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Sequential-typenonparametric test usingMann-Whitney statisticsUttam Bandyopadhyay & Atanu BiswasPublished online: 02 Aug 2010.

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Journal of Applied Statistics, Vol. 26, No. 3, 1999, 301± 308

Sequential-type nonparametric test usingMann± Whitney statistics

UTTAM BANDYOPADHYAY1

& ATANU BISWAS2,

1Department of Statistics,

University of Calcutta, India and2Applied Statistics Unit, Indian Statistical Institute,

Calcutta, India

ABSTRACT The paper provides a nonparametric test for the identity of two continuous

univariate distribution functions when observations are drawn in pairs from the popula-

tions, by adopting a sampling scheme which, using Mann± Whitney scores, generalizes the

existing inverse binomial sampling technique. Some exact performance characteristics of

the proposed test are formulated and compared numerically with existing competitors of

the proposed test. The applicability of the proposed test is illustrated using real-life data.

1 Introduction

The problem of testing the identity of two treatment eþ ects has been considered by

many authors in recent years (see, for example, Jennison & Turnbull, 1989, 1991,

1993; Kim & DeMets, 1987; Lan & DeMets, 1983; Pocock, 1977), by taking obser-

vations in pairs. In the present paper, we consider the problem under a univariate

set-up. The problem may be described as follows: Let F1 and F2 be two unknown

continuous univariate distribution functions. Then, by drawing independent obser-

vations from F1 and F2 in pairs, our object is to test the null hypothesis

H : F1 5 F2 (1)

against a class of one-sided alternatives

H a : F2> F1 (2)

when F2> F1 means F2(x) > F1(x) for all x with at least one strict inequality. In

particular, if we take F1 5 F(x) and F2 5 F(x 2 d ), 2 ` < x, d < ` , or F1 5 F(x)

and F2 5 F(x exp 2 ( d )), x> 0, 2 ` < d < ` , we have expressions (1) and (2) as

H : d 5 0 against H a : d > 0 (3)

Correspondence: U. Bandyopadhyay, Department of Statistics, University of Calcutta, 35 Ballygunge

Circular Road, Calcutta 700019, India.

0266-476 3/99/030301-0 8 $7.00 � 1999 Taylor & Francis Ltd

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302 U. B andyopadhyay & A. B iswas

For this problem, samples corresponding to F1 and F2 may be drawn by adopting

the following inverse scheme of sampling: Let X and Y denote the performance of

old and new treatments respectively. Suppose that X ~ F1 , Y ~ F2 and they are

independent. Here, X and Y are observed in pairs. Corresponding to the i th pair

(X i , Y i), a random variable Z i is de® ned by

Z i 5 {1, Y i> X i

0, Y i < X i

(4)

Obviously, the Z i terms are iid Bernoulli (p) random variables, where

p 5 p(F, d ) 5 ò F1 (x) dF2 (x)

This equals 12 under H and is greater than 1

2 under H a . Now, writing

Sn 5 Rn

i 5 1

Z i , n > 1

sampling is stopped at that n 5 N for which Sn 5 r is reached for the ® rst time,

where r is a predetermined positive integer. If

SÂ n 5 Rn

i 5 1

(1 2 Z i), n > 1

then SÂ N has the negative binomial (r, 12) distribution under H . Hence, given

a Î (0, 1), we can ® nd a non-negative integer c a that satis® es

PH {SÂ N < c a } < a < PH {SÂ N < c a + 1} (5)

Thus, a level a test rule that corresponds to the scheme of sampling mentioned

may be as follows: Stop sampling at the nth draw and accept H if

SÂ n> c a , 1 < n < N

Reject H if

SÂ N < c a (6)

This test is unbiased and consistent. The average sample number (ASN) of pairs

of the test is smaller than that of the test [SÂ N < c a ]. However, power functions of

the two tests will remain the same.

For the present problem, Bandyopadhyay & Biswas (1995) suggested some

nonparametric group-sequential-type tests using the inverse sampling scheme

mentioned here. A natural generalization of this sampling scheme is a scheme

which replaces {Sn } by Mann± Whitney statistics {Un }, and r by q(r), i.e. an integer-

valued quadratic function of r. In the next section, a test procedure is proposed

using this generalized inverse sampling scheme.

2 Proposed test procedure and its diþ erent performance characteristics

Here, the observations are drawn one by one and in pairs. After drawing the i th

pair, the diþ erences Y i 2 Xk (1 < j, k < i ) are observed instead of observing the

diþ erences Y i 2 X i . Then, an indicator variable Z ij is introduced as follows:

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Nonparametric test using Mann± Whitney statistics 303

Z i j 5 {1, Y j 2 X i> 0

0, Y j 2 X i < 0(7)

This leads to the following sequence of Mann± Whitney statistics:

Un 5 Rn

i 5 1Rn

j 5 1

Z i j , n > 1 (8)

The stopping variable N is then de® ned by

N 5 min{n: Un > q(r)} (9)

Now, writing

UÂ n 5 Rn

i 5 1Rn

j 5 1

(1 2 Z i j), n > 1 (10)

our proposed test is based on the random sequence

{UÂ n , 1 < n < N } (11)

Since UÂ n and N are both stochastically smaller under H a than under H , a level a

test is given by expressions (5) and (6) with SÂ n replaced by UÂ n , 1 < n < N , and c a

replaced by u a . Finally, to obtain an exactly size a test, the usual randomization

technique may be adopted. The test is based on Mann± Whitney statistics, so is

obviously exactly distribution free. The test can easily be shown to be unbiased

and consistent.

To obtain some exact performance characteristics of the proposed test, we

consider, for convenience, simply the non-randomized test. One of the character-

istics is measured by the power function of the test. To obtain such a function, we

take, for a given F, diþ erent d , and write h 5 (F, d ). The power function of the test

procedure is then

P ( h ) 5 P{UÂ N < u a ½ h } (12)

We also consider another characteristic which is the ASN function for the proce-

dure. This can be expressed as

A( h ) 5 R`

n 5 1

nPh (In e Jn ) (13)

where the events In and Jn are, respectively, given by

In 5 (U j< q(r), j 5 1, 2, . . . , n 2 1, Un > q(r)) (14)

Jn 5 (UÂ j < u a , j 5 1, 2, . . . , n 2 1, UÂ n > u a ) (15)

Note that equation (13) has an upper bound R t pN (t ½ h ), where pN (t ½ h ) is the

probability mass function (pmf ) of N.

Now, to ® nd the distribution of N , we start with the distribution of U t0 , where

t0 5 [(q(r))1/2] + 1 with [x] as the greatest integer contained in x, and write

U t0 + 1 5 U t0 + UÅ t0 + 1 (16)

where

UÅ t0 + 1 5 S t0 + 1 + SÂ t0 (17)

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304 U. B andyopadhyay & A. B iswas

FIG. 1. Power curves for the proposed test and its two competitors.

S t0 + 1 is the additional score for the inclusion of the variable X t0 + 1 in the system

{(X i , Y i), i 5 1, 2, . . . , t0 }, and SÂ t0 + 1 is that for the variable Y t0 + 1 in the system

{(X i , Y i), i 5 1, 2, . . . , t0 ; X t0 + 1}. Using this breakdown, we can ® nd, for any j, the

conditional pmf of U j + 1 , given U j 5 u j . We denote the pmf by p j + 1 (u j + 1 ½ u j). Finally,

we obtain

pN (t ½ h ) 5 P{U l< q(r), l 5 1, 2, . . . , t 2 1, U t > q(r) ½ h }(18)

5 Rq(r) 2 1

ut0 5 0

. . . Rq(r) 2 1

ut 2 1 5 0

[Pt0 (u t0 )pt0 + 1 (u t0 + 1 ½ u t0 ). . . p t 2 1 (u t 2 1 ½ u t 2 2 )P{U t > q(r) ½ u t 2 1 }]

Finally, in an illustrative computation, we ® rst ® nd the cut-o þ point u a , taking

a 5 0.065 and q(r) 5 21. Then, at diþ erent values of h , we compute the powers

and ASNs of the procedure by simulation. These are shown in Figs 1 and 2. The

distributions of N at diþ erent d values are shown in Table 1, taking normal, double

exponential and exponential parents.

The three values in each cell are those of the three respective parents.

3 Comparison of the proposed test procedure with some existing

competitors

In this section, we consider the following competitors of the proposed test

procedure:

(i) the test procedure discussed in Section 1;

(ii) ® xed-sample-size test, corresponding to the proposed test.

For competitor (i), the power function is given by

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Nonparametric test using Mann± Whitney statistics 305

FIG. 2. ASN curves for the proposed test and competitor (i).

P1 ( h ) 5 Rc a

k 5 0 (r + k 2 1

r 2 1 ) pr(1 2 p)k (19)

To obtain the ASN function, let W m be a random variable that denotes the number

of successes up to m failures. Then, writing

w (m) 5 E(W m ½ Wm < r) 5 Rr 2 1

s 5 0

sg(s) (20)

where

g(s) 5 (m + s 2 1

m 2 1 )(1 2 p)mp

s/ Rr 2 1

k 5 0(m + k 2 1

m 2 1 )(1 2 p)mp

k (21)

we have the expression for the ASN function as

A1 ( h ) 5 Rc a

u 5 0

(u + r) p h (u) + Ra

u 5 c a + 1

[c a + 1 + w (c a + 1)]p h (u) (22)

where p h (u) is the pmf of the negative binomial (r, p) distribution. Note that r /p is

an upper bound of A1( h ).

For competitor (ii), a level a test that corresponds to the proposed test is to

reject H if

UÂ n0 < n20 2 q(r) (23)

where

n0 5 á [(u a + q(r))1/2] ñ (24)

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306 U. B andyopadhyay & A. B iswas

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Nonparametric test using Mann± Whitney statistics 307

TABLE 2. Ovarian weight (grammes) of the soft-shelled turtle Lissemys punctata

punctata , using four treatments

A 1 : Control A2 : LH-treated A 3 : E-17 b -treated A4 : FSH-treated

28.0 30.0 30.0 50.0

27.4 26.7 26.7 43.4

32.4 27.2 27.2 44.9

25.4 28.0 28.0 45.0

32.2 51.3

Notes: LH, leutimizing hormone; E-17 b , estradiol-17 b ; FSH, follicle-stimulating

hormone. Dose, 15 l g per 100 g body weight for 15 days.

with á [x] ñ as the nearest integer to x. Let us denote the power function of the test

by P2( h ).

For competitors (i) and (ii), we have to take r 5 7 and n0 5 5 respectively. Each

test is suitably randomized to obtain the size 0.065. Powers and ASNs are computed

at diþ erent h values and are shown in Figs 1 and 2. Comparing the graphs, we

observe that the proposed test procedure is an improvement, in the sense that it

has almost the same power with a smaller ASN.

4 Illustrative example with real-life data

Here, we use an unpublished data set (Sarkar, 1993), which gives the eþ ects of

some treatments, including the control treatment, on the ovarian weight of the

soft-shelled turtle Lissemys punctata punctata. Because the turtle is an endangered

animal, from an ethical point of view, the smaller ASN is encouraging. The required

portion of the data is given in Table 2. We take a 5 0.065, q(r) 5 21 and u a 5 7.

While comparing A1 and A 2 , we have U3 5 2, UÂ 3 5 7, U4 5 7 and UÂ 4 5 9, which

accepts the null hypothesis of equality of treatment eþ ects at a cost of four pairs of

sample observations. While comparing A 3 and A4 , we have U3 5 9, U4 5 16 and

U5 5 25, yielding N 5 5. Here, UÂ 5 5 0, which rejects the null hypothesis.

5 Discussion

First, note that, in recent years, the group sequential method has often been

adopted for testing the equivalence of two treatments. In this method, a test is

carried out in K stages, yielding a maximum of K decision points, where K is a

predetermined positive integer. The proposed procedure can be easily extended in

this frame as follows.

For the kth (k 5 1, 2, . . . , K ) group, we de® ne

N k 5 min{n: Un > qk (r)} (25)

where one possible value of qk(r) may be k2q(r), since, after kn pairs of (X i , Y i )

terms, the next n pairs of (X i , Y i ) terms yield (2k + 1) n2 possible Z ij terms.

Then, de® ning UÂ n as in equation (10), a possible extension would be based on

{UÂ n , 1 < n < Nk }, 1 < k < K . Now, using these variables sequentially, a test may be

described as follows.

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308 U. B andyopadhyay & A. B iswas

Reject H at the kth stage (1 < k < K ) if

UÂ N l> c l , l 5 1, 2, . . . , k 2 1, UÂ Nk < ck

Accept H if

UÂ n > ck , for some n: 1 < n < N k , k 5 1, 2, . . . , K (26)

where {c1 , . . . , ck } are non-negative integers determined in such a way that the test

has overall level of signi® cance a . Diþ erent performance characteristics of the test

are under investigation.

Next, we indicate the applicability of the present approach to the following

decision-making problem:

d1 : d 5 0, d2 : d > 0, d3 : d < 0 (27)

Suppose that Z ij , Un and UÂ n are as in Section 2. Then, we set our stopping variables

N* as

N* 5 min{N *1 , N *2 } (28)

where

N *1 5 min{n: Un > q(r)}, N *2 5 min{n: UÂ n > q(r)} (29)

We stop sampling at the nth draw and accept d2 or d3 if UÂ n < a a or Un < a a , and

accept d1 if UÂ N * > a a and UN * > a a . Diþ erent performance characteristics of the

decision rule can be similarly set, and these are intended to be communicated in a

separate paper.

Acknowledgements

The work of the second author was supported by a Research Fellowship from the

Council of Scienti® c and Industrial Research (Sanction 9 /28(391) /94-EMR-I).

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