A PARTIALLY SEQUENTIAL SELECTION ...

COMMUN.

STATIST.-THEORY

METH.,

20(2), 703-710

(1991)

A PARTIALLY SEQUENTIAL SELECTION PROCEDURE FOR COHPARISON WITH A CONTROL David R. Bristol Pharmaceuticals Division CIBA-GEIGY Corporation Surrunit,NJ

H. H. Desu Department of Statistics State University of New York at Buffalo Buffalo, NY

ABSTRACT A distribution-free

selection rule based on a partially

sequential sampling scheme is presented for comparison of k treatments with a control for the family of continuous distribution functions.

A modification

of the selection rule is presented

for use when a specified quantile is used to define the preference pattern.

1.1

INTRODUCTION

In experiments that are performed to compare k experimental populations with a control population, one is often interested in selecting those experimental populations which are "better" than the control population. populations

Let n1, ..., Dk denote the k experimental and let nO denote the control population. Observations

from D. follow a continuous distribution defined by the cumulative 1

i=O,l, ... ,k. distribution function (cdf) F., 1 a stochastic ordering exists between Fi ( FO(x) "for all x or Fi(x) ~

and FO' so that Fi(x)

FO(x) for all x. 703

Copyright © 1991 by Marcel Dekker, Inc.

It is assumed that No further

704

BRISTOL AND DESU

restrictions, cdfs.

other than this very weak one, are imposed on the

The experimenter's

goal is to select a subset (possibly

empty) of the ~ experimental

populations

which contains all of

those which are better than DO'

A distribution-free

selection rule

for achieving the experimenter's

goal is proposed and studied.

This rule is based on data obtained through a partially

sequential

sampling scheme, which was introduced and discussed by \%lfe (1977a, 1977b) in the context of a two-sample median test. The proposed selection procedure is designed to satisfy the requirement

that the probability

of a correct selection

(CS) is not

less than p* (a specified constant such that (O.S)k < p* < 1). on P(CS) is referred to as the p~, - condition.

This requirement

A modification

for use when a particular

also presented.

quantile is of interest is

For each case, a CS occurs if the experimenter's

goal is achieved.

The proposed rules are designed to satisfy the

p* - condition. 1.2 THE SELECTION RULE fu~~ peeS) A partially

sequential

sampling scheme, considered by Holfe

in 1977, was later used by Costello and Hol£e (1980) to construct a distribution-free procedure.

treatments versus control multiple comparison

The proposed selection procedure

is based on observa-

tions obtained through a sampling scheme similar to that used by Costello and Wolfe.

Under this scheme, first a random sample of

specified size n = 2s + 1 is observed from the control population DO and then observations

are obtained sequentially

from each of the experimental sequences of observations mutually

independent

exceeds M. obtain

It is assumed that the

from .the experimental

populations

are

and each one of these sequences is independent

of the sample from DO' from DO'

populations.

one at a time

Let H be the median of the random sample

Sampling from Di is continued until exactly one observation Let N. be the number of observations l

exactly

one observation

greater

The selection rule is: Select Di

if

Ni < c,

from D. needed to 1

than M) i = 1, ... , k.

SEQUENTIAL

SELECTION

PROCEDURE

705

integer such that the p.", - condition

where c is the smallest positive is satisfied. Given ~1, the random variables N. fo l.l.owsa geometric 1 as F.(M) i 1. Thus

N1, ... , Nk are independent

distribution

I-F.(n),

with parameter

1

and

as long

1

P{N.1 -< tnl}

= 1 - {F.1(t1)}t,

= 1, 2, ...

t

and P(M ~ x) = I(FO(x); 5+1, s+I), where I(x;a,b) is the usual incomplete

Let J = J(F 0' F l'

beta function.

... , F k) denote the subset of subscripts

i such that Di is "better" than DO'

Since a CS occurs whenever

the selected subset contains all of the "better" experimental populations, P(CS) = P{N. < c for all i

=

I

J}

£

1

1

[1 - {G.(u)}c]) dI(u; s+1, s+l)

(n

o

i£J

1

where Gi = Fi(FOl) for i £ J. It may be noted that G is a i distribution function on (0,1). Clearly, the generalized least favorable

configuration

(GLFC) is obtained when J={l, 2, ... , k] ;

that is, when all of the experimental than DO'

populations

are better

Let E=(FO,F1,···,Fk)·

It is assumed that whenever Di is better than DO' Gi(u) < G(u) for a specified distribution

function G.

Thus k

1

P(CSiGLFC)

> I[l

-

u

{G(u)}cl

dI(u; s+l, s+l)

and hence inf peCS)

r

=f

which is a nondecreasing

k

1

[1 - {G(u)}cl

dI(u; s+l, s+l),

0

function of c for fixed k, s, and G.

the constant c is chosen such that the infimum of P(CS) is at least P"'.

Now

706

BRISTOL AND DESU

2.

GOAL I (SELECTING THOSE AT LEAST AS GOOD AS A CONTROL)

Here interest lies in the selection of those experimental populations which are at least as good as nO' as given by Definition 1. Definition

1:

An experimental

population n.1 is said to be ~

at least as good as nO if Fiex) ~ FO(x) for all x. Here G(u) = u, 0 < u < 1, so that the infimum of the probability of CS over

r

is

+ 1)Jl(1 inf P(CS I I) = ( s + 1) (2ss + 1 0 -uc)k u s(l -u )sd u

t = PCclI), say. Values of the smallest integer c such that P(cII) is not less than 0.95 are given in Table I for k

=

1(1)5 and s = 2(1)10(5)20,

with values of PCciI).

3.

GOAL II (COHPARISON I·IITHRESPECT TO A QUAIHILE)

In some situations, the experimenter

is interested in com-

paring a certain quantile of each experimental that of the control population

population

w i th

to judge whether it is better than

the control. Let q(a;F) denote the a-quantile of a cdf F. of the importance of certain a-quantiles,

In view

the following modified

sampling scheme is used to collect the data. A random sample of size n is observed from DO' vhe re l~(n+l)a~n.

Let t be the unique integer such that t~(n+l)a