o w i n g c o l o n s u r g e r y [9 ].

P e t o : A r e v i e w o f s ix tr ia l s th a t e v a l u a t e t h e e f f i c a c y o f a s p i r i n i n t h e p r e v e n t i o n

o f s e c o n d a r y m o r t a li t y i n p e r s o n s r e c o v e r e d f r o m m y o c a r d i a l in f a r c ti o n

[101.

C h a l m e r s : A r e v i e w o f a n u m b e r o f t r ia l s t h a t e v a l u a t e t h e e f f i c a c y o f a n t i-

c o a g u l a n t s i n t h e t r e a t m e n t o f a c u t e m y o c a r d i a l i n f a rc t i o n [ 11 ]. D a t a f r o m

1 8 s u r v e y s e m p l o y i n g h i s t o r i c a l c o n t r o l s ( H C T ) , e i g h t s t u d i e s e m p l o y i n g

a l t e r n a te l y a s s i g n e d c o n t r o l s ( A C T ) , a n d s ix r a n d o m i z e d c o n t r o l l e d tr ia ls( R C T ) a r e g i v e n . T h r e e e n d p o i n t s , t o t a l c a s e f a t a li t y r a t e s , c a s e f a t a li t y r a t e s

e x c l u d i n g e a r l y d e a t h s , a n d t h r o m b o e m b o l i s m r a t e s a r e c o n s i d e r e d , a l -

t h o u g h n o t a l l s t u d i e s r e p o r t a l l t h r e e e n d p o i n t s . H e r e w e c o n s i d e r t h r o m -

b o e m b o l i s m a n d t o ta l c a s e fa t a li ty r a te s i n t h e R C T s o n l y . T h e r e s u l t s f r o m

r a n d o m i z e d t ria ls ar e c o m p a r e d t o t h o s e o f n o n r a n d o m i z e d o n e s ( H C T s

a n d A C T s ) i n L a i r d a n d D e r S i m o n i a n [ 1 6 ] .

ME T HO D S

W e c o n s i d e r t h e p r o b l e m o f c o m b i n i n g i n f o rm a t i o n f r o m a s er ie s o f k

c o m p a r a t i v e c li ni ca l t ri al s, w h e r e t h e d a t a f r o m e a c h H a l c o n s i s t o f t h e n u m b e r

o f p a t i e n t s i n t r e a t m e n t a n d c o n t r o l g r o u p s , nT a n d n o , a n d t h e p r o p o r t i o n

o f p a t i e n t s w i t h s o m e e v e n t i n e a c h o f t h e g r o u p s , rT a n d r o L e t t i n g i i n d e x

t h e tr ia ls , w e a s s u m e t h a t t h e n u m b e r s o f p a t i e n t s w i t h t h e e v e n t in e a c h o f

t h e s t u d y g r o u p s a r e i n d e p e n d e n t b i n o m i a l r a n d o m v a r ia b l es w i t h a s s o c ia t e d

p r o b a b i l i t i e s pTi a n d P ci, i = 1 . . . . k . T h e b a s i c i d e a o f t h e r a n d o m e f f e c t s

a p p r o a c h is to p a r ce l o u t s o m e m e a s u r e o f t h e o b s e r v e d t r e a t m e n t e ff e ct i n

e a c h s t u d y , s a y y i, i n t o t w o a d d i t i v e c o m p o n e n t s : t h e t r u e t r e a t m e n t e f fe c t,

0~, an d t he s am pl i n g e r ro r , e~. Th e v a r i an ce o f e i i s t he s am pl e va r i an ce , s 2 ,

a n d i s u s u a l l y c a l c u l a t e d f r o m t h e d a t a o f t h e i th o b s e r v e d s a m p l e . T h e t r u e

t r e a t m e n t e f f ec t a s s o c i a t e d w i t h e a c h t r ia l w i ll b e i n f l u e n c e d b y s e v e r a l f a ct o rs ,i n c l u d i n g p a t i e n t c h a r a c t e r is t ic s a s w e l l a s d e s i g n a n d e x e c u t i o n o f t h e s t u d y .

T o e x p li c it ly a c c o u n t f o r t h e v a r i a ti o n i n t h e t r u e e f f e c ts , t h e m o d e l a s s u m e s

Oi = p. + ~i



M eta-An alys is in Cl inical Tr ia ls 181

w h e r e 0i i s t h e t r u e t r e a t m e n t e f f e c t i n t h e i t h s t u d y , W s t h e m e a n e f f ec t f o r

a p o p u l a t i o n o f p o s s i b l e t r e a t m e n t e v a l u a t i o n s , a n d 8 i i s t h e d e v i a t i o n o f t h e

i th s t u d y ' s e f f ec t f r o m t h e p o p u l a t i o n m e a n . W e r e g a r d t h e t r i al s c o n s i d e r e d

a s a s a m p l e f r o m t h i s p o p u l a t i o n a n d u s e t h e o b s e r v e d e f f ec t s t o e s t i m a t e p~a s w e l l a s t h e p o p u l a t i o n v a r i a n c e [ v a r( 8) = A 2 ]. H e r e , ~k r e p r e s e n t s b o t h t h e

d e g r e e t o w h i c h t r e a t m e n t e f fe c ts v a r y a c r o ss e x p e r i m e n t s a s w e l l a s t h e

d e g r e e t o w h i c h i n d i v i d u a l s t u d i e s g i v e b i a s e d a s s e s s m e n t s o f t r e a t m e n t e f -

f ec t s .

T h e m o d e l j u s t d e s c r i b e d c a n t h u s b e c h a r a c t e r iz e d b y t w o d i s t i n c t s a m p l i n g

s t a g e s . F i r st w e s a m p l e a s t u d y f r o m a p o p u l a t i o n o f p o s s i b l e s t u d i e s w i t h

m e a n t r e a t m e n t e f fe c t W a n d v a r i a n c e i n t r e a t m e n t e f f ec t s o f A2 . T h e n w e

s a m p l e o b s e r v a t i o n s i n t h e i t h s t u d y w i t h u n d e r l y i n g t r e a t m e n t e f fe c t 0~.

O n e i s s u e w h i c h d e s e r v e s s o m e a t t e n t i o n i s th e s p e c i fi c at io n o f t r e a t m e n t

e f f e c t, 0~. T h r e e c o m m o n l y u s e d m e a s u r e s a r e t h e r i s k d i f f e r e n c e , pr~ - p c /,

t h e r e l a t i v e r i s k , p T i / p c , , a n d t h e r e l a t i v e o d d s , [p Ti/(1 - p T i ) / p c J ( 1 - Pc/)] .

T h e r e l a ti v e o d d s i s p o p u l a r b e c a u s e o f i ts s u i t a b il i ty i n r e t r o s p e c t i v e o r c a s e

c o n t ro l s t u d i e s, a n d b e c a u s e i t h a s s o m e i n t e r e s ti n g m a t h e m a t i c a l p r o p e rt ie s .

I n p a r t ic u l a r , i f w e a s s u m e a c o n s t a n t r e l a ti v e o d d s (0i = ~ or A2 = 0 ) , t h en

t h e M a n t e l - H a e n s z e l s t at is t ic is o p t i m a l f o r te s t i n g H o: W = 1 , a n d t h e r e i s

c o n s i d e r a b l e l i t e ra t u r e o n e f f i c i e n t e s t i m a t e s o f V- a n d o n m e t h o d s f o r te s t i n g

H o: 01 = 02 = . . . = Ok. D e s p i t e t h e s e a d v a n t a g e s , t h e r e l a ti v e o d d s ( a n d

t h e c l o s e l y r e l a t e d r e l a t i v e r i sk ) s u f f e r s i n in t e r p r e t a b i l i t y . B y f a r t h e m o s t

i n t u i t i v e l y a p p e a l i n g m e a s u r e f o r t ri a ls o f c l in i ca l e f f ic a c y i s t h e r i sk d i f f e r e n c e ,

s in c e it m e a s u r e s a c t u al g a i n s w h i c h c a n b e e x p e c t e d i n t e r m s o f p e r c e n t a g e s

o f p a t i e n t s t r e a t e d . B e s i d e s r e l e v a n c e o f t h e m e a s u r e a n d s t at is t ic a l e f f ic i e n c y ,i t i s a l so d e s i r a b l e t o c h o o s e a m e a s u r e w h i c h i s n e a r l y c o n s t a n t o v e r s t u d i e s ,

s o t h a t t h e e f f e c t o f h e t e r o g e n e i t y i s m i n i m i z e d . U n l e s s t h e re i s n o t r e a t m e n t

e f f e c t a t a l l ( p T i = p c / f o r a ll i) , c o n s t a n c y o f t r e a t m e n t e f f e ct i n o n e s c al e ( s a y

p z J P c i = A f o r a l l i ) i m p l i e s v a r i a t i o n a c r o s s s t u d i e s i n a n o t h e r ( p T , - - P c / ,

s a y ) . T h u s i t i s c o n c e i v a b l e t h a t t h e w r o n g c h o i c e o f s c a le c o u l d i m p l y h e t -

e r o g e n e i t y i n t r e a t m e n t e f fe c ts w h i c h w o u l d n o t e x is t i f a d i f f e re n t m e a s u r e

w e r e u s e d . H o w e v e r , t h i s i s n o t li k e ly t o h a p p e n i n p r a c ti c e u n l e s s t h e r e is

a v e r y w i d e r a n g e i n t h e c o n t r o l r a t e s ( Pc ~) o r a ll t h e r a t e s a r e v e r y c l o s e t o

z e r o ( o r o n e ) . I n s u c h c a s e s , o n e m i g h t w a n t t o d o t h e a n a l y s i s i n b o t h t h e

r e l a t i v e o d d s a n d r i s k d i f f e r e n c e s c a l e s .

H O M O G E N E I T Y O F T R E A T M E N T E F FE C T

T o e v a l u a t e c o n s t a n c y o f t r e a t m e n t e f f e c t a c r o s s s tr a t a , w e u s e a l a rg e

s a m p l e t e s t b a s e d o n t h e s t a ti s ti c Q = £ w i ( y i - yw)2 , w h e r e y i i s t h e i t hl

t r e a t m e n t e f f e c t e s t i m a t e , ~ = E , w i y ~ w i i s t h e w e i g h t e d e s t i m a t o r o f t r ea t -

m e n t e f f e c t, a n d w i i s t h e i n v e r s e o f th e i th s a m p l i n g v a r i a n c e . T h e t e s t s ta t is t ic

Q is t h e s u m o f s q u a r e s o f th e t r e a t m e n t e f fe c t a b o u t t h e m e a n w h e r e t h e i ths q u a r e i s w e i g h t e d b y t h e r e c ip r o ca l o f t h e e s t i m a t e d v a r ia n c e . U n d e r t h e n u l l

h y p o t h e s i s , Q is a p p r o x i m a t e l y a x 2 s t at is t ic w i t h k - 1 d e g r e e s o f f r e e d o m ;

t h u s , w h e n e a c h s t u d y h a s a la r ge s a m p l e s i ze re l at iv e to t h e n u m b e r o f s t ra t a,

Q m a y b e u s e d t o t e s t H o: A2 = 0.




Whe n y~ is a difference in proport ions, r T i - - rc,, we estimate the sampling

variance in the ith study, s2, by

S 2 = r w i ( 1 - - r T i ) / n T i + r c i ( 1 - r c i ) / n c i , ( 1 )

and use Qw = E w ; ( y ~ - ~w)2 to test constancy of treatment effect.

The weights in Q may vary according to the assumptions made about the

sampling variances. For instance, when the sampling variances can be as-

sume d to be equal, then w ~ , i = 1 . . . . . k , is the inverse of a common sampling

variance s2. One review [9], which includes a qualitative assessment of ho-

mogeneity of treatmen t effect, uses the met hod of Gilbert et al. [15] to estimate

the magnitude of the variation across the differences in proportions. Since

the method of Gilbert et al. for estimating the variation in treatment effects

assumes a common sampling variance, we calculate Qu, the analogue of Qw,

assuming equal sampling variances. Here, the treatment effect is again thedifference in proportions, but

W i = S - 2 , i = 1 . . . . k,

where

s ~ = ~ s~/k,

i

and s2 is defined in equation (1).

We also use the Q statistic for testing homogenei ty in the relative odds

scale. In this scale,

Q L = ~ ,~ w i ( y i - y w ) 2

i

where

and

y i = In [ r T i ( 1 - - r c i ) / rc i (1 - r T i ) ] ,

W i ~ - S ~ 2 ' y w "~ ~ -- a w i y i / ~ W i ,

i i

S 2 = [ r l T i r T i (1 -- rTi)] -1 + [ncirci (1 - rci)] -1.

In the large sample case, QL is analogous to the goodness-of-fit test in

logistic models [17]. An alternate test statistic for assessing homogeneity is

the likelihood ratio test which is computational ly more cumbersome than the

Q statistic used here [18].

E S T IM A T I O N A N D C O M P U T A T I O N

Most of the reviews consider the differences in proportions as a measure

of treatment effect (Table 1). For estimating tL and A2 we also restrict our

attent ion to this scale.

When A2 ~ 0, Qw is used to derive a noniterative estimate of A2 by equating

the sample statistic with the corresponding expected value. This yields a

weighted estimator



M eta-An alys is in Cl inical Tr ia ls 183

A2 = m a x {0, {Q,, - (k - 1)} / [ ~ w i - ( ~_ ,w~ i / ~_ , wi)]},i i i

w h e r e Q w , ~ t w , w i a r e a s d e s c r i b e d a b o v e . T h e w e i g h t e d l e a s t s q u a r e s o r

C o c h r a n ' s [ 19] s e m i w e i g h t e d e s t i m a t o r o f i i s

~ w = ~ , w * y , / ~ , w * , (2 )i i

w h e r e

w * = ( w i - 1 + 4 2 ) - I (3 )

T h e a s y m p t o t i c s t a n d a r d e r r o r o f I~w i s

s.e . ( l~w) = ( ~ w*) -112. (4)i

W h e n t h e s a m p l i n g v a r ia n c e s ar e a s s u m e d t o b e eq u a l, t h e s e e q u a t i o n s r e d u c e

to :

Az, = m a x [0, {'~P~ (y , - y) 2 / (k - 1)} - s2],i

a n d

s . e . 0~ , ) = [ ( s 2 + A 2 ) / k ] 1<2,

w h e r e

.~ = ~ y i / k a n d s 2 = ~ s 2 / k .

i i

R a o e t a l. [ 20] d e r iv e A 2 f r o m a n u n w e i g h t e d s u m o f s q u a r e s p r o c e d u r e a n d

s h o w t h a t it is a ls o t h e M i n q u e e s t i m a t o r w h e n t h e s a m p l i n g v a r i a n c e s ar e

a ll e q u a l. T h e u n w e i g h t e d m e a n , ~ , , is e q u i v a l e n t t o t h e e s t i m a t e o f t h e

t r e a t m e n t e f f e c t i n r e v i e w s t h a t u s e t h e a v e r a g e d i f fe r e n c e i n p r o p o r t i o n s t o

a s s e s s t h e o v e r a l l t r e a t m e n t e f f i c a c y .

W i t h a n a d d i t io n a l a s s u m p t i o n t h a t y i i s N (0 ; , s 2 ) a n d 0 i i s N ( ~ , A 2), w ea ls o c o m p u t e m a x i m u m l i k e l ih o o d ( M L) a n d r e s tr i ct e d m a x i m u m l ik e l ih o o d

( RE M L ) e s t i m a t e s a n d c o m p a r e t h e m t o t h e n o n i t e ra t i v e o n e s . T h e m a x i m u m

l ik e l ih o o d e s ti m a t e s o f t h e u n k n o w n p a r a m e t e r s a r e t h o s e v a l u e s th a t m a x -

i m i z e t h e p r o b a b i l i ty d e n s i t y f u n c t i o n o f th e d a t a . I n R E M L e s t i m a t i o n , t h e

l i k e li h o o d t o b e m a x i m i z e d i s sl ig h t l y m o d i f i e d t o a d j u s t f o r ~ a n d A2 b e i n g

e s t i m a t e d f r o m t h e s a m e d a t a . T h e R E M L e s t i m a t o r s a r e t h e it e r a ti v e e q u i v -

a l e n t s o f t h e w e i g h t e d e s t i m a t o r s a b o ve . B o t h M L a n d R E M L e s ti m a t e s o f

a n d i t s s. e . t a k e t h e f o r m g i v e n i n e q u a t i o n s ( 2) a n d (4 ) w i t h w e i g h t s g i v e n

i n ( 3), b u t d i f f e r i n t h e w a y A2 i s e s t i m a t e d .

T h e M L e s t i m a t i n g e q u a t i o n s a r e g i v e n i n R a o et a l. [2 0] a n d t h e R E M L

e q u a t i o n s a r e r e v i e w e d b y H a r v i ll e [21]. F o r i m p l e m e n t i n g t h e M L o r R E M L

p r o c e d u r e s , w e u s e t h e E M a l g o r i t h m [ 22] w h i c h i s a n i t e ra t iv e p r o c e d u r e f o r

c o m p u t i n g m a x i m u m l ik e l i h oo d e s ti m a t es a p p r o p r i a t e w h e n t h e o b s e rv a t io n s

c a n b e v i e w e d a s i n c o m p l e t e d a t a .



184 R. DerS imon ian and N . La i rd

RESULTS

H o m o g e n e i t y o f T r e a tm e n t E f f e ct

W e p r e s e n t t h e s t a ti s ti c s f o r t e s t i n g h o m o g e n e i t y o f t r e a t m e n t e f f e c t i n

T a b l e 2. F o r t h e s e r e v i e w s Q w , t h e w e i g h t e d s t a t i s t ic i n t h e d i f f e r e n c e s c a l e,

a n d Q L, t h e a n a l o g o u s s t at is t ic in t h e l o g o d d s s c al e , im p l y s i m i l a r c o n c l u s i o n s

a b o u t t h e c o n s t a n c y o f t r e a t m e n t e f fe c t. T h e a s s u m p t i o n o f h o m o g e n e i t y h o l d s

i n t h e r e v i e w s b y D e S i l v a [7 ], S t j e r n s w a r d [8 ], P e t o [ 10 ], a n d i n C h a l m e r s '

[ 11 ] r a n d o m i z e d c o n t r o l l e d t r ia l s ( ca s e f a t a l i t y r a t e s) . I n t h e r e m a i n i n g f i v e

s e ts o f tr ia ls , t h e e v i d e n c e s u g g e s t s h e t e r o g e n e i t y o f t r e a t m e n t e f fe c t i rr e-

s p e c t iv e o f t h e s c a le o f m e a s u r e m e n t .

T h e r e v i e w b y P e t o [1 0] m e n t i o n s l a c k o f h e t e r o g e n e i t y i n t r e a t m e n t e f f e c ts

a c r o s s tr ia l s. F o r t h i s r e v i e w , t h e Q u s t at is t ic s u p p o r t s t h e h o m o g e n e i t y a s -

s u m p t i o n ( p v a l u e -- 0 .6 5 ), w h e r e a s b o t h Q ~ a n d QLs u p p o r t t h a t a s s u m p t i o n

o n l y m a r g i n a l l y ( p v a l u e = 0 .1 2 ). B a u m e t a l. [ 9] e s t i m a t e t h e v a r i a b i l i t y i n

t r e a t m e n t d i f f e r e n c e s u s i n g t h e m e t h o d o f G i l b e rt et a l. [1 5] a n d c o n c l u d e

t h a t r e la t iv e t o w i t h i n s t u d y v a r i a t i o n ( a s s u m e d e q u a l f o r a ll s t u d i e s ), b e t w e e n

s t u d y v a r i a t i o n is n e g li g i b le . T h i s q u a l i ta t i v e a s s e s s m e n t i s n o t c o n s i s t e n t w i t h

t h e r e s u l t s o f T a b l e 2 w h e r e a c o m m o n t r e a t m e n t e f f e c t a c r o s s th e t r i al s i n

t h is r e v i e w d o e s n o t s e e m t o h o l d in e i t h e r sc al e o f m e a s u r e m e n t . T h e t h i r d

r e v i e w w h i c h i n c l u d e s a t e s t o f h o m o g e n e i t y b e f o r e p o o l i n g t h e r e s u l t s [6]

u s e s a s l i g h t l y m o d i f i e d v e r s i o n o f Q ~ t o te s t th i s h y p o t h e s i s a n d t h e c o n c l u -

s i o n o f l a c k o f h o m o g e n e i t y a g r e e s w i t h t h e r e s u l t i n T a b l e 2.

A s i n t h e r e v i e w b y P e t o [ 10 ], Q u a n d Q ~ i m p l y d i f f e r e n t c o n c l u s i o n s a b o u tt h e h o m o g e n e i t y o f t r e a t m e n t e f fe c t i n t h e r e v i e w b y W i n s h i p [4 ]. I n th i s

r e v i e w a l s o , t h e m e t h o d a s s u m i n g e q u a l w e i g h t s ( Q u ) i m p l i e s h o m o g e n e i t y

o f e f fe c t w h i l e t h e w e i g h t e d o n e i m p l i e s th e o p p o s i te .

T h e s e r e s u l t s e m p h a s i z e t h a t t h e v a r i a t io n i n t h e t r e a t m e n t e f f ec t a c r o s s

s e v e r a l tr ia l s is o f t e n n o t n e g l i g i b le a n d s h o u l d b e i n c o r p o r a t e d i n t o t h e a n a l -

T a b l e 2 T e s t o f H o m o g e n e i t y ~

d ff Q J QL Q d

W inship 7 15 .2b 15.6b 7.9

C o n n 8 15 . 6 b 20.6 b 19.3bM iao 5 21.7b 18.8b 20.9b

DeSilva 5 9.1 4.4 7.3St jernsw ard 4 2 .1 2 .2 2 .7Ba um 25 40.4 b 35.2 b 3 5 . 3 b

Peto 5 9.0 9.2 3.5C h a l m e r s

Throm boem bol i sm 5 12.3b 10.3 b 10.6bCas e fatal i ty rates 5 3.5 2.4 1.8

aFigures in T ables 2-4 are based on data available at the time of review publication.

bp value <0 .10.

CDegrees of freedom .dQ statistic in difference scale (unequ al we ights).

~Q statistic in log o dds scale (unequal w eights).

fQ statistic in difference scale (equal we ights).



Meta-Analysis in Clinical Trials 185

ysis of the overall tr eatment efficacy. Lack of homoge neit y holds both whe n

the treatment effect is the difference in proportions and when it is the log

odds. The unweighted statistic which assigns an equal weight to each study

may not be appropriate for testing homogeneit y whe n differences in samplesizes and/or underlying proportions across studies are large.

Est imat ion

For all four methods of estimation we present estimates of ~ and its s.e.

in Table 3, and estimates of A2 in Table 4. The estimates of A2, and s.e. (~)

are quite similar in the weighted noniterative method, maximum likelihood,

and restricted maximum likelihood procedures. The A2s from these three

methods are zero or nearly so in the reviews by DeSilva [7], Stjernsward [8],

Peto [10], and Chalmers ' [11] randomized trials (case fatality rates). These

same reviews have Q statistics that are small relative to their degrees of

freedom (Table 2). The weighted met hod and the REML estimation procedures

consistently yield slightly higher values of A2 than the ML procedure. This

is because both these procedures adjust for p, and A2 being estimated from

the same data whereas the ML procedure does not. The estimates of ~ and

its s.e. from these three procedures are expected to be similar since the es-

timates of A2 are almost equal.

Comparing the unwe ight ed method of moment s with the other three meth-

ods, we find that the estimates for A2 from this meth od differ, and sometimes

differ widely, from the estimates of the other three m etho ds but wi thou t any

consistent pattern. The estimates of ~ and its s.e. from the unweighted methodalso differ from the estimates of the other three methods a nd these differences

are not necessarily due to the di fferences in A2s. In Chalmers' [11] randomi zed

trials (case fatality rates), for instance, even when A2 is zero for all four

methods, the est imate of ~ is 0.042 (s.e. = 0.024) for the unweight ed meth od

while it is 0.029 (s.e. = 0.012) for the other three methods. The original

reviewers report the unweighted average of the observed rate differences

T a b l e 3 Estimated Overall Effects and Their Standard Errors a

p,u b p,wc p,M a p,R e

Winship 0.406 (0.046) 0.389(0.058) 0.384 0.053) 0.387 0.056)Conn 0.102 (0.092) 0.075(0.072) 0.070 0.063) 0.073(0.069)Miao 0.077 (0.125) 0.094(0.111) 0.095 0.106) 0.093(0.118)DeSilva 0.026 (0.019) 0.027 0.019) 0.026 0.017) 0.027 0.019)Stjernsward 0.046 (0.020) 0.041 (0.018) 0.041 0.018) 0.041(0.018)Baurn 0.203 (0.031) 0.208(0.026) 0.208 0.025) 0.208(0.026)Peto 0.018 (0.008) 0.015(0.008) 0.014 0.008) 0.015(0.008)Chalmers

Thromboembolism 0.102(0.036) 0.079(0.020) 0.078 0.017) 0.078(0.020)Case fatality rates 0.042 0.024) 0.029(0.012) 0.029 0.012) 0.029(0.012)

aFigures in parentheses represent the standard errors of the corresponding estimates.

bNoniterative estimates with equal weights.

CNoniterative estimates with weights to reflect unequal variances.

dMaximum likelihood estimates.

eRestricted maximum ikelihood estimates.




T a b l e 4 Estimated Variation in the True Effects

A u2a A w2b /~M 2c A R2d

Wins hip 0.0020 0.0137 0.0096 0.0117

Conn 0.0442 0.0208 0.0112 0.0176Miao 0.0716 0.0540 0.0482 0.0638DeSilva 0.0007 0.0009 0.0006 0.0009Stjernsward 0 0 0 0Baum 0.0072 0.0062 0.0049 0.0057Peto 0 0.0002 0.0002 0.0002Chalmers

Thromboembolism 0.0041 0.0012 0.0007 0.0012Case fatality rates 0 0 0 0

aNoniterative estimates with equal weights.

bNoniterative estimates with weights to reflect unequal variances.

CMaximum ikelihoodestimates.eRestricted maximum ikelihood estimates.

(0.042) as an est imate of overall treatment efficacy. The weighted estimate of

the t reatment effect which weight s the observed effects in relation to sample

size is lower than the unweighted average, since some of the larger studies

have smaller estimated treatment effects.

DISCUSSION

We have used a simple random effects model for combining evidence, andapplied it to characterize the distribution of treatment effects in a series of

studies. The model is useful both in summarizing the data and in illustrating

the different kinds of results which one obtains from randomized and non-

randomized studies. In general, studies with greater potential for bias, such

as uncontrolled or nonr andomi zed ones, show greater treat ment effect as well

as greater heterogeneity [2,16].

One important finding that emerges from this investigation is that heter-

ogeneity of treatment effects across studies is common and should be incor-

porated into the analysis. The random effects model incorporates this het-

erogeneity, however small, in the analysis of the overall efficacy of the treatment.The method estimates the magnitude of the heterogeneity, and assigns a

greater variability to the estimate of overall treatment effect to account for

this heterogeneity . In principle, we can extend the model to include perti nent

covariate information [2]. Utilizing covariate information may substantially

reduce the he terogeneity of effects and thus allow for more specific therapeutic

recommendat ions. This is often difficult in practice, however, since covariate

information may be missing for some studies. Improvement in publication

standards for medical reporting a nd further methodological work for handling

missing covariate information are neede d to strengthen our ability to combine

results from clinical studies.

For estimating the overall treat ment effect and the variation of effects across

studies, our results suggest that the weighted noniterative method is an at-

tractive procedure because of the comparability of its estimates with those of

the maximum likelihood methods and because of its relative simplicity. On



Meta-Analys i s in Cl in ica l Tr i a l s 187

t h e o th e r h a n d , t h e u n w e i g h t e d m e t h o d w h i c h i g n o re s d i f f e re n c e s i n s a m p l e

s iz e s y i e ld s e s t i m a t e s t h a t o f t e n d i f f e r f r o m t h e e s t i m a t e s o f t h e o t h e r m e t h o d s .

A p r o b l e m i n p o o l in g d a t a w e h a v e n o t a d d r e s s e d h e r e i s t h a t o f p u b l ic a t io n

b ia s . T h i s p r o b l e m r e l at e s to s t u d i e s b e i n g e x e c u t e d , b u t n o t r e p o r t e d , u s u a l l yb e c a u s e t r e a t m e n t e f f ec t h a s n o t b e e n f o u n d . R e v i e w e r s g e n e r a l l y r e c o u n t

t h o s e s t u d ie s t h a t a p p e a r t o be w o r t h w h i l e a n d d i s c o u n t th o s e t h a t a re u n -

p u b l i s h e d o r a r e n o t in a g r e e m e n t w i t h a f a v o r e d g r o u p o f s tu d i es . T h e m e t h o d

w e d e s c r i b e h e r e r e p r e s e n t s a s y s t e m a t i c , q u a n t i t a t iv e p o o l i n g o f a v a i la b l e

d a t a t o r e s o l v e c o n t r o v e r s i e s a b o u t a t r e a t m e n t e f f ec t . W i t h e a c h i n d i v i d u a l

c o n t r o v e r s y , u n p u b l i s h e d i n f o r m a t i o n m a y b e e l ic i t ed a n d a l o n g w i t h r e c e n t

f in d i n g s t h e m e t h o d c a n b e u s e d t o u p d a t e t h e re s u lt s .

I n a ll o u r w o r k w e a s s u m e t h a t th e s a m p l i n g v a r i a n c es a re k n o w n , a l t h o u g h

i n r e a li ty w e e s t i m a t e t h e m f r o m t h e d a t a . F u r t h e r r e s e a r c h n e e d s t o b e d o n e

i n th i s a r e a a s t h e r e a r e a l t e r n a t i v e e s t i m a t o r s t h a t m i g h t b e p r e f e r a b l e t o t h e

o n e s w e u s e . F o r i n s t a n c e , i f t h e s a m p l e s i z e s in e a c h s t u d y a r e s m a l l , t h e n

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

m e n t a n d c o n t ro l g r o u p s m i g h t b e b e t t e r t h a n t h e o n e s b a s e d o n e s t im a t e s

o f p r o p o r t i o n s f r o m t h e i n d i v i d u a l s t u d i e s . A n o t h e r a l te r n a t iv e i s t o s h r i n k

t h e i n d i v id u a l p r o p o r t i o n s t o w a r d s a p o o l e d e s t i m a t e b e f o r e c a l c u la t in g t h e

v a r i a n c e s . F u r t h e r i n v e s t i g a t io n is n e e d e d b e f o r e o n e s i n g le m e t h o d e m e r g e s

a s s u p e r i o r .

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10. Pe to R: As pi r in af t e r m yo card ia l i nfarc t ion . La nce t i : 1172-1173, 1980 (un s ign ededi tor i a l )

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1 8 8 R . D e r S i m o n i a n a n d N . L a ir d

1 2. S t a m p f e r M J , G o l d h a b e r S Z , Y u s u f S , P e t o R , H e n n e k e n s C H : E f fe c t o f i n t r a v e n o u ss t r e p t o k i n a s e o n a c u t e m y o c a r d i a l i n f a r c t i o n : P o o l e d r e s u l t s f r o m r a n d o m i z e dtr ia ls . N En gl J M ed 3 07: 1180-1182, 1982

1 3. L o n g - t e r m a n d s h o r t - t e r m b e t a - b l o c k a d e a f t e r m y o c a r d i a l i n f ar c ti o n . L a n c e t i:

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1 9. C o c h r a n W G : A d j u s t m e n t s i n an a l y s is . In : P l an n i n g a n d A n a l y s is o f O b s e r v a t i o n a lS t ud i e s , M os e s L E , M os t e U e r F , E ds . N e w Y or k : W i l e y , 1983 , pp . 102 - 108

2 0. R a o P S, K a p l a n J, C o c h r a n W G : E s t im a t o r s f o r t h e o n e - w a y r a n d o m e f fe c ts m o d e lw i t h une qua l e r r o r va r i a nc e s . J A m S t a t A s s oc 76 : 89 - 97 , 1981

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