Application of statistical inference for partially ordered hypotheses

Pergamon

Computers ind. EngngVol. 27, Nos 1-4, pp. 35-38, 1994 Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0360-8352(94)00119-7 0360-8352/94 $7.00 + 0.00

A p p l i c a t i o n of s t a t i s t i c a l inference for p a r t i a l l y o r d e r e d h y p o t h e s e s Naoto Hoshino and Yoichi Seki

G u n m a University, Kiryu 376, Japan

have been obtained mainly for the simple order, simple tree and umbrella hypotheses[i,2,5,6]. We give graphical representations of these hypotheses in Table 1. In this table, an arc from i to j implies Pi >_ I~j. We have obtained the theorems which present computational methods using some numerical integration for the level probabilities of the two hypotheses: [/A1,... ,/~m--1] --~ ~/rn --~ [Pm+l .... , #k] and [/q,...,p,,] >_ [~,,+t . . . . . /~k](see, Table 1). For the sake of convenience, we refer to these hypotheses as type A and type B respectively. The result for type B permits us to select unordered s::~:~::~:m normal populat:,c:~. In this paper, we investigate a problem of ranking k normal populations aald selecting (ordered) superior m out of them. This problem corresponds to a test for the hypothesis: Pl _> " " _> #m _ [/*re+l,... ,~Lk](see, Table 1), which is referred to as type C. If we want to select inferior m, the inequality sign ">_" will be replaced by '_<'. The type C hypothesis is eonsidered as simple tree when m = 1 and simple order when m = k - 1, k. We show that the level probabilities for the type C can be reduced to that of the type A, and derive the computational method for the level probabilities. Finally, an example of practical applications is given to illustrate our results.

Abstract In quality control and decision support systems, ranking and selection are very important, because we often encounter situations in which we need to rank several objects and to select some out of them. We investigate a problem of ranking k(> 2) independent nonnai populations and selecting superior or inferior m(<_ k) out of then:. The purpose of this paper is to formulate the problem as that of statistical hypothesis test, and to give critical values for practical use. We present computational methods for the level probabilities, which are ~oo,te,! for fi:::tir_g t ~ ,~:t:?alvalues. K e y words : ranking and selection, level probability, partially ordered hypothesis, likelihood ratio test.

1

Introduction

The ranking and selection problems can be formulated as those of statistical hypothesis tests for partially ordered hypotheses, when we have some candidates for the choice from prior information, such as the experiences made to date. For example, consider that we have k independent normal populations with unknown meals Pi (i = 1,..., k) and common variance a 2 and believe from the prior information a population has the biggest mean. To confirm the belief we may collect samples from the populations, and test for the null hypothesis of homogeneity H0 against the partially ordered hypothesis Hi: Pl _> Pi (i = 2 , . . . , k ) , which is called simple tree and represented by the notation: Pl >_ [#'~,..., #,.]. If H0 is rejected then we can decide that the belief is true, and if Ha is accepted then we suspend the decision and collect filrther more smnplcs and so on. In the case of ranking all k populations simply, the hypothesis is of type: lq -> "" • -> #k called simple order. A theory of testing for partially ordered hypotheses was proposed by Bartholomew[2,3], and extended by Barlow et at[l], Robertson et a/.[5] and so on. In order to find critical values for such tests, we must compute level probabilities explained in Section 3. Formerly, the level probabilities

Table 1 Some types of partially ordered hypotheses Hypothesis

Graphical representation

simple order:

~:>..->l*k

, 1

~ . . . k2

simple tree:

/~.1 > [lL2. . . . . pk]

:-~3

umbrella: ~q _< " " <_ p~, _> "'" _> pk-

h ~.5- ' " . . . .

_

type A: [pl . . . . . pro+l] >

2 ~ . + 2

[/'/m+l . . . . . Ilk]

ra-1 ~

___ [.°,+, .....

I,,.]

35

,k]

,

m+l *

~ ~ : ~

type C:

m>"'->~L.,

,

kq

:

p,.

type B: [~,, ..... I~,,,1 -> [~,o,+~ .....

.k

2 k

2

" ::

m-,

m~.a::

~",

36

2

Selected papers from the 16th Annual Conference on Computers and Industrial Engineering Ci = Card(Bi). Then B t , . . . ,Bt are called level sets. Let £tk be the collection of all partitions of X into I level sets.

Likelihood ratio tests

Let Yij (i = 1 , . . . , k ; j = 1,...,n~) be a random sample of size n~ from k independent normal populations with unknown means /ai and common variance a 2. When a 2 is known, assuming a 2 = 1 without loss of generality, likelihood ratio tests for the null hypothesis Ho : /al . . . . . /ak -=/a against partially ordered alternatives Hi: partial order restrictions on components of # = (#l,...,/a~)', reject H0 for large values of

For a given partition, define £ ' on {1 . . . . . l} by i ~' j, if there exist s E B~ and t E Bj with s ~ t. On each B~, let £i denote the restriction of ~ to Bi. Furthermore, let P'(l, l; C1,..., Cl) be the level probability with partial order ~ ' and weight vector ( C h . . . , Cl), and let Pi(1, Ci) be the level probability with partial order £~ and equal weights. Then, for 1 < l < k, we have the following equation[I,5]: l

P(l,k)=

k

~ (BI,...,BI) E £tk

P'(l,l;C, ..... Ct)HPi(1,Ci ). (3) i=l

i=l

where/5 is the maximum likelihood estimator of # under Ho k n1 nl and given by/2 = Ei=t i Y i / E i = Ik ni and ~i = ;-T , ~i=t Yq, and t~ is the maximum likelihood estimator of/a, under Ht, that is, a value of #i mini,nizing Ei~=t w i ( y i -- /ai) 2 under Hr. The problem of computing ~' is that of a quadratic programming, and many algorithms have been proposed for such problem[I,4,5]. When a ~ is unknown, the likelihood ratio statistic is k

k

4

ni

For 1 _< l _< k, let P(l,k;w) be the probabilities that /5, takes exactly l distinct values when Ho is true. P(1, k; w) are called the level probabilities with weight vector w = (w~,...,w~);. Note that tim level probabilities depend on a partial order which determines Hr. Then the null distributions of these test statistics are respectively, for any real number c > 0

Computation of Pc(m)(1, k)

Let PA(m)(l, k; w) denote the level probabilities with weight vector w of the type A alternative. We obtain Pc(,o(l, k) of the type C alternative: /at _> ... _> #m >_ [/a,,+l,... ,/ak], utilizing the probability PA(,,)(k, k; w), which is computed as follows. Suppose that ~t,...,~k are independent normal variables with a common mean 0 and variances u,~t,..., w[ ~. Then we have that

PA(~)(k,k; w) =

k

Pr(2~ > c) = ~ , P(l, k; w)Pr(x~ >_c),

This recursive formula implies that the computation of P(I, k) with equal weights requires P'(1, l; Ct ..... Ct) with unequal weights. It is not possible in all cases of partial order to compute the level probabilities with unequal weights. In the next section, we derive the computational method for the level probabilities with equal weights, denoted by Pc(m)(l, k), of the type C alternative.

(1)

Pr{ min(~l . . . . . Y,,-1) > ,~,, > max(~,,+l . . . . . ~Ok)}.

l=l

where X~ is a chi-square variable with u degrees of freedom(x0~ = 0) and k

P r ( / ~ > e) = ~ P(l, k; vo)Pr{B(t-I)/~,(N-t)/2 >_c},

(2)

Note that PA¢,~)(k,k;w) don't depend on # and a 2. The level probability P.4cm)(k, k; w) is obtained by the numerical integration of

PA(,n)(k, k; to) =

I=1

where B,,b is a beta variable with parameters a and b(Bo,b = 0) and N = E~=I ni. Thus, to find critical values of 2~ and ~?~ statistics, we must compute the level probabilities P(I, k; w).

3

Level P r o b a b i l i t y

In this section, we consider the case of equal weights : wt = . . . . w~ in which all the sample sizes nl are equal to the same value n. If all the weights are equal then "P(I, k; w)" are abbreviated to 'P(l, k)'. In general, the level probabilities P(l, k) are computed recursively. First, P(k, k) is computed, and then for 1 < I < k, P(l, k) are computed using recursive formula. Finally, P(1, k) is computed from the equation E~=I P(l, k) = 1. For 1 < I < k, let Bt,...,B~ be a partition of X = {1,... ,k} with partial order ~ determining H1 such that /2~ takes the same value for any i E Bj, 1 <_j <_ l and let

i:i

~

I

KV w~ /

where (I, and ¢ are the distribution and density function of a standard normal variable respectively. Rewriting the level probabilities for the type C alternative according to the equation (3), we have the following result. Combinatorics is the basic ingredient of the proof of our theorem. Let X ~ = { 1 , . . . , s - 1}, s _< m be asubset o f X = {1,..., k} with partial order ~ of type C. The restriction of £ to X ~ is of simple order. Given a partition B1,..., Bt of X, let Bt .... , Bt (t < l) be a partition of X' with simple order into t level sets. Then there is a corresponding t-partition, denoted by [617~-.. 6]~], of the integer s - 1, where 7i expresses the number of C1,..., Ct which take the same value 6i, 61 > ... > 6x > 0, 71 + ' " + % = t and ")'t61 + ' " + 7x6x = s - 1. Conversely, given a t partition of the integer s - 1, there are correspondingly t!/(7~!... 7~!) partitions of X ' into t level sets. Furthermore, each of these

Selected papers f r o m the 16th Annual Conference on Computers and Industrial Engineering partitions of X' corresponds to a cardinal number vector (C,~o) . . . . ,C,(t)), where r is a permutation of {1 .... ,t} such that Ci Tt, C,,(1) for at least one i. For example, given the 3-partition [2211] of the integer 5, the corresponding three partitions of {1,2,3,4,5} are respectively {{1,2}, {3,4}, {5}}, {{1,2},{3}, {4,5}} and {{1}, {2, 3}, {4, 5}}. These partitions correspond to the ca'dinal number vectors (2, 2, 1), (2,1, 2) and (1,2, 2) respectively. Now we can obtain the theorem of the level probabilities. [Theorem I] The level probabilities Pc(,.o(l, k) satisfy

Pc(ml(k, k) = (k - m)!/k!, and for l < l < k

37

Since both £ ' and ~gt+l are of type C, from the equation (3) for 1 < I < k, we have that

Pc(m)(l,k) - E E s

E

P,(1,c,)

t (Bl,...,BOE.et,m_l

xPc
l- t- 1

x Pc(t+l)(l, l; C1,..., Ct, Ct+l, 1 , . . . , 1), where £t.,-1 is the collection of all partitions B b . . . , Bt of X' into t level sets and Pi(1, Ci) is the level probability with simple order and equal weights, and it is well known that P/(1, C~) = 1/Ci. Let S be the collection of all permutations 7r of {1,... ,t} such that Ci ~ C,(0 for at least one i. Then the summation over ,£t,~-1 becomes as follows,

Pc(,,)(l,k) =

k-m

8=rnax(1,m-k+l)

×E

t=max(Ii,>_2},rn-k+l-1 )

i:i. (~l

pc(,,,_.+l)(1,c,+~)

K~,s-I k i = l

x ~

~

Pc'( . . . . +1)(1, Ct+0

x ~ Pc(t+l)(l, l; C,O) . . . . . C,(t), Ct+l, 1. . . . . 1).

Kt,s-I

rES

XPA(t+I)(I, 1; C1,..., Ce, Ct+l, 1,..., 1), where I{~>2} denotes the indicator of the event {s >_ 2}, that is, I{,>2} --= 1 if s >_ 2 and I{~>2} = 0 otherwise, K,,~-I is the collection of all t-partitions [~' .-. 6~] of the integer s - l , and C,+1 = k - l - s + t + 2 . Pc(m)(1,k) can be "~: p , computed from the equation Et=l c(,,)~, ~:) = 1.

Letting St be the collection of all permutations r of {1,... ,t}, then the last line of the above equation can be transformed as follows,

Pc
[Proo~

I.i=l '7"") ~eS, Pc(t+1)(1, l; C~(1),. . . , C~(t), Ct+l, 1 , . . . , 1)

For 1 = k, we have that

Pc.(,,,)(k,k) =

=

Pr{ 91 > " " > 9m > 9~(m+1) > ' "

> 9,(k) },

lr

~.i=1 T i . )

Substituting this expression, we can obtain the theorem. []

where, in the case of equal weights, YI,..., 9k are independent random variables with the same normal distribution and r runs over all permutations of {m + 1,..., k}. Since for all fixed ~r, the probability in the right side equals I/k[,

Pctm)(k, k) = (k - m)!/k!. Next, the equation (3) is applied for 1 < l < k. Let B1 .... ,Bt be a partition of X = {1,...,k} into I level sets as shown in Fig.1. In this figure, B 1 , . . . , B t is the partition of X ~ = {1,... ,s - 1} into t level sets, B,+t is the level set containing m, and Bt+2,...,BI are the level sets of cardinality 1 and the number of ways for selecting B,+2 .... ,Bt is

I -kt--m 1

/ ' Thus

we

have that C,+1

/

k-(s-1)-(l-t-1)=k-l-s+t+2. Here, s i s a n y integer in the range [max(l, m - k + l ) , m] and t is any integer in the range [max(I{~_>2},m - k + l - 1), min(s - 1, I - 1)].

1

fii "25 l PA(t+l)(l,l;C1,...,Ct,Ct+l,1,...,1).

2

. . . . .

~

m+l m +:2

k-l+t+l

BI i BI,..., Bt

~'~.

k-l+t+2 ~

B,+2

k

Bt

t÷

~

Fig. 1 Typical level sets for the type C alternative

This theorem provides the reeursive method to compute the level probabilities for the type C alternative. We present the computed values of Pc(,~)(l, k) for k _< 7 in Table 2, where the probability PA(,,,)(k, k; w) is computed by Romberg's numerical integration. Since the type C alternative is of simple tree when m = 1 and of simple order when m = k - 1, k, the level probabilities of these cases are not tabulated. Pc(m)(1, k) is also excluded from the table, since it can be easily computed from the equation ~]~=1 Pc(~)(l, k) = 1. The entries in Table 2 permit us to Table 2 The level probabilities Pc(,,,)(l, k)

k m

I=2

I=3

1=4

1=5

1=6

1=7

4 2 5 2 3 2 3 4 2 3 4 5

0.4167 0.2500 0.3677 0.1147 0.2349 0.3313 0.0430 0.1210 0.2189 0.3024

0.3480 0.3951 0.3523 0.2956 0.3668 0.3510 0.1628 0.1893 0.3474 0.3465

0.0833 0.2500 0.1323 0.3520 0.2567 0.1659 0.3110 0.3290 0.2634 0.1903

0.0500 0.0167 0.1877 0.0781 0.0354 0.3089 0.2848 0.0991 0.0534

0.0333 0.0083 0.0028 0.1461 0.0500 0.0177 0.0074

0.0238 0.0048 0.0012 0.0004

38

S e l e c t e d p a p e r s f r o m the 16th A n n u a l C o n f e r e n c e on C o m p u t e r s a n d I n d u s t r i a l E n g i n e e r i n g

obtain critical values for testing against the type C alternative from the equations (1) and (2), and hence now we can solve the problem of selecting ordered superior m out of k independent normal populations.

5

An applied

5

significant level a

I

example

We present an example of solving the ranking and selection problem as that of ~2 test. In some factory, machines A, B, C, D, E, F and G have been used in the production processes of synthetic resins. Recently, a problem of selecting superior two machines which produce strong synthetic resins is discussed. According to the experiences made to date, we believe that the machine A seems to produce the strongest synthetic resin and the machine B seems to produce the second strongest one. An experiment to confirm our belief was made. Table 3 gives the data of the strength of synthetic resins produced using each machine. In each level, the sample size n is equal to 5. First, if we ignore the prior information then the usual F test is adopted for testing the null hypothesis of homogeneity H0 : /ZA = #B = PC = lZD = PS = P r = Pa against the unrestricted alternative : at least one of the equality of Ho does not hold. The F statistic is computed as F =

Table 4 Critical values of the E72 statistic for testing against the type C alternative when k = 7 and m = 2

7

-

= 1.94,

~i=l(Yi - / 5 ) 2 / 6

EL~ E~=~(~,j - ~)~/28 where the maximum likelihood estimate of # under H0 is computed as fi = 18.8. Since the computed F = 1.94 is smaller than the critical value 0.25 at the significant level 0.05 obtained from the F distribution with 6 and 28 degrees of freedom, the null hypothesis H0 is not rejected. However, taking account of the prior information, this example corresponds to the case of testing Ho against the type C alternative H1 : PA >-- # s >-- [flC,~D,~E,PF,PG]" We give the critical values of the E] statistic for testing against H~ in Table 4. From the data in Table 3, the maximum likelihood estimate of Iz under H1 is computed as /5] = ~A = 20.2, /5~ = /5~ = YB__+/)E _ 19.7, 2 / 5 ~ = y c = 1 8 . 6 , /5~)=y0=17.8, /5} = YF = 17.6 and /5~ = YG = 18.0.

From Table 4, the critical value at the significant level a = 0.05 and sample size n = 5 for H1 is 0.253. Since the computed/~2 = 0.290 > 0.253, the null hypothesis Ho is rejected. This result shows that synthetic resins have the largest strength when A is used and have the second largest strength when B is used. The example shows us that the ~2 and E~ tests are more powerful than the F test.

6

Conclusions

We have derived the recursive formula for the level probabilities Pc(m)(l, k), and presented the table of their values. Our results will be useful for solving the problem of se!ect'.ng superior or mt~r~::: m out of k independent '.:ormai populations as that of ~ a n d / ~ tests. The example given in Section 5 illustrates how to apply our results to the ranking and selection problems. Finally, we remark that the ~ a n d / ~ tests are unavailable without the prior information, because these tests may merely guarantee the significant level under the null hypothesis of homogeneity against the partially ordered alternative.

References [1] Barlow, R. E., Bartholomew, D. J., Brenmer, J. M. and Brunk, H. D.(1972). Statistical Inference under Order Restrictions, Wiley, New York. [2] Bartholomew, D. J.(1959a). '% test of homogeneity for ordered alternatives," Biometrika, Vol.46, pp.36-48.

Substituting these estimates, we have ,

~ = 5 ~(/5~ -/5)2 i=1

y,~ -/5)~ = 0.290. '=

'=

Table 3 The strength of synthetic resins (unit: kg/mm "~) ~ B

1

2

3

4

5

18.4 19.6 18.0 21.8 19.1 18.9

18.8 18.7 17.9 22.3 16.6 19.2

19.0 17.1 16.8 18.8 16.5 17.1

17.9 18.2 19.3 19.2 16.9 16.2

23.4 19.4 17.0 17.4 18.9 18.6

[ Yi I

[3] Bartholomew, D. J.(1959b). "A test of homogeneity for ordered alternatives II," Biomet~'ika, Vol.46, pp.328335. [4] Dykstra, R. L.(1983). "An algorithm for restricted least squares regression," J. Amer. Statist. Assoc.. Vol.78, pp.837-842. [5] Robertson, T., Wright, F. T. and Dykstra, R. L.(1988). Order Restricted Statistical Inference, Wiley. New York. [6] Shi, N. Z.(1988). "A test of homogeneity for umbrella alternatives and tables of the level probabilities," Comm. Statist., Theory and Methods Vol.17, pp.657-670.