Nonparametric bioequivalence tests for statistical ... - CiteSeerX

Nonparametric bioequivalence tests for statistical functionals and their ecient power functions Arnold Janssen

Abstract. The population bioequivalence of two measurements is considered via

dierentiable statistical functionals. This approach leads to ecient nonparametric bioequivalence tests given by the canonical gradient of the functional. The results are based on an asymptotic comparison of nonparametric power functions of rank tests. The bioequivalence regions are determined by implicit alternatives speci ed by the functional. They only depend on the functional and their ecient tests but not on any prior information concerning parametric submodels. Beyond our asymptotic solution of the bioequivalence testing problem also a nonparametric nite sample size solution is discussed when the power function can exactly be computed for a family of Lehmann's alternatives. It is shown that exact semiparametric solutions can serve as asymptotically nonparametric solutions. Special attention is devoted to the Wilcoxon functional P (X < Y ), the mean of cumulative hazards, and to the median functional which lead to the Wilcoxon test, the Savage test, and the median rank test, respectively.

AMS 1991 subject classi cations: 62G10, 62G15 Key words and phrases: Bioequivalence, population equivalence, equivalence test, testing statistical functionals, canoncial gradient, Wilcoxon test, Savage test, median test

1. Introduction

Bioequivalence tests for two sample trials are used to ensure that the distribution of two measurements X and Y are close together or at least that relevant parameters are close together. This is called population bioequivalence. A proposal for a nonparametric statistical de nition of bioequivalence was given by Wellek (1993, 1996) and Munk (1996) for the Wilcoxon functional P (X < Y ). Klinger (1995) obtained an exact nite sample solution for this functional under Lehmann's alternatives. In the present paper we will consider two-sided tests and their power functions for nonparametric functionals. Based on a bioequivalence criterion given by functionals we will establish a general class of asymptotically distribution free bioequivalence tests. In connection with applications and the methodology of the bioequivalence problem we refer to the survey article of Berger and Hsu (1996) and to the introduction of Munk (1996) who summarized the recent literature. Since it is hopeless to give an exact nonparametric solution at nite sample size we propose asymptotic solutions. The approach relies on an asymptotic approximation of the underlying model and their functionals. It goes back to the ideas and results of Pfanzagl and Wefelmeyer (1982, 1985) and Janssen (1997) for testing statistical functionals which also apply to the present problem. The key of this work is a linearization of the functional given by its canonical gradient, see Bickel et al. (1993) for applications in estimation theory. The second tool is a central limit theorem for statistical models labeled as local asymptotic normality (LAN) which is used to calculate power functions under local alternatives. The results of this paper can be summarized as follows. It is shown which type of rank tests belongs to a given functional. In addition their asymptotic power function is derived for implicit alternatives given by the functional. As application we get a principle for the construction and the comparison of bioequivalence tests. In a rst step the bioequivalence problem is solved for implicit alternatives along parametric submodels called bres. One obtains bioequivalence regions and 1

their tests which become asymptotically independent of the special parametric submodel. This approach establishes nonparametric asymptotically distribution free valid bioequivalence tests. These procedures turn out to be asymptotically ecient under certain circumstances, see section 4. Beyond our asymptotic solution we consider a nite sample size re nement in the sense of Klinger (1995) where the power function is known for a parametric submodel, for instance for Lehmann's alternatives. It is shown that all procedures have the same asymptotic performance, see Corollary 2.2. The results are not only restricted to the Wilcoxon functional. They work for a class of wider functionals which also includes R the Savage test and the functional ? log(P (Y > x)) dL(X )(x) which is just the average of the cumulative hazard function ? log(P (Y > x)). Throughout, the following two-sample testing model will be considered. Let X1; : : : ; Xn always denote independent real random variables with continuous distribution functions such that

X1; : : : ; Xn

1

are identically distributed with joint distribution P and let

Xn +1 ; : : : ; Xn 1

be also identically distributed with distribution Q where the second sample size is given by n2 := n ? n1 . Suppose that (P; Q) belongs to some family P of pairs of distributions, i.e. (P; Q) 2 P . The hypothesis of population bioequivalence is given by

P1 := f(P; Q) 2 P : P = Qg;

(1)

see again Berger and Hsu (1996) and Munk (1996) and references of these papers. Motivated by practical statistical problems we look for tests which are able to verify the signi cance of (1). However, there exists no reasonable test of level 2

 for P ?P1 against P whenever the model is rich enough. For these reasons the problem is often substituted by another testing problem, namely Hn against Kn

(2)

where Kn  P1 is a neighbourhood of P1 which may depend on the sample size n within the present aproach. The extended alternatives Kn are typically speci ed by parameters of the distributions or, more generally, by the values of statistical functionals of the distributions. For instance the nonparametric functional

(P; Q) = P

Q(f(x; y) 2 R 2

: y  xg) =

Z

FQ(x) dP (x)

(3)

(given by the distribution function FQ of Q) is often used which is related to stochastic ordering of the underlying random variables. Since (P; P ) = 1=2 the new alternative may be of the type

Z

Kn = f(P; Q) 2 P : j FQ(x) dP (x) ? 1=2j < ng

(4)

where the constant n > 0 may depend on the sample size n. As explained below the quantity n should depend on n within the asymptotic step up which re ects the increasing accuracy of the procedure if n tends to in nity. Although n ! 0 holds as n ! 1 one obtains { as usual { a reasonable approximation for nite sample size n. We refer to section 5 for the related discussion. The principle can be compared with the construction of ordinary con dence intervals for parameters given by the central limit theorem. Although these intervals usually have a lenght of order n?1=2 they are also used to get a good approximation of the probability that the parameter does not exceed a given threshold of tolerance at nite sample size. Before the basic ideas are introduced we will state the following observations which should be kept in mind. If n is any rank bioequivalence test then E(P;P )( n) = n is constant on the whole region P1 . Due to the invariance theory of tests the class of rank tests is well motivated for the underlying nonparametric model. 3

It turns out that the signi cance of (4) can be handled by general results of asymptotic statistics, see for instance Pfanzagl and Wefelmayer (1982, 1985) for the methodology. Their common set up will brie y be explained as follows. The statistician xes two error probabilities  and ,  < 1 ? , namely the type I error probabilitiy  and the type II error probability relative to P1 which are acceptable. Since we have P1 in mind we will consider tests n = n(X1 ; : : : ; Xn) such that

E(P;P )( n)  1 ?

(5)

holds for all (P; P ) 2 P1 . Within this concept the statistician obtains unbiased regions of distributions

K ( n) := f(P; Q) 2 P : E(P;Q)( n) > g  P1 :

(6)

We say that Hn = P ? K ( n) and P1 can be seperated within the given bounds of error probabilities. This approach ts in the concept (2) if Kn = K ( n ). Obviously, there is a trade of between the sample size and the region Hn. For a given type of test n the statistician can either determine P ? K ( n) or he can increase the sample size to ensure that P ? K ( n) is suciently large. It is important that Kn and n given by (4) may depend on the sample size. In the sequel an asymptotic solution of the testing problem will be proposed. The asymptotic model developed below serves as an approximation of the underlying nite sample problem. We follow the lines of Pfanzagl and Wefelmeyer (1982, 1985) and Janssen (1997) who introduced tests for statistical functionals. Basic work about tangents and local models can also be found in Strasser (1985) and Bickel et al. (1993).

4

2. Bioequivalence hypotheses speci ed by functionals, two-sided tests and their nonparametric power function In the sequel let

 : P ! R ; (P; Q) 7! (P; Q)

(7)

be a binary statistical functional such that (P; P ) = c is constant on diagonal elements for all (P; P ) 2 P . In view of (4) we are looking for a test 'n for

Hn = f(P; Q) 2 P : j(P; Q) ? cj  n g against Kn = f(P; Q) 2 P : j(P; Q) ? cj < ng;

(8)

where n may be adjusted via the sample size n. For a wide class of functionals we will introduce asymptotically valid level  tests for (8) taking the principle remarks of section 1 into account. Before the results are stated let us consider various examples of functionals where the concept works.

Example 2.1 R Let h : (0; 1) ! R denote a square integrable function with 0 < h2 (u) du < 1. A binary statistical function is de ned by

h(P; Q) =

Z

h(FQ(x)) dP (x)

(9)

which is given by the distribution function FQ of Q for the set of pairs P such that (9) exists. Since P has a continuous distribution function we have h(P; P ) = R 1 h(u) du for each function h. 0 (a) If we choose h = id to be the identity on (0; 1) we arrive at (3) which is called the Wilcoxon functional. (b) The choice h(u) = ? log(1 ? u) introduces the mean

h(P; Q) =

Z

Q(x) dP (x)

(10)

of the cumulative hazard function Q(x) := ? log(1 ? FQ(x)). (c) For 0 < q < 1 let FQ?1(q) be the left-sided continuous quantile function of FQ. 5

The nonparametric q-quantile functional is de ned by

(P; Q) = FP

(F ?1(q)) = P (x : F Q

Q(x) < q ) =

Z

1(0;q) (FQ(x)) dP (x)

(11)

for xed q and all pairs of continuous distributions. In the case of the uniform distribution P on (0; 1) we have just the q-quantile and for q = 1=2 the median is obtained. Further nonparametric functionals are discussed in Janssen (1997). In the next step two-sided tests and their power functions for functionals will be discussed rst along parametric submodels. As explained above an increasing sample size n can be compensated by a sequence of shrinking neighbourhoods Kn of P0 . As usual in asymptotic statistics the quality of the underlying tests will be studied along all regular one-parametric curves t 7! (Pt ; P0); t 2 U  R , at t = 0 which serve as distributions of the X 0s. They can be viewed as an arbitrary unknown one-parametric submodel. Suppose that ft : (Pt; P0) 2 Pg contains a neighbourhood of 0. Curves of this type are called to be locally (at 0) in P . The general testing problem now reduces to the parametric testing problem

f(Pt; P0) 2 P : j(Pt; P0) ? cj  ng against f(Pt; P0) 2 P : j(Pt; P0) ? cj < ng

(12)

It is now our concept to solve the testing problem rst for all bres (12). It is surprising that the solutions can be made independent from the special curves t 7! Pt and the bres. This concept can be carried out under the following regularity assumptions which are common in asymptotic statistics, see Pfanzagl and Wefelmeyer (1992), Strasser (1985) or Bickel et al. (1993). The curve t 7! Pt is always assumed to be L2 (P0)-dierentiable at t = 0 with R tangent g 2 L02(P0 ) := fh 2 L2 (P0) : h dP0 = 0g. The tangent is the L2 (P0)  dPt 1=2 derivate of t 7! dP at t = 0. This concept seems to be wide enough since 0

6

two arbitrary distributions can be connected by a L2-dierentiable curve. Suppose that the set of associated tangents (given by local curves in fP : (P; P0) 2 Pg at P0) is always convex. The regularity assumption concerning  is the following dierentiability. A statistical functional P 7! (P; P0) is called dierentiable at P0 with gradient _ 2 L02 (P0) if 1 ((P ; P ) ? (P ; P )) ! Z _ g dP for t ! 0 (13) 0 0 0 t t 0 holds for all local curves t 7! Pt where g is its tangent. There exists a gradient ~ R with smallest L2 (P0)-norm ( ~2 dP0)1=2 , which is called the canonical gradient. The meaning can be commented as follows. Assumption (13) is a linearization of the problem. Recall that if ~ is itself a tangent the associated curve de nes a least favorable family which yields the hardest parametric submodel (12). Sometimes it happens that the functional turns out to be linear at least for some submodel, for instance the functional (10) is linear for a proportional hazard model. Within this notion we xe our assumptions (14) - (21). Suppose that the relative sample size of the rst sample ful lls n2 =: d 2 (0; 1): lim (14) n!1 n A(i). For each P0 with (P0; P0) 2 P there exists a set PP such that 0

PP 3 P 7! (P; P0)

(15)

0

is dierentiable with canonical gradient ~(P0). A(ii). There exists a function ' : (0; 1) ! R such that ~(P0) is equal to

~(P0) = cP '  F0 ; ~(P0 ) 6= 0

(16)

0

for each P0, where F0 is the distribution function of P0 and cP is a positive constant. 0

Obviously, rank tests are appropriate candidates for the nonparametric bioequivalence model. They rely on linear rank statistics

X Sn = n?1 1=2 ( an(Rni) ? n1an ); i=1 n1

(17) 7

where Rni denotes the rank of Xi among X1; : : : ; Xn and an(i) are suitable scores, P 1  i  n; an := n1 ni=1 an(i). Suppose that the scores are L2(0; 1)-convergent

Z1 0

(an(1 + [nu]) ? '(u))2 du ! 0 as n ! 1

(18)

see Hajek and Sidak (1967) for details concerning rank tests. The concepts of shrinking neighbourhoods of P0 and one-parametric submodels naturally lead to implicit alternatives (19) for a given local curve t 7! Pt . The in uence of the shrinking neighbourhoods can be described by an additional local parameter # 2 R . For xed # consider a sequence tn = O(n?1=2 ) such that

(Ptn ; P0) = (P0 ; P0) + n?1 1=2 # + o(n?1=2 ):

(19)

As consequence of A(i) we have

#=

Z

~(P0) g dP0 lim sup(n11=2 tn)

(20)

n!1

where g is the tangent of the underlying curve and nlim (n11=2 tn) exists whenever !1 R ~(P ) g dP 6= 0 holds. (Notice that then (19) has at least nally always a 0 0 solution tn also when the term o(n?1=2 ) is cancelled.) These preparation lead to the following one-parametric local model for the joint distribution of the random variables. For xed # we introduce their distribution by the implicit parametrization (19) as

L(X1; : : : ; Xn) = Ptnn Pon =: Pn;# 1

(21)

2

The notation Pn;# of (21) is justi ed since the present asymptotic properties only depend on # and not on the special sequence tn.

Theorem 2.1

Under the assumptions (14)-(19) and (21) the asymptotic distribution of L(SnjPn;#) with respect to the convergence in distribution is a N (#dc?P 1 ; R R d 01 '2(u) du) normal distribution with mean #dc?P 1 and variance d 01 '2 (u) du. The asymptotic distribution only depends on the quantities of the functional and 0

0

8

P0 and it is independent of the special curve t 7! Pt and the choice of tn for the implicit representations. All proofs are given in the appendix. The present result is of some importance. It enables us to give a solution of the bioequivalence problem rst along the bres (12). We will see that in the case of Example 2.1 the asymptotic distribution of Sn is also independent of P0 since cP = 1 holds. A proper two-sided rank test is given by 0

8 > < >  n!1 Ptn P0

if (Ptn ; P0) 2 K~ n for all n

Sometimes it happens that the power function of our rank tests 'n can exactly be computed under semiparametric alternatives for each nite sample size, see section 5 for a concrete discussion concerning Lehmann's alternatives. Under our assumptions above we can introduce exact bioequivalence tests for these underlying semiparametric submodel. This solution turns out to be an asymptotic solution for all bres. Thus we obtain the slogan: exact semiparametric solutions are nonparametric solution in the asymptotic sense. 10

Corollary 2.2 R Let t 7! Ptn P0n denote a local bre such that g~(P0) dP0 6= 0 holds. Assume 1

2

that the exact power function of our two-sided Sn-test (22)

n(t) := EPtn P n (1(cn ;1)(jSnj)) 1

0

2

is known, where a suitable choice of critical value cn guarantees that n(0) = n, n ! , holds. Suppose that t+n , t?n are local parameters tn = O(n?1=2) with

n(tn) ?  ! 0 as n ! 1 and

n+ := (Ptn ; P0) ? c > 0 and n? := (Pt?n ; P0) ? c < 0: +

Then n+=n ! 1 and n?=n ! ?1 follows. As application of Corollary 2.2 the choice of Hn in (8) can be modi ed by setting

Hn = f(P; Q) 2 P : (P; Q) ? c  n? or (P; Q) ? c  n+g: If t 7! n(t) is unimodal and continuous with n(tn ) =  and n(0) = we have an exact solution along our local bre at nite sample size which is an asymptotic solution for all other bres. Questions concerning the quality and optimality of the sequence of test 'n are treated in section 4. Let us suppose that the constant cP of (16) is independent of P0 (which is actually true with cP = 1 for the functionals of Example 2.1). Then our choice of n (25) only depends on the value of the functional and we emphasize to insert n or n0 in (8) for a nite sample size. According to Theorem 2.2 the procedure works well for all local submodels lying in PP , see assumption A(ii). In the next section we will derive concrete bioequivalence tests. 0

0

0

3. Examples of nonparametric bioequivalence tests In this section concrete tests are obtained for the functionals which were considered in Example 2.1. Following section 2, we have to determine the canonical 11

gradient ~(P0) of the functional and the assumptions have to be checked. The technical details are included in Janssen (1997).

Example 3.1 (Full nonparametric model)

R

The assumptions A(i) and A(ii) hold for h(P; Q) = h(FQ(x)) dP0(x) under the following circumstances, see (9). The set P is the set of all pairs (P; Q) such R that h exists. Let K be a constant greater than 01 h(u)2 du. Then the choice

Z

PP = f(P; P0) 2 P : h2 (FP (x)) dP0(x)  K g 0

0

implies the dierentiability of P 7! h(P; P0) relative to PP with canonical gradient 0

~h(P0) = (h ?

Z1 0

h(u) du)  F0

(26)

where cP = 1. If h is bounded (as in Example 2.1 (a), (c))) we may choose PP = P to be the set of all pairs of continuous distributions. The construction above now leads to rank tests 'n (22) with asymptotic score function 0

0

'(u) = h(u) ?

Z1 0

h(v) dv:

(27)

Thus every two-sided linear rank test can be deduced from a statistical functional h. Observe that the critical distance n , de ned in (25),

n = n?1 1=2 d?1=2 #0

Z1 0

'(u)2 du

(28)

is independent of P0. In the sequel, three functionals are studied in detail. R (a) The Wilcoxon functional (3) id (P; Q) = FQ(x) dP (x) leads to the score i ? 1=2 yields the two-sided function '(u) = u ? 1=2. The scores an(i) = n+1 Wilcoxon test 'n (22) de ned by

Sn =

n X 1 ( Rni ? n1(n2+ 1) ): ? 1 = 2 (n + 1)n1 i=1 1

12

(29)

At nite sample size n = n1 + n2 the two-sided Wilcoxon bioequivalence test is now proposed for

Z Hn = f(P; Q) 2 P : j FQ(x) dP (x) ? 1=2j  n0 g against Z 0 Kn = f(P; Q) 2 P : j FQ(x) dP (x) ? 1=2j < ng

(30)

where

 n 1=2 n )1=2 (12)?1=2 #0 ( n + n = n n 1 1 2 0

(31)

is taken from (24)  for d = nn and the exact Wilcoxon variance c2n = nn+1 =12. (b) Next we study the functional given by the average of the cumulative hazards (10). The function h(u) = ? log(1 ? u) leads to the score function of the Savage test, see Hajek and Sidak (1967) for details. Similary to (25) and (30) the twosided Savage test can be used as a bioequivalence test. (c) The function h(u) = 1(0;q)(u) de nes the q-quantile functional (11). Thus the canonical gradient for the full model is given by 2

2

~(P0) = (1 ? q)1[0;q) ? q1(q;1])  FP : 0

(32)

The test statistic Sn is given by the scores

an(i) = (1 ? q)1(0;(n+1)q) (i) ? q1((n+1)q;1) (i)

(33)

for i = 1; : : : ; n, compare with (17) and (18). If q = 1=2 is taken the construction yields the median test. The rank tests 'n are asymptotically ecient within the local nonparametric model. We refer to Janssen (1997) and section 4 where special questions concerning the bioequivalence problem are discussed. Until now our assumptions lead to full statistical models where the set of tangents at P0 is dense in L02 (P0). Under this conditions the gradient is unique. However, if the statistician has more prior information concerning the model the 13

canonical gradient may change and consequently other procedures turn out to be ecient. (See Bickel et al. for the same eect when  is estimated.) To illustrate the concept let us consider the following semiparametric bioequivalence model depending on proportional hazards which is due to Munk (1996).

Example 3.2

R

Let id(P; Q) = FQ(x) dP (x) be the functional (3). For each continuous distribution let PP denote the set of Lehmann's alternatives 0

FP (x) = FP (x)1+t for t > ?1: 0

The tangent space of this restricted model at P0 is just the one-dimensional linear subspace of L02(P0 ) generated by g = ? log(1 ? FP ) ? 1. Observe that according to (26) the function _ id(P0) = FP ? 1=2 is still a gradient of id. Its canonical gradient ~id (P0) with respect to PP is the orthogonal projection of _ id(P0 ) on the tangent space. Thus ~id (P0)  g follows and our construction principle via functionals yields the Savage test for id under the restricted model which corresponds to g and the score function '(u) = ? log(1 ? u) ? 1. This approach establishes Munk's procedure. Notice that under proportional hazards the quantity n can be calculated exactly. This example shows that the tests depend on the geometry of the model which is represented by the set of tangents. 0

0

0

4. The optimality of bioequivalence tests It is well-known that the construction principle based on canonical gradients yields ecient tests or maximin tests. We refer to Pfanzagl and Wefelmeyer (1982, 1985) for parametric models and to Janssen (1997) for testing nonparametric functionals. In the sequel it is shown that the optimality also holds for our bioequivalence test. The optimality stated in Theorem 4.2 gives a justi cation for the recommendation of the sequence 'n of bioequivalence tests. Again the results are formulated in terms of the local parametrization which was explained 14

in the previous sections. Optimality is obtained for the full nonparametric model of Example 3.1 as well as for the semiparametric Example 3.2. In the sequel let us always assume that the bioequivalence testing problem is of the form (7) and (8) where the assumptions A(i), A(ii) and (14) hold for a single distribution P0. Below let P0 always be xed. Moreover, suppose that the model is rich enough and 1. and 2. hold. 1. Suppose that the set of tangents L of curves t 7! Pt , such that t 7! (Pt ; P0) 2 PP is a local curve in PP , is a convex set. In addition suppose that the set of tangents of L given by curves t 7! (Pt; P0) 2 PP \f(P; Q) : (P; Q) = cg, relative R to the set f(
;
) = cg, is L2(P0 )-dense in the set fg 2 L : g~(P0) dP0 = 0g. (Full tangent space condition for null hypothesis f(
;
) = cg.) 2. Let n = n((Rni)in) be a sequence of rank tests such that 0

0

0

n2 ( n ) = 1 ? lim E n1 n!1 Pn?1 1=2 t P0

(34)

holds for all sequences such that t 7! (Pt; P0 ) is a local curve in PP \ f(P; Q) : (P; Q) = cg. (Local asymptotic (1 ? )-similarity for f(
;
) = cg) Suppose that for each local curve t 7! (Pt ; P0) in PP and their associated implicit alternatives Pn;# of (21) the condition 0

0

lim sup EPn;# ( n )  1 ?

(35)

n!1

holds for # 6= 0. Again the optimality will be studied along all parametric bres (12) which are of local type.

Theorem 4.1

Under the condition stated above we have lim inf E ( )  P (#) n!1 Pn;# n 0

where P (#) is the asymptotic power function of the test 'n under consideration in (23). 0

15

Returning to our motivation of section 1 the previous result suggests that the unbiased regions (6) of 'n seem to be the smallest regions within the class of underlying rank tests n. A result of this type holds within a local model. Throughout, let An be a sequence of events. Then lim sup An = n!1

1 [ 1 \

m=1 n=m

An

1 is the event that in nitely many An occur. De ne lim inf A = [1 m=1 \n=m An . n!1 n Let P?1 be the inverse of the strictly decreasing function P j[0;1) : [0; 1) ! [0; 1]. 0

0

Theorem 4.2

Let n denote a sequence of locally unbiased bioequivalence rank tests in the sense of Theorem 4.1. For P0 consider a local curve t 7! (Pt ; P0) in PP with tangent g of t 7! Pt . Let K be any constant. De ne 0

H~ n( n) = ft 2 [?K; K ] : (Pn? = t ; P0) 2 P ; EP n? = 1

1

1 2

n1 1 2 t

P0n2 ( n )  g

Then inf H~ (' ) lim sup H~ n( n )  lim n!1 n n n!1

holds where the bar denotes the closure of a set. If < ~(P0); g >6= 0 then lim inf H~ (' ) = ft 2 [?K; K ] : jtj  j P?1()= < ~(P0); g > jg: n!1 n n 0

In case < ~(P0); g >= 0 the set is empty. Remark. The assumptions of Theorem 4.2 are satis ed for the functionals and tests given in the Examples 3.1 and 3.2. Observe that condition 1 trivially holds under the conditions of Example 3.2 since the tangent space has dimension one.

16

5. A comparsion of Wilcoxon power functions and conclusions The present asymptotic approach serves as a model approximation for nite size sample n. There are three possibilities to apply the results which rely on the asymptotic power function P (
) (23). The rst and second proposals come from asymptotic results. 1. Let and  be xed. Then the critical values of the ecient rank test 'n (22) can be determined and one obtains the nonparametric region (8), which is given by the bounds n or n0 considered in (24). 2. If  and n are xed values (where n can be viewed as given threshold of tolerance coming from a practical problem) then the critical values cn u(1+ )=2 of (22) can be tted such that 0

P (n ) =  0

holds. Then the test 'n may be applied and the value = P (0) is completely determined. Obviously, our asymptotic formulas include an estimate for the sample size n if a certain accuracy (expressed by xed ; n, and ) is required. 0

Corollary 2.2 includes a nite sample solution which may sometimes be better than the asymptotic approximation mentioned above. 3. Suppose that the statistician is able to compute the exact power function n(
) of two-sided Sn-rank tests for a parametric submodel. Then exact bioequivalence regions

n? < (P; Q) ? c < n+ can be proposed, see Corollary 2.2. This is also an asymptotically valid distribution free solution of the problem for all other bres. Within the nonparametric model each solution is asymptotically the same in our local sense. The program pointed out in 3 can be carried out for Lehmann's alternatives which is a proposal of Klinger (1995) for the Wilcoxon functional. Consider the 17

functionals of Example 3.2 and the corresponding linear rank statistics Sn given by the canonical gradient. Then the distribution of the ranks (Rni)in is well known for Lehmann's alternatives Pt of P0, see Davies (1971), Kalb eisch and Pentice (1980) and also Munk (1996). Thus L(SnjPtn P0n ) can be computed which gives as the power function n(
) for this bre. The quality of our approximation is now demonstrated for a concrete example at sample size n1 = n2 = 20 and the level  = 0:048 for the Wilcoxon functional (3). Fortunately, the exact power functions of the Wilcoxon test statistics can easily be computed under implicit alternatives. We refer to Klinger (1995) and Munk (1996) who expressed the functional in terms of Lehmann's alternatives Pt of P0. Let  = n?1 1=2 # = id (Pt; P0) ? 0:5 be the centered value of the functional (global parameter). Then gure 1 shows the power of the Wilcoxon bioequivalence test denoted by 1 ( + 0:5). The adjustment is done at  = 0:25 with 1

1(0:25) = 1(0:75) =  = 0:048:

2

(36)

This power function is now the banchmark for our approximation. If 1(
) would be unknown our asymptotic approximation yields the power function of gure 2 where 2( + 0:5) is given by Corollary 2.1 with the exact variance c2n = n2 =(12(n + 1)). The approximation is tted according to the proposal 1 above where := 1(0:5) is taken from the top of the power function of gure 1. We see that the power functions in the gures 1 and 2 almost coincide and we have here a good approximation.

Figure 1 and gure 2 about here

18

In this case we have = 0:7789 and

2(0:25) = 2(0:75) = 0:06 which is in comparison to (36) a little bit to high for Lehmann's alternatives. The nonparametric estimator for the threshold of tolerance (31) is

n0 = 0:26 which is much the same as n+ = ?n? = 0:25. As conclusion we see that here the nonparametric results are acceptable for Lehmann's alternatives.

Appendix In this appendix the results are proved. The proof of Theorem 2.1. This proof follows the lines of the proof of Theorem 4.2 of Janssen (1997). De ne

T0

n

=

n X i=1

cni'(F0(Xi))

where cni = n?1 1=2 ( nn 1f1;:::;n g (i) ? nn 1fn +1;:::;ng(i)). Standard asymptotic results for rank statistics, see Hajek and Sidak (1967), Chapt. V, yield 2

1

1

1

V arP n (Sn ? Tn0 ) ! 0 as n ! 1: 0

The asymptotic distribution of Tn0 will now be obtained under Pn;# by Le Cam's third Lemma, see Hajek and Sidak (1967), Chapt. VI. Notice rst that

V arP0n (Tn0 ) ! 12 := d

Z1 0

'(u)2 du:

If g is the tangent of the curve t 7! Pt then local asymptotic normality yields

Z n X dP n;# 2 log dP n = tn g(Xi) ? n1 tn g2 dP0=2 ? oP n (1): 0 i=1 1

0

19

On the other hand observe that n X

1=2 t cn1 n g(Xi)) = nlim n n n !1 1 n n1=2 1 1 i=1 1

n (T 0 ; t

lim CovPo

n!1

Z

g '  F0 dP = #dc?P 1 0

is independent of g and the underlying curve. Thus Le Cam's third Lemma (Hajek and Sidak (1967), p. 208) implies the result. 2 Corollary 2.1 immediatly follows from Theorem 2.1. The proof of Theorem 2.2. The asymptotic power function # 7! P (#) is unimodal and bell shaped. Thus P (#)   holds if and only if 0

j#j  d?1=2#0 cP (

Z1

0

0

0

'(u)2 du)1=2 = n11=2 n:

Consider now a sequence tn = O(n?1=2) with (Ptn ; P0) 2 H~ n. Without restrictions we may assume that n11=2 tn is convergent. (Otherwise take convergent subsequences.) This sequence de nes implicit alternatives Pn;# (21) where

#=

Z

~(P0) g dP0 nlim (n1=2 t ) !1 1 n

is speci ed by (20). Since j(Ptn ; P0) ? cj  n is assumed to be true statement (21) implies

j#j  d?1=2#

0 cP0 (

Z1 0

'(u)2 du)1=2:

Thus P (#)   holds, see (23). The second assertion follows similary. Sequences (Ptn ; P0) 2 K~ n belong to implicit alternatives with j#j  n11=2 n . 2 0

The proof of Corollary 2.2. The result is a consequence of Theorem 2.1 and Corollary 2.1. Without restrictions we may assume that n?1 1=2 t+n ! t0 is convergent. Then Ptnn P0n de nes implicit local alternatives with 1 +

n?1=2 ((Pt+n ; P0) ? c) = to lim n?1=2 n+ = nlim !1 1 n!1 1 20

Z

2

g ~(P0) dP0 =: # > 0:

Corollary 2.1 now implies

 = nlim

(t+) = P (nlim n?1=2 n+): !1 n n !1 1 0

On the other hand the choice of n (24) yields

 = P (n?1 1=2 n ) 0

for each n. Since P (
) is strictly decreasing on [0; 1) we have n+=n ! 1 as n ! 1. The sequence n? can be treated similary. 2 0

The proof of Theorem 4.1. The present proof follows the lines of the proof of Theorem 4.2 of Janssen (1997). In a rst step it is carried out for ordinary local alternatives. Afterwards the results can be applied to implicit alternatives (21). Suppose that t 7! Pt is an arbitrary curve of local type with tangent g which R leads implicit alternatives (21) for # 6= 0, i.e. g (P0 ) dP0 6= 0. R Next let h be any tangent with h (P0) dP0 = 0 of a local curve in PP \ f(
;
) = cg. For this pair of tangents g, h we will now introduce new local parameters s = (s1; s2; s3 ; s4)T 2 R 4 and new curves 0

t 7! t(s) with 0(s) = P0 ; (t(s); P0) 2 PP locally at t = 0, which have the tangents 0

(s1 + s3 )g ? (s2 + s4 )h at t = 0. Recall that the set L of tangents is assumed to be convex. Without restrictions we may assume that

t(s) 2 PP \ f(
;
) = cg 0

holds for s = (0; s2; 0; 0). After this preparation a further 4-parameter model is introduced. As distribution of (X1; : : : ; Xn) consider the distributons n;s := nn? = (s1 ; s2; 0; 0) nn? = (?s1 nn1 ; ?s2 nn1 ; s3 nn1 ; s4 nn1 ) 1

1

1 2

2

1

1 2

21

on R n which decribes the two sample problem by members of the curves t(s) at t = n?1 1=2 for various s-values. Consider the random variables

Zn = (Z1;n; Z2;n; Z3;n; Z4;n) n X n n n n X X X n n n 1 1 2 2 ? 1 = 2 = n1 n i=1 g(xi) ? n i=n +1 g(xi); n i=1 h(Xi) ? n i=n +1 h(Xi);  n n X n1 X n 1 n i=1 g(Xi); n i=1 h(Xi) 1

1

1

1

By the central limit theorem we have convergence of Zn under P0n = n;0 in distribution with 4-dimensional normal limit N (0; ) with

0 d =@ 1 R

0

0 2

1 A

and

0 1 2 jjgjj2 < g; h > A 1 = 2 = @ R

< g; h >

jjhjj22

where jjgjj22 = g2 dP0 and < g; h >= g h dP0. We have local asymptotic normality for the family n;s, namely log dn;s = sZn ? sT  s=2 + Rn(s) dn;0 where Rn(s) ! 0 in n;0 -probability. The limit experiment is thus the 4-parameter Gaussian shift (R4 ; B4; fs : s 2 R 4 g) given by normal distribution s = N (s; ) and ds (z) = zT s ? sT  s=2; zT = (z ; z ; z ; z ) 2 R 4 : log d 1 2 3 4 0 Notice that dn;s j ) ! L(log ds j ) L(log d n;s do 0 n;0 converges in distribution. For details concerning local asymptotic normality we refer to Strasser (1985), Ch. 13. The following arguments follow the proof of the weak compactness theorem of Ruschendorf (1988), p. 157. Since ( n; Zn) is tight under n;0 there exists for each subsequence a further subsequence fmg such that

L(( m; Zm)jm;0 ) ! Q0 22

in distribution on [0; 1]  R 4 . By Le Cam's third Lemma we have

L(( m; Zm)jm;s) ! Qs in distribution on [0; 1]  R 4 where dQs (z ; z ; z ; z ; z ) = exp(zT s ? sT  s=2): dQ0 0 1 2 3 4 Let now j (z0 ; : : : ; z3 ; z4) = zj be the projection for j = 0; : : : ; 4. Notice that 0 is a limit test of m within the model. Obviously we have the asymptotic power function lim E ( ) = m!1 m;s m

Z

0 exp((1 ; 2; 3; 4 )s ? sT  s=2) dQ0 =: (0; s):

It will be shown that this power function is independent of (s3 ; s4). According to Hajek and Sidak (1967), sect. V1.5, we may choose linear rank statistics Z10 ;n 0 ? Zj;n ! 0 in n;0 -probability for j = 1; 2. Consequently and Z20 ;n with Zj;n (0; 1 ; 2) and (3; 4 ) are Q0 -independent since ( m ; Z10 ;m; Z20 ;m) only depends on ranks and (Z3;m; Z4;m) is a function of order statistics which are independent under m;0 . Applying conditional expectations we see that

R

0 exp((1; 2; 3 ; 4)s ? sT  s=2) dQ0 Z = E (0 exp((1; 2 ; 3; 4)s ? sT s=2) j 0; 1; 2 ) dQ0 = =

Z

Z

0 exp(s1 1 + s2 2 ? (s1; s2)1 (s1 ; s2)T =2)


E (exp(s33 + s44 ? (s3; s4)2 (s3; s4)T =2) j 0; 1 ; 2) dQ0 0 exp(s11 + s22 ? (s1; s2)1 (s1; s2)T =2) dQ0

is independent of the coordinates (s3; s4). Our assumptions imply (0; s)  for all s 2 R 4 and (0; s) = for s = (0; s2; 0; 0) and s = (0; 0; s3; s4). Hence (0; s) = follows for all s = (0; s2; s3; s4). Within this class of unbiased tests there exists a solution 0 which 23

minimizes the Qs-power, namely

8 > < > 2 j V arQ (1? < ga; h > 2)1=2 u(1+ )=2 0=> > :0  0

In this de nition the quantities of the orthogonal decomposition g = ga + gb of g given by < ga; h >= 0 and gb = h for some 2 R are needed. Standard results involving testing normal distributions with nuisance parameters imply

EQs (0 )  EQs ( 0 ) for all s 2 R 4 ; compare with Strasser (1985), sect. 28, for instance. The lower bound is equal to

EQs ( 0 ) = (s1 d1=2jjgbjj2 + u(1+ )=2 ) + (s1 d1=2 jjgbjj2 ? u(1+ )=2 ) since the asymptotic relative eciency of 0 in direction (s1 ; 0; 0; 0) with respect to 1fj j ~(P0) / jj~(P0 )jj22. By assumption 1 we may choose a sequence of tangents hn with < hn; ~(P0) >= 0 and jjhn ? g2jj2 ! 0. Since the 24

corresponding sequence gb = gb(n) converges to g1 in L2(Q0 ) we obtain the bound

K  (s1d1=2 < ~(P0 ); g > jj~(P0 )jj?2 1 + u(1+ )=2 ) +(s1d1=2 < ~(P0 ); g > jj~(P0)jj?2 1 ? u(1+ )=2 ): This inequality establishes the lower bound for local alternatives. From this bound the result can be deduced for implicit alternatives Pn;#, see (21). According to (20) and the ULAN-discussion in Janssen (1997) we may choose s1 = # < ~(P0); g >?1 which implies tn = n?1 1=2 s1 + o(n?1 1=2 ). For this choise of s1 our lower bound is the bound for EPn;# ( n ). According to (15) this is just the desired result. 2 The proof of Theorem 4.2. In a rst step we will establish the bound for H~ n('n). Corollary 2.1 together with statement (19) implies n2 ('n ) = P0 (t <  lim E n1 ~(P0); g >) n!1 Pn?1 1=2 t P0

for xed t. Since P is strictly decreasing on [0; 1) we have 0

ft 2 [?K; K ] : jtj > j P?1()= < ~(P0 ); g > jg  lim inf H~ (' ) n!1 n n 0

and

ft 2 [?K; K ] : jtj  j P?1()= < ~(P0); g > jg  lim sup H~ n('n) n!1

0

whenever < ~(P0); g >6= 0. Otherwise these sets are empty. Consider now the sequence of rank tests n . Theorem 4.1 implies n ( n )  P (t <  ~(P0); g >): lim inf E n n!1 Pn ? = t P 1

1

1 2

0

2

0

If < ~(P0); g >= 0 holds then lim supn!1 H~ n( n ) is empty since P (0) = > . If the inner product is not zero the same arguments as above show that 0

lim sup H~ n  ft 2 [?K; K ] : jtj  j P?1(d)= < ~(P0 ); g > jg n!1

0

holds. 2 25

Acknowledgment. I like to thank my colleague H. Klinger for his introduction to the bioequivalence problem and the computation of the power function given in gure 1 along the lines of his 1995 notes.

References Berger, L.R., and Hsu, J.C. (1996), "Bioequivalence trials, intersection-union tests and equivalence con dence sets", Statistical science, 11, 283{302. Bickel, P., Klaassen, C.A.J., Ritov, Y., and Wellner, J.A. (1993), Ecient and Adaptive estimation for Semiprametric Models, Johns Hopkins Ser. Mathem. Science, Johns Hopkins Univ. Press, Baltimore and London. Davies, R.B. (1971), "Rank tests for Lehmann's alternative", Journal of the American Statistical Association, 66, 879{883. Hajek, J.A., and Sidak, Z. (1967), Theory of rank tests, Academic Press, New York. Janssen, A. (1997), "Testing nonparametric statistical functionals with application to rank test", Preprint. Kalb eisch, J.D., and Pentice, R.L. (1980), The statistical analysis of failure time data, Wiley, New York. Klinger, H. (1995), "Ein Rangsummentest fur das A quivalenzproblem bei zwei unabhangigen Stichproben", Notes of a talk given at the conference Biometrisches Kolloquium, Hohenheim, March 1995. Munk, A. (1996), "Equivalence and interval testing for Lehmann's alterative", Journal of the American Statistical Association, 91, 1187{1195. Pfanzagl, J., and Wefelmeyer, W. (1982), Contributions to a general asymptotic statistical theory, Lecture Notes in Statistic, Vol. 13. Springer-Verlag, Berlin/Heidelberg. Pfanzagl, J., and Wefelmeyer, W. (1985), Asymptotic Expansions for general Statistical Models, Lecture Notes Stat. 31 Springer, Berlin-Heidelberg. 26

Pfanzagl, J. (1994), Parametric statistical Thoery, De Gruyter Textbook, BerlinNew York. Ruschendorf, L. (1988), Asymptotische Statistik, Teubner Skripten zur Mathematischen Stochastik. Teubner, Stuttgart. Wellek, S. (1993), "Basing the analysis of comparative bioavailability on an individualized statistical de ntion of equivalence", Biometr. Journ., 35, 47{55. Wellek, S. (1996), "A new approach to equivalence assessment in standard comporative bioavailability trials by means of the Mann{Whitney statistics", Biometr. Journ., 38, 695{710.

A. Janssen, Mathematical Institute, University of Dusseldorf, D-40225 Dusseldorf, Germany.

27