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8/2/2019 Shklov, Powell (1988) Distributional Bases for Individual Cell Significance Tests
1/19
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D I S T R I B U T I O N A L B A S E SF O
I N D I V I D U A L C E L L S I G N I F I C A N C E T E S T S
I
C O N T I N G E N C Y T A B L E S :
B I N O M I A L , M U L T I N O M I A L ,
H Y P E R G E O M E T R I C A N D U N I F O R M
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N o r m S h k l o v a n J a P o w e l l
U n i v e r s i t y o W i n d s o rW i n d s o r , O n t a r i o , C a n a d a
T h i s p a p e r h a b e e n p r e p a r e d f o p r e s e n t a t i o n t t h P s y c h o m e t r i c S o c i e t y i L o A n g e l e s , C a l i f o r n i a o J u n e 2 7 1 9 8 8 .
C J C P o w e l l , 1 9 8 8 .\ P A P E R S \ S H K L 0 V 6
8/2/2019 Shklov, Powell (1988) Distributional Bases for Individual Cell Significance Tests
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I N T R O D U C T I O NI i u s u a l f o e d u c a t o r s t d e f i n e l e a r n i n g i t e r m s o c h a n g e s i
b e h a v i o r . T h m e a s u r e m e n t o t h e s e c h a n g e s h a b e e n a c c o m p l i s h e d
u s i n g f r e q u e n c y c o u n t s w i t h i n p r e d e f i n e d c a t e g o r i e s o o b s e r v a b l eb e h a v i o r . O n e x p e c t s t o b s e r v e a n i n c r e a s e i f r e q u e n c y a m o n g t h o s eb e h a v i o r s d e e m e d d e s i r a b l e o r d e c r e a s e i f r e q u e n c y a m o n g t h o s e
b e h a v i o r c a t e g o r i e s d e e m e d u n d e s i r a b l e . S t a t i s t i c a l a n a l y s i s i c o n d u c t e d u s i n g t h a s s u m p t i o n t h a t t h e s e f r e q u e n c i e s , a l t h o u g h i
t h e m s e l v e s d i s c r e t e , r e p r e s e n t u n d e r l y i n g c o n t i n u o u s v a r i a b l e s .
T h e r e i a n o t h e r q u i t e d i f f e r e n t w a i w h i c h c h a n g e m i g h t b
o b s e r v e d a n d m e a s u r e d . B d e f i n i n g a n t w c a t e g o r i e s o b e h a v i o r a
s t a t e o n e a n d s t a t e t w a n o b s e r v i n g a c r o s s t i m e i n t e r v a l , t h
c h a n g e s w h i c h o c c u r b e c o m e o b s e r v a b l e a c a t e g o r y i n t e r a c t i o n s w h i c h
a p p e a r a s s i n g l e c e l l s i n c o n t i n g e n c y t a b l e s .
F o t h t w b t w t a b l e , t h e r e a r e s e v e r a l p o s s i b l e a n a l y t i c
a p p r o a c h e s f r o m w h i c h t s e l e c t . I t h m o r e g e n e r a l c a s e , w h e r e m o r e
t h a n t w o i n t e r a c t i n g c a t e g o r i e s a r e i n v o l v e d , h e r e t o f o r e a s a t i s f a c t o r y
m e t h o d f o t h e i n t e r p r e t a t i o n o s i n g l e c e l l s i n c o n t i n g e n c y t a b l e s h a
n o b e e n a v a i l a b l e .
T h i s p r e s e n t p a p e r i t h m o s t r e c e n t i s e r i e s b t h p r e s e n t
a u t h o r s i n t e n d e d t d e v e l o p m a t h e m a t i c a l l y v a l i d m e t h o d f o t h
s t a t i s t i c a l i n t e r p r e t a t i o n o s i n g l e c e l l s i c o n t i n g e n c y t a b l e s . T h
p r a c t i c a l p u r p o s e b e h i n d t h i s s e a r c h h a s b e e n t e n a b l e t h s e c o n d
a u t h o r , f o r e s e a r c h r e a s o n s , t d e f i n e l e a r n i n g a s i g n i f i c a n t c o n t i n g e n t
c h a n g e s i b e h a v i o r a c r o s s t i m e . T h i s a p p r o a c h w o u l d e n a b l e t h r e s e a r c h e r t r e t a i n t h q u a l i t a t i v e a n d i s c r e t e c h a r a c t e r i s t i c s o t h i n i t i a l d a t a s e w h i l e c o n d u c t i n g q u a n t i t a t i v e a n a l y s i s .
8/2/2019 Shklov, Powell (1988) Distributional Bases for Individual Cell Significance Tests
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A l p r e v i o u s a t t e m p t s b G o o d m a n ( * * ) , F a b e r ( * * ) , F u c h s & K e n e t t( 1 9 8 0 ) a n t h p r e s e n t a u t h o r s ( * * * ) h a v e b e g u n w i t h t h p r i n c i p l e sb e h i n d t h c e l l v a l u e s ( [ E ] 2 / E ) i t h X ( C h i S q u a r e ) d i s t r i b u t i o n
T h i s p r e s e n t p a p e r h a u s e d d i f f e r e n t a p p r o a c h , a t t e m p t i n g i n s t e a d t d e t e r m i n e w h i c h o f o u r d i s t r i b u t i o n s :
1 B i n o m i a l
2 M u l t i n o m i a l
3 H y p e r g e o m e t r i c
a n 4 U n i f o r m
b e s t f i t t e d r a n d o m l y g e n e r a t e d f r e q u e n c y d i s t r i b u t i o n s o s i n g l e c e l l s i n
c o n t i n g e n c y t a b l e s l a r g e r t h a n t w b t w w h e n t h m a r g i n a l t o t a l s a r c o n s t r a i n e d .
B A S I C D I S T R I B U T I O N S
I a n a l y z i n g t h e c o n t i n g e n c y t a b l e , a d e s c r i p t i o n m u s t b m a d e
c o n c e r n i n g t h u n d e r l y i n g d i s t r i b u t i o n w h i c h i m o s t a p p r o p r i a t e . T h i s
p r o b l e m w a f i r s t t r e a t e d b G A B a r n a r d a n d b y E . S . P e a r s o n f o t a b l e s i B i o m e t r i k a 3 4 1 9 4 7 .
W c o n s i d e r f o u r p o s s i b l e c a s e s o u n d e r l y i n g d i s t r i b u t i o n , v i z . ,
1 ) b i n o m i a l , 2 m u l t i n o m i a l , 3 h y p e r g e o m e t r i c a n 4 t h u n i f o r m . T h e i r
r e s p e c t i v e p r o b a b i l i t y f u n c t i o n s a r e :
( X , S ) p q S " X , ( 1
V , s
P k k l/ ' " p Y ^ ( 3
(
a n ( x = _ i _ , ( 4
8/2/2019 Shklov, Powell (1988) Distributional Bases for Individual Cell Significance Tests
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I t h c o u r s e o o u i n v e s t i g a t i o n s i w a f o u n d t h a t t h e l a s t t w d i s t r i b u t i o n s , n a m e l y t h h y p e r g e o m e t r i c a n t h e u n i f o r m w e r e n o a p p r o p r i a t e f o t h p u r p o s e i h a n d , s w c o n c e n t r a t e d o u c o n s i d e r a t i o n s o t h f i r s t t w o , n a m e l y t h b i n o m i a l a n m u l t i n o m i a l .T h w a i w h i c h t h d i s t r i b u t i o n e n t e r s i n t o t h e p i c t u r e i i t h c a l c u l a t i o n o t h e x p e c t e d v a l u e i t h c e l l o c o n c e r n . T h v a l u e s o
w i l l b c a l c u l a t e d i t h s a m e m a n n e r a t h t e s t f o r h o m o g e n e i t y a n w i l l b e t r e a t e d a s i n v a r i a n t f o t h e p r o b a b i l i t y c a l c u l a t i o n .
I w a f o u n d , a w i l l b e c o m e e v i d e n t i t h r e m a i n d e r o t h
p a p e r , t h a t t h m u l t i n o m i a l i s i m o s t c a s e s , a J a n n e r o n e h a s h o w n f o t h t w o - b y - t w o c a s e , t h d i s t r i b u t i o n m o s t s u i t a b l e f o c a l c u l a t i n g t h e s e e x p e c t e d v a l u e s .
O p a r t i c u l a r i n t e r e s t i t h e m u l t i n o m i a l d i s t r i b u t i o n w h i c h w a s
u s e d i s p e c i a l w a y . T h e p r o c e d u r e e m p l o y e d i d i s c u s s e d i d e t a i l .
M A T H E M A T I C A L C O M P A R I S O NI a e a r l i e r a p p r o a c h ( P o w e l l S h k l o v , 1 9 8 7 ) t h p r o c e d u r e w a s
t u s t h a s s u m p t i o n t h a t :
X - 1 0 . . E ) / V E . ( 5 w i t h E . ( R o w S u m . ) ( C o l u m n S u . ) / N
a n E . . / N p ( 6
t r e w r i t e , a l g e b r a i c a l l y , t h e d i f f e r e n c e b e t w e e n p r o p o r t i o n ss t a t i s t i c .
B s u b s t i t u t i o n a n r e a r r a n g e m e n t , w o b t a i n e d t h f o l l o w i n gr e s u l t .
/ . . . . ( 7 V / E / N \ ( E ) / N ,
8/2/2019 Shklov, Powell (1988) Distributional Bases for Individual Cell Significance Tests
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E q u a t i o n ( 7 w a r e s c a l e d t a c c o m m o d a t e t h p r o b l e m t h a t g r o u ps i z e i c o r r e l a t e d w i t h X b t r a n s f o r m i n g t h r a t i o s i n t o t h e i r a r c s i n ,a f o l l o w s .
Z i s . . L . Q . . s . . . i . n . L a Z N } z , . , . l n . L E Z K X / . N ( 8 V j ^ Y / n ) * a r c s i n ( ( N - E ) / N )
T h i s m o d i f i c a t i o n r e d u c e d s u b s t a n t i a l l y t h c o r r e l a t i o n b e t w e e nt h i s c e l l s t a t i s t i c c o m p a r e d t t h X s t a t i s t i c ( f r o m . 7 4 8 ; . 0 0 0 0 2t o . 4 7 2 ; . 0 1 2 ) . I d i n o e l i m i n a t e t h i s e f f e c t .
s e c o n d p r o b l e m e m e r g e d . T h Z o e q u a t i o n ( 8 w a n o
s t a n d a r d s c o r e a s i t s t a n d a r d d e v i a t i o n w a l e s s t h a n 1 . 0 0 . I f a c t ,t h i s s t a n d a r d d e v i a t i o n w a d i f f e r e n t f o e a c h t a b l e c o n f i g u r a t i o n ,
r e q u i r i n g a r e n o r m a l i z i n g f a c t o r t b e a p p l i e d f r o m a a r r a y w h i c h
r e f l e c t e d t a b u l a r d i m e n s i o n s .
A t h i s p o i n t , a a l t e r n a t i v e a p p r o a c h w a a t t e m p t e d . T h
u n d e r l y i n g i d e a i p r e s e n t e d i F i g u r e
I N S E R T F I G U R E H E R E
T h e t w o - b y - t w o t a b l e s h o w n i F i g u r e i c o l l a p s e o a n
r e a s o n a b l e p a r t i t i o n o f a n d e s i r e d s e g m e n t o a n c o n t i n g e n c y t a b l e i w h i c h t h p a r t i t i o n h a m o r e t h a n t w o c a t e g o r i e s i o n d i r e c t i o n a n a
l e a s t t w i t h o t h e r .
T h t o l e f t ( l e a d ) c e l l ( c o n t a i n s t h r e l a t i o n s h i p f o w h i c h
t h c u m u l a t i v e p r o b a b i l i t y o t h o b s e r v e d f r e q u e n c y i d e s i r e d w i t h t h
o b s e r v e d m a r g i n a l t o t a l s a n t a b l e t o t a l a s s u m e d t b i n v a r i a n t .
I t h c e l l t t h r i g h t i p l a c e d t h s u o a l t h o t h e r c e l l s i t h r o i w h i c h t h l e a d c e l l o c c u r s . I t h c e l l b e l o w t h
8/2/2019 Shklov, Powell (1988) Distributional Bases for Individual Cell Significance Tests
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F I G U R E
D a t a R e d u c t i o nP r o c e d u r e s
L e a dC e l l
S u o r e s t o
c o l u m n
t h t h
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C 2 , 2S u o t h b a l a n c o
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8/2/2019 Shklov, Powell (1988) Distributional Bases for Individual Cell Significance Tests
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t
c o r r e s p o n d i n g r e s i d u a l c o l u m n s u t h t a b l e r e s i d u a l i p l a c e d i c e l l
C 2 Z 'T h m a r g i n a l c o n s t r a i n t s i t h i s p r o c e d u r e f i t h p r o b a b i l i t i e s ( o
e x p e c t e d v a l u e s ) f o e a c h c e l l s u c h t h a t :
P . ( 9
w i t h c o r r e s p o n d i n g v a l u e s f o t h o t h e r c e l l s .
T h m a x i m u m v a l u e i c e l l i t h l e s s e r o R a n d T T h
r a n g e o v a l u e s w h i c h c a o c c u r i C i t h m i n i m u m o R , S , T o p l u s a n t h m i n i m u m v a l u e i c e l l i s
m i ( R , T ) m i ( R , S , T , o U B e c a u s e t h p a r t i t i o n i n g a n t h e c o n s t r a i n t s m a k e t h c u m u l a t i v e
p r o b a b i l i t y o t h i s s i t u a t i o n l e s s t h a n o n e , t h c a l c u l a t i o n s n e e d t b
r e s c a l e d o r n o r m a l i z e d u s i n g t h e p o s s i b l e s u o a l t h p r o b a b i l i t i e su n d e r t h e s e c o n s t r a i n t s , a s f o l l o w s :
p < 1 0 >w h e r e p . S ( & * 2 V i \ S * S l \ m 2 ( . $ . . * . I ) m J ( & * J l \ m 4m 1 ! m 2 ! m ^ m A ? \ P U ^ \ ^ ( ' f o r t h r a n g e m i t m a
T h v a l u e s o h a n a r c h o s e n a c h a n g e s s t h a t t h
r o a n c o l u m n s u m s a r i n v a r i a n t .I t h e n f o l l o w s t h a t t h a d j u s t e d c u m u l a t i v e f r e q u e n c y f r o m m i
t 0 ( t h e o b s e r v e d f r e q u e n c y t b t e s t e d ) w i l l b e
n t m x n
R o w s o c o l u m n s w i t h z e r o ( 0 s u m s a r m e r e l y d r o p p e d f r o m t h a n a l y s i s . C e l l s c o n t a i n i n g z e r o , b c o n t r a s t , a r p o t e n t i a l l y m e a n i n g f u l .
8/2/2019 Shklov, Powell (1988) Distributional Bases for Individual Cell Significance Tests
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F o t h o n t a i l e d t e s t s , w i l l b s t a t i s t i c a l l y s i g n i f i c a n t i i
i l e s s t h a n o e q u a l t . 0 o g r e a t e r t h a n o r e q u a l t . 9 5 . I t h r e s u l t s r e p o r t e d l a t e r t h o s e p r o b a b i l i t i e s a t h u p p e r e n o t h r a n g e( P X 1 - 5 0 ) w e r e r e s c a l e d s u c h t h a t P t h u s a v o i d i n g
i n t e r p r e t a t i o n c o n f u s i o n .
A l t h a t r e m a i n s i t s h o w t h a t e q u a t i o n ( 1 1 ) f i t s t h e s i m u l a t e d
o u t c o m e f o r a n d o m o b s e r v a t i o n s u n d e r t h e s e c o n s t r a i n e d c o n d i t i o n s .
A P P L I C A T I O N
W i t h t h e a i o r a n d o m n u m b e r g e n e r a t o r t w s e t s o f f i v e r u n so 1 0 t w o - b y - t w o c o n t i n g e n c y t a b l e s , e a c h c o n t a i n i n g 5 d i s c r e t e
e v e n t s , w e r e p r o d u c e d . I t h f i r s t s e t , n f u r t h e r c o n s t r a i n t s w e r e
i m p o s e d . I t h s e c o n d s e t , o n l y t h o s e t a b l e s i w h i c h t h m a r g i n a l
t o t a l s w e r e 2 w e r e c o l l e c t e d f o r s t u d y .
I t h f i r s t s e t , t h e o b s e r v e d f r e q u e n c y f o e a c h c e l l v a l u e a m o n g
t h 1 0 r u n s o t a b l e s o 5 e v e n t s w a c o l l e c t e d f o r a l f o u r c e l l s i e a c h t a b l e . T h i s c o l l e c t i o n w a m a d e w i t h o u t r e g a r d f o w h i c h c e l l w a i n v o l v e d . T h r e s u l t i n g f r e q u e n c y d i s t r i b u t i o n s w e r e a v e r a g e d t t a k ea d v a n t a g e o c e n t r a l t e n d e n c y a n t h e n c o m p a r e d t t h b i n o m i a l
d i s t r i b u t i o n . T h r e s u l t s o f t h i s c o m p a r i s o n i g i v e n i T a b l e 1
I n s e r t T a b l e H e r e
A s w o u l d b e e x p e c t e d f r o m t h e o r y , t h i s d i s t r i b u t i o n i s w i t h o u t
d o u b t , b i n o m i a l .
I t h s e c o n d c a s e , w h e r e i n t h m a r g i n a l f r e q u e n c i e s w e r e
c o n s t r a i n e d , t h e d i s t r i b u t i o n o t h f r e q u e n c i e s f o c e l l w e r e
c o l l e c t e d . A b o u t l / 8 0 t h o a l t h i t e r a t i o n s e m e r g e d w i t h t h i s
8/2/2019 Shklov, Powell (1988) Distributional Bases for Individual Cell Significance Tests
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T A B L E
B i n o m i a l F i F o r C e l l F r e q u e n c i e s
C * R a n d o m l y G e n e r a t e dC o n t i n g e n c y T a b l e s .
N u m b e r o fC e l l E v e n t s
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A v e r a g e s o s e t s o f f r e q u e n c i e s .
T H C H S Q U A R E V A L U E I 1 . 0 4 7 ; d 1 5 . 9 9 5
8/2/2019 Shklov, Powell (1988) Distributional Bases for Individual Cell Significance Tests
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P I G U 1 ED a t a R e d u c t i o n P r o c e d u r e s
M O T H E RC E L L S u o t h r e s t o t h R O
S u o t h r e s t o t h C O L U M N
C o l u m n . , S u
S u o t h b a l a n c e o t h T A B L E
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8/2/2019 Shklov, Powell (1988) Distributional Bases for Individual Cell Significance Tests
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C 2 . 1S u o r e s t o
c o l u m n
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C Z , 2S u o t h r e s t o t h r o
C 2 . 2S u o t h b a l a n c e o
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8/2/2019 Shklov, Powell (1988) Distributional Bases for Individual Cell Significance Tests
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T A B L E
B i n o m i a l f i f o r c e l l f r e q u e n c i e s
o n r a n d o m l y g e n e r a t e dc o n t i n g e n c y t a b l e s .
N u m b e r o C e l l E v e n t s
1
1
1
1
1
1
1
1
1
1
2
2
O b s e r v e dF r e q u e n c i e s
.
.
3 .
4 .
9 .
8 .
1 1 . 3
1 2 . 6 5
1 2 . 4
1 1 . 3 5
8 . 6 . 4 5
4 .
3 . 2 5
1 .
. 6
. 6
E x p e c t e dF r e q u e n c i e s
. 7
1 . 2 3 4
2 . 5 8 6
4 . 6 3 4
7 . 2 0 9
9 . 8 5 2
1 1 . 9 4 1
1 2 . 9 3 7
1 2 . 6 0 5
1 1 . 1 0 4
8 . 8 8 46 . 4 7 8
4 . 3 1 8
2 . 6 3 9
1 . 4 8 2
. 7 6 5
. 6 2 2
0 . 0 0 0
0 . 1 5 3
0 . 0 6 6
0 . 0 0 4
0 . 4 9 6
0 . 0 9 2
0 . 0 3 4
0 . 0 0 6
0 . 0 0 3
0 . 0 0 5
0 . 0 2 60 . 0 0 0
0 . 0 0 0
0 . 1 4 1
0 . 0 0 0
0 . 0 1 7
0 . 0 0 1
T o t a l 1 . 0 4 7
T H C H S Q U A R E V A L U E I 1 . 0 4 7 ; d 1 5 . 9 9 5
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T A B L E
B i n o m i a l f i f o r f r e q u e n c y d i s t r i b u t i o n
o f C e l l C w i t h m a r g i n a l c o n s t r a i n t s .
6 . 9 3 4
2 . 0 1 3
0 . 1 1 2
0 . 4 2 7
9 . 8 0 6
5 . 8 6 1
1 . 3 6 7
0 . 0 2 6
1 . 6 5 9
8 . 2 9 1T o t a l 3 6 . 4 9 6
T H C H S Q U A R E V A L U E I 3 6 . 4 9 6 ; d 9 . 0 0 1
N u m b e r o C e l l E v e n t s
1
1
1
1
1
1
1
1
O b s e r v e dF r e q u e n c i e s
,
3 .
8 .
1 4 . 2
2 4 . 2
2 1 . 2
1 5 . 0
8 .
3 .
.
E x p e c t e dF r e q u e n c i e s
8 . 4 5 8
7 . 2 0 9
9 . 8 5 2
1 1 . 9 4 1
1 2 . 9 3 7
1 2 . 6 0 5
1 1 . 1 0 4
8 . 8 8 4
6 . 4 7 8
9 . 8 2 6
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T A B L E
M u l t i n o m i a l f i f o f r e q u e n c y d i s t r i b u t i o n
o C e l l l f w i t h m a r g i n a l c o n s t r a i n t s .
N u m b e r o C e l l E v e n t s
1
1
1
1
1
1
I
1
O b s e r v e dF r e q u e n c y
.
3 .
8 .
1 4 . 2
2 4 . 2
2 1 . 2
1 5 . 0
8 .
3 .
.
E x p e c t e d X F r e q u e n c y
1 . 1 3 3 0 . 0 9 8
3 . 3 0 2 0 . 0 0 3
8 . 4 5 3 0 . 0 1 4
1 5 . 7 1 7 0 . 1 4 6
2 1 . 3 9 3 0 . 3 6 8
2 1 . 3 9 3 0 . 0 0 2
1 5 . 7 1 7 0 . 0 3 3
8 . 4 5 3 0 . 0 0 0
3 . 3 0 2 0 . 0 0 3
1 . 1 3 3 0 . 0 9 8
T o t a l 0 . 7 6 5
T H C H S Q U A R E V A L U E I 0 . 7 6 5 ; d 9 . 9 9 5
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T A B L E
C o m p a r i s o n o t h n u m b e r o f s i g n i f i c a n t c e l l s i n e a c h 4 x t a b l e f r o m M u l t i n o m i a la n d A r c s i n p r o c e d u r e s .
A L C E L L S W I T H O U T P R I N C I P L ED I A G O N A L C E L L S
G a i n e d 0 > E 0 < L o s t # S i g G a i n e d O > E 0 < E L o s t S i g
< 9 9
1 0 1 0 1 1 1 1 1 2 1 2 1 3 1 3 1 4 1 4 1 5 1 5 1 6 1 6 1 7 1 7 1 8 1 3 1 9 1 9 2 0 2 0 2 1 2 1 2 2 > 2 2 5
' :
2 3 3
1 0 6 9 9 9
1 1 9
1 1 9 8 9 7
1 1 1 8 1 9 2 0 1 7 1 6 1 4 1 1 . 3 8 5 4 3
t
P
e
: -
L 1 2 5 3 6 6 6 6 6 7 5 4 5 3 5 8 9
1 0 7
7 7 5 4 2 3 1 1
. -
l i 2 8 1 0 1 4 i - 1 2 L 4 5 7 1 3
A L C E L L Sa g e t s i g . 5 9 3 ; s i g . a t t h e . 0 l e v e l .
W I T H O U T S T A B L Ea g e f s i g - - 2 5 4 ; n o s i g . a . 0 l e v e l .
A G E V S G R O U P S I Z Ea g j . 2 5 3 ; n o s i g . a . 0 l e v e l .
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T A B L E
D i s t r i b u t i o n o P r p o s t A n s w e r
S e l e c t i o n s f o A g e L e v e l 1 7
P r e T e s t S e l e c t i o n
1 = 8 . 2p = . 0 0 0 3
0 = 1 7 = 2 1 p = . O 5 1 6
T O T A L S
0 = = 3 . 6 6p = . 4 7 9 9
0 = " = 8 . 9 2p = . 2 7 6
0 = = 1 5 . 1 3p = . 0 0 0 4
6 = 4 0 . 7 6p = . 0 0 0 0
0 = = 6 . 7p = . 2 6 1 5
0 = = 1 6 . 3 9p = . 0 0 1 6
= 1 . 7 2P 2 = . 4
0 = = 4 . 6 4p = . 2 1 7 8
I T O T A L S
0 = 1 3 = 1 0 . 9 1p = . 1 4 8 8
1 = 2 9 . 4 1P = . 0 0 0 1
m u m t . m k j m < m . d | _
0 = = 0 . 7p = . 0 0 3 5
0 = = 1 . 8 7p = . 2 7 9
C e l l s 0 > a t . 0 5 l e v e l
C e l l s 0 < a . 0 l e v e l
0 = = 4 . 8 5p = . 3 4 6
1
2 = 1 1 . 8 p = . 0 0 0 0
1 8
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T A B L E
E m p i r i c a l T e s t o f t h e M u l t i n o m i a l P r o c e d u r eW i t h S t a b l e A n s w e r s R e m o v e d
A g i M o n t h s
o P r e s t e s t
( a
< 9 9
1 0 1 0 1 1 1 1 1 2 1 2 1 3 1 3 1 4 1 4 1 5 1 5 1 6 1 6 1 7 1 7 1 8 1 8 1 9 1 9 2 0 2 0 2 1 2 1 2 2
> 2 2 5
T o t a l s
* *
5
: ;
-
6
W r o n gA n s w e r C h a n g !
D
' .
6
.
5
3
2
7
'
<
1 f
3 ! L
1
1 1
1
6
1 9
" ^ 1 1
1 1 = r
1
1 1
1 8
* >
6
S t a b l eH r o n g
A n s w e r s
1
1 1 1 1 1 , '
^
1 2 2 1 1 1
i
3 3
1 1
1 1 1 X 1 1 1 1 1 1 1 1
1 1
5 6
1 )
^ <
1
:
1 3
R i g h tA n s w e r C h a n g e s
"
"
1 1 1
6
*
-
6
2 O 2 1 1
2 0
1
1 1
2 1 1 1 1
2 0
1 1 1 1 1 2 1 3 6 5 7 6 5 4 3 3 3 1 2 1
6 5
M
1 1 2 5 3 6 6 6 6 6 7 5 4 5 3 5 8 9
1 0 7 7 7 5
1 0 2 3 1 1
1 4 5 7
E N L A R G E D N U M B E R S a r s i g n i f i c a n t a t t h e . 0 l e v e l .I f r o n t o n u m b e r s i n d i c a t e s
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T A B L E
D i s t r i b u t i o n o P r p o s t A n s w e r
C h a n g e s f o rA g e L e v e l 1 7
P r T e s t S e l e c t i o n
1
T O T A L S
0 = 1 3 = 8 . 0 5p = . 0 0 2 4
= 1 . 3 8P = . 4
0 = 1 7 = 1 1 . 4 8p = 0 . 0 0 2 4
0 = 1 7 = 1 4 . 8 7p = 1 3
0 = = 4 . 0 3p = . 3 7 5 0
0 = = 2 . 8 5p = . O 3 1 9
0 = = 5 . 2 3p = . 4 2 9 O
0 - = 5 . 2 8p = . 0 9 8 7
0 = = 3 . 7 1p = . 0 0 1 6
0 = = 1 . 1 7p = . 0 8 6 0
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T A B L E
D e n s i t y o S i g n i f i c a n t E v e n t sW i t h i n A n t i c i p a t e d C h a n g e P a t t e r n s
C H A N G EP A T T E R N
-
- *
- *
-
A g e r a n g e s a r e e s t i m a t e d f r o m r e p o r t e d r e a s o n i n g .
1 0 . 5
1
A G R A N G E
1 2 . 5 Y e a r s
1 6 . 5 Y e a r s
2 0 Y e a r s
2 0 Y e a r s
P R O P O R T I O NO S I G N I F I C A N T
C E L L S
. 6
. 7
. 7
. 8