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Comparison of procedures for testing the equality of survival distributions Joe, Harry 1979

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COMPARISON OF PROCEDURES FOR TESTING THE EQUALITY OF SURVIVAL DISTRIBUTIONS by HARRY SUE WAH|jOE B . S c , The U n i v e r s i t y of V i c t o r i a , 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES (Department of Mathematics) We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1979 MASTER OF SCIENCE i n Harry Sue Wah Joe, 1979 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e 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 r e e 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 a n d s t u d y . I f u r t h e r a g r e e 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 t h e Head o f my D e p a r t m e n t 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 n o t 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 . D e p a r t m e n t o f HorfKe.^ <\H-'' c- s T h e 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 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 D a t e £ * . f 4 >>. i i ABSTRACT A common problem i n l i f e t e s t i n g i s assessing whether two s u r v i v a l d i s t r i b u t i o n s are i d e n t i c a l . Sequential procedures based on the Mantel-Haenszel s t a t i s t i c (Muenz, Green and Byar, 1977), the Wilcoxon s t a t i s t i c (Davis, 1978) and the Savage s t a t i s t i c ( K o z i o l and Petkau, 1978) have been proposed f o r t e s t i n g the e q u a l i t y of two s u r v i v a l d i s t r i b u t i o n s . These procedures, and t h e i r f i x e d p o i n t censoring counterparts, are numerically compared on the b a s i s of power under Lehmann a l t e r n a t i v e s and expected p r o p o r t i o n sampled. Among the s e q u e n t i a l procedures, those based on the Wilcoxon and Savage s t a t i s t i c s seem p r e f e r a b l e to that based on the Mantel-. Haenszel s t a t i s t i c . E a r l y d e c i s i o n r u l e s are introduced and shown to be of value i f used i n conjunction w i t h e i t h e r the s e q u e n t i a l or f i x e d p o i n t censoring procedures. F i n i t e sample c o r r e c t i o n s to the asymptotic c r i t i c a l values of the s e q u e n t i a l procedure proposed by K o z i o l and Petkau are presented. i i i TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENT v i i Chapter 1 INTRODUCTION 1 Chapter 2 DESCRIPTION OF PROCEDURES 6 2.1 INTRODUCTION 6 2.2 PROCEDURES BASED ON THE SAVAGE STATISTIC 6 2.3 PROCEDURES BASED ON THE WILCOXON STATISTIC 8 2.4 PROCEDURES BASED ON THE MANTEL-HAENSZEL STATISTIC ... 9 2.5 THE MODIFIED MGB SEQUENTIAL PROCEDURE 11 Chapter 3 NULL DISTRIBUTION RESULTS 13 3.1 INTRODUCTION 13 3.2 THE KP PROCEDURE, WITH EQUAL SAMPLE SIZES 13 3.3 THE DAVIS, MGB AND MODIFIED MGB PROCEDURES, WITH EQUAL SAMPLE SIZES 27 3.4 THE SEQUENTIAL PROCEDURES, WITH UNEQUAL SAMPLE SIZES.. 29 3.5 THE FIXED POINT CENSORING PROCEDURES, WITH EQUAL SAMPLE SIZES 33 Chapter 4 COMPARISON OF PROCEDURES 35 i v TABLE OF CONTENTS Chapter 5 EARLY DECISION RULES 44 Chapter 6 EXAMPLES 50 6.1 THE FIRST EXAMPLE 50 • 6.2 THE SECOND EXAMPLE 52 Chapter 7 DISCUSSION 55 BIBLIOGRAPHY 57 V LIST OF TABLES TABLE I Summary of N u l l D i s t r i b u t i o n of S*(.p) = *., * , 1/2 max S,/{var(S„ E L ) } i n the case m = n = N/2 15 l<k<N P k NP' 0 I I Summary of N u l l D i s t r i b u t i o n of S(p) = max S, / { v a r C S „ H.)} i n the case m = n = N/2 17' l<k<Np k N p 0 I I I S i mulation Estimates of p r { S + ( p ) > x01} 2 0 CO IV Simulation Estimates of pr{S(p) > x a} 21 00 V F i t t e d C r i t i c a l Values f o r KP Procedures 24 VI S i m u l a t i o n Estimates of A c t u a l Levels Corresponding to F i t t e d C r i t i c a l Values f o r KP Procedures 28 V I I Simulation Estimates of A c t u a l Levels Corresponding to One-Sided F i t t e d C r i t i c a l Values i n the Unbalanced, Case, w i t h N =? 100, a = .05 31 V I I I Simulation Estimates of A c t u a l Levels Corresponding to Two-Sided F i t t e d C r i t i c a l Values i n the Unbalanced Case, w i t h N = 100, a = .05 32 IX Simulation Results f o r Operating C h a r a c t e r i s t i c s of the One-sided Tests Under Lehmann A l t e r n a t i v e s , with m = n = N/2, a = .05 .. 36 X S i m u l a t i o n Results f o r Operating C h a r a c t e r i s t i c s of the Two-Sided Tests Under. Lehmann A l t e r n a t i v e s , w i t h m = n = N/2, a = -05 .. 39 XI Expected Prop o r t i o n s Sampled Using E a r l y D e c i s i o n Rules i n the One-Sided Case, w i t h a = .05, p = l 48 X I I Expected Proportions Sampled Using E a r l y D e c i s i o n Rules i n ; t h e Two-Sided Case, with a - .05, p = 1 . 49 X I I I S t a t i s t i c s f o r the F i r s t Example ••• 51 XIV Mouse S u r v i v a l Data 53 v i • LIST OF FIGURES FIGURE 1 Estimated c r i t i c a l values f o r the one-sided s e q u e n t i a l KP procedure at nominal l e v e l s .10, .05 and .01 25 2 Estimated c r i t i c a l values f o r the two-sided s e q u e n t i a l KP procedure at nominal l e v e l s .10, .05 and .01 26 v i i ACKNOWLEDGEMENT I would l i k e to thank Profes s o r A. John Petkau f o r h i s guidance and as s i s t a n c e i n the producing of t h i s t h e s i s . I al s o appreciate the a s s i s t a n c e of Professor James A. K o z i o l . I would l i k e to thank Janet C l a r k and Mary D a i s l e y f o r the typ i n g of t h i s t h e s i s . F i n a l l y , I would l i k e to thank the N a t i o n a l Research C o u n c i l of Canada f o r supporting me w i t h an NRC Postgraduate S c h o l a r s h i p . - 1 -Chapter 1 INTRODUCTION A common s i t u a t i o n i n l i f e t e s t i n g i s assessing whether two treatments are e q u a l l y e f f e c t i v e i n prolonging l i f e . We suppose that N subjects are to be tre a t e d (m subjects w i t h the f i r s t treatment, n subjects w i t h the second treatment, N = m + n) and that a l l are a v a i l a b l e at the same time to begin experimentation. The observations are lengths of s u r v i v a l or times u n t i l f a i l u r e and are n a t u r a l l y ordered. We assume that the f a i l u r e times from the two treatments have.respective continuous d i s t r i b u t i o n f u n c t i o n s ( s u r v i v a l d i s t r i b u t i o n s ) F^ and F^. We then want to t e s t the n u l l hypothesis HQ that the s u r v i v a l d i s t r i b u t i o n s are equal ( i . e . F^ = F^) against e i t h e r the a l t e r n a t i v e hypothesis that f a i l u r e times from the f i r s t treatment are s t o c h a s t i c a l l y l a r g e r ( i . e . F^ < F^, the f i r s t treatment i s superior) or the a l t e r n a t i v e hypothesis H that the s u r v i v a l d i s t r i b u t i o n s are unequal ( i . e . F^ f ^2}' ^ a n y nonparametric t e s t s are a v a i l a b l e f o r the s i t u a t i o n where a l l f a i l u r e times have been observed, the Savage and Wilcoxon rank t e s t s being perhaps most commonly used. However i n some l i f e t e s t s , such as c l i n i c a l t r i a l s , i t i s d e s i r a b l e to make a d e c i s i o n concerning the r e l a t i v e e f f i c a c i e s of competing treatments before a l l the f a i l u r e times are known. Two d i f f e r e n t types of procedures that permit e a r l y d e c i s i o n s are the f i x e d p o i n t censoring procedure and the s e q u e n t i a l or pro g r e s s i v e censoring pro-cedure . In a f i x e d p o i n t censoring procedure., a d e c i s i o n i s made a f t e r observing, say, the f i r s t r f a i l u r e times (where r i s p r e s p e c i f i e d ) . Gastwirth (1965) and Johnson and Mehrotra (1972) have i n d i c a t e d how the Savage, Wilcoxon and other l i n e a r rank t e s t s can be a p p r o p r i a t e l y modified f o r t h i s f i x e d p o i n t - 2 -censoring s i t u a t i o n , In a s e q u e n t i a l or progressive censoring procedure, at most r f a i l u r e times are observed and a d e c i s i o n i s made whether to continue or terminate the experiment a f t e r observing each f a i l u r e time. Thus, the experiment can be terminated e a r l i e r i f evidence accumulates against the n u l l hypothesis. C h a t t e r j e e and Sen (1973) i n v e s t i g a t e d a general c l a s s of s e q u e n t i a l procedures based on l i n e a r rank s t a t i s t i c s and, among other r e s u l t s , determined asymptotic c r i t i c a l values f o r these pro-cedures. Subsequently, s p e c i f i c s e q u e n t i a l procedures have been i n v e s t i g a t e d by Muenz, Green and Byar (1977), Davis. (1978) and K o z i o l and Petkau (1978); these procedures are r e s p e c t i v e l y based on the Mantel-Haenszel, Wilcoxon and Savage s t a t i s t i c s . The theory of C h a t t e r j e e and Sen provided asymptotic c r i t i c a l values f o r the second and t h i r d of these procedures. In l i f e t e s t i n g , e x p o n e n t i a l , and more g e n e r a l l y W e i b u l l , d i s t r i b u t i o n s are o f t e n used to model e m p i r i c a l s u r v i v a l d i s t r i b u t i o n s ; t h e r e f o r e any non-parametric procedure that i s to be used i n p r a c t i c e should perform w e l l when the underlying d i s t r i b u t i o n f u n c t i o n s F^ and are exponential or W e i b u l l . Thus, d i f f e r e n t competing procedures are o f t e n compared on the b a s i s of power under Lehmann a l t e r n a t i v e s (subclasses of which, are the exponential and W e i b u l l s c a l e s h i f t a l t e r n a t i v e s ) . Sequential procedures are a l s o compared on the b a s i s of expected p r o p o r t i o n sampled, the average p r o p o r t i o n of the N subjects observed before a d e c i s i o n i s made; f o r a f i x e d p o i n t censoring procedure w i t h censoring p o i n t . r , the expected p r o p o r t i o n sampled i s r/N. Using a procedure w i t h low values of expected p r o p o r t i o n sampled w i l l l i k e l y reduce the cost of an experiment but there could be an increased r i s k of an i n c o r r e c t d e c i s i o n . Both the power and expected proportion, sampled should t h e r e f o r e be considered i n choosing a procedure. Muenz, Green and Byar. (1977, h e r e a f t e r r e f e r r e d to as MGB) presented - 3 -no asymptotic r e s u l t s r e l e v a n t to t h e i r s e q u e n t i a l , procedure based on the Mantel-Haenszel s t a t i s t i c . Instead^, they numerically examined the n u l l d i s t r i b u t i o n s of t h e i r proposed s t a t i s t i c and the usual Mantel-Haenszel s t a t i s t i c and determined approximate c r i t i c a l values fo r . t h e s e s t a t i s t i c s i n the s p e c i a l case of m=n = N/2, r = N. A l s o , they i n d i c a t e d how the p o s s i b i l i t y of e a r l y acceptance of the n u l l hypothesis could be incorporated i n t o t h e i r s e q u e n t i a l procedure and the corresponding f i x e d p o i n t censoring procedure. They num e r i c a l l y compared the two procedures on the b a s i s of power under Lehmann.alternatives and expected p r o p o r t i o n sampled ( i n the case m = n = N/2, r = N) . For the sequential.procedure based, on the Wilcoxon s t a t i s t i c , Davis (1978) examined the accuracy of the asymptotic c r i t i c a l values (provided by the theory of Chatterjee and Sen) f o r f i n i t e samples. He numerically compared t h i s procedure with the f i x e d p o i n t censoring pro-cedure due to H a l p e r i n and Ware (1974) f o r the case of exponential s c a l e s h i f t a l t e r n a t i v e s . The l a t t e r procedure has a c u r t a i l e d form which i n c o r -porates the p o s s i b i l i t y of e a r l y acceptance of the n u l l hypothesis. For the s e q u e n t i a l procedure based on the Savage s t a t i s t i c , K o z i o l and Petkau (1978, h e r e a f t e r r e f e r r e d to as KP) presented asymptotic d i s t r i b u t i o n s of t h e i r proposed test, s t a t i s t i c s i n the case of contiguous exponential s c a l e s h i f t a l t e r n a t i v e s (they l a t e r discovered that s i m i l a r asymptotic r e s u l t s hold i n the more general case of contiguous W e i b u l l s c a l e s h i f t a l t e r n a t i v e s ) . In a d d i t i o n , they compared the asymptotic p r o p e r t i e s , power and expected pro-p o r t i o n sampled, under contiguous exponential s c a l e s h i f t a l t e r n a t i v e s , of t h e i r s e q u e n t i a l procedure with those of the corresponding f i x e d p o i n t cen-soring, procedure. The numerical s t u d i e s of MGB and Davis and the t h e o r e t i c a l i n v e s t i g a t i o n of KP demonstrated that the d i f f e r e n c e s i n power between the s e q u e n t i a l procedures and t h e i r f i x e d p o i n t censoring counterparts were q u i t e s mall f o r l a r g e r sample s i z e s ; moreover, the s e q u e n t i a l procedures o f f e r - 4 -considerable r e d u c t i o n i n the expected p r o p o r t i o n sampled. Although these r e s u l t s e s t a b l i s h the a t t r a c t i v e n e s s of the s e q u e n t i a l procedures, no d e t a i l e d comparisons among the d i f f e r e n t s e q u e n t i a l procedures, have yet been c a r r i e d out. On the basis of the l i m i t e d numerical r e s u l t s presented by both MGB.and Davis, one cannot d e f i n i t i v e l y compare t h e i r competing s e q u e n t i a l procedures. A l s o , comparison.,of the asymptotic r e s u l t s of KP to the Monte Carlo r e s u l t s of MGB and Davis seems i n a p p r o p r i a t e . Considering the nature of the stopping r u l e s of the s e q u e n t i a l procedures based on the Savage and Mantel-Haenszel s t a t i s t i c s (see S e c t i o n 6.1 f o r f u r t h e r e x p l a n a t i o n ) , KP conjectured that the s e q u e n t i a l procedure based on the Savage s t a t i s t i c w i l l g e n e r a l l y be more powerful (under Lehmann a l t e r n a t i v e s ) than that based on the Mantel-Haenszel s t a t i s t i c , w h i l e the l a t t e r procedure w i l l g e n e r a l l y be quicker to come to a d e c i s i o n . A l s o , s i n c e the Savage rank t e s t i s the l o c a l l y most powerful rank, t e s t under Lehmann a l t e r n a t i v e s (Savage, 1956) and the s e q u e n t i a l procedure.proposed by Davis i s i d e n t i c a l . t o that proposed by KP except Wilcoxon scores are used i n s t e a d of Savage sco r e s , one might expect that the l a t t e r procedure w i l l be more powerful than the former. The purposes of t h i s t h e s i s are to determine the sample s i z e r e q u i r e d before the asymptotic c r i t i c a l values f o r the s e q u e n t i a l procedure proposed by KP provide a.reasonable approximation, and more im p o r t a n t l y , to compare i n d e t a i l the s e q u e n t i a l procedures proposed by MGB, Davis and KP. We f o r m a l l y d e s c r i b e a l l the procedures under c o n s i d e r a t i o n i n Chapter 2. In Chapter 3, we examine the accuracy of the asymptotic c r i t i c a l values f o r f i n i t e samples f o r .the s e q u e n t i a l procedure based on the Savage s t a t i s t i c . There we i n d i c a t e : how we nu m e r i c a l l y determined approximate c r i t i c a l values f o r the s e q u e n t i a l procedures. In.Chapter 4, we nu m e r i c a l l y compare the s e q u e n t i a l procedures, to each other and to t h e i r f i x e d p o i n t censoring counterparts.. We i n d i c a t e i n Chapter 5 how e a r l y d e c i s i o n r u l e s , which - 5 -i n c l u d e the p o s s i b i l i t y of e a r l y acceptance of the n u l l hypothesis, can be incorporated i n t o the procedures, based on the Savage and Wilcoxon s t a t i s t i c s . We provide some i n d i c a t i o n of the r e d u c t i o n i n the expected p r o p o r t i o n sampled which can be r e a l i z e d w i t h t h e i r implementation. We present two examples i n Chapter 6, and discuss our r e s u l t s i n Chapter 7. - 6 -Chapter 2 DESCRIPTION OF PROCEDURES Sec t i o n 2.1 INTRODUCTION Let X..,...,X and X ,X be independent random samples of s i z e s 1 m m+1 N . m and n (n-N-m). from populations i r ^ and n w i t h r e s p e c t i v e continuous d i s -t r i b u t i o n f u n c t i o n s F^ and F^. We are i n t e r e s t e d i n t e s t i n g the n u l l hypo-t h e s i s HQ : F _ = F 2 a 8 a l n s t e i t h e r the one-sided a l t e r n a t i v e : F^ < or the two-sided a l t e r n a t i v e : F^ / F^. Let X = (X^^,...,X^^) denote the vector of order s t a t i s t i c s , and l e t D = (d ,...,d^) denote the vector of i n d i c a t o r v a r i a b l e s , where d, =1 i f the i t h order s t a t i s t i c , X,.., of the i ( i ) combined sample i s from po p u l a t i o n IT^, d^ = 0 i f i t i s from IT^ . In Sections 2.2, 2.3 and 2.4, the f i x e d p o i n t and progressive censoring procedures based on the Savage, Wilcoxon and Mantel-Haenszel s t a t i s t i c s are described. In Sec t i o n 2.5, a m o d i f i c a t i o n of the s e q u e n t i a l procedure proposed by MGB i s o u t l i n e d . Section 2.2 PROCEDURES BASED ON THE SAVAGE STATISTIC When a l l observations are a v a i l a b l e , the Savage s t a t i s t i c can be w r i t t e n as ( c f . Savage, 1956; Hajek and Sidak, 1967, p. 97) SN ' I d i a N ( ± ) 1=1 where a.-(i) = b„(i) - 1 and the scores b„,(i) are defined by b„(i) N N N N i I I (N-j+1) . I t is well-known that E(S N|H Q) = 0 and 3=1 V a r ( SN I V = S C l - b N ( N ) / N ) . * N 1 - 7 -I f only the f i r s t k of the N order s t a t i s t i c s are to be observed, the appropriate m o d i f i c a t i o n of the Savage s t a t i s t i c can be w r i t t e n as ( c f . Gastwirth, 1965; Johnson and Mehrotra, 1972) N I k k where a ^ ( i ) = b j j ^ ) ~ a n c* b N ( i ) = t> N(i) i = 1, ,.k N = I b (j ) / ( N - k ) i = k+l,...,N. j=k+l Note that S* . = S*' = S„. I t i s e a s i l y v e r i f i e d that E.(S*|H.) = 0 and N-1 N N J- k 1 0 -The f i x e d p o i n t censoring procedure w i t h censoring p o i n t r = Np (0 < p <_ 1) i s to. r e j e c t HQ f o r the one-sided a l t e r n a t i v e i f *• . * i 1/2 S r/{var (S^lHg)} i s too l a r g e , or f o r the two-sided a l t e r n a t i v e i f |S*|/{var ( S * | H 0 ) } 1 / 2 i s too l a r g e . Since under H Q |s*/{var(S* | H Q)} 1 / 2 i s a s y m p t o t i c a l l y (as min (m,n) °°) d i s t r i b u t e d as a standard normal random v a r i a b l e , the asymptotic c r i t i c a l values a t the a - l e v e l of s i g n i f i c a n c e are z. and z. ,„ r e s p e c t i v e l y f o r the one-sided and two-sided t e s t s . . 1-a 1-0./1 (Note: z s a t i s f i e s ' ' $ ( z ) = B , where $ i s the standard normal cumulative p p d i s t r i b u t i o n f u n c t i o n ) . The pr o g r e s s i v e censoring procedure can be described as f o l l o w s . For the oner-sided a l t e r n a t i v e , p r o g r e s s i v e l y determine f o r every k, 1 <_ k <_ r , so t .1/2 "k & 1 / 2 whether S '/{var (S i E L ) } ' exceeds, some c r i t i c a l value x; i f s erminate k r 1 0 observation and r e j e c t H^, i f not, continue. I f S ^ / t v a r ( S r | H ^ ) } " 1 < k < r , are a l l l e s s than x, then H Q can be accepted ( i . e . H Q i s r e j e c t e d - 8 -*~ir * l i l / 2 i f and only i f max S /{var (S |H )} i s too l a r g e ) . For the two-sided l _ k _ r a l t e r n a t i v e , a s i m i l a r procedure based on |S | i s appr o p r i a t e . I t has been K. shown by Chatterjee and Sen that as min (m,n) 0 0 and r/N p(0 < p <_ 1) , l i m pr{{max S*/[var (S* [H ) ] 1 / 2 _ x} = 2 [ l - $ ( x ) ] (2.1) N-*» I f J ^ r 00 l i m p r f m a x | S* | / [var (S* | H ) ] 1 / 2 _ x} = 4 J ( - l ) k [ l - $ ( [2k+l]x) ] (2.2) 'N-**> l<k<r r k=0 fo r every x > 0. For the l a r g e r values of x, the r i g h t hand s i d e of (2.2) can be a c c u r a t e l y approximated by 4[1 - $ ( x ) ] (see KP, 1978, Table 1). The asymptotic c r i t i c a l value at the a - l e v e l of s i g n i f i c a n c e i s z, f o r the 1-a/2 one-sided t e s t ; f o r the two-sided t e s t , i t i s approximately z^ f o r a l e s s than .30. Section 2.3 PROCEDURES BASED ON THE WILCOXON STATISTIC The Wilcoxon s t a t i s t i c can be w r i t t e n as ( c f . Wilcoxon, 1945; Hajek and Sidak, 1967, p.. 87) W N " ! d i 3 N ( i ) 1=1 where a^d) = b^ (i) - (N+i)/2 and the scores have the form b^ (i) = i. I t i s well-known that E(W |H )• = 0 and var (WN|H ) = mn(N+l)/12. I f only the f i r s t k order s t a t i s t i c s are to be observed, the appropriate m o d i f i c a t i o n of the Wilcoxon s t a t i s t i c can be w r i t t e n as < - \ dA^ 1=1 where a j j ( i ) = bk(i) - (N+l)/2 and - 9 -b N ( 1 ) = b N ( 1 ) i = 1 » • • • » k N = I b N ( j ) / ( N - k ) i = k+1 , . . . , N. j=k+l 7\ I t i s e a s i l y v e r i f i e d that E(W H ) = 0 and var (Wk|HQ) = mn(N+1) 12 [ l - { N2 ( l - p ) 3 - ( l - p ) } / ( N 2 - l ) ] where p = k/N. In both the f i x e d p o i n t and progressive censoring s i t u a t i o n s , the procedures based on the Wilcoxon s t a t i s t i c s are completely analogous to those based on the Savage s t a t i s t i c s and i n each case the asymptotic c r i t i c a l values are i d e n t i c a l . Remark. The f i x e d p o i n t censoring procedure based on the Wilcoxon s t a t i s t i c which i s being considered here i s d i f f e r e n t from the f i x e d p o i n t censoring procedure due to H a l p e r i n and Ware (1974). Their procedure p r e s c r i b e s termination of the experiment as soon as a p r o p o r t i o n p of the f a i l u r e times has been observed i n one of the two treatment groups; t h i s procedure would be somewhat quicker to terminate, p a r t i c u l a r l y when the n u l l hypothesis i s Section 2.4 PROCEDURES BASED ON THE MANTEL-HAENSZEL STATISTIC The Mantel-Haenszel s t a t i s t i c can be motivated as f o l l o w s . A f t e r the i t h f a i l u r e time i n the combined sample, f o r the 2 x 2 t a b l e f a l s e . - 10 -group I group I I t o t a l f a i l e d d. 1-d. x 1 survived m - £ d. n - i + £ d. N - i j = l J j = l J i - 1 i - 1 t o t a l m- I d. n - i + l + I d. N-i+1 j = l J j = l J 0 (by convention, £ d. = 0 ) . Under the n u l l hypothesis, the expected value i - 1 of D. given (d.,...,d. , ) • i s e. = (m. - Y d.)/(N-i+1) and the variance of x 1 i - i 1 . n i J = l D. given (d,,...,d. ..) i s v. = e . ( l - e . ) ; . the q u a n t i t i e s e. and v. l 1 x-1 x x x x x are suggested from the hypergeometric d i s t r i b u t i o n a s s o c i a t e d w i t h the 2 x 2 t a b l e . . The Mantel-Haenszel s t a t i s t i c ( c f . Mantel and Haenszel, 1959; Mantel, 1966), when a l l the observations are a v a i l a b l e , i s N H XN = tl J M i - V J - i r v r v. . x=l x=l Under HQ t h i s s t a t i s t i c i s a s y m p t o t i c a l l y chi-squared w i t h one degree of freedom. I f only the f i r s t k order s t a t i s t i c s are to be observed, the m o d i f i c a t i o n of the Mantel-Haenszel s t a t i s t i c i s x=l x=l MGB consider the Mantel-Haenszel s t a t i s t i c without the c o n t i n u i t y c o r r e c t i o n ; that i s , they use X=l 1=1 - 11 -We use the unsquared v e r s i o n of t h i s s t a t i s t i c , 1=1 1=1 1=1 The l a s t e q u a l i t y can be e a s i l y v e r i f i e d . The f i x e d p o i n t censoring procedure w i t h censoring p o i n t r i s to r e j e c t HQ f o r the one-sided a l t e r n a t i v e i f M^'(or s i g n { £ ( e_~d^)}M^. . , which i s i = l of the form of the one-sided s t a t i s t i c . M G B use) i s too l a r g e , or f o r the two-sided a l t e r n a t i v e i f |M'| i s too l a r g e . KP have shown _ v. to be a 1 = 1 1 ft. N c o n s i s t e n t (as N •+ oo and m/N •+ \, 0 < \ < 1) estimator of var (S r|Hg), therefore the f i x e d p o i n t censoring procedures based on the Mantel-Haenszel and Savage s t a t i s t i c s are a s y m p t o t i c a l l y equivalent. The progressive censoring procedure w i t h at most r order s t a t i s t i c s to be observed i s to r e j e c t H f o r the one-sided a l t e r n a t i v e i f max K i s too l<k<r l a r g e , or f o r the two-sided a l t e r n a t i v e i f max j ML | i s too-large ( c f . S e c t i o n l<k<r 2.2). MGB d i d not provide asymptotic c r i t i c a l values f o r these t e s t s , but i n s t e a d provided c r i t i c a l value curves (obtained by smoothing Monte Carlo c r i t i c a l values) i n the s p e c i a l case of equal sample s i z e s (m=n = N/2) w i t h r = N. S e c t i o n 2.5 THE MODIFIED MGB SEQUENTIAL PROCEDURE We consider a s l i g h t m o d i f i c a t i o n of the procedures based on the Mantel-Haenszel s t a t i s t i c . This m o d i f i c a t i o n , M^ , c o n s i s t s of r e p l a c i n g k- & ft £ v. i n the d e f i n i t i o n of M^, as given i n (2.3), by v a r ( S k | H Q ) ( i . e . M^ = i = l ft *• • 1/2 S,/{var(S. • EL)} ' ). The t e s t s t a t i s t i c s f o r the s e q u e n t i a l procedure are k k' 0 - 12 -now max and max |M^ | f o r the one-sided and two-sided t e s t s l<k<r -klHl r k * r e s p e c t i v e l y . Since £ v^ i s a c o n s i s t e n t estimator of v a r C S j j H g ) , t h i s i = l m o d i f i c a t i o n has no e f f e c t a s y m p t o t i c a l l y . In the f i x e d p o i n t censoring case, the modified procedure i s i d e n t i c a l to the procedure based on the Savage s t a t i s t i c . - 13 -Chapter 3 NULL DISTRIBUTION RESULTS Section 3.1 INTRODUCTION We w i l l now r e f e r to the s e q u e n t i a l procedures as the KP, Davis, MGB and modified MGB procedures. In t h i s chapter, one o b j e c t i v e i s to examine the accuracy of the asymptotic c r i t i c a l values f o r the KP procedure f o r f i n i t e samples. I t was found that use of the asymptotic c r i t i c a l values f o r the KP procedure r e s u l t e d i n somewhat.conservative t e s t s , even f o r moderate-sized samples. Therefore, i n the equal sample s i z e case, e m p i r i c a l c r i t i c a l values were simulated and f i t t e d by smooth curves. The f i t t e d c r i t i c a l values and curves are presented i n Se c t i o n 3.2; a l s o presented are exact c r i t i c a l values i n the equal sample s i z e cases N = 10, 12, 14, 16, 18, 20 w i t h p= .5 and 1. S i m i l a r methods were used to. determine approximate c r i t i c a l values i n the equal, sample s i z e case f o r the Davis, MGB and modified MGB procedures; t h i s i s discussed, i n S e c t i o n 3.3. In Se c t i o n 3.4, we study the e f f e c t of unequal sample s i z e s (m =f n) on the c r i t i c a l values f o r the four s e q u e n t i a l procedures. In S e c t i o n 3.5, we examine the accuracy of the asymptotic c r i t i c a l values f o r the f i x e d p o i n t censoring procedures based on the Savage, Wilcoxon and Mantel-Haenszel s t a t i s t i c s . S e c t i o n 3.2 THE KP PROCEDURE, WITH EQUAL SAMPLE SIZES We consider the case i n which at most the f i r s t r = Np (0 < p <_ 1) of the N order s t a t i s t i c s are to be observed and examine the n u l l d i s t r i b u t i o n s of the f o l l o w i n g s t a t i s t i c s : - 14 -+ * * i 1/2 S (p) = max S /{var (S H )} l<k<N P k N P ° S"(p) = min S*/{var(S* |H ) } 1 / 2 l<k<Np k N P U S(p) = max | s * | / { v a r ( S ^ | H 0 ) } 1 / 2 . l<_k<Np P For s m a l l values of Np, the n u l l d i s t r i b u t i o n s of these s t a t i s t i c s can be determined by complete enumeration of the D-vectors. We have done t h i s i n the case of equal sample s i z e s , m = n = N/2, f o r N = 10(2)20 and p= .5,1. Some r e s u l t s f o r the n u l l d i s t r i b u t i o n of S +(p) are summarized i n Table I . For each N and each nominal l e v e l a, pr{S +(p) > x} i s ta b u l a t e d f o r those values of x y i e l d i n g l e v e l s equal to or c l o s e s t to the nominal l e v e l . A l s o tabulated are the a c t u a l s i g n i f i c a n c e l e v e l s corresponding to the asymptotic c r i t i c a l values. The corresponding r e s u l t s f o r the n u l l d i s t r i b u t i o n of S(p) are summarized i n Table I I . When examining the d e t a i l e d l i s t i n g s of these n u l l d i s t r i b u t i o n s , the e q u a l i t y + P r{S(p) > x} = 2 pr{S (p) _ x} was found to hold f o r every value of x greater than 1.20. For the case of equal sample s i z e s , we have p r { S + ( p ) _ x} = pr{S (p) < -x} (3.1) and consequently pr{S(p) _ x} = 2pr{S +(p) _ x} - pr{S +(p) _ x, s " ( p ) <_ -x} . (3.2) Since one would expect the second term on the r i g h t of (3.2) to be small Table I. Summary of Null D i s t r i b u t i o n of S +(p) = max S*/{var(S* |H ) 1 / 2 i n the case m = n = N/2 iskfNp k Np1 0 A. Case p = 1/2 Nominal Level a »01 .10 .20 Asymp. c r i t . value x_; 2.5758 1.9600 1.6449 1.2816 N = 10 x P(x) p(x!) 2.93525 2.17754 .00397 .02381 .00397 1.87445 1.74455 .04365 .06349 .02381 1.63090 1.52987 .08333 .10317 .06349 1.13838 1.02472 .18254 .22222 .14286 N = 12 x P(x) 2.53578 2.29918 .00758 .01407 .00108 1.94556 1.91469 .04004 .05628 .03355 1.51214 1.42933 .08874 .10498 .05628 1.07443 1.05372 .19156 .20779 .15909 N = 14 x P(x) 2.50633 2.43675 .00845 .01049 .00641 1.79897 1.74010 .04516 .05128 .02681 1.49701 1.45310 .09819 .10431 .07372 1.28083 1.24120 .17162 .20221 .16550 N = 16 x P(x) P(x a) 2.36928 2.30429 .00940 .01158 .00723 1.82886 1.81493 .04856 .05074 .03333 1.59055 1.58721 .08990 .10404 .07902 1.17269 1.14726 .18780 .21717 .14646 N = 18 x P(x) p(x!) 2.38616 2.37675 .00983 .01057 .00465 1.80557 1.79755 .04994 .05167 .03723 1.48015 1.47812 .09831 .10004 .06981 1.17599 1.16673 .19852 .20025 .14842 N = 20 x P(x) p(x!) 2.45476 2.43631 .00988 .01013 .00631 1.80666 1.80114 .04950 .05015 .03197 1.53333 1.52844 .09926 .10289 .07701 1.11532 1.10848 .19856 .20107 .15927 Table I. (continued) B. Case p = 1 Nominal Level a .01 .05 .10 .20 Asymp. c r i t . value a x 2. 5758 1.9600 1.6449 1.2816 N - 10 x P(x) P(x«) 2.16068 2.04176 .00794 .01190 0 1.72068 1.70878 .04365 .05159 .01587 1.43442 1.41063 .09921 .10317 .05556 1.17279 1.14023 .19841 .20238 .13492 N = 12 x P(x) P(x") 2.10872 2.10159 .00974 .01082 0 1.70214 1.69424 .04978 .05087 .02056 1.47539 1.46955 .09957 .10065 .05411 1.11111 1.10933 .19805 .20022 .13853 ON I N = 14 x P(x) P(x a) 2.15557 2.14459 .00991 .01020 .00058 1.73078 1.72705 .04983 .05012 .02127 1.44734 1.44708 .09994 .10023 .05944 1.11729 1.11612 .19843 .20047 .15181 N = 16 x P(x) P(x a) 2.17564 1.17368 .00995 .01002 .00101 1.71965 1.71940 .04996 .05004 .02378 1.46137 .10 .06208 1.14658 1.14577 .19992 .20008 .15058 N = 18 x P(x) P(x a) 2.19731 2.19719 .009996 .01002 .00144 1.73829 1.73822 .04994 .05016 .02489 1.47292 .10 .06399 1.14808 1.14804 .19914 .20134 .15121 N " 2 0 x 2.22034 2.22033 1.74874 1.74872 1.47955 1.47947 1.15242 1.15238 P(x) .009997 .010002 .049996 .050001 .099997 .100008 .19999 .20003 P(x a) .00178 .02621 .06716 .15533 Note: P(x) = pr{S +(p) >_ x) . Table I I . Summary of N u l l D i s t r i b u t i o n of S(p)=max | S | /{var (S | H Q)} i n the case m = n - N/2 l<k<Np P A. Case p = 1/2 Nominal L e v e l ,01 ,05 .10 .20 Asymp. c r i t . value x a 2.8070 2.2414 1.9600 N = 10 x 2.93525 2.17754 2.17754 1.87445 1.87445 1.74455 P(x) .00794 .04762 .04762 .08730 .08730 .12698 , c u .00794 .00794 .04762 1.6449 1.63090 1.52987 .16667 .20635 .12698 N = 12 N = 14 N = 16 N = 18 N = 20 x P(x) p<*!> CO x P(x) P<<> CO x P(x) P<x"> C O x P(x) P<^> CO x P(x) P ( x a ) 3.24559 2.53578 2.19567 2.10365 1.94556 1.91469 1.51214 lm*2*l\ .00216 .01515 .04113 .05411 .08009 .11255 .17749 .ZUyyb .02814 .06710 .11255 .00216 2.66789 2.58285 .00874 .01282 .00466 2.26397 2.00885 .04138 .05361 .04138 1.79897 1.74010 .09033 .10256 .05361 2.69470 2.64703 .00886 .01010 .00513 2.06598 2.06382 .04926 .05796 .02751 1.82886 1.81493 .09713 .10148 .06667 2.58815 2. 54005 .00930 .01078 .00485 2.13379 2.12408 .04879 .05027 .03348 1.80557 1.79755 .09988 .10333 .07445 2.66495 2.64515 .00970 .01019 .00450 2.05488 2.04124 .04997 .05127 .03292 1.80666 1.80114 .09901 .10031 .06393 1.49071 1.45310 .19639 .20862 .14744 1.59055 1.58721 .17980 .20808 .15804 1.48015 1.47812 .19663 .20008 .13961 1.53333 1.52844 .19851 -20578 .15402 Table I I . (continued) B. Case p = 1 Nominal Level .01 .05 .10 .20 Asymp. c r i t . value x51 2.8070 2.2414 1.9600 1.6449 N = 10 x P(x) P(x ) 2.30338 2.16068 .00794 .01587 0 . 1.86338 1.85064 .04762 .05556 .00794 1.72068 1.70878 .08730 .10317 .03175 1.43442 1.41063 .19841 .20636 .11111 N = 12 x P(x) P(x") 2.28068 2.23712 .00866 .01082 0 1.91484 1.91002 .04978 .05195 .00866 1.70214 1.69424 .09957 .10173 .04113 1.47539 1.46955 .19913 .20130 .10823 N = 14 x P(x) p d ! ) 2.27755 2.27221 , .00991 .01049 0 1.91882 1.91064 .04953 .05012 .01224 1.73078 1.72705 .09965 .10023 .04254 1.44734 1.44708 .19988 .20047 .11888 N = 16 x P(x) P ( x a ) 2.31515 2.31212 .00995 .01010 .00031 1.94290 1.94095 .04988 .05004 .01430 1.71965 1.71940 .09992 .10008 .04755 1.46137 .20 .12416 N = 18 x P(x) P(x") 2.34872 2.34758 .009996 .01004 .00049 1.95843 1.95775 .04998 .05002 .01645 1.73829 1.73822 .09988 .10033 .04977 1.47292 .20 .12797 N = 20 x P(x) PO£) 2.37883 2.37850 .00999 .010002 .00076 1.97456 1.97427 .04998 .050001 .01823 1.74874 1.74872 .09999 .100002 .05242 1.47955 1.47947 .19999 .20002 .13433 Note: P(x) - pr { S(p)> x } . - 19 -f o r l a r g e r values of x, the above e m p i r i c a l r e s u l t i s not unexpected. The r e s u l t s i n Tables I and I I c l e a r l y i n d i c a t e t h a t , at l e a s t f o r these sample s i z e s , the use of the asymptotic c r i t i c a l values f o r the KP procedure w i l l r e s u l t i n t e s t s which are unduly conservative. Although the cost of determining the n u l l d i s t r i b u t i o n s e x a c t l y becomes p r o h i b i t i v e as Np i n c r e a s e s , the d i s t r i b u t i o n s can s t i l l be e a s i l y simulated by random sampling of the D-vectors. A s i m u l a t i o n experiment was c a r r i e d out i n order to determine how l a r g e the sample s i z e must be before the asymptotic c r i t i c a l values provide a reasonable approximation. Table I I I summarizes the r e s u l t s f o r the one-sided t e s t at nominal l e v e l s .10, .05 and .01. Each entry i n the t a b l e i s based on 1500 simulated samples; e n t r i e s f o r a f i x e d N but d i f f e r e n t values of p and <* were obtained from the same set of simulated samples. In c a l c u l a t i n g the s i m u l a t i o n estimates of p r { S + ( p ) >_ x}, (3.1) was used to form two d i f f e r e n t estimates which were then averaged to form the f i n a l estimate; the standard e r r o r of any 1/2 i n d i v i d u a l entry P i n Table I I I i s estimated to be at most {P(l-P)/3000} The corresponding r e s u l t s f o r the two-sided case are presented i n Table IV (the same set of simulated samples as i n the one-sided case was used). The r e s u l t s i n Tables I I I and IV c l e a r l y i n d i c a t e t h a t , even f o r what would i n most instances be thought of as l a r g e sample s i z e s , the use of the asymptotic c r i t i c a l values f o r the KP procedure w i l l r e s u l t i n t e s t s which are somewhat conservative. Note a l s o the tendency of the t e s t s to be more conservative f o r the case p = 1.0 than f o r the cases p=.50, .75 and .90. These r e s u l t s are i n c o n t r a s t to the corresponding r e s u l t s provided by Davis (1978; Table I I I , p. 394) who found that f o r the one-sided Davis procedure, the. asymptotic c r i t i c a l value (at nominal l e v e l .05) provides an adequate approximation (although s t i l l y i e l d i n g a s l i g h t l y conservative t e s t ) f o r - 20 -Table I I I . Simulation Estimates of pr {S (p) > x01} . m=n P .25 .50 .75 .90 1.0 a = .10 10 .039 .065 .068 .063 .059 20 .085 .079 .085 .084 .075 30 .080 .083 .088 .087 .085 40 .084 .084 .083 .088 .081 50 .087 .092 .097 .094 .088 70 .090 .096 .093 .089 .085 100 .079 .085 .086 .087 .085 200 .090 .093 .099 .099 .099 a = .05 10 .015 .027 .031 .026 .022 20 .042 .040 .037 .036 .033 30 .045 .041 .042 .041 .035 40 .038 .041 .039 .041 .039 50 .047 .048 .047 .046 .044 70 .044 .049 .044 .044 .042 100 .038 .044 .042 .041 .036 200 .043 .047 .041 .051 .051 a = .01 10 .000 .005 .004 .002 .001 20 .006 .006 .005 .005 .004 30 .006 .007 .010 .008 .007 40 .005 .008 .008 .008 .006 50 .008 .011 .009 .007 .006 70 .008 .010 .008 .006 .006 100 .009 .010 .007 .005 .006 200 .006 .012 .011 .011 .009 *Each entry i s based on 1500 simulated samples. - 21 -Table IV. Simulation Estimates of pr { S(p) _ .a,* x } . 0 0 m=n P .25 .50 .75 .90 1.0 a = .10 10 .030 .054 .061 .052 .045 20 .084 .079 .074 .073 .066 30 .089 .083 .083 .083 .069 40 .077 .083 .079 .083 .078 50 .095 .096 .093 .091 .089 70 .088 .099 .088 .087 .083 100 .077 .089 .083 .082 .072 200 .085 .095 .103 .102 .102 a = .05 10 .030 .029 .033 .021 .019 20 .025 .040 .033 .029 .025 30 .034 .035 .042 .037 .033 40 .034 .044 .039 .039 .040 50 .041 .049 .049 .044 .037 70 .043 .050 .043 .044 .038 100 .036 .045 .041 .035 .031 200 .042 .059 .056 .057 .053 a = .01 10 .000 .003 .002 .000 .000 20 .010 .007 .007 .004 .003 30 .003 .008 .009 .008 .007 40 .003 .007 .007 .009 .005 50 .009 .013 ..009 .005 .004 70 .009 .010 .005 .007 .003 100 .009 .010 .006 .007 .005 200 .005 .012 .014 .011 .009 *Each entry i s based on 1500 simulated samples. - 22 -s i t u a t i o n s i n which Np > 60. That the asymptotic r e s u l t s , are much slower to take hold w i t h the KP procedure than w i t h the Davis procedure might be explained upon e v a l u a t i o n of the moments of the modified.Savage and Wilcoxon s t a t i s t i c s . L e t t i n g SS = S*/{var(S * | H )}1/2 and SW = W*/{var(W*[H 0)} 1 / 2, c a l c u l a t i o n s (Hajek r r r 1 0 r r r 1 J and Sidak, 1967, p. 81-82) i n the equal sample s i z e case y i e l d E(SS 3) = E(SW 3) = 0 r r E.(S.S*> = 3 - f s ( | , | ) / N + 0(N" 2) E(SWS = 3 - f (|,|)/N + 0 ( N - 2 ) r w IN z where f s ( p , X ) = 6 + p" 2[6-x" 1(1-X)" 1][-3p 2+9p-6(1-p)In 2(1-p)+8(1-p)In(1-p)J, f (p,X)= 6 +.6p[6-X _ 1(l-X)" 1][l-(l-p) 3]~ 2[15-105p+210p 2-180p 3+63p 4-5p 5]. w f (p , i b evaluated at p = .25, .50, .75, .90, 1.0 i s r e s p e c t i v e l y 4.847, s 2.288, 3.892, 7.597, 18.000. S i m i l a r l y , f (p,-^) evaluated at p = .25, .50, w 2 .75, .90, 1.0 i s r e s p e c t i v e l y 5.375, 3.086, 3.385, 3.578, 3.600. The usual (Edgeworth) c o r r e c t i o n to the asymptotic d i s t r i b u t i o n f u n c t i o n s of these s t a t i s t i c s then become (Hajek and Sidak, 1967, p. 146; see a l s o B i c k e l and van Zwet, 1978; Robinson, 1978) pr{SS r < x} ^ , * ( x ) - $ ( 4 ) ( x ) f s ( | , | ) / 2 4 N (3.3) pr{SW < x} ^  $(x) - $ C 4 ) - ( x ) f ( ~ ) / 2 4 N . (3,4) r — w JN z 1 1 For the l a r g e r values of p = r/N, because ^(p,-^) < ^g^P'^' a n c* (3.4) i n d i c a t e that the asymptotic d i s t r i b u t i o n provides a b e t t e r approximation f o r - 23 -SW/_ than f o r SS^, One might then conjecture f o r these values of p = r/N that the asymptotic d i s t r i b u t i o n s (2,1), (.2,2) provide better, approximations * *, 1/2 f o r the Davis s t a t i s t i c s , max W /{var(W |H )} and l<_k<r *'t-i * i 1/2 max jW, |/{var(W JH )} than f o r the corresponding KP s t a t i s t i c s , l < k t r max ' S</{var(S | H j } ' and max |S |/{var(S |H )} ' . l<k<r k r U l<k<r k r U - « Since the asymptotic c r i t i c a l values are of l i t t l e use f o r the sample s i z e s of usual i n t e r e s t , the set of simulated samples used to produce Tables I I I and IV was a l s o used to determine e m p i r i c a l c r i t i c a l values f o r the KP procedure. These e m p i r i c a l c r i t i c a l values f o r the one-sided t e s t are p l o t t e d i n Figure 1 f o r the nominal l e v e l s .10, .05 and .01. C r i t i c a l values f o r a given value of N but d i f f e r e n t values of p and a were obtained from the same set of 1500 simulated samples. The "exact" c r i t i c a l values (values y i e l d i n g l e v e l s c l o s e s t to the nominal l e v e l ) f o r the case N = 20 are a l s o p l o t t e d on t h i s f i g u r e . S i m i l a r l y , the analogous r e s u l t s f o r the two-sided t e s t are p l o t t e d i n Figure 2. To remove some of the sampling v a r i a b i l i t y evidenced i n the f i g u r e s , smooth curves were f i t to the e m p i r i c a l c r i t i c a l v a lues. Guided by the asymptotic theory, various simple curves depending on N but not on p were + 1/2 te s t e d . For the one-sided case, curves of the form S„, = a -b exp ( -ce N ) N,ct a a a where a i s the asymptotic c r i t i c a l value appeared to perform w e l l and were a f i t t e d by non - l i n e a r l e a s t squares. S i m i l a r l y f o r the two-sided case, 1/2 curves of the form S„ = A -B exp(-C N ) where A i s the asymptotic N,ct a a a a c r i t i c a l value were f i t t e d . The values of the f i t t e d c o e f f i c i e n t s and the r e s u l t i n g "smoothed'' e m p i r i c a l c r i t i c a l values are provided i n Table V, and the curves are depicted i n Figures 1 and 2, - 24 -Table V.' F i t t e d C r i t i c a l Values f o r KP Procedures* m=n N,a a = .10 .05 a = .01 a = .10 a = .05 ot = .01 10 1.5113 1. 7956 2.3296 1.7956 2.0360 2.5432 15 1. 5268 1.8194 2.3719 1.8194 2.0692 2.5907 20 1.5384 1.8368 2.4018 1.8368 2.0930 2.6241 25 1. 5477 1.8504 2.4245 1.8504 2.1112 2.6492 30 1.5554 1.8613 2.4424 1.8613 2.1257 2.6689 40 1.5676 1.8781 2.4692 1.8781 2.1477 2.6980 50 1.5770 1.8906 2.4883 1.8906 2.1635 2.7185 60 1.5844 1.9002 2.5026 1.9002 2.1755 2.7337 70 1.5906 1.9078 2.5137 1.9078 2.1849 2.7454 80 1.5958 1.9141 2.5225 1.9141 2.1925 2.7546 90 1.6002 1.9193 2.5297 1.9193 2.1986 2.7619 100 1.6040 1.9236 2.5355 1. 9236 2.2037 2.7680 150 1.6171 1.9378 2.5536 1.9378 2.2198 2.7862 200 1.6249 1.9454 2. 5623 1.9454 2.2279 2.7947 OO 1.6449 1.9600 2.5758 1.9600 2.2414 2.8070 * F i t t e d c r i t i c a l values are defined by S M = a -b exp(-c N ) and J N,a a a a 1/2 • S„ = A - B exp(-C N ) , where the values of these c o e f f i c i e n t s are N,a a a a 0.10 1.6449 .23087 .12231 1.9600 .33036 .15599 0.05 1.9600 .33036 .15599 2.2414 .44989 .17537 0.01 2.5758 .56852 .18717 2.8070 .63821 .19757 - 25 " FIGURE 1 E s t i m a t e d c r i t i c a l v a l u e s f o r t h e o n e - s i d e d s e q u e n t i a l KP p r o c e d u r e a t n o m i n a l l e v e l s . 1 0 , . 0 5 , a n d . 0 1 . 0 . 0 4 . 0 a.o vfN 1 2 . 0 1 6 . 0 2 0 , 0 ) - 27 -The e m p i r i c a l c r i t i c a l values f o r the two-sided t e s t at nominal l e v e l .10 were i d e n t i c a l i n a l l the cases that were simulated to the e m p i r i c a l c r i t i c a l v a l u - s f o r the one-sided t e s t at nominal l e v e l .05 ( r e c a l l that the same set of simulated samples was used i n the one-sided and two-sided cases). This i s a r e f l e c t i o n of the f a c t t h a t , f o r these v a l u e s , the second term on the r i g h t on (3.2) i s n e g l i g i b l e . Therefore, the f i t t e d curve at the two-sided .10 l e v e l i s i d e n t i c a l to that at the one-sided .05 l e v e l (see Table V). A s i m u l a t i o n experiment was c a r r i e d out to assess the accuracy of these f i t t e d c r i t i c a l values f o r the. values of N employed i n the Monte Carlo comparison study to f o l l o w i n Chapter 4. The r e s u l t s are summarized i n Table V I . Each entry i n the t a b l e i s based on 1500 simulated samples ( d i f f e r e n t from those which were used to construct Tables I I I and I V ) ; e n t r i e s f o r a f i x e d N but d i f f e r e n t values of p and a were obtained from the same set of simulated samples. Although considerable sample v a r i a b i l i t y i s evident i n Table VI, i t i s nonetheless c l e a r that the f i t t e d c r i t i c a l values provide a s u b s t a n t i a l improvement over the asymptotic c r i t i c a l v a lues. Section 3.3 THE DAVIS, MGB AND MODIFIED MGB PROCEDURES, WITH EQUAL SAMPLE SIZES For a f i x e d value of N, the set of simulated samples used to construct Tables III-and IV was a l s o used to determine e m p i r i c a l c r i t i c a l values f o r the other three s e q u e n t i a l procedures. For the Davis procedure, curves of e x a c t l y the same form as described i n Secti o n 3.2 were f i t t e d ; a separate curve f o r the case p = ,25 was used because of slower convergence of the e m p i r i c a l c r i t i c a l , values to the asymptotic c r i t i c a l values.. Davis i n d i c a t e d that the asymptotic, c r i t i c a l values are u s e f u l f o r Np > 60, but curves were f i t f o r the Davis procedure so that the Monte Carlo comparison study would - 28 -Table VI. S i m u l a t i o n Estimates of A c t u a l L e v e l s Corresponding to F i t t e d C r i t i c a l Values f o r KP Procedures*. m=n .25 .50 .75 .90 1.0 1-sided case a = .10 15 .124 .105 .109 .107 .097 30 .080 .096 .097 .100 .097 50 .105 .114 .111 .113 .111 a = .05 15 .051 .052 .055 .049 .045 30 .053 .049 .050 .053 .052 50 .056 .055 .058 .054 .052 a = .01 15 .017 .015 .009 .008 .006 30 .009 .014 .013 .015 . O i l 50 . .012 ' .010 .009 .014 .013 2-sided case a = .10 15 .108 .119 .111 .109 .099 30 .097 .100 .097 .101 .097 50 .105 .103 .097 .096 .092 -a = .05 -•• - • • ...... — -—, 15 .053 .059 .056 .049 .043 30 .040 .053 .053 .053 .044 50 .053 .051 .057 .054 .049 a = .01 15 .009 .018 .013 .008 .007 30 .011 .011 .012 .011 .010 50 .008 .008 .011 .009 .007 *Each entry i s based on 1500 simulated samples. be as f a i r as p o s s i b l e . The s i t u a t i o n was somewhat l e s s t r a c t a b l e f o r the MGB and modified MGB procedures s i n c e asymptotic c r i t i c a l values and other appropriate asymptotic theory are u n a v a i l a b l e f o r these procedures. The e m p i r i c a l . c r i t i c a l values f o r the MGB and modified MGB procedures appeared to depend on Np rather than N, Many d i f f e r e n t curves were t e s t e d , and e v e n t u a l l y curves s i m i l a r i n form to those used by MGB were used to f i t the e m p i r i c a l c r i t i c a l values as f u n c t i o n s of Np. The f i t t e d curves are v a l i d only f o r the range of Np s t u d i e d . The curves presented by MGB, used as f u n c t i o n s of Np, f i t t e d the e m p i r i c a l c r i t i c a l values reasonably w e l l , but MGB d i d not present curves f o r t h e i r one-sided t e s t . The simulated samples which had been'used to t e s t the accuracy of the f i t t e d c r i t i c a l values f o r the KP procedure were.also used to t e s t the accuracy of the f i t t e d values f o r the Davis, MGB and modified MGB procedures. In a l l cases, the f i t t e d c r i t i c a l values appeared adequate and were.therefore used i n the Monte Carlo comparison study. Section 3.4 THE SEQUENTIAL PROCEDURES, WITH UNEQUAL SAMPLE SIZES To t h i s p o i n t we have r e s t r i c t e d ourselves to the case of balanced experiments (m=n). Although the asymptotic c r i t i c a l values f o r both the KP and Davis procedures are a l s o v a l i d i n the unbalanced case (m -f n) , l a r g e r sample s i z e s might be required before the asymptotic c r i t i c a l values provide u s e f u l approximations. Hence, i t i s of i n t e r e s t to examine the accuracy of the f i t t e d c r i t i c a l values i n the unbalanced case. In the unbalanced case, w i t h X = m/nv the d i s t r i b u t i o n s of the modified Savage and Wilcoxon s t a t i s t i c s are no longer symmetric about zero; c a l c u l a t i o n s (Hajek and Sidak, 1967, p. 81-82) show that - 30 -E(SS 3) = ( 1 - 2A)[NX(1-A ) 1 1 / 2 g s(|) + 0(.N~ 3 / 2) (3.5) E(SW 3) = ( 1 - 2 X ) [ N A ( 1 - X ) J 1 / 2 g ( J ) + 0 O T 3 / 2 ) (3.6) r w L N where g_(p) = p ~ 3 / 2 [ 2 p + 3 ( l - p ) l n ( l - p ) ] , g (p.) = - 3 3 / 2 { l - ( l - p ) 3 ] " 3 / 2 p ( l - p ) 3 . s w g_(p) evaluated at p = .25, .50, .75, .90, 1.0 i s r e s p e c t i v e l y -1.178, -.112, .709, 1.299, 2.000; s i m i l a r l y , g (p) evaluated at p = .25, .50, .75, w .90, 1.0 i s r e s p e c t i v e l y -1.247, -.397, -.062, -.005, 0.0. Upon s u b s t i t u t i n g these values i n t o (3.5) and (3.6), one might be able to p r e d i c t whether the c r i t i c a l values f o r the one-sided t e s t s should be increased or decreased f o r d i f f e r e n t values of X. A s i m u l a t i o n experiment was c a r r i e d out to examine the accuracy of the f i t t e d c r i t i c a l values i n the unbalanced cases X = .25, .33, .40, .60, .67, .75 f o r a l l the s e q u e n t i a l procedures under c o n s i d e r a t i o n . For the one-sided t e s t s , the r e s u l t s f o r the case N = 100, a = .05 are summarized i n Table V I I , each entry i s based on 1500 simulated samples. The r e s u l t s are s i m i l a r f o r other values of N and a. For the one-sided MGB and modified MGB t e s t s , there are obvious trends, i n the simulated s i g n i f i c a n c e l e v e l s which are i n c r e a s i n g i n X. For the KP t e s t s , there i s a s l i g h t decreasing trend f o r p = .75, .90 and 1.0. I t appears that g (p) and g (p) are i n d i c a t o r s of s w p o s s i b l e trends i n the one-sided KP and Davis t e s t s . For a l l the one-sided t e s t s , there are s l i g h t trends i n p which are i n c r e a s i n g or decreasing depending on whether X < 1/2 or X > 1/2. These trends might be due to the f a c t that the moments (3.5) and (3.6) are i n c r e a s i n g i n p when X < 1/2 and decreasing when A > 1/2. For the two-sided t e s t s , the r e s u l t s f o r the case N = 100, a = .05 are summarized i n Table V I I I ; the t e s t s t a t i s t i c s have the same d i s t r i b u t i o n - 31 -Table V I I . S i m u l a t i o n Estimates of A c t u a l Levels Corresponding to One-Sided F i t t e d C r i t i c a l Values i n the Unbalanced Case, w i t h N = 100, a = .05. X .25 .33 .40 .60 .67 .75 KP P = .25 .047 .043 .050 .055 .046 .061 P = .50 .057 .041 .055 .055 .051 .046 p = .75 .057 .049 .057 .059 .042 .035 p = .90 .065 .047 .063 .051 .048 .028 p = 1.0 .062 .051 .063 .047 .043 .021 MGB P = .25 .020 .024 .037 .071 .072 .117 P = .50 .028 .028 .043 .075 .083 .113 P = .75 .029 .031 .044 .078 .084 .111 P = .90 .029 .031 .045 .076 .082 .085 P = 1.0 .030 .031 .046 .059 .066 .082 Mod MGB p = .25 .023 .034 .040 .068 .071 .117 p = .50 .037 .034 .046 .073 .078 .107 p = .75 .043 .035 .048 .073 .079 .078 p = .90 .045 .035 .049 .057 .078 .074 p = 1.0 .047 .035 .049 .056 .062 .071 Davis p = .25 .046 .045 .050 .062 .053 .061 p = .50 .053 .037 .047 .051 .050 .050 p = .75 .051 .044 .056 .055 .051 .042 p = .90 .053 .043 .055 .057 .052 .041 p = 1.0 .053 .044 .055 .057 .052 .041 * Each entry i s based on 1500 simulated samples. - 32 -Table V I I I . Simulation Estimates of A c t u a l Levels Corresponding to Two-Sided F i t t e d C r i t i c a l Values i n the Unbalanced * Case, w i t h N = 100, a = .05. U - . 5 | 0 .10 .17 .25 KP p = .25 .053 .052 .043 .049 p = .50 .051 .053 .048 .043 p = .75 .057 .054 .054 .050 p = .90 .054 .057 .052 .049 p = 1.0 .049 .052 .047 .046 MGB p = .25 .048 .048 .054 .061 p = .50 .055 .057 .058 .069 p = .75 .058 .060 .055 .072 p = .90 .049 .057 .057 .072 p = 1.0 .049 .057 .055 .071 Mod MGB p = .25 .049 .048 .059 .059 p = .50 .057 .057 .059 .066 p = .75 .058 .062 .055 .071 p = .90 .049 .058 .059 .072 p = 1.0 .048 .057 .058 .071 Davis p = .25 .055 .055 .047 .052 P = .50 .048 .051 .042 .041 p = .75 .049 .052 .044 .043 p = .90 .049 .051 .044 .043 p = 1.0 .049 .051 .044 .043 Each entry i n the f i r s t column i s based on 1500 simulated samples, each entry i n the other columns i s based on 3000 simulated samples. - 33 -f o r A and 1-A, th e r e f o r e two estimates of the a c t u a l s i g n i f i c a n c e l e v e l s were averaged to obtain the e n t r i e s i n the t a b l e . The observed l e v e l s are reasonably c l o s e to the nominal l e v e l f o r the two-sided KP and Davis t e s t s , w h i l e they tend to be high f o r the two-sided MGB and modified MGB t e s t s , though not as extreme as f o r the one-sided case. From t h i s s i m u l a t i o n experiment, i t appears that the e f f e c t of unequal sample s i z e s on the l e v e l of the t e s t i s reasonably s m a l l f o r the KP and Davis procedures; the f i t t e d . . c r i t i c a l values appear to be u s e f u l f o r the KP procedure provided .33 _ A __ .67, and f o r the Davis procedure provided . 25 <_ A __ .75. The e f f e c t of unequal sample s i z e s on the l e v e l of the t e s t seems q u i t e s i g n i f i c a n t f o r the MGB and modified MGB procedures; p a r t i c u l a r l y i n the one-sided case where the f i t t e d c r i t i c a l values appear to be u s e f u l only f o r A near .5. Sec t i o n 3.5 THE FIXED POINT CENSORING PROCEDURES, WITH EQUAL SAMPLE SIZES Although the comparison among the d i f f e r e n t s e q u e n t i a l procedures i s of primary i n t e r e s t , we w i l l a l s o compare the s e q u e n t i a l procedures w i t h t h e i r f i x e d p o i n t censoring counterparts. The standardized s t a t i s t i c s f o r the f i x e d p o i n t censoring procedures (with censoring p o i n t r) based on the Savage, Wilcoxon and Mantel-Haenszel s t a t i s t i c s are r e s p e c t i v e l y S 5 7 { v a r ( S * | H 0 ) } 1 / 2 (or Mj.) , W*/{var(W*|H 0)} 1 / 2 and M*. Each one of these s t a t i s t i c s i s a s y m p t o t i c a l l y d i s t r i b u t e d as a standard normal random v a r i a b l e . We examined the accuracy of the asymptotic c r i t i c a l values f o r these f i x e d p o i n t censoring procedures i n both the one-sided and two-sided cases, using the. set of simulated samples used to construct Tables I I I and IV. The asymptotic c r i t i c a l values appeared to be. adequate i n the Savage and Wilcoxon cases provided Np > 30, but i n the Mantel-Haenszel case they l e d to t e s t s which r e j e c t e d the n u l l hypothesis too o f t e n ( t h i s l a s t r e s u l t - 34 -agrees w i t h MGB.,. 1978, p, 619). MGB provided f i n i t e sample c o r r e c t i o n s to the asymptotic c r i t i c a l values i n the case p = 1, Assuming that c o r r e c t i o n s should be s i m i l a r f o r the other values of p, we tested'the accuracy, of these f i t t e d c r i t i c a l . v a l u e s using the simulated samples used to construct Table VI. The f i t t e d c r i t i c a l , values seem adequate provided Np i s not too s m a l l . Therefore, i n the Monte Carlo comparison study, asymptotic c r i t i c a l values were used f o r the f i x e d p o i n t censoring procedures based on the Savage and Wilcoxon s t a t i s t i c s , and the c r i t i c a l values provided by MGB were used f o r the f i x e d p o i n t censoring procedure based on the Mantel-Haenszel s t a t i s t i c . - 35 -Chapter 4 COMPARISON OF.PROCEDURES In t h i s chapter, we compare the procedures under c o n s i d e r a t i o n i n terms of both power and expected p r o p o r t i o n sampled. We examine the p r o p e r t i e s of the procedures f o r the case of Lehmann a l t e r n a t i v e s s p e c i f i e d by 1 - F 2 ( t ) = [ l - F 1 ( t ) ] Y f o r 0 < y < 0 0 • The n u l l hypothesis i s now y = 1; the one-sided and two-sided a l t e r n a t i v e s are r e s p e c t i v e l y y > 1 and y ^ 1. A s i m u l a t i o n experiment was c a r r i e d out to i n v e s t i g a t e these p r o p e r t i e s i n the cases m=n = 15, 30, 50. The r e s u l t s f o r the one-sided procedures w i t h a = .05 are summarized i n Table IX; each entry i n the t a b l e i s based on 500 simulated samples (The D-vectors can be simulated q u i t e e a s i l y i n the case of Lehmann a l t e r n a t i v e s , f o r example, see MGB, 1977, p. 624.). Results f o r the dither values of, a were q u a l i t a t i v e l y q u i t e s i m i l a r . The analogous r e s u l t s f o r the two-sided procedures are. presented i n Table X. I t was a n t i c i p a t e d t h a t , f o r l a r g e r sample s i z e s , , the f i x e d p o i n t censoring procedures based on the Savage and Mantel-Haenszel s t a t i s t i c s would . . - . r •' have e s s e n t i a l l y i d e n t i c a l power f u n c t i o n s , s i n c e £ v. i s a c o n s i s t e n t 1=1 1 estimator of var(S r|HQ). I t seems from the tables that t h i s does h o l d , even f o r q u i t e small sample s i z e s . The f i x e d p o i n t censoring procedure based on the Wilcoxon s t a t i s t i c appears to be e s s e n t i a l l y equivalent i n power to the other two f i x e d p o i n t censoring procedures f o r the smaller values of p and l e s s powerful than the other two procedures f o r the l a r g e r values of p. Comparison of the s e q u e n t i a l procedures i s l e s s s t r a i g h t f o r w a r d than comparison of t h e i r f i x e d p o i n t censoring counterparts because both power and expected p r o p o r t i o n sampled must be considered. The tab l e s i n d i c a t e that the - 36 -Table IX. Simulation Results for Operating C h a r a c t e r i s t i c s of the One-sided Tests Under Lehmann Al t e r n a t i v e s , with m = n = N/2, a = .05*. Fixed Point Progresive Censoring Censoring Power Power Exp. Prop. Sampled S** MH W KP MGB MMGB D KP MGB MMGB D N = 30, p = .90 1.00 .07 .06 .05 .05 .06 .06 .05 .89 .87 .87 .88 1.25 .15 .15 .14 .14 .12 .12 .13 .87 .84 .84 .84 1.50 .27 .26 .26 .28 .25 .24 .26 .82 .77 .78 .77 2.00 .55 .54 .48 .52 .45 .42 .46 .74 .68 .69 .68 2. 50 .74 .73 .68 .73 .60 .59 .62 .67 .60 .60 .60 3.00 .88 .87 .79 .84 .74 .72 .72 .61 .53 .53 .54 3.50 .91 .90 .84 .89 .83 .81 .81 .56 .46 .47 .48 N = 30, p = 1.0 1.00 .06 .06 .05 .05 .06. .06 .05 .95 .93 .93 .94 1.25 .16 .16 .14 .13 .12 .12 .13 .93 .90 .90 .90 1.50 .28 .27 .26 .26 .24 .24 .26 .87 .83 .83 .82 2.00 .55 .55 .49 .51 .44 .42 .46 .79 .73 .73 .72 2.50 .75 .75 .68 .71 .60 .58 .62 .71 .63 .63 .63 3.00 .88 .88 .79 .83 .74 .72 .72 .63 .55 .55 .56 3.50 .91 .91. .84 .88 .82 .81 .81 .57 .48 .48 .49 N = 60, p = .50 1.00 .03 .03 .03 .04 .03 .03 .03 .50 .49 .49 .49 1.20 .12 . 12 .13 .14 .10 .10 .12 .48 .47 .47 .48 1.35 .20 .19 .20 .20 .16 .16 .18 .48 .46 .46 .47 1.50 .30 .28 .29 .31 .23 .23 .27 .46 .45 .45 .45 1.75 .38 .37 .37 .38 .30 .30 .34 .45 .43 .43 .44 2.00 .59 .58 .57 .60 .49 .49 .55 .41 .39 .39 .40 2.25 .70 .69 .70 .71 .61 .60 .65 .39 .36 .36 .38 *Each Entry is based on 500 simulated samples. **S = Savage; MH = Mantel-Haenszel; W = Wilcoxon; KP = Kozi o l and Petkau; MGB = Muenz, Green and Byar; MMGB =. modified MGB, D = Davis. - 37 -KP MGB MMGB D KP MGB MMGB D Table IX. (continued) Y S MH W 1.00 .04 .04 .03 1.20 .16 .15 .14 1. 35 .23 .22 .21 1.50 .38 .37 .37 1.75 .54 .53 .49 2.00 .74 .74 .70 2. 25 .83 .82 .78 N = 60, p = .75 .04 .03 .03 .04 .14 .12 .12 .12 .23 .18 .18 .20 .36 .30 .29 .32 .53 .42 .41 .44 . 72 .63 .62 .66 .81 .73 .73 .76 .74 .74 .74 .74 .72 .70 .70 .71 .70 .67 .67 .68 .67 .64 .64 .64 .63 .59 .60 .61 .56 .51 .51 .52 .52 .46 .46 .47 1.00 .05 .05 .03 1.20 .18 .18 .15 1.35 .27 .27 .23 1. 50 .43 .43 .38 1.75 .62 .62 .53 2.00 . 79 .79 .71 2.25 .89 .89 .82 N = 60, p = 90 .04 .04 .03 .04 .18 .13 .12 .13 .27 .19 .18 .20 .41 .32 .31 .33 .62 .46 .46 .46 .78 .68 .67 .68 .87 .79 .78 .77 .89 .88 .88 .88 .86 .83 .83 .84 .83 .80 .80 .81 .79 .75 .75 .75 .72 .68 .68 .69 .64 .57 .57 .57 .57 .50 .50 .51 1.00 .05 .05 .03 1.20 .19 .19 .15 1.35 .29 .29 .22 1.50 .45 .46 .38 1.75 .63 .64 .54 2.00 .81 .82 . 72 2.25 .90 .90 .82 N = 60, p = 1.0 .04 .03 .03 .04 .17 .13 .12 .13 .25 .18 .18 .20 .42 .32 .31 .33 .60 .47 .45 .46 .78 .67 .66 .68 .87 .79 .78 .77 .97 .96 .96 .96 .93 .91 .91 .92 .90 .87 .87 .87 .85 .80 .81 .80 .77 .73 .73 .74 .67 .59 .60 .60 .60 .52 .52 .53 N = 100, p = .25 1.00 .06 .06 .06 .06 .05 .05 .07 .25 .24 .24 .25 1.15 .08 .08 .09 .09 .08 .08 .10 .24 .24 .24 .24 1.25 .15 .14 .15 .14 .10 .10 .13 .24 .24 .24 .24 1. 50 .26 .26 .26 .26 .22 .22 .27 .23 .23 .23 .23 1.75 .37 .36 .38 .37 .28 .28 .37 .23 .22 .22 .23 2.00 .51 .50 .51 .50 .43 .43 .52 .22 .20 .20 .21 - 38 -Table IX. (continued) Y S MH W KP MGB MMGB D KP MGB MMGB D N = 100, p = .50 1.00 .06 .06. .07 .05 .06 .06 .05 .49 .48 .48 .49 1.15 .14 .14 .13 .13 .11 .12 .12 .49 .47 .47 .48 1.25 .20 .20 .20 .19 .16 .16 .18 .47 .46 .46 .47 1.50 .39 .38 .38 .38 .32 .32 .34 .45 .41 .41 .43 1.75 .62 .62 .62 .59 .45 .46 .53 .41 .39 .38 .40 2.00 .81 .81 .80 .79 .66 .66 .75 .37 .33 .33 .35 1.00 .06 .06 .06 1.15 .14 .14 .15 1.25 .26 .26 .24 1.50 .54 .54 .48 1.75 .76 .76 .73 2.00 .89 .89 .87 N = 100, p = .75 .06 .05 .05 .05 .15 .13 .13 .14 .24 .20 .20 .21 .52 .39 .39 .43 .75 .60 .60 .65 .89 .83 .82 .84 .74 .72 .72 .73 .72 .69 .69 .71 .70 .67 .67 .68 .63 .59 .59 .61 .56 .52 .52 .52 .48 .41 .41 .42 1.00 .06 .06 .06 1.15 .15 .15 .14 1.25 .30 .30 .24 1.50 .58 .59 .53 1.75 .84 .84 .77 2.00 .94 .04 .90 N - 100, p = .90 .06 .05 .06 .05 .15 .14 .13 .14 .27 .20 .19 .21 .59 .45 .44 .46 .79 .68 .67 .69 .92 .86 .85 .85 .89 .87 .86 .87 .86 .83 .83 .84 .83 .80 .80 .80 .73 .68 .68 .69 .64 .58 .58 .58 .53 .44 .44 .45 N = 100, p = 1.0 1.00 .05 .05 .07 .06 .05 .05 .05 .98 .95 .95 .96 1.15 .16 .16 . 15 . 15 .13 .13 .14 .94 .91 .91 .91 1.25 .28 .29 .25 .26 .19 .19 .21 .91 .87 .87 .87 1.50 .63 .64 .53 .58 .45 .45 .46 .79 .73 . 73 .74 1.75 .86 .87 .77 .81 .69 .67 .69 .67 .61 .62 .61 2.00 .95 .95 .90 .93 .87 .86 .85 .56 .45 .45 .46 Table X. Simulation Results for Operating Characteristics of the Two-sided Tests Under Lehmann Alternatives, with m =n = N/2, a = .05*. Fixed Point Progressive Censoring Censoring Power Power Exp. Prop. Sampled Y s** MH W KP MGB MMGB D KP MGB MMGB D N = 30, p = .90 1.00 .06 .06 .05 .06 .06 .05 .06 .89 .87 .87 .88 1.25 .09 .09 .08 .09 .07 .06 .07 .88 .87 .87 .87 1.50 .18 .18 .21 .19 .17 .15 .20 .85 .82 .82 .81 2.00 .42 .42 .38 .41 .33 .31 .34 .79 .75 .76 .75 2.50 .59 .59 .53 .56 .47 .45 .49 .73 .69 .69 .69 3.00 .75 .75 .66 .73 .63 .60 .62 .67 .62 .62 .62 3.50 .84 .83 .77 .81 .74 .72 .72 .62 .55 .55 .55 N = 30, p = 1.0 1.00 .05 .05 .05 .06 .06 .05 .06 .95 .94 .94. .94 1.25 .08 .09 .08 .07 .07 .06 .07 .95 .93 .93 .94 1.50 .18 .18 .21 .17 .17 .15 .20 .91 .87 .88 .87 2.00 .41 .42 .38 .39 .33 .29 .34 .84 .80 .81 .80 2.50 .58 .59 .53 .53 .47 .43 .49 .78 .72 .73 .72 3.00 .75 .76 .66 .70 .62 .60 .62 .71 .64 .65 .65 3.50 .83 .84 .77 .80 .73 .70 .72 .64 .57 .58 .57 N = 60, p = .50 1.00 .04 .03 .03 .03 .03 .03 .03 .50 .49 .49 .50 1.20 .08 .07 .07 .08 .06 .06 .07 .49 .49 .49 .49 1.35 .13 .12 .14 .14 .11 .10 .12 .49 .48 .48 .48 1.50 .18 .18 .18 .18 .16 .16 .17 .48 .46 .46 .47 1.75 .28 .26 .26 .28 .20 .20 .24 .47 .46 .46 .47 2.00 .48 .46 .46 .47 .36 .36 .41 .44 .42 .42 .43 2.25 .59 .59 .58 .60 .49 .49 .55 .42 .40 .40 .41 *Each Entry is based on 500 simulated samples. **S = Savage; MH = Mantel-Haenszel; W = Wilcoxon; KP MGB = Muenz, Green and Byar; MMGB = modified MGB, D = Koziol and Petkau; = Davis. - 40 -Table X. (continued) Y S MH W KP MGB MMGB D KP MGB MMGB D N = 60, p = .75 1.00 .05 .05 .04 .05 .04 .04 .04 .74 .74 .74 .74 1.20 .09 .09 .09 .10 .07 .07 .08 .73 .72 .72 .72 1.35 .15 .15 .15 .15 .12 .12 .13 .72 .70 .70 .71 1.50 .28 .27 .24 .28 .22 .22 .23 .70 .67 .67 .68 1. 75 .43 .42 .37 .42 .32 .31 .34 .67 .65 .65 .65 2.00 .62 .61 .59 .61 .51 .50 .53 .61 .57 .57 .59 2.25 .73 .73 .70 .74 .64 .63 .66 .57 .52 .52 .53 1.00 .05 .05 .04 1.20 .11 .11 .10 1.35 .17 .17 .15 1.50 .30 .30 .26 1.75 .48 .49 .38 2.00 .70 .70 .62 2.25 .80 .80 . 72 N = 60, p = .90 .05 .04 .04 .04 .11 .09 .08 .08 .16 .14 .13 .14 .30 .24 .24 .23 .47 .35 .33 .35 .67 .57 .53 .55 .79 .68 .66 .68 .89 .88 .88 .88 .88 .86 .86 .86 .86 .84 .84 .84 .82 .79 .79 .80 .78 .75 .75 .75 .70 .65 .65 .66 .64 .58 .58 .58 1.00 .05 .06 .04 1.20 .12 .12 .09 1.35 .18 .18 .16 1.50 .33 .34 .26 1. 75 .49 .50 .39 2.00 .70 . 72 .62 2.25 .82 .83 .72 N = 60, p = 1.0 .05 .04 .04 .04 .10 .08 .08 .08 .15 .13 .13 .14 .29 .24 .23 .23 .44 .34 .32 .35 .66 .56 .53 .55 .79 .68 .65 .68 .97 .96 .96 .96 .95 . 94 .94 .94 .94 .91 .91 .91 .90 .85 .86 .87 .84 .81 .81 .81 .74 .69 .69 .70 .68 .61 .61 .61 N = 100, p = .25 1.00 .06 .06 .06 .05 .05 .05 .06 .25 .24 .24 .25 1.15 .07 .07 .07 . .06 .05 .05 .06 .25 .24 .24 .25 1.25 .11 .11 .09 .08 .07 .07 .09 .25 .24 .24 .24 1.50 .20 .20 .18 .18 .14 .14 .20 .24 .23 .23 .24 1.75 .30 .29 .27 .24 .18 .19 .25 .24 .23 .23 .24 2.00 .42 .42 .40 .35 .29 .29 .37 .23 .22 .22 .23 - 41 -Table X. (continued) Y S MH W KP MGB MMGB D KP MGB MMGB D N = 100, P = .50 1.00 .03 .03 .04 .04 .05 .05 .04 .50 .49 .49 .49 1.15 .08 .08 .08 .07 .06 .07 .07 .49 .48 .48 .49 1.25 .12 .12 .12 .13 .11 .11 .11 .48 .47 .47 .48 1.50 .28 .27 .26 .27 .22 .22 .24 .47 .44 .44 .46 1.75 .46 .46 .45 .44 .34 .34 .40 .44 .42 .42 .44 2.00 .70 .70 .68 .69 .55 .55 .62 .40 .38 .37 .39 N = 100, P = .75 1.00 .05 .05 .05 .05 .05 .05 .04 .74 .73 .73 .74 1.15 .10 .10 .09 .10 .08 .08 .08 .73 .72 .72 .73 1.25 .18 .18 .17 .17 .13 .13 .14 .72 .70 .70 .71 1.50 .41 .41 .38 .39 .30 .30 .32 .67 .63 .63 .65 1.75 .66 .66 .59 .64 .50 .50 .54 .61 .57 .57 .59 2.00 .85 .84 .81 .83 .74 .73 .77 .53 .47 .47 .49 N = 100, P = .90 1.00 .04 .05 .05 .04 .05 .05 .04 .89 .87 .87 .88 1.15 .09 .10 .09 .09 .09 .09 .08 .88 .86 .86 .87 1.25 .19 .19 .18 .19 .15 .15 .15 .86 .83 .83 .84 1.50 .48 .48 .41 .47 .35 .34 .34 .79 .74 .74 .76 1.75 .72 .73 .65 .71 .57 .56 .55 .70 .65 .66 .66 2.00 .89 .90 .83 .89 .80 .80 .80 .60 .52 .52 .53 N = 100, P = 1.0 1.00 .04 .04 .04 .04 .05 .05 .04 .98 .96 .96 .97 1.15 .09 .10 .09 .09 .09 .09 .09 .97 .94 .94 .95 1.25 .20 .20 .18 .19 .15 .15 .15 .94 .91 .91 .91 1.50 .49 .51 .41 .46 .35 .34 .34 .85 .80 .81 .81 1.75 .75 .77 .65 - .72 .58 .56 .56 .74 .70 .70 .70 2.00 .91 .91 .84 .89 .80 .79 .80 .63 .54 .54 .55 - 42 -MGB and modified MGB procedures are e s s e n t i a l l y equivalent and that the Davis procedure tends to dominate both of them. The three procedures are equivalent i n power near p = 1, but. the Davis, procedure i s more powerful than the MGB and modified MGB procedures, when p i s removed, from 1. Although the MGB procedure appears to have a s l i g h t advantage i n terms.of expected p r o p o r t i o n sampled, the d i f f e r e n c e s with the modified MGB and Davis procedures.are n e g l i g i b l e . Thus, on the b a s i s of the cases considered, one might p r e f e r the Davis pro-cedure to the MGB and modified MGB procedures. The KP procedure i s more powerful than the Davis procedure; the d i f f e r e n c e s i n power being l a r g e r i n the cases where p i s c l o s e to 1. On the other hand, the Davis procedure has the advantage i n terms of expected p r o p o r t i o n sampled, and t h i s advantage i s most pronounced i n those cases where the KP procedure has the grea t e s t advantage i n power. In choosing between these two s e q u e n t i a l procedures i n any p r a c t i c a l s i t u a t i o n , the r e l a t i v e importance of power and expected p r o p o r t i o n sampled, should be taken i n t o c o n s i d e r a t i o n . I t i s a l s o of i n t e r e s t to compare the s e q u e n t i a l procedures to t h e i r f i x e d p o i n t censoring counterparts; the s e q u e n t i a l procedures are, of course, l e s s powerful. The t a b l e s i n d i c a t e that the MGB procedure s u f f e r s the most l o s s i n power r e l a t i v e to i t s f i x e d p o i n t censoring counterpart. One might not be w i l l i n g to t o l e r a t e such lo s s e s i n power f o r gains i n expected p r o p o r t i o n sampled. The l o s s e s i n power f o r the Davis procedure are more reasonable, w h i l e f o r the KP procedure the los s e s i n power are q u i t e s m a l l . Of course, a l l the s e q u e n t i a l procedures o f f e r s u b s t a n t i a l gains i n the expected p r o p o r t i o n sampled., p a r t i c u l a r l y f o r the case i n which there are la r g e d i f f e r e n c e s between the two treatments under c o n s i d e r a t i o n . For the s p e c i a l case of contiguous exponential s c a l e s h i f t a l t e r n a t i v e s , K o z i o l and Petkau (1978; see a l s o Lesser, 1979) were able to determine the - 43 -asymptotic properties, of both the f i x e d p o i n t and pr o g r e s s i v e censoring procedures based on the Savage s t a t i s t i c . While the asymptotic r e s u l t s do not provide accurate approximations f o r the sample s i z e s under considera-t i o n , the conclusions to which one i s l e d on the b a s i s of the asymptotics are i n f a c t v e r i f i e d by the Monte Ca r l o r e s u l t s ; that i s , f o r the procedures based on .the Savage s t a t i s t i c , w h i l e the power of the s e q u e n t i a l procedure i s only s l i g h t l y s m a ller than that of the f i x e d p o i n t censoring procedure, s u b s t a n t i a l reductions i n the expected p r o p o r t i o n sampled can r e s u l t from the use of the s e q u e n t i a l procedure. In summary, i t appears t h a t , at l e a s t i n the case of Lehmann a l t e r -n a t i v e s , the choice i s narrowed to the f i x e d p o i n t censoring procedure based on the Savage s t a t i s t i c (or the equivalent procedure based on the Mantel-Haenszel s t a t i s t i c ) and the s e q u e n t i a l procedures based on the Savage and Wilcoxon s t a t i s t i c s . I f the s i t u a t i o n i s such that maximal power i s e s s e n t i a l , one would use the f i x e d p o i n t censoring procedure. I f one i s w i l l i n g to e n t e r t a i n minimal l o s s e s i n power i n exchange f o r s u b s t a n t i a l r eduction i n expected proportion; sampled, one would use the KP procedure. Somewhat marginal f u r t h e r gains i n expected p r o p o r t i o n sampled can be achieved at the expense of l o s s e s i n power by the use of the Davis procedure. - 44 -Chapter 5 EARLY DECISION RULES The r e s u l t s i n Chapter 4 show that use of the s e q u e n t i a l procedures can r e s u l t i n s u b s t a n t i a l time savings. Further time savings can be achieved through the use of e a r l y d e c i s i o n r u l e s ; t h i s w i l l be i n v e s t i g a t e d i n t h i s chapter. By an e a r l y d e c i s i o n r u l e we mean a r u l e f o r terminating the experiment which has the property that i f the experiment had been continued (to the l i m i t s imposed by the de s i g n ) , the decison made would have been no d i f f e r e n t . While such r u l e s can be expected to be most e f f e c t i v e when used i n conjunction w i t h f i x e d point censoring procedures, they can a l s o be expected to improve the behaviour of the s e q u e n t i a l pro-cedures i n those instances where one would u s u a l l y be l e d to accept the n u l l hypothesis. Consider the modified form of a l i n e a r rank s t a t i s t i c based on the non-decreasing scores { a ^ ( i ) } and defined by * k T k = I d i a N ( i ) + fa-VV10 ( 5- 1 } i = l k _ N where ro^ = £ d^ and a^(k) = £ a N ( j ) / ( N - k ) . A l s o l e t n^ = k-m^. Note i = l j=k+l that T>T , = T„ = T„, the usual form of the l i n e a r rank s t a t i s t i c . Now, f o r N-1 N N k < N-1, Tk+1 = \ " I d k + 1 - (m-m k)/(.N-k)][a N(k+l) - a ^ k + D J . Since a N ( k + l ) - a^k+1) >_ 0, upper and lower bounds f o r T^^. i n terms of Tfc are obtained by s e t t i n g d v + 1 equal to 0 and 1 r e s p e c t i v e l y i n t h i s expression. - 45 -Noting f u r t h e r that the value of no longer changes (with k) a f t e r a l l the order s t a t i s t i c s corresponding to e i t h e r group have been observed, i t i s easy to determine upper and lower bounds on the p o s s i b l e values of T (r > k) i n terms of m^ and T . In p a r t i c u l a r L (k) < T* < U (k) r — r — r where U r(k) = T* + (m-n^) [ ^ ( r ) - ^ ( k ) ] i f n-n k > r-k ft = + (m - m^ ) [ a N ( n + m^ ) - a^Ck) ] i f n-n^ < r-k and - i -L (k) = \ ~ ( n - n k ) [ a N ( r ) - a ^ k ) ] i f m-rn^ _ r-k ft = T k - (n - nfc) [a N(m + n k) - a N ( k ) ] i f m-rn^ < r-k. These bounds can now be used to de f i n e e a r l y d e c i s i o n r u l e s to be used i n conjunction w i t h both the f i x e d p o i n t and progressive censoring procedures ft based on { a ^ ( i ) } . In p a r t i c u l a r , f o r the one-sided t e s t based on T , the e a r l y d e c i s i o n r u l e i s as f o l l o w s . For each k, 1 < k < r , compute L^(k) and U_(k) . Stop and accept at the f i r s t k f o r which U r(k) i s l e s s than or equal to the c r i t i c a l value f o r the t e s t based on T^; stop and r e j e c t a t the f i r s t k f o r which L^(k) i s greater than t h i s same c r i t i c a l value. ft S i m i l a r l y , f o r the one-sided t e s t based on max T , the e a r l y d e c i s i o n r u l e l<k<r i s as f o l l o w s . For each k, 1 <_ k <_ r , compute t l _ ( k ) . Stop and accept at the f i r s t k f o r which. U (k) i s l e s s than or equal to the c r i t i c a l value f o r * ft the t e s t based on max T ; stop and r e j e c t H at the f i r s t k f o r which T l<k<r - 46 -exceeds t h i s same c r i t i c a l v a l u e , The e a r l y d e c i s i o n r u l e s are defined s i m i l a r l y i n the two-sided case. The e a r l y d e c i s i o n r u l e based on | T | i s to accept HQ a t the f i r s t k f o r which max {|L (k) | , [ U r (k) | } i s l e s s than or equal to the c r i t i c a l value f o r the t e s t based on | T^| , and to r e j e c t H^ at the f i r s t k f o r which e i t h e r .L(.k) or -U^Ck) i s greater than t h i s c r i t i c a l ft value. The e a r l y d e c i s i o n r u l e based on max I T, I i s to accept a t the -11 k 0 l<_k_r f i r s t k f o r which max { (k)| , | ( k ) | } i s l e s s than or.equal to the c r i t i c a l ft value f o r the t e s t based on max |.T | , and to r e j e c t at the f i r s t k f o r l<k<;r i * i which | T^| exceeds t h i s same c r i t i c a l v alue. Since both the Savage and Wilcoxon s t a t i s t i c s are of form (5.1), the above serves to define e a r l y d e c i s i o n r u l e s f o r the f i x e d p o i n t and pro g r e s s i v e censoring procedures based on these s t a t i s t i c s . On the other hand, the Mantel-Haenszel s t a t i s t i c s ft M^ are not of form (5.1). I t appears, however, that ( f o r 1 < r < N) ft M* * * M'-h. > M i f d , =0 and M < M i f d , =1. I f t h i s were the case, r+1 — r r+1 r+1 — r r+1 e a r l y d e c i s i o n r u l e s of the general form above could be defined f o r the f i x e d point and pro g r e s s i v e censoring procedures based on the Mantel-Haenszel ft s t a t i s t i c s . - The appropriate upper bound, U^(k), would be the value M^ would achieve based on the D-sequence which has been observed up to and i n c l u d i n g the kth order s t a t i s t i c and extended by adding f i r s t a l l the remaining O's followed by the remaining l ' s . The appropriate lower bound, ' L (k),-would be determined s i m i l a r l y w i t h the l ' s added before the O's. MGB, i n f a c t , use these bounds i n the case r = N f o r t h e i r e a r l y d e c i s i o n r u l e s . For the modified MGB s t a t i s t i c s , M^, i t appears that > M^ . i f = 0 and M' .... < M' i f d ,.=1, provided not a l l of the order s t a t i s t i c s have been r+1 — r r+1 r observed, i n any one of the two groups. I f so, bounds U_;(.k) and l ^ O O can be found, and e a r l y d e c i s i o n r u l e s e x i s t f o r the modified MGB procedure. - 47 -In order to examine the gains provided by these e a r l y d e c i s i o n r u l e s , the cases (with p =1) d i s p l a y e d i n Tables IX and X were redone (using the same simulations) f o r the procedures based on the Savage, Wilcoxon and Mantel-Haenszel s t a t i s t i c s . The r e s u l t i n g values of the expected p r o p o r t i o n s sampled are summarized i n Table XI and X I I . I t i s c l e a r that w h i l e s u b s t a n t i a l gains are p o s s i b l e f o r the f i x e d point censoring procedures, most of the f u r t h e r gains f o r the s e q u e n t i a l procedures are made i n the neighbourhood of the n u l l hypothesis. Of course, any savings i n time which can be achieved at no expense i n power are noteworthy. I f one were planning to use one of these f i x e d point or progressive censoring procedures, the e a r l y d e c i s i o n r u l e s should be employed. - 48 -Table XI. Expected Proportions Sampled Using E a r l y Decision Rules i n the One-Sided Case, with a = .05, p = 1*. Fixed Point Censoring Progressive Censoring Y S _ MH W KP MGB D N = 30 1.00 .59 .61 .45 .55 .47 .40 1.25 .67 .68 .52 .61 .52 .45 1.50 .71 .72 .55 .62 .51 .45 2.00 .77 .77 .61 .64 .52 .47 2.50 .77 .78 .61 .62 .49 .47 3.00 .76 .76 .60 .58 .47 .45 3.50 .72 .72 .58 .54 .42 .42 N = 60 1.00 .70 .71 .53 .66 .59 .48 1.20 .79 .80 .61 .71 .63 .53 1.35 .80 .81 .63 .71 .61 .53 1.50 .84 .84 .66 .72 .61 .53 1.75 .85 .85 .68 .69 .60 .54 2.00 .83 .83 .66 .63 .52 .48 2.25 .81 .80 .64 .58 .47 .45 N = 100 1.00 .78 .78 .59 .73 .65 .53 1.15 .82 .83 .65 .75 .67 .56 1.25 .86 .86 .68 .77 .67 .57 1.50 .88 .89 .72 .72 .63 .56 1.75 .86 .86 .70 .65 .56 .51 2.00 .82 .82 .65 .55 .43 .42 * Each entry : is based on 500 simulated samples. ** S = Savage, MH = Mantel -Haenszel, W ! = Wilcoxon > KP = Ko z i o l and Petkau, MGB = Muenz, Green and Byar, D = Davis. - 49 -Table XII. Expected Proportions Sampled Using E a r l y Decision Rules i n the Two-Sided Case, with a = .05, p = 1*. Y Fixed Point Censoring W S** MH 1.00 .64 .66 .52 1.25 .65 .67 .53 1.50 .69 .71 .56 2.00 .73 .74 .59 2.50 .76 .76 .62 3.00 .76 .76 .61 3.50 .73 .73 .59 Progressive Censoring N = 30 KP MGB D .60 .54 .47 .61 .54 .47 .63 .54 .48 .64 .55 .49 .64 .54 .50 .63 .52 .48 .59 .48 .46 N = 60 1.00 .75 .76 .58 .71 .64 .53 1.20 .77 .78 .60 .72 .64 .54 1.35 .79 .80 .62 .73 .64 .54 1.50 .81 .82 .65 .73 .63 .55 1.75 .84 .84 .67 .72 .63 .56 2.00 .84 .84 .67 .68 .58 .53 2.25 .82 .82 .65 .64 .53 .50 N = 100 1.00 .81 .83 .64 .77 .70 .59 1.15 .82 .83 .66 .77 .70 .60 1.25 .84 .85 .67 .78 .69 .59 1.50 .88 .88 .71 .76 .67 .59 1.75 .88 .87 .72 .70 .62 .56 2.00 .84 .83 .68 .61 .51 .49 * Each entry i s based on 500 simulated samples. ** S = Savage, MH = Mantel-Haenszel, W = Wilcoxon, KP = Koziol and Petkau, MGB = Muenz, Green and Byar, D = Davis. - 50 -Chapter 6 EXAMPLES Sec t i o n 6.1 THE FIRST EXAMPLE MGB consider a . h y p o t h e t i c a l t r i a l to i n v e s t i g a t e the e f f e c t on s u r v i v a l of a new adjuvant therapy f o r lung cancer, the therapy to be administered at the time of i n i t i a l treatment. Thirty-two p a t i e n t s are i n i t i a l l y e n r o l l e d i n the study, 16 i n the c o n t r o l group, (standard therapy) and 16 i n the t e s t group (standard plus adjuvant). The D-vector corresponding to the ranked lengths of s u r v i v a l i s (00000100010001101111111 010111001), where. 1 denotes a p a t i e n t from the c o n t r o l group, and 0 denotes a p a t i e n t from the t e s t group. We wish to t e s t the e q u a l i t y of the under- -l y i n g s u r v i v a l d i s t r i b u t i o n s against the two-sided a l t e r n a t i v e , using the various procedures discussed p r e v i o u s l y . The s t a t i s t i c s S*'/{ var ( S j H )} 1 / 2 , W*/{ var (W | H )} 1 / 2 , * £ and M^ (1 <_ k <_ 32) are tabulated i n Table X I I I . The values of the t e s t s t a t i s t i c s i n the uncensored case are | S N | / { v a r ( S N | H 0 ) } 1 / 2 = 1.502, | wj/{var(W N| H 0 } 1 / 2 = 2.337 and = 1.514; the c r i t i c a l values at the a = .05 l e v e l are approximately 1.96, 1.96 and 2.06 r e s p e c t i v e l y . From these values, one can see that the Wilcoxon s t a t i s t i c i s more s e n s i t i v e to departures from the n u l l hypothesis evidenced i n the data than e i t h e r the Savage or Mantel-Haenszel s t a t i s t i c s ; only the t e s t based.on the Wilcoxon s t a t i s t i c w i l l r e j e c t H„ at the a = .05 l e v e l . J 0 However, each of the s e q u e n t i a l procedures (with p= 1) w i l l r e j e c t H^ at the a = :.05 l e v e l . The c r i t i c a l values f o r the KP, Davis, MGB and modified MGB t e s t s t a t i s t i c s are approximately 2,07, 2.12, 2,60 and 2.60 r e s p e c t i v e l y . Thus, the KP procedure w i l l r e j e c t H n at the 13th observation and the Davis, - 51 -b l e XIII k S t a t i s t i c s f o r d k the F i r s t K Example K * M; k I 0 0.186 0.302 1.000 1.000 2 0 0.378 0.603 1.437 1.437 3 0 0.577 0.905 1. 791 1.790 4 0 0.782 1.206 2.105 2.103 5 0 0.995 1.508 2.399 2.393 6 1 0.844 1.300 1.861 1.852 7 0 1.058 1.583 2.163 2.152 8 0 1.282 1.866 2.455 2.439 9 0 1.515 2.148 2.742 2.718 10 1 1.385 1.998 2.387 2.359 11 0 1.622 2.261 2.671 2.635 12 0 1.870 2.525 2.959 2.910 13 0 2.131 2.789 3.255 3.187 14 1 2.033 2.695 3.012 2.932 15 1 1.930 2.601 2.775 2.690 16 0 2.192 2.827 3.063 2.961 17 1 2.099 2.751 2.862 2.752 18 1 2.000 2.676 2.660 2.550 19 1 1.894 2.601 2.458 2.351 20 1 1.779 2.525 2.254 2.155 21 1 1.655 2.450 2.046 1.958 22 1 1.519 2.374 1.834 1.758 23 1 1.371 2.299 1.616 1.553 24 0 1.577 2.393 1.817 1.752 25 1 1.438 2.337 1.622 1.567 26 0 1.651 2.412 1.823 1.767 27 1 1.526 2.374 1.655 1.607 28 1 1.377 2.337 1.465 1.427 29 1 1.191 2.299 1.243 1.217 30 0 1.315 2.318 1.350 1.327 31 0 1.502 2.337 1.514 1.502 32 1 1.502 2.337 1.514 1.502 Note'? S i = S * / { v a r(S N| v>1/2  K - V { var(W N|H 0)} 1/2 - 52 -MGB and modified MGB procedures w i l l r e j e c t H^ at the 9th observation. In t h i s example, the d e c i s i o n s r e s u l t i n g from the KP and MGB procedures are d i f f e r e n t (at the a =,05 l e v e l ) from those r e s u l t i n g from t h e i r f i x e d p o i n t censoring counterparts. There i s t h i s r i s k i nvolved i n one wishes to make a d e c i s i o n w i t h incomplete s u r v i v a l , i n f o r m a t i o n . I t i s not s u r p r i s i n g that the Davis procedure r e j e c t s e a r l i e r than the KP procedure, given the r e l a t i v e performances of the Wilcoxon and Savage s t a t i s t i c s w i t h the o v e r a l l s u r v i v a l data. That the MGB procedure r e j e c t s HQ before the KP procedure i s to be expected here, given the nature of the two procedures (see KP, 1978, p. 622) . Consider the scheme of monitoring the course of the t r i a l by p l o t t i n g the p o i n t s (k, S^) s e q u e n t i a l l y and connecting by s t r a i g h t l i n e s . Under H ,• the r e s u l t i n g curve w i l l vary about the x - a x i s . The MGB and KP procedures impose square root and h o r i z o n t a l boundaries r e s p e c t i v e l y on t h i s graph; would be r e j e c t e d (and the t r i a l would cease) were the curve to cross the appropriate boundary. In t h i s example, the c o n s i s t e n t p a t t e r n of e a r l y deaths from the t e s t group w i l l lead to e a r l i e r r e j e c t i o n of by the MGB procedure than the KP procedure, s i n c e the square r o o t boundary i s i n i t i a l l y w i t h i n the h o r i z o n t a l boundary. Secti o n 6.2 THE SECOND EXAMPLE Ma r t i n (.1946) measured the length of s u r v i v a l of mice i n f e c t e d w i t h t u b e r c l e b a c i l l i ; h i s experimental r e s u l t s ( M o s t e l l e r and. Tukey, 1977, p. 506) are given as f o l l o w s i n Table XIV. The t o t a l sample s i z e i s 362; the s i z e of group A i s 226, the s i z e of group B i s 138. - 53 -Table XIV Mouse S u r v i v a l Data S u r v i v a l ( i n days a f t e r i n o c u l a t i o n ) Group 14+ 15+ 16+ 17+ 18+ 19+ 20+ 21+ 22+ A 7 6 12 23. 23 43 37 23 17 B 1 0 0 3 2 7 12 14 16 23+ 24+ 25+ 26+ 27+ 28+ 29+ 30+ __31 A 6 7 8 5 1 2 1 0 3 B 16 24 12 4 0 0 4 4 19 We wish to t e s t the e q u a l i t y of the underlying s u r v i v a l d i s t r i b u t i o n s of the two groups against the two-sided a l t e r n a t i v e , using the v a r i o u s procedures discussed p r e v i o u s l y . Randomization was used to a s s i g n ranks to t i e d observations; t h i s technique precludes any changes i n the l e v e l s of s i g n i f i c a n c e of the various procedures a t t r i b u t a b l e to the d i s c r e t e n e s s of the data (Hajek and Sidak, 1967, S e c t i o n I I I . 8 . 1 ) . The values of the t e s t s t a t i s t i c s i n the uncensored case are | S N | / { v a r ( S N | H 0 ) } 1 / 2 = 8.892, | wj / { v a r ^ y } 1 7 2 = 9.791 and \^\ = 9.321. Because these values are so h i g h l y s i g n i f i c a n t (P , < .001), one can conclude obs that the two underlying s u r v i v a l d i s t r i b u t i o n s are not i d e n t i c a l . The s e q u e n t i a l procedures (with p =1) would have terminated the study q u i t e e a r l y (at the a = .05 l e v e l ) ; the KP procedures would have r e j e c t e d at the 60th obs e r v a t i o n , the Davis procedure at the 37th o b s e r v a t i o n , and the MGB and modified MGB procedures at the 19th observation. In terms of time, the procedures would r e j e c t between the 16th and 18th days ( i n c l u s i v e ) of the study; the time saving afforded by the s e q u e n t i a l procedures i s s u b s t a n t i a l . - 54 -Remark. In these two examples, the use. of the e a r l y d e c i s i o n r u l e s described i n Chapter 5 would not a f f e c t the s e q u e n t i a l procedures (at the a = .05 l e v e l ) . Use of the e a r l y d e c i s i o n r u l e s (at the a = .05 l e v e l ) would r e s u l t i n e a r l y d e c i s i o n s f o r the uncensored t e s t s . In the f i r s t example, the t e s t s based on the Savage and Mantel-Haenszel s t a t i s t i c s would accept H^ at the 28th observation and the t e s t based on the Wilcoxon s t a t i s t i c would r e j e c t H "at .the 24th observation.. In the second example, the t e s t s based on the Savage, Wilcoxon and Mantel-Haenszel s t a t i s t i c s would r e j e c t H n at the 219th, 151st and 216th observations r e s p e c t i v e l y . - 55 -Chapter 7 DISCUSSION The s i m u l a t i o n s t u d i e s show that of the procedures considered i n t h i s t h e s i s , the "best" procedures (under Lehmann a l t e r n a t i v e s ) are the f i x e d point censoring procedure based on the Savage s t a t i s t i c , the s e q u e n t i a l KP procedure and the s e q u e n t i a l Davis procedure. I f power against Lehmann a l t e r n a t i v e s i s of predominant concern, the. f i x e d p o i n t censoring procedure based on the Savage s t a t i s t i c i s to be p r e f e r r e d . I f the expected p r o p o r t i o n sampled i s of major concern, the Davis procedure i s to be p r e f e r r e d . A good compromise i s the KP procedure, which loses l i t t l e power w i t h respect to.the f i x e d point censoring procedure and i s only m a r g i n a l l y worse i n terms of expected p r o p o r t i o n sampled w i t h respect to the Davis procedure. In a c l i n i c a l t r i a l s e t t i n g , where the d e c i s i o n to be made w i l l a f f e c t f u t u r e p a t i e n t s , i t would appear that power i s of primary importance, and one would be w i l l i n g to s a c r i f i c e s m a l l l o s s e s i n power only f o r s u b s t a n t i a l gains i n the expected p r o p o r t i o n sampled. From t h i s p o i n t of view, one might p r e f e r the KP procedure. I f one decided to use the KP procedure, the c r i t i c a l values presented i n Table V should be used, as use of the asymptotic c r i t i c a l values f o r the KP s t a t i s t i c s r e s u l t i n somewhat conservative t e s t s (at the s i g n i f i c a n c e l e v e l s of u s u a l i n t e r e s t ) , even f o r moderate-sized samples. The other two s e q u e n t i a l procedures under c o n s i d e r a t i o n , the MGB and modified MGB procedures, tend to be dominated by the Davis procedure, the Davis procedure i s generally, more powerful and i s n e g l i g i b l y d i f f e r e n t i n terms of expected p r o p o r t i o n sampled (with the e a r l y d e c i s i o n r u l e s , the Davis procedure appears to be s u p e r i o r i n terms of expected p r o p o r t i o n sampled, see Tables XI and X I I ) . The MGB and modified MGB procedures a l s o have the - 56 -disadvantages of having no asymptotic theory and having c r i t i c a l values which change considerably (at l e a s t f o r the one-sided t e s t ) as X = m/N v a r i e s from 1/2). For any'of the suggested procedures above, the e a r l y d e c i s i o n r u l e s discussed 1 i n Chapter 5 can be p r o f i t a b l y i n corporated. There are gains i n expected p r o p o r t i o n sampled, together w i t h no l o s s of power. Some a d d i t i o n a l computational labour i s required f o r i t s implementation, however. A l s o , f o r the s e q u e n t i a l procedures, e s s e n t i a l l y no gains i n power accrue as p increases from .9 to 1; e x p l o i t a t i o n of t h i s observation would lead to time savings w i t h e s s e n t i a l l y no attendant l o s s of power. I f an experiment were to be truncated at a. p a r t i c u l a r p r e s p e c i f i e d p o i n t i n time T, rather than at an order s t a t i s t i c , the r e s u l t s i n t h i s t h e s i s s t i l l h o ld; " r our of N" censoring i s a s y m p t o t i c a l l y equivalent (under H^) to t r u n c a t i o n at T, when F~ 1(r/N) + T as N -»- °° ( c f . C h a t t e r j e e and Sen, 1973). Further i n f o r m a t i o n on the a n a l y s i s of s u r v i v a l data i n c l i n i c a l t r i a l s can be found i n Armitage (1975) and Peto et a l (1976, 1977). - 57 -BIBLIOGRAPHY Armitage, P. (1975). Sequential Medical T r i a l s . 2nd Ed. John Wiley, New York. B i c k e l , P.J. and van Zwet, W.R. (1978). Asymptotic expansions f o r the power of d i s t r i b u t i o n - f r e e t e s t s i n the two-sample problem. Ann. S t a t i s t . 6, 937-1004. Cha t t e r j e e , S.K. and Sen, P.K. (1973). Nonparametric t e s t i n g under progressive censorship. C a l . S t a t i s t . B u l l . 22, 13-50. Davis, C.E. (.1978). A two-sample Wilcoxon. t e x t f o r p r o g r e s s i v e l y censored data. Commun. S t a t i s t . Theor. Meth. A7(4), 389-98. Gastwirth, J.L. (1965) . A s y m p t o t i c a l l y most powerful rank t e s t s f o r the two-sample problem w i t h censored data. Ann. Math. S t a t i s t . 36, 1243-7. Hajek, J . and Sidak, Z. (1967). Theory of. Rank Tests. Academic Press, New York. H a l p e r i n , M. and Ware, J . (1974). E a r l y d e c i s i o n i n a censored Wilcoxon two-sample t e s t f o r accumulating s u r v i v a l data. J . Amer. S t a t i s t . Assoc. 69, 414-22. Johnson, R.A. and Mehrotra, K.G. (1972). L o c a l l y most powerful rank t e s t s f o r the two-sample problem w i t h censored data. Ann. Math. S t a t i s t . 43, 823-31. K o z i o l , J.A. and Petkau, A.J. (1978). Sequential t e s t i n g of the e q u a l i t y of two s u r v i v a l d i s t r i b u t i o n s using the modified Savage s t a t i s t i c . B iometrika 65, 615-23. Lesser, M.L. (1979). Two-sample nonparametric s e q u e n t i a l l i f e t e s t i n g using the p r o g r e s s i v e l y censored Savage s t a t i s t i c . Unpublished Ph.D. d i s s e r t a t i o n , Rutgers U n i v e r s i t y . Mantel, N. (1966). E v a l u a t i o n of s u r v i v a l data and two new rank order s t a t i s t i c s a r i s i n g i n i t s c o n s i d e r a t i o n . Cancer Chemotherapy Reports 50, 163-170. Mantel, N. and Haenszel, W. (1959). S t a t i s t i c a l aspects of the a n a l y s i s of data from r e t r o s p e c t i v e s t u d i e s of disease. J o u r n a l of the N a t i o n a l Cancer I n s t i t u t e 22, 719-48. M a r t i n , A.R. (1946), The use of mice i n the examination of drugs f o r chemotherapeutic a c t i v i t y against mycobacterium t u b e r c u l o s i s . J , P a t h o l . B a c t e r i d . 58, 580-5. Muenz, L.R., Green, S.B. and Byar, D.P, (.1977), A p p l i c a t i o n s of the Mantel-Haenszel s t a t i s t i c to the comparison of s u r v i v a l d i s t r i b u t i o n s . B i ometrics 33, 617-26. - 58 -M o s t e l l e r , F.-and Tukey, J.W. (.1977). Data A n a l y s i s and Regression. Addison-Wesley, Reading, Pa. Peto, R., P i k e , M.C., Armitage, P., Breslow, N.E., Cox, D.R., Howard, S.V., Mantel, N. , McPherson, K., Peto, J . , and Smith., P.G. (1976,77) . Design and a n a l y s i s of randomized c l i n i c a l t r i a l s r e q u i r i n g prolonged observation of each p a t i e n t I , I I . Br. J . Cancer 34, 585-612; 35, 1-39. Robinson, J . (1978). An asymptotic expansion f o r samples from a f i n i t e p o p u l a t i o n . Ann. S t a t i s t . 6, 1005-11. Savage, I.R. (1956). C o n t r i b u t i o n s to the theory of rank order s t a t i s t i c s : the two-sample case. Ann. Math. S t a t i s t . 27, 590-615. Wilcoxon, F. (1945). I n d i v i d u a l comparisons by ranking methods. Biometrics B u l l . 1, 80-83. 

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