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A transformation to stabilize the variance of binomial distributions Green, Virginia Beryl (Berry) 1967

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A TRANSFORMATION TO STABILIZE THE VARIANCE OP BINOMIAL DISTRIBUTIONS VIRGINIA,BERYL (BERRY) GREEN B. Sc., P o r t l a n d S t a t e C o l l e g e , i 9 6 0 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF ; . MASTER OF ARTS i n the Department .•' ' ; ' . o f • ;--MATHEMATICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e . r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA May, 1967 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g ree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y , I f u r t h e r ag ree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s , I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada D a t e ( i i ) . ;• ' ABSTRACT A transformation i s sought for a binomially d i s t r i b u t e d random variable f such that the transformed v a r i a t e Y(f) exhibits a homogeneous variance, E{(Y(f )-E{Y(f)}) 3 = 1,;' and an unbiased mean, E{Y(f)) = Y(p), f o r the family of binomial d i s t r i -butions of given sample size generated by p . Y(f) i s expanded i n a Taylor series about p , conditions are set corresponding to the above requirements, and the r e s u l t i n g non l i n e a r d i f f e r e n t i a l equations! i n Y(p) are solved numerically. The success of the transformation i s comparable to. published'transformations. ( i l l ) ACKNOWLEDGEMENT I would l i k e t o e x p r e s s my th a n k s t o Dr. S t a n l e y Nash f o r s u g g e s t i n g t h i s p r o b l e m and f o r h i s h e l p and p a t i e n c e I n p u r s u i n g i t . I would a l s o l i k e t o thank t h e N a t i o n a l R e s e a r c h C o u n c i l f o r i t s f i n a n c i a l a i d and The U n i v e r s i t y o f B r i t i s h . Columbia Computing' C e n t r e f o r the use o f t h e i r f a c i l i t i e s . .. • ' ( i v ) ' ; TABLE OP CONTENTS . ' , "• ; Page CHAPTER I I n t r o d u c t i o n 1 CHAPTER I I F o r m u l a t i o n o f t h e C o n d i t i o n s f o r Homogeneous ., V a r i a n c e and U n b i a s e d Mean. 8 CHAPTER I I I The Second Order A p p r o x i m a t i o n s 14 CHAPTER IV The N u m e r i c a l S o l u t i o n 19 CHAPTER V A D i s c u s s i o n o f t h e N u m e r i c a l . r e s u l t s 23 BIBLIOGRAPHY . ' '.. 71 TABLES Page TABLE I . N u m e r i c a l s o l u t i o n compared w i t h an e x a c t s o l u t i o n 21 TABLE I I N u m e r i c a l s o l u t i o n compared w i t h M a c l a u r i n s e r i e s s o l u t i o n 22 TABLE I I I The b i a s i s d i s p l a y e d f o r v a r i o u s t r a n s f o r m a t i o n s 29 -31 TABLE IV S t a r t i n g v a l u e s t o t h e n u m e r i c a l s o l u t i o n . 4 l TABLE V E f f e c t i v e range o f t h e t a b u l a t e d t r a n s f o r m a t i o n . 42 TABLE V I The b e s t t r a n s f o r m a t i o n s a t t a i n e d f o r . ; n = 4 -to n = 30 . '4 -3-70 ( v i ) GRAPHS GRAPHS 1, 2y 3 , ^ Comparison c f t h e a b i l i t y o f t h e t r a n s f o r m a t i o n s o f Tukey Anaeombe and Ta b l e V I t o p r o v i d e a s t a b l e v a r i a n c e Page 33-36 GRAPHS 5, .6, 7 Response i n t h e v a r i a n c e t o a v a r i a t i o n o f the t e r m i n a l v a l u e s o f T a b l e V I • \ 37-39 GRAPH 8 Response i n t h e v a r i a n c e t o t h e s t a r t i n g v a l u e s used i n t h e s o l u t i o n 40 • / Chapter I I n t r o d u c t i o n F o r many s t a t i s t i c a l a p p l i c a t i o n s t h e c a l c u l a t i o n s a r e g r e a t l y s i m p l i f i e d i f t h e p o p u l a t i o n s under c o n s i d e r a t i o n have e q u a l v a r i a n c e s . F u r t h e r m o r e the homogeneity c o n d i t i o n i s n c e s s a r y f o r many s t a n d a r d s t a t i s t i c a l t e s t s . Thus a common pr o c e d u r e i n s t a t i s t i c s i s t o t r a n s f o r m t h e v a r i a t e s i n s u c h a way t h a t e q u a l o r a p p r o x i m a t e l y e q u a l v a r i a n c e s a r e o b t a i n e d . A t r a n s f o r m a t i o n a p p l i c a b l e t o b i n o m i a l l y d i s t r i b u t e d v a r i a t e s was I n t r o d u c e d by F i s h e r ( 5 ) ' i n 1922. A number o f s u c h t r a n s -f o r m a t i o n s f o r b i n o m i a l l y d i s t r i b u t e d random v a r i a b l e s a r e a v a i l a b l e ; w e l l known ones i n c l u d e : F i s h e r ' s (5) Y = a/n S i n " 1 Jxjxi , Y ' i n r a d i a n s V a r i a n c e (Y) = 1 + o ( l ) a s n - » » (2jn S i n " 1 J^n. x = 0 2jn S i n " 1 J^n~ x = 1, 2, . . . , n - l 2jn S i n " 1 Jl- -Jn . x = n / f o r Y i n r a d i a n s , V a r i a n c e (Y) ^ 1 •+ - ^ 1 1 + 2^§. \ / a t p = 1/2 n . nd Anscorabe's ( l ) Y/= 2(n+±) ' S i n " " 1 ( — | ) -l V 1 / 2 .,„-l 1/2 Y i n r a d i a n s ? V a r i a n c e (Y) = 1 + O ( ^ ) 2A n" n 2. 1/2, 1/2 Tukey's (6) Y = (n + | ^ [ S i n " 1 ^ ) - + S i n " 1 ( f g ) ]' Y i n radians, Variance (Y) = 1 + .06 f o r almost a l l cases f o r which — <: p _< ~ — . •/ The transformations are various modifications of Fisher's arcsine transformation. The forms given here are not a l l exactly as o r i g i n a l l y given by these authors. They are given i n t h e i r present form for uniformity's sake. Bartlett's'and Tukey's versions are obtained empi r i c a l l y . Anscombe.introduces three parameters into the^Fisher equation: y - 2 ( n + ( i l ) l / 2 S i ^ g f - ) 1 7 , 2 ' finds the asymptotic moments of y through Taylor s e r i e s expansion, and evaluates the parameters such that the variance i s optimally stable. / • . • -In Eisenhart (4) pg. 410 we f i n d a simple d e r i v a t i o n of Fisher's transformation based on the general developments of Curtiss ( J ) . " In this thesis we have recognized that Eisenhart's expressions may be extended to higher orders of approximation. S p e c i f i c a l l y we f i n d an expression f o r the variance of the trans-formed binomial variate Y . Variance ( Y) = * S Y * + ESL$f2l Y y + f i J ^ l ? + P q ( l - f M > ) Y^Y + 5 ( p q ) ^ q - P ) y Y + ii«lZ Y . v ,. { i m l ! •+ Pq(l;6pq)) Y? 6n3 1 4 W> 1 5 2n^ W " + ^ J J ^ I ! y 2 + 1 (pq) 2(q-p) Y 2 Y 5 + ifi§£ Y 2 Y 4 2 . 1 1 + terms i n — j r , — f > • • • • n rP The d e t a i l e d d e r i v a t i o n o f t h i s e q u a t i o n i s t o be f o u n d i n c h a p t e r I I o f t h i s t h e s i s . B r i e f l y t h i s e q u a t i o n i s o b t a i n e d by e x p a n d i n g • Y ( ^ ) i n a T a y l o r s e r i e s about p , t a k i n g t h e m a t h e m a t i c a l e x p e c t a t i o n E ( Y ( x / n ) ) , f o r m i n g ( Y ( x / n ) - E ( Y ( x / n ) ) ) and d e t e r m i n i n g i t s e x p e c t e d v a l u e . S e t t i n g t h e v a r i a n c e e q u a l t o u n i t y we have t h e n t o s o l v e a non l i n e a r d i f f e r e n t i a l e q u a t i o n ( a c c o r d i n g t o our n o t a t i o n Y^ i s t h e k t h d e r i v a t i v e o f Y) . -k Note t h a t t r u n c a t i o n o f terms c o n t a i n i n g n . k > 1 g i v e s t h e a n g u l a r t r a n s f o r m a t i o n o f F i s h e r . F o r improved r e s u l t s t h e e q u a t i o n -k a r i s i n g f r om t r u n c a t i o n o f terms c o n t a i n i n g n k > 2 must be c o n s i d e r e d : 2 'I nY^ + ( q - p j Y ^ + p q Y ^ ' + -|pqYf = |_ " ! No c l o s e d f orm s o l u t i o n c o u l d be found t o t h e non l i n e a r d i f f e r -e n t i a l e q u a t i o n . A c l o s e d f o r m a p p r o x i m a t e s o l u t i o n r e s e m b l i n g . Anscombe's and e q u i v a l e n t i n i t s a b i l i t y t o p r o v i d e a s t a b l e ' 4 . v a r i a n c e was f o u n d : / _ n -• 1/2 0 Y = - V V 2 S i n " 1 ( ( - ^ 4 ) ( ~ - 1)}' • ( s e e e q n T ) (n -/ ' S i n c e a n u m e r i c a l . s o l u t i o n was e v e n t u a l l y employed, we chose t o c o n s i d e r a' h i g h e r o r d e r o f a p p r o x i m a t i o n : t r u n c a t i o n o f terms c o n t a i n i n g n ~ k k > 3 • T h i s l a t t e r e q u a t i o n s h a l l he r e f e r r e d t o as the t h i r d o r d e r a p p r o x i m a t i o n and t h e f o r m e r as t h e second o r d e r a p p r o x i m a t i o n . The t a b u l a t e d t r a n s f o r m a t i o n s appended a r i s e ' 7 f r o m the n u m e r i c a l s o l u t i o n t o t h e t h i r d o r d e r / a p p r o x i m a t i o n w h i c h was s u p e r i o r t o t h e second o r d e r r e s u l t s . D i f f e r e n t i a l e q u a t i o n s have i n g e n e r a l f a m i l i e s o f s o l u t i o n s . I n c h a p t e r I I I we argue f o r a one p a r a m e t e r ^ f a m i l y o f s o l u t i o n s t o t h e d i f f e r e n t i a l e q u a t i o n a r i s i n g f r o m t h e second o r d e r a p p r o x i m a t i o n and a two p arameter f a m i l y o f s o l u t i o n s f o r . t h e t h i r d o r d e r a p p r o x i m a t i o n . We may t a k e t h e p a r a m e t e r s as Y ^ ( l / 2 ) i n t h e second o r d e r a p p r o x i m a t i o n and as Y ^ ( l / 2 ) and Y ^ ( l / 2 ) i n t h e t h i r d o r d e r a p p r o x i m a t i o n . A l l v a l u e s o f t h e parameters do not g i v e r i s e t o an e q u a l l y s t a b i l i z e d v a r i a n c e . Thus th e known approximate t r a n s f o r m a t i o n s were u s e d t o i n i t i a l l y f i x t h e p a r a m e t e r s . A s e a r c h i n t h e n e i g h b o u r h o o d o f t h e s e i n i t i a l v a l u e s was 'conducted u n t i l t h e c r i t e r i o n o f s t a b i l i t y was met. Two q u e s t i o n s a r e now r a i s e d . F i r s t what i s t h e maximum s t a b i l i t y o f v a r i a n c e t h e t r a n s f o r m a t i o n can a c h i e v e w i t h o u t any a p p r o x i m a t i o n ? S e c o n d l y what c r i t e r i o n o f s t a b i l i t y i s t o be adopted? I n answer t o the f i r s t q u e s t i o n , i t can be seen u s i n g a method due t o C u r t i s s (3) t h a t p e r f e c t s t a b i l i z a t i o n c a n n o t be o b t a i n e d . The o n l y t r a n s f o r m a t i o n g i v i n g a s t a b l e v a r i a n c e i s t h e t r i v i a l mapping o f x/n onto a c o n s t a n t ( w h i c h o f c o u r s e s t a b i l i z e s t h e v a r i a n c e t o z e r o ) . P r o o f : L e t Y ( k ) be a t r a n s f o r m e d b i n o m i a l v a r i a t e i n d e p e n d e n t o f p and whose v a r i a n c e i s in d e p e n d e n t o f p . F o r a g i v e n n t h e v a r i a n c e o f Y ( k ) i s Y 2 ( k ) ( > k ( l - p ) n - k - [ * ? Y ( k ) ( > k ( l - p ) n - k ] 2 = C k=o K k=o * where C may depend on n b u t n o t on p. We e x t r a c t t h e c o n s t a n t t e r m on the l e f t hand s i d e . Y 2 ( o ) ( l - p ) n + e Y 2(k)(£)p k(l-p) n- k - [ Y ( o ) ( l - p ) n + k = l ' k' S Y ( k ) ( 2 ) p k ( l - P ) n " k ] 2 = C k = l * Y 2 ( o ) - Y 2 ( o ) E ( - l ) k ( " ) p k ' + E Y 2 ( k ) ( £ ) p k ( l - P ) n - k k = l K / k = l . " .' • / . • .. • . ' -[Y(o)+Y(o) ? (-l) k(g)p k +... Z Y ( k ) ( ^ ) p k ( l - p ) n - k ] 2 = C . . k=l' / k = l K /• Here we see t h a t t h e ^ c o n s t a n t terms c a n c e l ; t h e n C = 0 and t h e i t r i v i a l mapping f o l l o w s . I n f a c t t h e v a r i a n c e may be e x p r e s s e d a s . a p o l y n o m i a l i n /p . w i t h c o e f f i c i e n t s d e p e n d i n g upon t h e t r a n s -6. formed v a r i a t e . The degree o f t h e p o l y n o m i a l i s a t most 2n so t h a t s t a b i l i z a t i o n cannot be improved beyond th e a c c u r a c y w i t h w h i c h a non z e r o c o n s t a n t can be a p p r o x i m a t e d , on t h e i n t e r v a l [ 0 , 1 ] by a p o l y n o m i a l o f degree 2n whose c o n s t a n t t e r m i s z e r o . We f u r t h e r o b s e r ve t h a t t h e v a r i a n c e i s z e r o f o r p = 0 and p = 1 and i s symmetric about p = 1/2 . These p r o p e r t i e s s h o u l d be p r e s e r v e d by t h e t r a n s f o r m e d v a r i a t e , a d d i n g t o t h e d i f f i c u l t y o f s t a b i l i z a t i o n . To answer the second q u e s t i o n we chose t h e somewhat a r b i t r a r y c r i t e r i a t h a t t h e v a r i a n c e l i e w i t h i n . + 2°/o o f one f o r as l a r g e a range o f p as p o s s i b l e . F o r a d e t a i l e d c o m p a r i s o n o f o u r t r a n s f o r m a t i o n w i t h Tukey's and Anscombe's we r e f e r the r e a d e r t o graphs 2, 3 , 4 w h i c h a r e r e p r e s e n t a t i v e o f t h e r e s u l t s f o r n between 10 and 15, 16 and 25, and over 2.5 r e s p e c t i v e l y . Comparison o f t h e t h r e e t r a n s f o r m a t i o n s f o r /n = 4 shows t h e g r e a t e s t s i g n i f i c a n t i m p r o v e -I ment f o r our t r a n s f o r m a t i o n . Q u a l i t a t i v e l y i t may be s a i d t h a t t h e v a r i a n c e o f o u r / t r a n s f o r m e d v a r i a t e has a s m a l l e r range o f / s t a b i l i t y under Tukey's 6°/o c r i t e r i o n b u t a l a r g e r range under a 2°/o c r i t e r i o n . . Anscombe's t r a n s f o r m e d v a r i a t e g i v e s r i s e t o a. " f l a t " variance./but a t a c o n s i d e r a b l e s a c r i f i c e o f r a n g e . We have computed the v a r i a n c e f o r a l l t h r e e t r a n s f o r m a t i o n s f o r n = k t o n =/30 . I n a l l c a s e s e x c e p t n = k we f i n d Tukey's extended range r e s u l t s f r o m a 6°/o "hump" n e a r t h e ends o f h i s s t a b l e range. T h i s e f f e c t can be i n d u c e d i n our t r a n s f o r m a t i o n by a p p l y i n g an end p o i n t " c o r r e c t i o n " t o our t r a n s f o r m a t i o n , t h i s i s i l l u s t r a t e d i n graphs 5, 6, and 7 . A r e q u i r e m e n t o f t h e t r a n s f o r m a t i o n i n a d d i t i o n t o s t a b i l i t y o f v a r i a n c e ' i s E { Y ( x / n ) } = Y(P) . T h i s c o n d i t i o n d i d not e n t e r e x p l i c i t l y i n t o our s o l u t i o n b u t we f i n d our t r a n s f o r m a t i o n , s t a t i s f i e s t h e c o n d i t i o n more c l o s e l y t h a n Tukey's b u t n o t q u i t e as w e l l as Anscombe's t r a n s f o r m a t i o n . 8. Chapter I I F o r m u l a t i o n o f t h e C o n d i t i o n s f o r Homogeneous V a r i a n c e and U n b i a s e d Mean I n t h i s c h a p t e r the t r a n s f o r m e d v a r i a t e Y ( f ) i s expanded i n a T a y l o r s e r i e s about p . C a l l i n g upon t h e c e n t r a l moments o f the u n t r a n s f o r m e d v a r i a t e we f o r m t h e moments o f t h e t r a n s f o r m e d v a r i a t e and s e t down t h e c o n d i t i o n s on t h e f u n c t i o n Y ( f ) .' / . L e t f be^/a random v a r i a b l e h a v i n g a b i n o m i a l d i s t r i -b u t i o n f u n c t i o n B ( f : p , n ) , f t h e r e l a t i v e f r e q u e n c y o f s u c c e s s , p t h e p r o b a b i l i t y o f s u c c e s s i n a s i n g l e t r i a l and n t h e number o f t r i a l s . We desire a f u n c t i o n Y ( f ) i n d e p e n d e n t o f p , monotone i n c r e a s i n g , a n t i s y m m e t r i c about t h e p o i n t ( 1 / 2 , Y ( l / 2 ) ) , h a v i n g e x p e c t a t i o n ,Y(p)/ and a v a r i a n c e o f u n i t y . We t a k e Y ( o ) = 0 . . L e t Y ( f ) be d e f i n e d and have d e r i v a t i v e s up t o and / i n c l u d i n g t h e / ( k + l ) t h o r d e r f o r 1 a l l f e [ 0 , 1 ] . E x p a n d i n g Y ( f ) i n a T a y l o r s e r i e s about p € (0,1) we have: Y ( f ) = Y(p) + ( f - p ) Y 1 ( p ) + ( f - p ) 2 Y 2 ( p ) / 2 J + ... • , ... + ( f - p ) k Y k ( p ) / k i + ( f - p ) k + 1 Y k + 1 ( 0 ) / ( k + l ) i Y^(p) i s u n d e r s t o o d t o be t h e k d e r i v a t i v e o f Y e v a l u a t e d a t f = p . The l a s t t e r m i s t h e L a Grange r e m a i n d e r . 9 -e ( p , f ) and 6 depends upon f . 9. T a k i n g t h e e x p e c t a t i o n o f Y ( f ) s E ( Y ( f ) ) , w h e r e u k = E ( ( f - p ) k ) E ( Y ( f ) ) = Y ( p ) + u 2 Y 2 ( p ) / 2 J + n 5 Y 5 ( p ) - + . . . I • . . . + u k Y k ( p ) / k i + E ( ( f - p ) k + 1 Y k + 1 ( e ) ) / ( k + l ) ! The moments o f a b i n o m i a l d i s t r i b u t i o n _ a r e " o = 1 E ( f ) = P u x = 0 = SaLgSl = w ( n - 2 }  J n ,, - 3(pq) , p q ( l - 6 p q ) / - 2 \ n vi = I 0 ( p q ) 2 ( q - P ) + p q ( l - 1 2 P q ) ( q - p ) 0 w ( n - 3 ) n . n ^ = 1 5 ( p q ) 5 ... 5 (pq ) 2 ( ' 5 - 2 6 p q ) + p q ( l - 3 0 p q + 1 2 0 ( p c j 2 ) = w ( n - 3 ) n n ^ u 2 k - l = w ( n " k . ) ^2k = w ( n _ k ) - k \ k w h e r e u . = w ( n ) means t h a t l i m n u . = C j some f i n i t e n o n z e r o c o n s t a n t . 10. Assuming E ( Y k + 1 ( 9 ) ) and E ( Y 2 + 1 ( 6 ) ) a r e 0 ( 1 ) we f i n d 2 E( Y ( f ) ) = Y(p) YJv) + Y,(p) Y^(p) + .... + • ^ ^ . 6n on ' • _k + '£t \ ( P ) + °(n 2 ) T r u n c a t i n g terms beyond n - 2 and r e q u i r i n g t h a t E ( Y ( f ) ) = Y ( p ) results i n the d i f f e r e n t i a l e q u a t i o n 1*) 3pqY 4(p) + 4(q-p)Y 5(p) + 1 2 n Y 2 ( p ) = 0 . To s a t i s f y the c o n d i t i o n t h a t . E ( Y ( f ) - E ( Y ( f ) ) ) 2 = 1 we f o r m the d i f f e r e n c e , • co { ( f - p ) k - u . } Y ( f ) - E( Y ( f ) ) = ( f - p ) Y ] _ ( p ) + s ' k i ' K Y k ( p ) , k— 2 the square o f the d i f f e r e n c e , o o o 0 0 { ( f - p ) k + 1 - ( f - p ) u v ] ( Y ( f ) - E ( Y ( f ) ) ) 2 = ( f - p ) 2 Y 2 ( p ) + 2 E ^ Y ^ p j Y ^ p ) ' k—2 / 2 .+ E r ( f - p ) 2 k - 2 ( f - p ) V M 2 } ^ k=2 K k ( k ! r + 2 ^ ' - ' ^ - ( ^ V i ^ ^ ^ S ^ t h e n E ( Y ( f ) - E ( Y ( f ) ) r = u 2 Y j ( p ) + ' ^ ( p ) Y 2 ( p ) - > . J k ( p ) Y L ( p ) = Y 2 ( p ) + £a(|=2l Y j p j Y ^ p ) + { - k § £ + P q ( ^ 6 p q ) } Y 1 ( p ) Y ( p ) n -1- n n , ... J n ^ ^ • + ^pq)^(q-p) Y ^ P J Y ^ P ) + Y 1 ( P ) Y 5 ( P ) 611^ 1 4 . % i + { i p a i ! + ( p q ) ( i - 6 p q ) } Y 2 ( P ) + f / s a f Y 2 ( P ) + | ( P q ) ^ ( q - P ) Y 2 ( p ) Y 5 ( p ) + - ^ - Y 2 ( p ) Y 4 ( p ) + terms i n ' n " 4 + N e g l e c t i n g terms c o n t a i n i n g n ~ k , k > 2_, and s e t t i n g E ( Y ( p ) - E ( Y ( p ) ) ) 2 = 1,.we o b t a i n 2 2*) n Y 2 ( p ) >• ( q - p ) Y 1 ( p ) Y 2 ( p ) + pqY ]_(p.)Y 5(p) + | p q Y 2 ( p ) . = | p q S i m i l a r i l y t h e c o n d i t i o n E( Y ( f )-Y(p) ) 2 = 1 ... where Y 2 ( p ) • • Y , ( p ) Y T ( p ) E ( Y ( f ) - Y ( p ) . ) 2 = Z H 2 k 7 ^ 2 + 2 2 E ^k+L - \ ^ ? L k = l • ( k J ) k = l L=k+1 K + l j K , i j -12. = £2 Y 2(p);"+. ( i L E a i ! + p q ( l - 6 p q ) ) Y | ( p ) + ^ I j E S ^ Y 2 ( p ) + PSl2l£l Y l ( p ) Y p ( p ) + + P ^ - f o l ? ) Y 1 ( p ) Y 3 ( p ) + 5 ( P q ) 2 ( q - P ) Y 1 ( p ) Y 4 ( p ) + ^ Y 1 ( p ) Y ( . ( p ) + 5(pq ) 2 ( q,-p ? Y ( p ) Y + K ^ a L Y 2 ( p ) Y 4 ( p ) + terms i n + .... , 8n r e s u l t s i n . * 9 2 3*) n Y 2 ( p ) + (q-p>Y 1(p)Y 2(p)--+ p q Y - ^ p ^ p ) + |pqY|(p) = | ^ . We can more e a s i l y a p p l y t h e symmetry c o n d i t i o n by t r a n s f o r m i n g e q u a t i o n s 1 * ) , 2*), and a c c o r d i n g t o : t = 2 p - l U ( t ) = Y(p) - Y ( l / 2 ) L e t t i n g be u n d e r s t o o d t o be the k t h d e r i v a t i v e o f U ( t ) s e q u a t i o n 1*) becomes: 1) 3 n U 2 - 2 t U 3 + | ( l - t 2 ) U 4 = 0 . The s i m i l a r i t y o f e q u a t i o n s 2*) and 3*) s u g g e s t s a s i m u l t a n e o u s e x p r e s s i o n o f them t h r o u g h a par a m e t e r m . • : 2) m = 1/2 \ 13. n i l 2 - 2 t U , U 0 + ( l - t 2 ) U - , U , + m ( l - t 2 ) U 2 = — 1 1 2 v 1 5 2 ( 1 - t 2 ) 3) m = 3 A • S i n c e the e a s i e s t approach t o t h e above d i f f e r e n t i a l e q u a t i o n i s a p p a r e n t l y t h r o u g h a n u m e r i c a l s o l u t i o n , we f i n d i t p o s s i b l e t o c o n s i d e r a h i g h e r o r d e r o f a p p r o x i m a t i o n t o t h e r e q u i r e m e n t s o f 1 * ) , 2*),- and 3*) • F o r t h e c o n d i t i o n on 1*.) t r u n c a t i o n o f terms c o n t a i n i n g n f o r k > 3 g i v e s r i s e to:. • 2 4) 3 n 2 U 2 - 2 n t U ? + | n ( l - t 2 ) u y - t ( l - t 2 ) U 5 + - j j ( l - t 2 ) Ug = 0 * v • • . .. • \ f o r t he c o n d i t i o n on 2*') } ' ' ' 1 2 n 2 U 2 - 24ntU ] LU 2 + ( I 2 n ( l - t 2 ) + 8 ( 3 t 2-l)')U- LU 5 - 20t( l - t 2 ) ^ ^ 5) + 3 ( . l - t c ) U XU 5•+ ( 6 n ( l - t c ) + 6 ( 3 t * - l ) ) U 2 + 5 ( l - t * ) u | 2 N2 T T ... + 2 r , .,.^2 , ^ T T 2 , J . 2 N 2 2 2 x 2 T T 1 2 n 5 - 3 6 t ( l - t * ) U p U , + 6(1-1^) U pU, = ± 2 * 2 5 2 4 ( 1 - t 2 ) A g a i n the e q u a t i o n r e s u l t i n g f r o m t h e r e q u i r e m e n t o f 3*) i s s i m i l a r . The c o e f f i c i e n t o f U 2 i s r e p l a c e d by 9 n ( l - t 2 ) + 6 ( 3 t The c o e f f i c i e n t o f UpU^ i s r e p l a c e d by - 4 o t ( l - t 2 ) and t h e P 2 c o e f f i c i e n t o f UgU^ • i s r e p l a c e d by 7 - 5 ( l - t ) , ( e q u a t i o n 6) 14. Chapter I I I The Second Order A p p r o x i m a t i o n s . ^ . I n t h i s c h a p t e r we f i n d an a p p r o x i m a t e s o l u t i o n t o t h e second o r d e r a p p r o x i m a t i o n , e q u a t i o n s 2 and 3, we f i n d an e x a c t s o l u t i o n t o a n o n l i n e a r d i f f e r e n t i a l e q u a t i o n o f t h e same f o r m as e q u a t i o n s 2 and 3 3 and vie examine t h e M a c L a u r i n s e r i e s s o l u t i o n t o e q u a t i o n s 2 and J>. As n becomes l a r g e , t r u n c a t i o n o f terms beyond n"*3' would seem t o be a v a l i d a p p r o x i m a t i o n and l e a d s t o t h e r e s u l t U ( t ) = n 1 / / 2 S i n _ 1 t w h i c h i s t h e f a m i l i a r a n g u l a r t r a n s f o r m a t i o n : o f F i s h e r s i n c e 2 S i n _ 1 , / f = + S i n _ 1 ( 2 f - l ) . But we n o t e t h a t under the supposed a p p r o x i m a t i o n t h e d e r i v a t i v e s c o n t a i n s i n g u l a r -i t i e s o f i n c r e a s i n g o r d e r n e a r t h e e n d p o i n t s t = + 1 . Thus the h i g h e r d e r i v a t i v e s w i l l i n c r e a s e " f a s t e r " n e a r t h e e n d p o i n t s overcoming a p p r o x i m a t i o n s based on t h e magnitude o f t h e i r c o e f f i -c i e n t s . T h i s s u g g e s t s t h a t s u b s t i t u t i o n , o 1/2 ^ ( t ) = g ( t ) / ( l - t 2 ) w i l l s e r v e t o remove t h e s i n g u l a r b e h a v i o r as t a p p r o a c h e s + 1 . The e q u a t i o n s 2) and 3) become under t h i s s u b s t i t u t i o n , ' / ' • 1 2 ((n+1) - ( n - m ) t 2 ) g 2 +/2mt( l - t 2 ) g S l + ( l - t 2 ) (gg 2+mg 2) = ( l - t 2 ) n 2 . • / /. . . . / • 2 N o t i n g t h a t l - t " a l m o s t " f a c t o r s o u t , i n t h a t n+1 n-m. f o r l a r g e n , . we may argue f o r t h e d o m i n a t i o n o f t h e l e a d i n g term and w r i t e t h e a p p r o x i m a t i o n ^ 15-((n+1) - ( n - m ) t 2 ) g 2 = ( l - t 2 ) n 2 , v;hich y i e l d s t h e approximate s o l u t i o n 7) U ( t ) = ( n / ( n - m ) 1 / 2 ) S i n " 1 {((n-m)/(n+l) ) ' l / 2 t ] . Here we f i n d a s i m i l a r i t y t o the B a r t l e t t c o r r e c t i o n i n t h a t the argument o f a r c s i n e does n o t r e a c h i t s f u l l v a l u e and an even s t r o n g e r resemblance t o t h e t r a n s f o r m a t i o n o f Anscombe. T h i s a p p r o x i m a t i o n and t h e t r a n s f o r m a t i o n o f Anscombe a r e f o u n d t o ! s t a b i l i z e the v a r i a n c e e q u i v a l e n t l y . R e t u r n i n g t o t h e d i f f e r e n t i a l e q u a t i o n i n g 3 we p ' >•-c o n s i d e r t h e a r t i f i c i a l case m = - 1 . A f a c t o r 1-t may b e \ 1/2 removed and the e x a c t s o l u t i o n g = n / ( n + l ) i s . e v i d e n t \ 1/2 — 1 whereupon U ( t ) = n / ( n + l ) S i n " t i s an e x a c t s o l u t i o n t o e q u a t i o n s 2) and 3 ) . T h i s c i r c u m s t a n c e i s u s e f u l f o r t e s t i n g t h e proposed n u m e r i c a l s o l u t i o n . A Macl.aurin s e r i e s s o l u t i o n i s o f l i t t l e p r o f i t b u t does demonstrate s e v e r a l s a l i e n t f e a t u r e s , a degree o f f r e e d o m i n the s o l u t i o n t o e q u a t i o n s 2) and 3 ) , and t h a t t h e r e q u i r e m e n t s o f monotone i n c r e a s i n g a n t i s y m m e t r i c s o l u t i o n do n o t p r e c l u d e a s o l u t i o n . We r e q u i r e s o l u t i o n s t o t h e d i f f e r e n t i a l e q u a t i o n n U 2 - 2 t U 1 U 2 + ( l - t 2 ) ^ ^ + m ( l - t 2 ) U 2 = n 2 / ( l - t 2 ) and we t a k e U(0) = 0 . I f U ( t ) i s a n t i s y m m e t r i c t h e n t h e odd 16. d e r i v a t i v e s o f U ( t ) a r e symmetric and t h e even d e r i v a t i v e s a r e antis.ymmetric. I n s p e c t i o n o f t h e terms i n t h e d i f f e r e n t i a l e q u a t i o n above shows each t o be symmetric i f U ( t ) i s a n t i s y m m e t r i c . I t i s c o n v e n i e n t t o e s t a b l i s h t h e k c o n d i t i o n on t h e M a c l a u r i n c o e f f i c i e n t s by a p p l i c a t i o n o f t h e o p e r a t o r d/dt k t i m e s and s e t t i n g t = 0 . S i n c e U ( t ) a n t i s y m m e t r i c r e q u i r e s e a ch t e r m \k i n t h e d i f f e r e n t i a l e q u a t i o n t o be symmetric t h e a c t i o n o f ( d / d t ) upon each term g i v e s some a n t i s y m m e t r i c f u n c t i o n f o r k odd whose v a l u e a t t = 0 i s z e r o . F o r k = 0 ,2 , 4 , 6 , 8 , and 10 t h e f o l l o w -i n g c o n d i t i o n s o b t a i n 7 r e s p e c t i v e l y . « Q2' n B „2 . _ d*TJ(t) n a l + a l a 5 = n *k ~ d ^ k / t=o ( 2 n - 6 ) a 1 a ^ + a 1 a ^ + ( l + 2 m ) a 2 = 2 n 2 (2n-20)a ] La 5+a 1a 7+(2n-20-2 il-m)a 2 + (7+8m)a^a 5 = 4 i n 2 ( 2 n - 4 2 ) a x a 7 + 1 0 ( 3 n - 3 9 - 2 4 m ) a 3 a 5 + a x a g + 4 ( 4 + 3 m ) a 5 a ? + + ( l 5 + 2 6 m ) a | = 6.'n2 ( 2 n - 7 2 ) a 1 a g + 5 6 ( n - 5 6 ) a ^ a 7 + ( 4 2 0 n - l 4 0 0 - l 4 5 6 m ) a 2 + a ^ a ^ + + ( 2 9+l6m)a-a 9 + (98+136m)a 5a 7 = 8 » n 2 ( 2 n + 7 0 ) a 1 a 1 1 + 9 0 ( n - 1 0 ) a 3 a 5 + 10(42n+8400-12240m)a 5a 7 + 17. + a 1 a 1 ^ + ( ^ 6 + 2 0 m ) a 5 a n + (225+276m)a 5a^ + ( 2 1 0 + 3 1 2 m ) a 2 + + ( 2 6 l 0 - l 4 4 0 m ) a - a o = 1 0 J n 2 . The s o l u t i o n i s d e t e r m i n e d i f an a d d i t i o n a l c o n s t r a i n t such as the i n i t i a l s l o p e U-^0) i s s p e c i f i e d . The c o n d i t i o n o f a n t i s y m m e t r y does n o t e x c l u d e a s o l u t i o n and f o r a t l e a s t c e r t a i n c h o i c e s o f ui(°) w e f i n d U ( t ) monotone. F o r example we may t a k e U^(0) = 0 = a^ and n = 10 , m = 1/2 3 f i n d i n g . U ( t ) = a l = 3.1623 a 5 = 63.246 a 7 = 7 5 8 . 9 5 "9 = 40.477 a l l = I . 3 3 8 1 1 . I O a 1 5 = - I . 5 2 6 0 . I O 3 . l 6 2 3 t + . 5 2 7 0 5 t 5 + . I 5 0 5 8 6 t 7 + . 0 0 0 1 1 1 5 t 9 + . 0 3 3 5 2 2 t i ; L . 2 4 5 0 6 6 t 1 5 • Under t h e s e c o n d i t i o n s ( a ^ = 0) t h e dependence upon m i s s h i f t e d t o t h e c o e f f i c i e n t o f t 9 . I t i s a p p a r e n t t h a t convergence on t h e i n t e r v a l t = ( - 1 , 1 ) w i l l r e q u i r e many more terms t h a n a r e g i v e n h e r e . At t h i s p o i n t a b r i e f r e v i e w o f what i s known and unknown about t h e r e q u i r e d s o l u t i o n w i l l s e r v e as a r a t i o n a l e t o a n u m e r i c a l s o l u t i o n . 13. 1) We have no r e c u r s i o n r e l a t i o n 2) We have a degree o f freedom i n t h e s y s t e m o f c o n d i t i o n s and we must t h e n s e a r c h f o r t h e i n i t i a l s l o p e w h i c h b e s t s t a b i l i z e s t h e v a r i a n c e . 3) We do not know how many terms w i l l be r e q u i r e d t o i n s u r e convergence on t h e i n t e r v a l t = (-1,1). 4) We have an e x a c t s o l u t i o n f o r t h e a r t i f i c i a l s i t u a t i o n m = - 1 . 5) We have a M a c l a u r i n s e r i e s s o l u t i o n w h i c h may be presumed t o converge over about h a l f t h e range o f t . I t would seem t h a t 4 and 5 would s e r v e a s a s u f f i c i e n t t e s t f o r the a c c u r a c y o f any n u m e r i c a l s o l u t i o n we may a t t e m p t . 19-Chapter IV The N u m e r i c a l S o l u t i o n I n t h i s c h a p t e r we d e v i s e a n u m e r i c a l s o l u t i o n s u i t e d to- the e q u a t i o n s t o be s o l v e d and demonstrate i t s a c c u r a c y f o r c e r t a i n s p e c i a l c a s e s . A wide v a r i e t y o f n u m e r i c a l t e c h n i q u e s — a r e known; many o f f e r t h e advantage o f some i n d i c a t i o n o f t h e e r r o r i n v o l v e d . W e l l known t e c h n i q u e s use p r e d i c t o r - c o r r e c t o r methods w h i c h a r e ^ a l t e r n a t e l y a p p l i e d u n t i l i d e n t i c a l r e s u l t s a r e o b t a i n e d . I t was thought t h a t none o f t h e s e s u i t e d the p r o b l e m a t hand. A t e c h n i q u e w h i c h seemed a p p r o p r i a t e t o our p r o b l e m and l e n t i t s e l f t o a computer s o l u t i o n was d e v i s e d as f o l l o w s . We n o t e t h a t e q u a t i o n s 2 and 3 can be w r i t t e n i n t h e form, U 3 ( t ) = F ( t , n ; U 1 , U 2 ) , • ' which can be d i f f e r e n t i a t e d t o g i v e U 4 ( t ) = F'C.t.n.U^Ug.Uj) • > . We can a l s o w r i t e f o r d s m a l l i ' • . ' \ . . . U(t+d) = U ( t ) + U 1 ( t ) d + U 2 ( t ) d 2 / 2 . ' + . . . + U 4 ( t ) d V 4 J l l j t + d ) = U 1 ( t ) + .. . + U 4 ( t ) d 3 / 3 i U 2 ( t + d ) = U 2 ( t ) + . . . + U 4 ( t ) d 2 / 2 J 20, For s t a r t i n g v a l u e s jwe may t a k e U(0) = 0 . By symmetry we have U 2 ( 0 ) = H ^ ( 0 ) j - 0 . U^(0) i s f i x e d by t h e f i r s t o f t h e above e q u a t i o n s and an assumed v a l u e f o r U,(0) . The p o i n t by / • p o i n t e x t r a p o l a t i o n i s o b v i o u s as i s t h e e x t e n t i o n t o t h e s o l u t i o n / - • - "5 o f t he t h i r d o r d e r e q u a t i o n ( . i n c l u d i n g terms i n n J ) . A c c o r d i n g t o T a b l e ^ I we f i n d t h i s p r o c e d u r e a c c u r a t e l y r e p r o d u c e s t h e e x a c t s o l u t i o n o b t a i n e d f o r t h e case m = -1 f o r 99°/o of the range o f t and t h a t the e n d p o i n t s a r e somewhat u n d e r e s t i m a t e d . T a b l e I I demonstrates agreement w i t h t h e M a c l a u r i n s e r i e s . i n the r e g i o n where t h e s e r i e s i s t h o u g h t t o be v a l i d and a r e l a t i v e i n s e n s i t i v i t y t o t h e s i z e o f d . We t a k e t h i s l a t t e r o b s e r v a t i o n as a f u r t h e r d e m o n s t r a t i o n o f t h e v a l i d i t y o f t h e n u m e r i c a l s o l u t i o n . The t h i r d o r d e r e q u a t i o n s f r o m w h i c h we o b t a i n maximum s t a b i l i z a t i o n were n o t s u b j e c t e d t o t h e same s c r u t i n y as t h e second o r d e r e q u a t i o n s . 21. TABLE I E q u a t i o n 2 f o r m = -1 has as an e x a c t s o l u t i o n U ( t ) = n / ( n + l ) 1 / 2 S i n _ 1 t T h i s t a b l e compares the e x a c t and n u m e r i c a l s o l u t i o n . n = 5 e x a c t n u m e r i c a l 0.000 0.200 0.400 0.600 0.800 0.990 1.000 0.00000 0.41102 0.84001 1.31354 1.89283 2.917458 3.20637 0.00000 0.41102 0.84000 1.3135 1.8923 2.9172 3.1643 n = 10 e x a c t n u m e r i c a l 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.300 0.900 0.990 1.000 0.00000 0.30202 0.60712 0.91868 1.24077 1.57871 1.94023 2.33791 2.79590 3.37623 4.30937 4.73613 0.00000 0.30202 0.60712 0.91868 1.2408 1.5787 1.9402 2.3379 2.7959 3.3762 4.3091 4.6742 I n t h e n u m e r i c a l s o l u t i o n we have t a k e n d = 10 -3 and f o u n d no s i g n i f i c a n t improvement upon t a k i n g a f i n e r d i v i s i o n . 22. TABLE I I T h i s t a b l e demonstrates t h e e f f e c t o f t a k i n g d i f f e r e n t s i z e s o f d, the i n t e r v a l a p p e a r i n g i n th e n u m e r i c a l s o l u t i o n . These a r e com-pared t o the M a c l a u r i n s e r i e s d e t e r m i n e d on page 1 7 . We f i n d e x c e l l e n t agreement over the range o f v a l i d i t y o f t h e M a c l a u r i n s e r i e s . n = 10 U x ( 0 ) = ( 1 0 ) 1 / 2 m = 1/2 e q u a t i o n 2 t d = 1 0 " 2 d = 1 0 " 5 -4 d = 10 s e r i e s 0.000 0.00000 0.00000 0.00000 0.00000 0.100 0 . 3 1 6 2 3 0.31623 0 . 3 1 6 2 3 0.31623 0.200 0.63262 0.63262 0.63262 0.63261 0.300 0 . 9 ^ 9 9 9 0.94999 0.94998 0.94998 0 . 4 0 0 1.2706 , 1.2706 1.2705 1.27049 0.500 1 . 5 9 8 8 . 1.5988 1 . 5 9 8 8 1 . 5 9 8 8 5 0.600 1 . 9 4 2 9 1 . 9 4 2 9 1 . 9 4 2 9 0.700 2.3166 2.3166 2.3165 0.800 2.7^39 2 . 7 4 3 9 2 . ,7437 0.900 3 . 2 7 5 ^ 3 . 2 7 5 4 3 . 2 7 5 2 1.000 4.1733 4 . I954 4 . 1 9 6 9 R e c a l l i n g the f a c t t h a t t h e s t a r t i n g v a l u e U-^O) must be v a r i e d t o f i n d t h e optimum s t a b i l i z a t i o n o f t h e v a r i a n c e , we f i n d t h a t we can make a l a r g e g a i n i n t h e t i m e r e q u i r e d t o e x e c u t e a n u m e r i c a l s o l u t i o n by.making a s m a l l s a c r i f i c e i n a c c u r a c y . The l o s s i n accuracy, t a l c i n g d t o be o f the o r d e r o f 1/1,000 s h o u l d be i n s i g n i -f i c a n t i n th e computed v a r i a n c e . 23. Chapter V A - D i s c u s s i o n o f t h e N u m e r i c a l R e s u l t s The n u m e r i c a l r e s u l t s a r e d i s c u s s e d and r e p r e s e n t a t i v e cases s u b j e c t e d t o s c r u t i n y . The n u m e r i c a l p r o c e d u r e r e q u i r e s t h e s e t o f s t a r t i n g v a l u e s { 1^(0),U 2(o) ,U--(o) . 1^(0)} f o r t h e second o r d e r a p p r o x -i m a t i o n s and t h e s e t o f s t a r t i n g v a l u e s {U- L(o)_,U 2(o),U^(o) J,U2|.(q), Uj-(o)} f o r t h e t h i r d o r d e r a p p r o x i m a t i o n s . A c c o r d i n g t o t h e . c o n d i t i o n o f symmetry imposed upon t h e s o l u t i o n , we f i n d U^(o) = 0 f o r k even. F o r k odd we have no c o n d i t i o n e x c e p t t h e d i f f e r -e n t i a l e q u a t i o n e v a l u a t e d a t t = 0 . I n t h e second o r d e r a p p r o x i -mations (eqns. 2 and 3) we f i n d nU 2(o). + 1 ^ ( 0 ) 1 ^ ( 0 ) = n 2 w h i c h g i v e s U-^(o) i n terms o f U-^(o) . We have no c o n d i t i o n . on U^(o) e x c e p t t h a t we s h o u l d e x p e c t i t t o have a v a l u e n e a r . y n s i n c e U 1 ( o ) - n as n - » . Thus as a f i r s t a p p r o x i m a t i o n U-j^o) was t a k e n t o be y n . Graph 8 f o l l o w i n g shows t h e e f f e c t on the v a r i a n c e w h i c h r e s u l t s f r o m t a k i n g U-^(o) d i f f e r e n t f r o m y n as a s t a r t i n g v a l u e f o r the n u m e r i c a l s o l u t i o n t o t h e second o r d e r e q u a t i o n 2, f o r / n = 5 . We see t h a t as t h e i n i t i a l s l o p e u\(o) i s d e c r e a s e d (the v a r i a n c e becomes l e s s s t a b l e , t h a t i s humps are i n t r o d u c e d ^ but we o b s e r v e an e x t e n s i o n o f t h e range o f approximate s t a b i l i t y . I t i s t o be n o t e d t h a t f o r U-^o) = 1 .800 a s t a b i l i z a t i o n m e e t i n g Tukey's c r i t e r i o n i s o b t a i n e d . F o r most / 24. v a l u e s o f n. a s o l u t i o n m e e t i n g Tukey's c r i t e r i o n c o u l d be o b t a i n e d by v a r y i n g t h e i n i t i a l s l o p e U-Jo) i n t h e second o r d e r , a p p r o x i m a t i o n (eqn. 2 ) . -'However t h e s e s o l u t i o n s c o n t a i n e d a f a u l t n o t found i n Tukey's t r a n s f o r m a t i o n ; t h e y d i d n o t i n g e n e r a l have a r e l a t i v e l y f l a t r e g i o n i n t h e m i d d l e . The a b i l i t y t o meet Tukey's c r i t e r i o n by u s i n g a second o r d e r a p p r o x i m a t i o n made i t seem p r o f i t a b l e t o examine a h i g h e r o r d e r o f a p p r o x i m a t i o n ; however we found t h a t we w e r e ' n e i t h e r a b l e t o improve s i g n i f i c a n t l y t h e range o f Tukey's s t a b i l i t y ( k e e p i n g t h e same d e v i a t i o n o f + 6°/o) n o r were we a b l e t o r e t a i n t h e range r e p o r t e d by Tukey and d e c r e a s e the f l u c t u a t i o n t o s i g n i f i c a n t l y l e s s t h a n + 6°/o . There b e i n g l i t t l e p o i n t i n m e r e l y d u p l i c a t i n g Tukey's t r a n s f o r m a t i o n , we chose t o p r e s e n t an a l t e r n a t e t r a n s f o r m a t i o n , one x^hich w o u l d n o t attempt t o o f f e r t h e range o f s t a b i l i t y o f Tukey but r a t h e r t o o f f e r the maximum p o s s i b l e range f o r a s i g n i f i c a n t l y more s t r i n g e n t c r i t e r i o n o f s t a b i l i t y namely + 2°/o. Graphs 2, 3, and 4 a r e r e p r e s e n t a t i v e o f the b e s t t r a n s f o r m a t i o n s a t t a i n a b l e under t h e t h i r d o r d e r a p p r o x i m a t i o n , e q u a t i o n 5, s a t i s f y i n g t h e + 2°/o c r i t e r i o n . The two a r b i t r a r y s t a r t i n g v a l u e s U-^o) and U^(o) r e q u i r e d f o r t h e s o l u t i o n o f t h e t h i r d o r d e r a p p r o x i m a t i o n , e q u a t i o n 5, were f i r s t t a k e n as 7. Ul(°) = 'JffFT From t h e a p p r o x i m a t e s o l u t i o n f o u n d t o e q u a t i o n 2 . n 2 - n U 2 ( o ) U^(o) = -—fj~ 7 J c 7 )— the c o n d i t i o n on U-^(o) a r i s i n g f r o m t h e second o r d e r a p p r o x i m a t i o n eqn. 2 a t t = 0. 25-UV(o) was t a k e n t o s a t i s f y t h e t h i r d o r d e r a p p r o x i m a t i o n , e q u a t i o n 5 a t t = 0 . 1 2 n 2U 2(o) + ( 1 ^ - 8 ) ^ ( 0 ) ^ ( 0 ) + 31^(0 )1^(0) •+ 5u|(o) = 1 2 n 5 . S m a l l v a r i a t i o n s i n U-^o) were t r i e d u n t i l a maximum range r e t a i n i n g 2°/o s t a b i l i t y was a t t a i n e d . A c o r r e s p o n d i n g v a r i a t i o n on U-(o) independent o f U-^o) was n o t a t t e m p t e d . A t a b l e o f v a l u e s o f . 1^(0) used f o r t h e t r a n s f o r m a t i o n s appended t o t h e t h e s i s has been i n c l u d e d ( T a b l e IV) f o r the sake o f c o m p l e t e s p e c i f i c a t i o n o f the n u m e r i c a l p r o c e d u r e . Graph 1, (n=4) comparing t h i s work w i t h t h e Anscombe and Tukey t r a n s f o r m a t i o n s , r e p r e s e n t s an anomaly, s i n c e the t r a n s f o r m a t i o n s o b t a i n e d by t h e p r o c e d u r e o u t l i n e d i s a s t r i k i n g improvement t o t h e p u b l i s h e d t r a n s f o r m a t i o n s . The r e m a r k a b l e r e s u l t o b t a i n e d s u g g e s t s t h a t f o r o t h e r v a l u e s o f n we have n o t a t t a i n e d t h e optimum s o l u t i o n . Perhaps a more d e t a i l e d s e a r c h o f s t a r t i n g v a l u e s t o t h e t h i r d o r d e r a p p r o x i m a t i o n (eqn 5) w i l l p r o v e f r u i t f u l . A two p a r a m e t e r v a r i a t i o n i s s u g g e s t e d . R e c a l l the n u m e r i c a l s o l u t i o n t o t h e non l i n e a r d i f f e r -e n t i a l e q u a t i o n (m = -1 i n our second o r d e r a p p r o x i m a t i o n ) f o r w h i c h an e x a c t s o l u t i o n was found. The f a i l u r e o f t h e n u m e r i c a l s o l u t i o n a t t h e e n d p o i n t s o f t h e i n t e r v a l f o r t h i s e q u a t i o n was apparent but n o t s e r i o u s , (see T a b l e I ) 26. i The f a i l u r e i m p l i e s some i n a c c u r a c y i n t h e p r o c e d u r e , but o n l y near the e n d p o i n t s o f t h e i n t e r v a l . ' T h i s c i r c u m s t a n c e s u g g e s t s a' v a r i a t i o n i n the t e r m i n a l v a l u e s o f t h e t r a n s f o r m e d v a r i a t e t o examine t h e i r e f f e c t upon t h e v a r i a n c e . Graphs 5, 6, and 7 d i s p l a y t h e e f f e c t - A "hump" i n t h e e n d p o i n t r e g i o n s i s g e n e r a t e d upon the i n c r e a s e o f Y ( l ) '= - Y ( o ) , t h e t e r m i n a l v a l u e s . An i m p o r t a n t f e a t u r e to.be n o t i c e d i s t h e l i n e a r r e s p o n s e i n t h e v a r i a n c e t o the v a r i a t i o n o f the t e r m i n a l v a l u e s o f t h e t r a n s f o r m e d v a r i a t e , AV(p) = k AY(1) where k , t h e c o n s t a n t o f p r o p o r t -i o n a l i t y , i n c r e a s e s as p approaches z e r o o r one. We use t h e l i n e a r r e s p o n s e t o produce c e r t a i n e f f e c t s i n t h e v a r i a n c e i n a c o n t r o l l e d manner. F o r a g i v e n change i n t h e t e r m i n a l v a l u e s we can e s t i m a t e - w i t h r e a s o n a b l e a c c u r a c y t h e e f f e c t s u c h a change w i l l produce i n the v a r i a n c e , whereas a v a r i a t i o n i n t h e s t a r t i n g v a l u e s o f the n u m e r i c a l s o l u t i o n p r o d u c e s .an e f f e c t i n t h e v a r i a n c e d i f f i c u l t t o e s t i m a t e . F o r n = 20 and n = 30 t h e t r a n s f o r m a t i o n s a r e not g r e a t l y improved by v a r i a t i o n o f t h e t e r m i n a l v a l u e s . However f o r n = 10 (Graph 7) we can i m m e d i a t e l y see t h a t a t e r m i n a l v a l u e o f 4 . 0 5 t o 4 .07 w i l l s i g n i f i c a n t l y e x t e n d t h e range and y e t r e t a i n t h e + 2°/o c r i t e r i o n . We suggest t h a t t h e t a b l e s appended may be ' i m p r o v e d somewhat by a s i m p l e v a r i a t i o n o f t h e t e r m i n a l v a l u e s , however we' b e l i e v e t h e b e s t r e s u l t s would be o b t a i n e d t h r u i n d e p e n d e n t v a r i -a t i o n o f the two independent s t a r t i n g v a l u e s o c c u r i n g i n t h e n u m e r i c a l s o l u t i o n t o t h e t h i r d o r d e r a p p r o x i m a t i o n , e q u a t i o n 5« 27-Up t o t h i s p o i n t l i t t l e a t t e n t i o n has been p a i d t o t h e a u x i l i a r y r e q u i r e m e n t E ( Y ( f ) ) = Y(p) . Chapter I I c o n t a i n s t h e development o f the c o n d i t i o n t h a t t h i s b e - s a t i s f i e d ^ 0 = E( Y( f ) ) - Y(p) = |g Y 2 ( p ) + P^1" P ) V p ) V P> + ( P q ) 2 ( q - P ) Y _ ( p ) + Y g ( p ) + terms i n \ 9 ^ , . . . 12n^ . 3 43n^ n n^ A somewhat t r i v i a l s o l u t i o n i s Y ( f ) = a + b f , Y k ( p ) = 0 f o r k > 1 ,, the l i n e a r form i s not the g e n e r a l c o n d i t i o n b u t does s a t i s f y t h e re q u i r e m e n t e x a c t l y . • I n view o f the r o u g h l i n e a r i t y d i s p l a y e d by the t r a n s f o r m a t i o n s l i s t e d i n t h e a p p e n d i x we s h o u l d f i n d t h i s r e q u i r e m e n t a p p r o x i m a t e l y s a t i s f i e d . A d d i t i o n a l l y f o r p = 0 and p = 1 we f i n d t h e c o n d i t i o n i n h e r e n t l y s a t i s f i e d and on a c c o u n t o f t h e symmetry a s c r i b e d t o t h e t r a n s f o r m e d v a r i a t e we f i n d t h e c o n d i t i o n i s s a t i s f i e d e x a c t l y f o r p = 1/2 . Thus t h e c o n d i t i o n i s s a t i s f i e d e x a c t l y f o r p""= 0, p = 1/2 and p = 1 r e g a r d l e s s o f the t r a n s f o r m a t i o n i f i t i s a n t i s y m m e t r i c about p = 1/2 . C o n s i d e r t h e b i a s = -|E (Y) - Y(p) | . I n T a b l e I I I f o r n = 10 the maximum b i a s o b s e r v e d and t h e sum o f t h e b i a s e s , the sum b e i n g over p = f = 0, .1, .2,...,1.0 , a r e t a b u l a t e d f o r t he v a r i o u s t r a n s f o r m a t i o n s c o n s i d e r e d . We f i n d Tukey's t r a n s f o r m a t i o n t o be the,most b i a s e d and e q u a t i o n 7 (m=3/4) t o be the l e a s t b i a s e d . T h i s l e a s t b i a s e d e x p r e s s i o n i s n o t recommended i n v i e w o f i t s i n a d e q u a c y i n s t a b i l i z i n g t h e v a r i a n c e ; f u r t h e r i t . • / ' " 2 8 . does not p r o v i d e a s i g n i f i c a n t improvement o v e r t h e Ansc.ombe form. The t r a n s f o r m a t i o n found i n the t a b l e s appended g i v e s a As a f u r t h e r e x a m i n a t i o n o f t h e b i a s we have- s o l v e d e q u a t i o n 6 ( t h e t h i r d o r d e r a p p r o x i m a t i o n t o t h e c o n d i t i o n E ( ( Y ( f ) - Y ( p ) ) ) w i t h the same n u m e r i c a l p r o c e d u r e and s t a r t i n g v a l u e s u sed t o gen e r a t e the t a b l e s appended. T h i s s o l u t i o n d i s p l a y e d a s l i g h t l y more f a v o r a b l e b i a s t h a n i t s a n a l o g but the s o l u t i o n i n t r o d u c e d a r e s t r i c t e d range o f s t a b i l i t y . b i a s i n t e r m e d i a t e between Anscombe's and Tukey's t r a n s f o r m a t i o n s . / / / / TABLE III A comparison o f t h e b i a s f o r v a r i o u s t r a n s f o r m a t i o n s . vie c o n s i d e r n = 10 and p.e [ 0 , 1 / 2 ] , p e [ 1 / 2 , 1 ] f o l l o w s on account o f the a n t i s y m m e t r y o f Y ( f ) about f = 1 / 2 . ANSCOMBE1S TRANSFORMATION 0 0 .1 O.d 0 . 3 0.4 0-5 maximum b i a s sum o f b i a s e x p e c t e d v a l u e 1.2176 2.2368 3.0670 3-7889 4.4523 5.0900 0.1335 0.669 2 Y(p) ' 1.2176 2.3703 3.1712 3.8545. 4.4826 5.0900 TUKEY'S TRANSFORMATION 0 0 .1 0 . 2 0 . 3 0.4 0 . 5 maximum b i a s sum o f b i a s e x p e c t e d v a l u e 0.9925 2.1361 3.0723 3.8050 4.4625 5.0900 0.2338 0.9434 Y(p) 0.9925 2.4199 3.2079 3-3779 4.4949 5.0900 30. TABLE I I I CONTINUED THIS WORK EQUATION 5 0 0. 0, 0. 0. 0, 1 2 3 4 5 maximum b i a s sum o f b i a s e x p e c t e d v a l u e -3.9931 -2.8936 -2.0372 -1.0375 -0.6407 0.0000 0.1813 0.7768 Y(p) -3 .9931 -2.7123 -1 .9223 -1.2440 -0.6l.20 0.0000 THIS WORK EQUATION 6 0 0 .1 0 . 2 0 . 5 0.4 °-5 maximum b i a s sum o f b i a s e x p e c t e d v a l u e -3 .9631 -2 .8839 - 2 , 0 3 3 3 -1 .3064 - 0 . 6 4 0 5 0.0000 0.1739 0.7532 Y(p) - 3 . 9 6 3 1 -2 .7100 - 1.9220 -1.2440 - 0 . 6 1 2 0 0.0000 EQUATION 7, m, = 1/2 0 0. 0. 0. 0. 0. maximum b i a s sum o f b i a s e x p e c t e d value. - 3 . 8 6 9 2 -2 .8^21 -2.0227 -1.3.011 - 0 . 6 3 7 3 0.0000 0.1325 O .6656 Y(p) - 3 . 8 6 9 2 -2 .7196 -1 .9190 - 1 . 2 3 5 7 - 0 . 6 0 6 6 0.0000 TABLE I I I 'CONTINUED EQUATION 7, m = 3/4 p e x p e c t e d v a l u e Y ( p) 0 - 3 - 8 l c ; 3 -3 .8158 0 .1 - 2 . 8 2 8 2 - 2 . 7 0 8 0 0.2 -2.0127 -I . 9 1 5 6 0.3 -1.2972 - 1.2349 0.4 -0.6365 -O .6065 0 . 5 ,0.0000 0.0000 maximum b i a s 0.1202 sum o f b i a s / 0 .6192 / / / / / / / 32. GRAPHS .Graphs 1, 2, 3 , and 4 compare t h e e f f e c t i v e n e s s i n s t a b i l i z i n g the v a r i a n c e f o r the t r a n s f o r m a t i o n s o f Tukey, / Anscombe and the t a b l e s appended. Graphs $J 6 , and 7 i l l u s t r a t e t h e v a r i a n c e f o r t h e t r a n s f o r m e d v a r i a t e • / . • " /' / Y ( f ) where 'Y(f) a r e found i n the t a b l e s appended and x i s a v a r i a t i o n on the t e r m i n a l v a l u e s o f t h e t r a n s f o r m e d v a r i a t e . Graph 8 i l l u s t r a t e s a t y p i c a l r e s p o n s e i n t h e v a r i a n c e t o v a r i a t i o n o f t h e s t a r t i n g v a l u e s i n t h e second o r d e r a p p r o x i -m a t i o n , e q u a t i o n 2. GRAPH 1 3 3 * {::: i^iii-.:' i ' : ; ' i i i i ; ' | iii l • i:i- M i i ; ; i i i :ii: :;ii i i i i H i ! • • - :' i i ; i i : i ~ : ; lifciyf B iiiiiiii - J 1 p i \: ;: till i : : iiii '. , i : f- •::i-;t- • tig: . . . . TT— ~~h •' . .. ..n.' : ' •tif • rfi'frHr :_. l::r; t::x: i: lHJYr'IV i H ii!: j] [ i P iii: Rmmj fl'jljljj iihk;;: :.:: H i • '..:....__L: . • :;..r-^ .;!:.:.; . 7 | i : i I :|=g p l l ! ;iii ': '.'-'..: • • lit" m i: a :i:i iili MS • i | i •iiifl-' ' * ] • P s r: ; !:! : :: ::~ ir:: :r.;i:r.: i ' • • ' ^: ' :'i;i: ffrf iffii ? iiii ! ii|:i/i i . ._: l i l i i f i i i : 'ii ; rf., 7& •rr;r li:i.i ii:u :»;: j »; ; Fjr; b •! I:!::.:.: : <-.) ; iiii |i j j i jjjp j || fef i | | i: Iiii iiii ii i F j i i i i iii-'i : i Ll i 1 j ; , .. . ; ii i - "• is f ::!.' —— — -I - • ' | T 1 B l l B p i i l i i i i l • i . i ^ !1 Ii .: ! ! i : ::l :::: 1 • : - - -iii: iiriiiiii —Qhf-m : \ • • • i i.:... — i";i- '-'•"'• ,;:V. ':- ,-: .... : ; ; i •pfr •iiii "" *'. :::: -I: ; s :: :i .. i ,> # i i i i i i i ^ :; V: ::\,:Ii -\ "; i .rH s i : ' ! .iji: "CliCiTp i : :: :::: . :;::F;::: n: ri '< p ! .-1 - n ','{•', 'trkx ff 1 ; : :''::• |3||¥* Pi iii ^mteiH^iini|!ii i i i i [ l i i i i l i l p i i l l l i i tm Hi! • ii'i iiii iiii rrfr i tH. iiii • !>: : .• • I • * • mm jiff rrrr Nil •IP 1 .  a_^ -i:J. + iii ii— iii -|- i._ -i-;--}ii i . . i ........ -N.. ; . . . . . . .... . , i... . : ' ' :- ' i-'i -rrrh BE an •r.;: 111) iiii iiii v • i is: • ;- i--\i! '• i; :ni l+i. rtf| . : : ::: :• •.' ' i: ii:i • i ii. :;;.i i ii iiii i.ii:viii > • rtj. .., .t... i . iiii'iiii ill: ji , > [: • 1 " 'J' ' 1 ;• i - -':• . ::r. ::;: 1:1 .... ii : r ii Eri. -*4' i j • — | if iiii i: ii . r > ^ ^ i - j | i iii iiii iiii i - - l - r ^ - " i iiiijiiii ; ; ; ; i'rri j Ht "n wrrf j -1 • ~-—I.,..- * ; i ; ; :::: . rr** 1x1 : i«H i. :.::::: ::: :!::r.:;r; z ii IHjH : i : : -s 8ft ii O- . iii: i 1 4ii i i'i |;1'' 0 .„:.:.: ..... i : i ; ii :ii--i. d ; i :it-: • '/: :, :: ••; ••• i • ; , ; ii : ' VS Mrr 111' MM'jj Tjj1 "Hf1^ GRAPH h R E S P O N S E OF THE VARIANCE TO VARIATION OF THE LhlD-POMTS n = s o GRAPH The variance at s point' increases a s x /ncr&ases. The aster is k indicates the terminal ve/ue. near-est +o t'rrdt in the ia'ol<3s appended, X = 7 . 6 0 0 ?. 5 9 0 0 7, 58O0 ' # 7. S10O 7.5600 IS 50 7.5^00 7.5300 1.5ZOO 7, SI 00 1.50OO 7. ^900 7 </80Q 7^/700 7 Y600 7 V 5 0 O 7 'Moo 7.3600 .4-41. TABLE IV S t a r t i n g v a l u e s to' the n u m e r i c a l s o l u t i o n o f e q u a t i o n 5 N / 1^(0) U 5 ( 0 ) 4 / 1.3416 6.5597 5 7 1.8000 4.8889 6 2.1000 4.5429 7 2.4000 3.6167 8 2.6500 2.9509 9 2.3460 2.8460 10 3-0440 2.4115 11 3-1300 3 .0703 12 3.3500 2.7851 13 3.5000 2.8757 14 3.6500 2.5936 15 • • . 3-7500 . 3.7500 16 . 3.8900 3.5693 17 '•• 4.0100 3.3993 18 4.129 5 4.1295 19 4.2485 4.2435 20 4.3610 4.5021 21 4.4751 4.5632 22 4.5350 ' 4.6916 23 4.6924 4.8103 24 4.7974 4.9274 25 4.9000 5.0510 26 5.0009 ' 5.1523 27 5 .0996 5 .2632 23 5.1965 5.3688 29 5 . 29 1 5 5 .480 6 30 5.3355 5-5504 \ TABLE V Range o f t h e t r a n s f o r m a t i o n g i v i n g a v a r i a n c e w i t h i n ~2°/o o f u n i t y 1 + . 0 2 1 + . 06 N Minimum P • Minimum T h i s Work Tukey 4 . 25 . ( .20 5 .23 .20 6 • . 21 .17 7 .20 .14 8 . .20 .13 9 . .19 . 1 1 10 ' .18 .10 11 .15 .09 12 • 15 .03 13 . 13 . 03 14 .12 . 07 15 .12 . 07 16 .10 . 0 6 17 .10 . 0 6 18 .0Q' . 0 6 19 .09 . 0 5 20 .09 ~~ . 0 5 21 . 08 . 0 5 22 .08 . 0 5 23 .08 .04 24 .08 .04 25 .07 .04 26 .07 .04 27 • • 07 .04 28 . .07 .04 29 . 06 . 0 3 30 . 06 . 0 3 43-TABLE V I These t a b l e s c o l l e c t t o g e t h e r t h e b e s t a t t a i n a b l e t r a n s f o r m a t i o n o b t a i n e d t h r o u g h e q u a t i o n 5 and the s t a r t i n g v a l u e s l i s t e d i n Table IV. The a p p r o x i m a t e range o f s t a b l e v a r i a n c e a c c o r d i n g t o t h e + 2°/o c r i t e r i o n i s g i v e n i n T a b l e V. These t a b l e s do not c o n t a i n t h e end p o i n t a d j u s t m e n t s p r e v i o u s l y d i s c u s s e d . p 0.0000 0.25C0 0. 5CCC 0 .7500 1. C000 U ( ? ) -.2. 32 81 -0.7 9 99 . O.COOO 0.7 999 2.32 81 p E ( U ( X ) - E ( U ( X ) ).)**2 E ( U (X ) ) 0.010 0.C902 -2.267 4 0.C20 0.1741 -2.2076 0.030 0. 2 5 2 1 - 2 . I 4 8 6 0 .04 0 0. 3245 -2.0904 0. 050 0.3915 -2.03 3 0 0. 060 0.4 535' -1.9764 0.070 0.5107- -1.92 06 0.000 0.5635 - 1 . 8 6 5 6 0.090 • ' 0.6119 -1.8112 0. 100 0.6563 -1.7 576 0. 150 0 . b 2 6 1 -1 . 4 9 9 7 0-200 0.9 271 - 1 . 2 5 7 0 0. 250 / '0.93 03 - 1 . 0 2 7 5 0. 300 , .1 . 0027 '/ 1.0074 • / , 1.00 41 -0. 8 0 8 9 0.350 -0 . 5 9 9 0 0.400 -0.395 7 0.450 /' 0.99 94 -0.1968 0.500 0.9 97 5 0.0000 7 TABLE V I N = 5 lie; P .0.0000 0.2C00 0.4000 0.6000 0.8000 1 .0000 U ( P ) -2 . 6526 -1.2514 -0.3665 0.3665 1.2514 2.6526 P 0.010 0.020 0.030 0.04 0 / 0.0 50/ 0.06 0 .0.070 0.080 0.090 0.100 0. 150 0. 200 0.250 0.300 0. 3 50 0.4C0 0 .450 0.5C0 )-E (U(X n.)**2 E ( U ( X ) ) 0.0946 -2.5831 0. IS24 -2.5145 0.2 6 37 -2.4470 0.3390 . -2.3 804 0.4 084 -2.3148 0.4 724 -2.2501 0.5313 -.2. 1863 0.5854 -2.1233 0.6349 -2.0613 0.6802 -2.0000 0.8514 -1.7057 0.9511 -1.4292 1.0021 - 1 . 1679 1.0223 -0.9193 1.0252 -0.6807 1.0207 -0.4496 1.0153 -0.2235 1..0 131 0.0000 ( TABLE V I N = 6 P 0 .COCO G. 166 7 0.2333 0 .5000 0 . 666 7 0. 8 33 3 1. COCO U( P ) -2.9495 -1.6212 -0.72 79 O.COOO C.7 2 79 1.6212 2.94 95 p E ( U ( X ) - E ( U ( X ) ) J * * 2 E ( U ( X ) ) 0.010 0.1017 -2.8704 0.020 0.1954 -2.792 7 0 .030 0.2817 -2.7161 0.04 0 0.3609 -2.6408 0.050 0.4336 -2.5666 0.060 0.5001 -2.4936 0.070 0.5 6 08 -2.4218 0.080 0. 61. 62 -2.3510 0.090 0.6 6 65 -2.2812 0. ICO 0.7 121 -2. 2.12 5 0.150 ; 0.8800 -1.8 834 0.200 0.9 741 -1.5757 0.2 50 1.0181 -1.2861 0.300 .1 .0 32 2 -1.0114 0.350 1.0305 • -0.7484 0-4 00 1.0228 -0.4941 0.4 50 .1.0158 -0.2456 0.500 1.01.30 . 0.0000 TABLE V I IT = 7 P 0 . C O C O G . 1429 0 . 2 8 5 7 G .4 286 0 . 5 7 1 4 0 . 7 1 4 3 0 . 8571 1. COCO U ! P ) - 3 . 2 1.54 - 1 . 9 4 04 - 1 . 0 7 6 2 - 0 . 3 4 46 0 . 3 4 4 6 1 .0762 i . 9 4 04 3 . 2 1 5 4 P 0 . 0 1 0 O . C 2 0 0 . 0 3 0 0 . 0 4 0 0 . 0 5 0 0 .060 0 . 0 7 0 0 . 0 8 0 0 . C 9 0 0 . 100 0 . 150 0 . 2 0 0 0 . 2 5 0 0 . 300 0 . 350 0 .400 0 . 4 5 0 0 . 5 0 0 ) - E ( U ! X ) ) ) * * 2 E ( U ( X ) ) 0 . 1088 - 3 . 1270 0 . 2 0 79 - 3 . 0 4 0 3 0 . 2 9 8 2 -2 . 9552 0 . 3 8 02 - 2 . 8 7 1 6 0 . 4 5 4 7 - 2 . 7 8 9 5 0 . 5 2 2 1 - 2 . 7 0 8 9 0 . -5 8 3 2 - 2 . 6 2 9 7 0 . 6 38 2 - 2 . 5 5 1 9 0 . 6 8 7 8 - 2 . 4 7 53 0 . 7 32 3 - 2 . 4 000 0 . 8 9 30 - 2 . 0 4 0 9 0 . 9 7 8 6 - 1 . 7 0 6 9 i . o i a i: - 1 . 3 9 3 4 1 .0311 - 1 . 0 9 6 2 1 .0307 - 0 . 8 1 1 5 1. 0253 - 0 . 5 3 6 1 1.0 20 3 •; . - 0 . 2666 1 .0183 O.COOO O O O O O O O C O O C C O O Q O Q O o u i o u i o u i o u i o o o o o o ^ o o o o o o o o o o >s e ^ o- u i w \ ! H o o o o o o o c o o o o t - O O O O O O O O O CS O ' J l 'JJ IV o o u i - o m n v."! o vTi o o o o o o o o o o X t— ( - ' f - ' t — t— i — o o o o o o o o o o o o - -. . . . I c o o o o o o -o -s.' - J o •> m ^ J> u ro r - m ho ro ro ro i— co i—• o i v - j »— o"> c : t— ro ro ro —-vO O w C> 3 - o u i w m W X- O C O W o l C) o c ' J l - J CO c> w •?• * - ' J l U -S (> ? U - J CO - J — X I I I I u ) N f - o o O i - r c w <- i v UJ o o C- U> i v C O O ^ J O O O - J C - C O i-3 6 w <J HI II CO r o I I I I I I I I I I I I I I I I I m O O O O H K H W N N M W W W W U W W -. c O W U l f f i ( - > B I - ' , J 1 | > ^ O O O O H W W -O U1 W CO W vO 'Ol ^ UJ 3> O Ui N H W ^ I - CO w 0 > C O U ) J S C O ^ ' J l O N C U i C O v D v O O N O -co TABLE VI N = 9 p U { P ) 0.coco -3.7385 0.11 1 I -2.4753 0 .2222 -1.6661 0. 3 3 3-3' -0.9664 0.4444 -0.3169 0.5556 0.3169 0.6667 0.9664 0.7778 1 .6661 0.0889 2.4 7 53 1.0000 3.7385 P E ( U ( X ) - t ( U ( X ) ) ) * * 2 E ( U ( X ) ) 0.010 0. 1350 -3. 6264 0.020 ; 0.2540 -3.5174 0.03C 0.3 58 7 -3.4114 0.040 0.4507 -3.3081 0.050 0.5314 -3.2075 C.060 0.6020 -3.1094 0.070 0.6 63 7 -3.0137 0.080 0.7 174 -2. 9201' 0.090 0.7642 -2.8280 0 . 100 0.8047 -2.7394 0. 150 0.9 3 73 -2.3192 0.200 0..9959 -1.9352 0.2 50 1.0179 -1.5784 0. 300 1.02 32 -1 .2418 0. 350 1.0220 -0.9198 0.400 1.0192 -0.6079 0.450 1.0 170 -0.3024 0.500 1.0162 0-0000 o o o o o o o o o o o o c o o o o o on j> \jj u-' ro ro i — r~ o O O 1,1 O J1 O U ! O Ol o o o» o o o o o o c o c o o o o o o o o o VJ M h-o o o o o o o o o o o o o o o o o o O O O O O G O O O O O T 5 o o o o o o o o o o o o o o o o o o o o o o r n c o o O O C o o o o o o * « • • • » • • e « • 0 • 0 fr « e • 1 o O o o - r> CC o 'J l r : r - r n »—< •—' t—< (—' o 01 X- r—-. i—' •Jc ^) r." '•ji —-U1 U1 U l c r o- o CO o-i co ro 1—• CO J> ro c co ro - J - J OJ o OJ OJ M U l CTJ Ol •—j — X — ro i — »— o o I o i i >— r o -s O - j v C - r v o o ^ r o y j - J ^ c — o >\> 4- .— o — j J> ro s—• o ~ ' - ' O J O J O O C i O O O J O J t - ' l-J > 1-1 H O I I I I O O O O i-i I I I I I 11 I I I I I I m ro ro ro ro O J oJ O J O J O J O J O J O J — • O O J O ^ J O J Cr O CT! vC o <-> ro 4> u*. o -^i co — O P i> 51 O J W +" O w C O r" M f X o co o - J O J O J o ^ o ot ro O J on ~*j — O v 0 - J O U l W M i C 0 , O O 3 v H 0 v ^ ' J l J > W -U i o TABLE V I N = 11 p I H P ) -0000 -4.2300 . 0909 -2.9589 . 10 L8 -2. 1676 .2727 -1.4948 .3636 -0.87 n ' .4545 -0.2895 . 54 5 5 0 . 2 6 9 5 .6 364 0 .8777 .7 2 7 V 1 .'+948 .8182 2.167 6 .9091 2.95 89 .0000 4.2 3 00 P E ( U ( X ) - [ ; ( U ( X ) ) J**2 E ( J ( X ) ) 0.010 0. 164 5 -4.0928 0.020 0.3 04 7 -3.9604 0.0 30 0.42 42 -3.8327 0.040 0.52 56 -3.7094 0.050 0.6 114 -3.5900 0. 06 0 0.6839 -3.4 744 0.070 0.7449 -3.362 3 0.08 0 0.7961 -3.2534 0.090 0.8388 -3. 1477 0- 100 0.8 744 -3.0448 0-150 0.9773 -2.5670 0.200 1.0113 -2. 1371 0-250 1 .01.8 5 -1.7412 0.300 ... 1.0171 -1.3693 0-350 1.0141 -1.0142 0.400 1.0118 -0.6704 0.4 50 1 .0.104 -0-33 35 0.500 1.0 100 0.000 0 / /' / oo r-< o oo <t- co ro cc i- l <r o —. i n o t—4 r - CO ro ro <j" Co vO CO 'X! o CD o X i r - ro co <r oo o Co cc o- CM r-H CM i n o <J- o ~— ro r—i o O LP. CM i—i o CM XT o r - ro o —> 9 • • • • • • • * * ft e ft 9 « ft 00 ro ro o^ ro ro co CM ;\j r-l r-l r—1 o o o U.I I I I I I I I I I I I I I I I I X t n oo 00 i n o i n —* ro ro i n o i n CO o~ IA o o 4- O o r-l CO O o ro o o <r o o o <f- O nj <r LP co r - s0 CO r\; o-j 00 r—( o 0> o o •o CO cco -o <^  c co co o : / J ro io. r-j 00 ro r - o co .—1 .—1 Q O —\ -1 . 1 t r " IT > ro LL! .—i OO - r— >• CO co o <T0 o O o O O ) <s- ro r\J .—I r-l c o o r-l Cxi 1 c o o o c o o O o o O r-< —1 - 1 r-l .—i r-l I I I I I I X c o o r - o r o r - c r o r - o o o r - o o o o o o o o o o o o o o o o o o o O ro >C O ro O o ro <C O ro »C O —< CM ro sj- L->. o >- co Co O i n O i n C i.o O i n o C L O C O v O L n r O — I O C O - J 3 L O . r 0 r - » O Q . O O O O O O O O O — i r H O J O s l r o r O s J - v j - i n C O r - i r v i r O v r m i n o r - c o o c • • • • • • • » • • • • • • • • • < . . . • • • • . . . . • • O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o - * o o o o o o o o o o o o o o o o o u i j > 4 > w u > i \ j r \ ; i ~ ' f — o o o o c c o o o o o o o o o o o o o o o o c o o f - O O O O O O O O O O O O O o • • o "o o w ^ a ' ^ r - u o - c o w u i - j o t ; i— O O J C > O r o o n C O > - ' J > - ~ J O O J O - 0 o o o o o o o o o o o m cr. X I O O O O O O O O V O ^ C O J C C - N I iT- c -O O O O O t — r O W O C ' N O > > f l O « ; c o - < c r - - i H - ^ - o - ' U J 4 > c o i — O J i ~ ro O-. (— OJ vO X i> O J ro I o i i (-• o o o cc f\j r v c . vo C' I i ro O J o o O J o o >^ O - O J U i OJ OJ vO O O J O J O J O J O O -P- J> 4> O J on 4> O O i-3 rS iri < £3 II H V)! I I I I I I I I I I I I I I I I I m O O O H H r , W W W W U ) W W W 4 > 4 > J > 4 > — • « • • • » • « • • • • • < _ : C O ^ M O O C S I - ' C S i - U J U l ^ O W - J H O I V ' X o r o c o o c r c 3 C O v D c o j > o J ^ u i c r ; c r ^ D v O c r — O J I W W O O t S O v d O I K W M O O N O - W -o o o o o o o o o o o o o o o o o o W J> w u; M M r - i— 0 0 0 0 0 0 0 0 0 - c O o> O U1 O 101 O \J1 O vC CC - J C J> (V) i— O O O O O O O O O O O O O O O O O O C O O O O O O O O O O O O O O c o - ^ i - ^ o o ^ u o ^ ' o j r o M r - o o O r\J vjl C 3 t - ^ O ro u-i CO .Cv O "C O t N N 1 w >B J> O P H >l W ^ O m r - i - l r — I — ' r - i r - O O O O O O O O O O — I O O O O O O O O s O v O C B C 3 - J - v l f > U i * J J N 3 m O O O O O O »- - >— o"i LO >o W O I V 'o-' no Co i— •— U ^ ' J l w W - v l J l L v C O N C O ^ - ^ C r C ' W ^ r - C • > | O W O O W O * N * U l i f 1 N - J h ' O u l O - < l ' -X u> ro ro i— o o-c I I i— ro I I I vC CT- CD O r N -< -fr- ro O J> O J o o vn o o on ro o o u> ro o vT. O fO O'l ro o--vl t—• • c O co ro ro co co O i i i i i i i i I I i i i i I I i m • « • • • ' • cr o w - J r - u i v O J - K - r m o c o o H W ^ L n ^ i — / TABLE VI N = 15 / . / p I 1 U ( P ) -0000 -5.0965' .0667 -3.8229 .13 3 3 . / -3.0416 .2000 / -2.3958 .2667 / -1.8155 .3333 / -1.2/34 .4000 / -0.7550 .4667 / -0.2502 .5333 / 0.2 5 02 .6000 / 0./5 50 .666 7 . ./ 1.2734 .7333 , . 1.8 155 .0000/ . 0667 2.3958 3.0416 .9 333 3.8229 .0000 5.096 5 P E ( U ( X ) - E ( U ( X ) ) ) * * 2 E ( U ( X ) ) 0.010 0.2188 -4.9105 0.020 0.3 94 5 -4.7 3 39 0.030 0.535 0 -4.5658 0.040 0.64 69 •-4.4 05 6 0.050 0.73 55 -4.2525 0.060 0.805 3 -4.1058 0.070 0.8599 -3.9651 0.080 0.9022 -3.8298 0.090 0.934 7 -3.6994 0.100 0.9594 -3.5735 0. 150. 1.0128 -2.9989 0.200 1.0174 -2.4917 0.2 50 1.0124 -2.0286 0.300 1.0081 -1.59 51 0.350 1.0057 -1 . .1814 0. 400 1.0047 -0.7810 0.450 1.0043 -0.3886 0.500 1.00 4 2 0.0000 TABLE V I N = 16 P U ( P ) .0000 -5.2976 . 062 5 -4.0 2 36 . 1250 -3.2378 .1875 -2.5956 .2500 -2.0236 .3125 -1.49 0 8 . 37 50. -0.9813 . 43 /5 -0.4374 .5 000 0.0000 . 5625 0.43 7 4 .62 50 0.98 13 .6875 1.49 0 8 . 7500 2.0 2 36 .8125 2.5956 . 8 750 3.2 3 78 . 9375 4.02 36 .0000 5.2976 P E ( U ( X ) - t ( U ( X ) ) ) * * 2 E ( J ( X ) ) 0 .0 10 0.2321 -5.09 94 0.020 0.4161 -4.9119 0.0 30 0.5612 -4.7340 0.04 0 0.67 52 -4.564 8 0.050 0.7 64 0 -4.4-0 35 0.060 0.8326 -4.2495 0.070 0.8353 -4.1019 0 .080 0.9252 -3.9603 0.090 0.9551 -3.8241 0.100 0.97 71 -3.6929 0. 150 1.0187 -3.0962 0.200 1.0165 -2.5716 0.250 1.008 9 -2.0937 0.300 1.004 5 -1.6466 0.350 1.0032 -1.2199 0 .400 1.0036 -0.8066 0 .450 1.0042 -0.4014 0.500 1.0045 0.0000 TABLE V I N ='17 p U ( P ) 0.0000 -5.4065 0.0 5 38 -4.2158 0.1176 -3.4334 0.1765 -2.7937 0.2353 -2.2252 0. 294 1 -1.6977 0.3529 -1.1961 0.4113 -0.71 12 0.4 706 -0.2360 0.5294 0.2360 0.5882 0.7112 0.6471 1.1961 0.7059 1.6977 0.7647 2.2 2 52 0.8235 2.7937 0.8824 3.4334 0.9412 4.2158 1.0000 5.4865 P E(U(X)'-E(U(X) ) )**2 E ( U ( X ) ) 0.010 0.2 436 -5.2769 0.020 0.4339 -5.07 93 0.030 0.5819 -4.8924 0 .040 0.6962 -4.7152 0.050 0.7839 -4.5467 0.060 0.85 06 -4.3861 0.070 0.9009 -4.2327 0.080 0.938 3 -4.0856 0.090 0.9657 -3.9444 0. 100 0.9854 -3.808 4 0. 150 1.0196 -3. 1919 0. 200 1.0 157 —2. 65 09 0.2 50 1.0084 -2. 1581 0.300 1.004 4 -1.6971 0.350 1.00 31 -1.2572 0.400 1.0031 -0.8311 0.4 50 1.00 33 . -0.4135 0.500 1.0034 0.0000 TABLE V I N = 18 5 8 . p U ( P ) .0000 -5.6713 .0556 -4.4025 .1111 -3.6220 . 1667 -2.98 49 .2222 -2.4200 . 2 778 -1.8975 . 3333 -1.4023 . 3889 -0.3 252 .44 4 4 ... -0. 4598 .5000 0.0000 .5556 0.45 9 8 .6111 0.9252 .6667 1.4023 . 7222 1.8975 .7778 2.4200 .8333 2.9 8 49 .8889 3.6220 .9444 4.4025 .0000 5.6713 —_ P E ( U ( X ) - E ( U ( X ) ) ) * * 2 E ( U ( X ) ) 0.010 0.25 54 -5.4501 0.020 0.4521 -5.2 42 3 0.030 0.6027 -5.0 464 0.040 0.7172 -4.86 12 0.050 0.S037 -4.6857 0.060 0.86 8 3 -4.5186 0.070 0.9161 -4.3593 0.080 • 0.9509 -4-. 2 0 7 0 0.090 0.97 5 3 -4.0608 0.100 0.9933 -3.9203 0. 150 1.0203 -3.2 845 0.200 1.0144 -2.7276 0.250 1.0074 -2.2205 0.300 1.003 9 -1.7461 0. 3 50 1.0028 •• -1.2934 0.400 1.0026 . -0.8550 0.450 1.002 7 -0.4254 0. 500 1.0027 0.0000 o o o o o o c o o o o o o o o o o o v J 1 ^ 4 > O J O J r o r o r - - H - 0 0 0 0 0 0 0 0 0 _ 0 o a i O u i O ' j i O u i O v C c a ^ O ' U i ^ w W ' -o o o o o o o o o o o o o o o o o o f - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 O +- o J> cs O J CD w - J w - J w o - 1 H o> H o ITI o t i rn o o o o o o o o o o o o o o c o o o o o o ro ro >\J ro oo o O J 4> W W M O N l\J X o I O v O * o c o f f - » i f f . j > w r n X on ^ 'OJ O J ro ro o o o o I I I I ro ro O J O J co o i co J — cr C J o »— o n c o o o o ^ J o o j C O r o ro c> i — O O -o ro r o -o oo o vO OJ O J U J O J >-* o o 0> - 4 ^1 C> O W C C J CD — O cr o co on x> CO Ul U i H w W U l r - J> 4> •K-ro I I I I I I I I I I I I I I I I I m « . . « » • • • • c o + ^ c c o j - ^ r o c c O J O r - O J j > c ^ c c O ' - i - ^ c r - - -O - J K C 3 W » - M + , - C S 4 > J > W ' N I O W 0 - H K — O C W O->lOvOl>.'OOvOUHriCOiM > H C O -o o o o o o o o o o o o o o o o o o Ul >^ J- w w w w >-o m o uV o o i o u i o o o o o o o o i — o o o o o o O vO K p 'ji-^-o o o o o o . o o o o -v W M o o o r - O O O O O O O O O O C O O O O O O O O o o o o O C C 3 - J Ul o w c w o o o o c o o o o o o o on m vn O ui O O O o o JO U! O Ul o o o o o o o o o u> u> ro ro O v . n o o o o o o o o r- O O O <_h O O O o o o o TO •- ' r-'t-'h- ' r— r - ' H - ' l - ' l - ' O O O O O O O O O o O O CD O O O O O o r - ^ ro ro ro <o o ro >.o O r -ro o O O vO i— O CO cc ro so CO 'jJ co co —I O CP CO •oo m on ro -O O J> U l l » CO i— ~ J v-> -0 CO U l X o — .• I ro ro —v! ' co cr JO- ~ X o -P- O J bJ ro ro i— o o I I I I ro ro bJ U' O ro co o CO !- O co ->i ro - J o^ cc m >u -j o ro -^ i O ro o: V-o co co ro o >— u> - J ro -»> o U J o o o vO O CO CD ..C ro I CO -v.' ro - J oo vo -o O -vl <c vn cr cr w c N o w H VO p- o oo O r— o ro H3 > tr1 <{ H ro o ro I I I I I I I I I I I I I I I I I m I O 0 3 H U J O O . ' ' H W W N ' O O O C ! C 0 O - J " OO, o TABLE V I ' N = 21 0 0 o. 0 0 p 0.0000 0.0476 0.0952 0.142 9 0.1905 .2381 ,2857 3333 .3810 .42 86 0.4762 0.52 38 0.5714 0.6190 0.6667 0.7143 0 . 76 19 0.8095 0.8571 0.9048 0.9524 1.0000 P 0.010/ 0.020' 0.0*0 0.040 0.050 0.060. 0.070 0.080 0.090 0. 100 .150 ,2 00 ,250 , 300 , 3 50 ,400 0, 0. 0 . 0. 0. 0, 0. 50 0.500 U ( P ) -6.1918 -4.9357 ,' -4.1574 -3.5250 -2 .9692 -2.4593 - 1.9794 - 1 . 5202 -1.0758 -0.6415 -0.2132 0.2 132 0.6415 1.0758 1.5202 • 1.9 794 2.4593 2.9692 3.5 2 50 4.1574 4.9 357 6.1918 (U(X)-.E(Ul X) ) )**2 E ( J ( X ) ) 0.2866 -5.9376 0.4988 -5.7010 0.6547 -5.4800 0.763 4 -5.2725 0.8504 -5.0771 0.9088 -4.8922 0.949 7 -4.7168 0.97 78 -4.5496 0.9965 -4.3898 1.0084 -4.2365 1.0193 -3. .5469 1.0102 -2.9452 1.0041 -2.3 97 8 1.0020 -1.8355 1.00 17 -1.3 966 1.00 17 -0.9232 1.0017. -0.4593 1.0017 0.0000 o o o o o o o o o o o o o o o o o o ui 4> o u. o o 4- W U J l\) W r -O Ul O UV O Ul O O O O O O O o o - J O U l o o o o o o o J> ro >— o o o o r - o o o o o o o o o o o o o o o o o o o o o o o O O CO CO - J o ui o 0 S >— -J o J > » J co isj o ui r— c- ro UJ a vo Ul >- o o -j ro CD uJ vO J> Ul U . > 4> U U W N o> G Ul O O cr o ui o J> vO j.- co ro -o >— o o ui O 4> o u i o u i r - r > r o - - j u i C 3-f s o u i o "U o o o o o X o o o — • • • • I O O O O O O O O O O vO vO -i) Cr. - j O Ul Is-' m O O O O O O O f — i— O C3 'U> ro Cr- CO -O. r— o — I — r - r - 1 ) — ^ U vD W W Ul O O > 4 s ^ S 1 C uiuiui-F-crj^i— o - r - r o J > u i v o c r J > ^ i — i — . X C > u n 4 > o J U J r o r o r - ' r - - , o o I I I I I I I O o o >- i — -ro ro O J I I on Cr 1>J r— '»0 CT r— CT r— -0 W O W o 4 s W Ui o co ui -o o ro ui o ui o c r r o u i O J > - J s D 4 > J > iv co or o o t-' o oo C I V i - < l r - ' C r r - C r W M L J — H O r 'jl J O on -i> vO ro O 'Ul "O - J C - J C S ' O i O U ' i M C ' ^ U l C D -4 > o < > ^ i s 4 > o > i 4 s o u i N c r r - H ro ro ro I I I I I I I I I I I I I i I I I m O O O l - ' H N U I W I ' f X - l ' J I U l U l U l U i O ' ' -0 4 > d ^ J - l > 0 0 ' W J > C > o ; o W J > ^ C O O ' -C - j 4 > N N U i r - U ) W o t U O h - ' O O i - J 4 ' v O X 0 O J > i f l * 0 - r + ' ' O > i t N o o o r - u J ^ - j w ^ O r ' C 3 ^ N i J + s W O U , + s W i C ^ l ) ; i ! W W - ' ro TABLE VI N = 23 p U( P ) 0.0000 -6.5215 0.0435 -5.2708 0.0870 -4.4940 0.130 4 -3.8632 0.1739 -3.3113 0.2174 -2.8075 0.2609 -2.3353 0.304 3 -1.88 51 0.347 8 -1.4 509 0.39 13 -1.02S3 0.4348 -0.6 L38 0.4 78 3 -0.2041 0.5217 0.2041 0.5652 0.6138 0.6007 1.0283 0.652 2 1.4509 0.6957 1.8851 0.7391 2.3353 0.7326 2.8075 0.8261 3.3113 0.3696 3.8632 0.9130 4.49 40 0.9565 5.2708 1.0000 6. 5215 P E ( U ( X ) - E ( U ( X ) ) ) * * 2 E ( J ( X ) ) 0.010 0.3074 -6.2453 0.020 0. 52 88 -5.9898 0.030 0.68 71 -5.7524 0.040 0.7992. -5.5307 0.050 0.8776 -5.3227 0.060 0.9 316 -5. 1266 0.070 ! 0.963 1 -4.9410 0.08 0 0.9919 -4.7646 0.090 1.0069 -4.5 95 3 0. 100 1.0157 -4.4351 0.150 1.0186 -3.7119 0.200 1.0082 -3.0820 0.250 1.0027 -2.5091 0.300 1.0012 -1.9730 0.350 1.0012 -1.4614 0.400 1.0013 -0.9660 0.450 1.0014 -0.4806 0.500 1.0014 0.0000 T A B L E V I N = 24 P 0.0000 0.0417 0.0833 0.1250 0.1667 0.2083 0.2500 0.2917 0.3333 0.3 75 0 0.4167 0.4 58 3 0.5000 0.54 1 7 0.58 3 3 •0. 6250 0.6667 0.7083 0.7500 0.7917 0.3333 0.8750 0.9167 0.9583 1 .0000 P 0.010 0.020 0.030 0.040 0.050 0.0 60 ./ 0.070 / 0.08 0 / 0.090 0. 100 , 0. 150/ 0.2 00 0.250 0. 300 0.350 0 .400 0.450 0.500 U ( P ) -6.6309 -5 . 4 J 2 7 .. - 4 . 6569 -4.0269 -3.4 764 -2.9750 -2.5061 -2.0597 -1.62 99 -1.2 122 -0.8034 -0.4003 0.0000 0.4003 0.8034 1.2122 1.6299 2.0597 2.5061 2.97 5 0 3.4764 4.02 69 4.6569 5.4327 6.6309 E ( U ( X ) - E ( U ( X ) ) ) * * 2 0.3174 0.54 3 0 0.7020 0.3130 0.8895 0.9413 0.9 755 0. 9975 1.0108 1.0182 .1.0 180 1.007 4 1. 002 2 1.0009 1.0010 1.00 12 1.0012 1.0012 E( U ( X ) ) -6.393 7 -6.1290 -5.8837 -5.6551 -5.4411 -5.2396 -5.0492 -4.0684 -4.6960 -4.5311 -3.7917 -3.140 2 -2.5630 -2.0154 -1.4928 -0.9867 -0.4909 0.0000 p 0.0000 0.0400 0.0800 0.1200 0.1600 0.2000 0.2400 0.2000 0.3200 0.3600 0.4000 0 .4400 0.4 800 0. 5200 0.5600 / 0.6 000 / 0.6400./ 0.6800 0.7 200/ 0.7600 0.8000 0.8400 0.8800 0.9200 0.9600 1.0000 TABLE VI N = 25 / U( P ) -6.8342 -5.5910 -4.8 164 -4.1870 -3.6379 -3.1337 -2.6726 -2.2293 -1.0038 -1.390 6 -0.9868 -0.5395 -0.1961 0. 1961 0.5395 0.9868 1.3906 1.8038 2.2298 '2.6726 3.1387 3.5379 4.1870 4.8164 5.5910 6.8342 P E ( U ( X ) - E ( U ( X ) ) ) * * 2 E ( 'J ( X ) ) 0.010 0.32 61 -6.5368 0.020 0.5 549 -6.2634 0 .030 0.7142 -6.0107 0.040 0.8240 -5.7756 0.050 0.8985 -5.5559 0.060 0.9483 -5.3494 0.0 70 0.9306 . -5.1544 0 .080 1.0009 -4.9694 0.090 1.0128 -4.7932 0. 100 1.0192 -4.6247 0. 150 1.0169 -3.8698 0.200 1.0 0 64 -3. 2131 0.250 1.0017 -2.6158 0.300 1.0007 -2.0569 0.350 1.0009 -1.5235 0.400 1.0011 -1.0070 0.450 1.0011 -0.5010 0. 500 1.0011 0.0000 TABLE V I N = 26 P 0 . 0 0 0 0 0 . 0 3 8 5 0 . 0 7 6 9 0 . 1 1 5 4 0 . 1 5 3 8 0 . 192 3 0 . 2 308 0 . 2 6 9 2 0 - 3 0 7 7 0 . 3 4 62 0 - 3846 0 . 4 2 3 1 0 . 4615 0 . 5 000 0 . 5 3 8 5 0 . 5769 0 - 6 154 0 . 6 5 3 8 0 . 6 9 2 3 0 . 7 3 03 0 -7692 0 . 0 0 7 7 0 . 8 4 6 2 0 . 8 8 4 6 0 . 9 2 31 0 .961 5 1 .0000 P E ( U ( 0 . 0 1 0 0 . 0 2 0 0 . 0 3 0 0 .04 0 0 . 0 5 0 0 .060 0 . 0 7 0 0 . 0 8 0 0 . 0 90 0 . 1 0 0 0 . 1 5 0 0 . 2 0 0 0 .2 50 0 . 3 0 0 0 . 3 5 0 0 . 4 0 0 0 . 4 5 0 0 . 5 0 0 U ( P ) - 6 . 9 8 7 0 - 5 . 7 4 6 3 - 4 . 9 7 3 0 - 4 . 3 4 4 3 - 3 . 7 9 6 1 - 3 . 2 9 8 7 - 2 . 3 3 5 2 - 2 . 3 9 55 -1 . 9732 - 1 - 5 6 4 0 - 1 . 1 6 4 6 - 0 . 7 7 2 5 - 0 . 3 8 5 1 0 . 0 0 0 0 0 . 3 8 5 1 0 . 7 7 2 5 1 . 1646 1 .564 0 1 .9732 2 . 3 9 5 5 2 . 8 3 5 2 3 . 2 9 0 7 3 . 7 961 4 . 3 4 43 4 . 9 7 30 5 . 7 4 6 3 6 . 9 3 7 0 ) - E ( U ( X ) ) ) * * 2 E ( U ( X ) ) 0 . 3 3 5 6 - 6 . 6 7 3 8 0 . 5 68 0 - 6 . 3 9 6 4 0 . 7 2 7 5 - 6 . 1 3 6 1 0 . 8 3 5 8 - 5 . 8 9 4 6 0 . 9 0 0 3 - 5 . 6 6 9 1 0 . 9 5 5 9 - 5 . 4 5 7 6 0 . 9 8 6 2 - 5 . 2 5 8 1 1 .0048 - 5 . 0 6 89 1 .0154 - 4 . 8 8 8 9 1-0207 - 4 . 7 1 6 8 1.01 63 - 3 . 9 4 6 5 1 . 0 0 5 9 - 3 . 2 7 6 7 1 .0013 - 2 . 6 6 7 6 1 . 0 0 0 4 - 2 . 0 9 7 6 1 . 0 0 0 7 - 1 . 5 5 3 6 1 . 0 0 1 0 - 1 . 0 2 7 0 1-0010 - 0 . 5 1 0 9 1-00 10 0 . 0 0 0 0 0 0 0 0 ' 0 6 0 0 0 * 1 0 0 5 * 0 9 C 2 5 * 0 - 6 0 0 0 * 1 0 5 ^ * 0 5 9 V 0 * 1 - 6 0 0 0 * 1 o o v o z e o s - T - 9 0 0 0 * 1 0 5 £ * 0 5 1 £ 1 * 2 - £ 0 0 0 * 1 0 0 £ * 0 ^ 8 1 A * Z - 0 1 0 0 * 1 0 5 2 * 0 1 6 £ £ *e- £ 5 0 0 * 1 0 0 2 * 0 Z I Z O ' - V - 9 5 1 0 * 1 0 5 1 * 0 0 Z 0 8 *+?- 8 T 2 0 * 1 0 0 1 * 0 9 2 8 6 * b - 5 A 1 0 " 1 0 6 0 * 0 ^ 9 9 1 • g = - 1 9 0 0 * 1 0 8 0 * 0 S 6 S £ * 5 - 2 1 6 6 * 0 O/.O* 0 S £ 9 5 * 5 - 8 2 9 6 * 0 0 9 0 * 0 OOOA * 5 - 5 A 1 6 * 0 0 5 0 * 0 0 1 "10 ' 9 - i • / o * c 0»70 * 0 6 8 5 2 * 9 - bO^/l * 0 o e o * o 8 9 2 5 - 9 - 8 0 8 5 * 0 0 2 0 * 0 1 3 1 8 * 9 - 2 5 + / £ * 0 0 1 0 * 0 ( ( X ) n ) 3 Z#*(( ( X ) n ) 3 - ( X ) f l ) 3 d u n v 0 0 0 0 * 1 £ 3 6 8 * 5 0 £ 9 6 * 0 9 9 2 1 * 5 0 5 2 6 * 0 5 3 6 ' ? * •? 6 8 8 8 * 0 2 1 5 6 * £ 6 1 5 8 * 0 8 ^ 1 8 * 0 1 V 6 6 * Z SLLL'O £ Z 5 5 * 2 / . 0*7l * 0 £Q£J'Z- / . £ 0 Z * 0 LZCL'l A 9 9 9 * 0 5 i £ £ * T 9 0 2 9 * 0 0 0 5 6 * 0 9 2 6 5 * 0 8 Z 9 5 * 0 9 5 5 5 * 0 6 8 8 1 * 0 5 8 1 5 * 0 6 8 8 1 * 0 - 5 1 8 * 0 8 Z 9 5 * 0 - V / ^ * 0 0 0 S 6 * 0 - W_ 0 V * 0 5 Z £ £ * 1 - VOiE'O Z . Z C I ' 1 - £ £ £ £ * 0 £ 8 £ 1 * 2 - £ 9 6 2 * 0 £ Z 5 5 * 2 - £ 6 5 2 * 0 T V 6 6 * 2 - 2 2 2 2 * 0 ess*/ *£- 2 5 8 1 * 0 2 1 5 6 * £ - 1 8 V 1 - 0 5 8 6 '/ * V - 1 1 1 1 * 0 9 9 2 1 * 5 - 1 W . 0 * 0 £ 8 6 8 * 5 - 0 1 £ 0 * 0 \ L £ l ' L - 0 0 0 0 * 0 ( d ) n d LZ = N IA TABLE V I N = 28 p U ( H ) C-0000 -7.2844 0.0357 -6.0473 0.0714 -5.27 73 0. 107 1. -4.6499 0. 1429 -4.1034 0. 1786 -3.6089 0.2143 -3.1497 0.2 500 -2.7154 0.205V -2.2994 0.3214 -1.8 972 0.3571 -1.5057 0.3929 -1.1224 0.4286 -0.74 50 0.4643 -0.3715 0.5000 0.0000 0.5357 0.3715 0.5714 0.7450 0.6071 1.12 2 4 0.6429 1.5057 0.6786 1.0972 0.7 143 2.2994 0.7 500 2.7154 0.7857 3.1497 0.8214 3.6089 0.3571 4. 10 34 0.8929 4.649 9 0.9286 5.2773 0.964 3 6.0473 1.0000 7.28 44 P E ( U ( X ) - E ( U ( X ) ) ) # * 2 E ( U ( X ) ) 0.010 0.3 546 -6.9546 0.020 0.5933 -6.6545 0.030 0.7 526 -6.3793 0 .040 0.8577 -6. 1250 C .050 0.92 5 8 -5.8886 0.060 0. 959 0 -5.6673 0.0/0 0-99 5 5 -5.4591 0.080 1 . 0 i 0 9 -5.2620 0.090 1-0190 -5.0746 0.100 '"' 1.02 25 -4.8956 0. 150 1.0 149 -4.0956 0. 200 .1.00 48 ' -3.4004 0.250 1.0007 -2.7683 0.300 1..000 1 -2. 176 7 0.350 1.0005 -1.6122 0.400 1.0008 -1.0657 0.450 1.0008 -3.5302 0. 500 1.0008 0.0000 TABLE V I N = 29 0.0000 0.0345. 0.0690 0.1034 0.13 79 0.172 4 0.2069 0.2414 0.2759 3103 3448 3793 413 8 0 0 0. 0. 0.44 83 0.4828 0.5172 5517 5 36 2 6 2 07 6552 6897 7 241 7586 7931 8 276 8621 8966 9310 9655 0000 P 0.010 0 .020 0.030 0 . 040 0.050 0.060/ 0.070 0.0 80 0.090 0 . 100 0. 150 0. 200 0.250 0.300 . 3 5 0 400 450 ,500 0, 0, 0. 0, U ( P ) . -7.4293 -6.1935 -5.4 252 -4.7 9 06 -4.2 52 8 -3.7 5 95 -3 . 3021 -2.8702 -2 . 4 569 -2.0578 -1.6697 -1 .2902 -0.9170 -0.5484 -0.10 25 0. 132 5 0.5484 0.9170 1.2902 1.6697 2.0578 2.4569 2.87 02 3. 30 21 3.7 595 4.2528 4 .7985 5.4252 6.1935 7.4 2 93 X)-E(UIX).) )**2 E(U(X) ) 0.3641 -7.0888 0.6055 -6.7799 0.7645 -6.4973 0.8677 -6.2369 0.9 3 36 -5.9951 0-974 5 -5.7692 0.9993 -5.5567 1.0132 -5.3558 1.0203 -5.1649 1.02 3 0 -4.9826 L.0142 -4. 1681 1.00 43 -3.4506 1 .0005 -2.8173 1.0000 -2.2153 1.0005 -1.6407 1.00 0 8 -1.0045 1.0008 -0.5395 1.0007 0.0000 O O O O O O O O O O O O O O O O O O U 1 4 > J > U J O j r o r o r - ~ > - 0 0 0 0 0 0 0 0 o u i o u i o u i o m o *o co - u co u i J > U J ro o o o o o o o o o o o o o o o o o O f r - O O O o o o o o o o o o o o o o o c o o o o o o o o o o o O v O O » o a c 3 c ^ ^ N ' p a > F ' j , ^ u i ^ . - > J > w w w N ) W \ ! ! - r - H o o o O P U 1 o ? w o ^ u c i > w o o ' i i ) O C » w o C ' U ' O o - w o r > w o c - o J O C CO CO O O CO O O Cv O CO CO O CO CO O O O CT- O J O 0 - VJJ O co 10 o co co o O -<i U O >1 W O ^ W O - N ' W C v ' W O - J W O - J W C -v' W C -vl W O w o I — • ( — I — ' l — ' O l — ' r- 1 r— r - r— I — ' r - C O O X O O O — o o o o o o O O O O vD O O O O O vO O -M -vi -vj O J v£j 4> O O o o o O r - N N H 4> U> W r— U . ro o co co co CO 4> o ro CO o 00 I ~ J co U J m -vi - J I - — CO -0 vO Ul c CO CC v i o -X s I C - ' J l ^ X - W W W N N i - ' H f - ' O O o o o I I ! I I ro ro I I OJ O-' U l CO O J X- cr U l CO vO 4> Ul O Ul - j ro J> O o o u i ro O -0 r— I—" O Ul Ul cr M C O J v O N W O W s l O J - 0 ; M f f ' C 4 > 0 + s i O U l .— ro 'ui co ro u i o u i ro co u-. O J f— r- O J u i co o - o >OJ , » ' O W . S O O O > O C + S W V O + S C H H V I . O I J 1 0 0 • g c o i s u i w w o w w u i c o c c - j a c u ' u i o N v 1 co I I I U l CO -0 • • • c CO Ul " 'OJ — J "0 J> — U l H3 > tr1 1—1 o •M-ro I I I I I I I I I I I I I I I I I m • • • • (_ ouor-c^rocxjuirooroaococoi— co co vo ro — O I ' O C i ' ^ C M - W 0 1 U l + N U l C s O ^ H O W X o c o w c s L v \ j i v o o c o w c o w v O O - « j j > j > r j w O C O C C O H 4 > v O v l W v O W O C M i J + N l - , W 4 > -o 71. / BIBLIOGRAPHY Anscombe, F.J;., / B a r t l e t t , M-.S. / / B a r t l e t t , M.S., B a r t l e t t , M.S., C u r t i s s , J.H., "The t r a n s f o r m a t i o n o f P o i s s o n , .binomial, and n e g a t i v e b i n o m i a l d a t a " , B i o m e t r i k a , V o l . 3 5 ( 1 9 4 8 ) , pp. 246-254. "Some examples o f s t a t i s t i c a l methods o f r e s e a r c h i n a g r i c u l t u r e and a p p l i e d b i o l o g y " , J o u r n a l o f the R o y a l S t a t i s t i c a l S o c i e t y , S u p p l ; , V o l . 4 ( 1 9 3 7 ) , pp. 137-170. "The square r o o t t r a n s f o r m a t i o n i n a n a l y s i s o f v a r i a n c e " , J o u r n a l o f t h e R o y a l S t a t i s t i c a l S o c i e t y , S u p p l . , V o l . 3 ( 1 9 3 6 ) , pp. 6 8 - 7 8 . "The use o f t r a n s f o r m a t i o n s " , B i o m e t r i c s , V o l . 3 ( 1 9 4 7 ) , pp. 39-52. "On t r a n s f o r m a t i o n s u sed i n t h e a n a l y s i s o f v a r i a n c e " , A n n a l s o f .M a t h e m a t i c a l S t a t i s t i c s " , . V o l . 14 ( 1 9 4 3 ) , pp. 107-122. E i s e n h a r t , F i s h e r , R.A. F i s h e r , R.A. C h u r c h i l l , " I n v e r s e s i n e t r a n s f o r m a t i o n of p r o -p o r t i o n s " , pp. 3 9 7 - 4 1 6 , i n T e c h n i q u e s o f S t a t i s t i c a l A n a l y s i s , e d i t e d by C. E i s e n h a r t , M. H a s t a y , W. W a l l i s . ' M c G r a w - H i l l , New Y o r k and London, 1 s t ed. ( 1 9 4 7 ) . "On the dominance r a t i o " , P r o c e e d i n g s o f th e R o y a l S o c i e t y o f E d i n b u r g h , V o l . 42 ( 1 9 2 1 - 1 9 2 2 ) , pp. 3 2 1 - 3 4 1 . "The d i s t r i b u t i o n o f gene:' r a t i o s f o r rare m u t a t i o n s " , P r o c e e d i n g s o f t h e R o y a l S o c i e t y o f E d i n b u r g h , V o l . 50 ( 1 9 3 0 ) , pt>. 2 0 4 - 2 1 9 . Tukey, John W., and Freeman, M.F. " T r a n s f o r m a t i o n s r e l a t e d t o t he a n g u l a r and t h e square r o o t " , A n n a l s o f M a t h e m a t i c a l S t a t i s t i c s , V o l . 2 1 ( 1 9 5 0 ) . pp. 6 0 7 - 6 1 1 . 

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