On nonparametric profile analysis of several multivariate samples

Journal

of Statistical

Planning

and Inference

9 (1984) 285-296

285

North-Holland

ON NONPARAMETRIC PROFILE ANALYSIS SEVERAL MULTIVARIATE SAMPLES

OF

Vasant P. BHAPKAR Department of Statistics, University of Kentucky, Lexington, KY 40506, USA Received

3 June

1982; revised manuscript

Abstract: A unified development of several

multivariate

U-statistics analysis

14 January

profiles

are discussed

1983

is offered for asymptotically distribution-free profile analysis This includes as special cases procedures based on generalized

and also those based on linear rank statistics.

of location

properties

samples.

received

and also scalar profiles.

for tests of hypotheses

Furthermore,

Finally,

it includes

asymptotic

and subhypotheses

power

as special cases and consistency

of interest.

AMS Subjecf Classification: 62H15, 62GlO. Key words andphrases: Generalized tency;

Asymptotic

U-statistics;

Linear

rank statistics;

Profile

analysis;

Consis-

power.

1. Introduction and notation Let X.’ = (x!” XJ2’ t = 1,2 , . . . , n;, be independent random vectors from rt 9-.*y X.(P)), It It the i-th populition with continuous c.d.f. Fj. Assume that we have such independent random samples from k populations for i = 1,2, . . , , k with a total sample of size N= Cj ni on p variables. In the normal parametric model it is assumed that the populations are p-variate normal with means&=(#) I ’ p!*),...,,~!~)) 9 i= 132,**-, k, and common nonsingular I covariance matrix Z.‘The hypothesis df homogeneity of populations is then

H,:

&=y2=...=~k,

(1.1)

while the hypothesis of parallelism of population H,:

profiles is

~!1)-jll(‘)=~(!2)-~~2)=...=iu!P)_~u1(P),

(1.2)

for i=2, . . . . k. H, may be interpreted as the hypothesis of no interaction between variables and k populations. Such a hypothesis formulated in the form (1.2) is meaningful only when the p variables are commensurable. This is the case, for example, when thep variables represent repeated measurements on the same subject at p different time points, or when they represent the scores for the same subject on p tests in a battery, with more or less identical scale of measurement for each variable.

p

0378-3758/84/$3.00 0 1984, Elsevier Science Publishers

B.V. (North-Holland)

V.P. Bhapkar / Profile analysis

256

There are several well-known (see, e.g. Kshirsagar (1971), Morrison (1976)) MANOVA procedures available for testing the hypothesis H, and, similarly, for the hypothesis H,. The need for developing suitable nonparametric analogs has been recognized for quite some time. For the homogeneity hypothesis H,*: Fi =F,=...=F,,

(1.3)

asymptotically distribution-free procedures have been developed by a number of workers (see Bhapkar (1965), Suguira (1965), Tamura (1966)). A nonparametric analog of hypothesis H, was developed by Bhapkar and Patterson (1977, 1978) for the location problem and they offered a class of asymptotically distribution-free procedures. Similarly, an analog was developed by Bhapkar (1979) for the scalar problem and a suitable class of asymptotically distribution-free procedures was offered Let XI = (X!” , , . . . ,X!@) represent independent random vectors with c.d.f. Fi, i = 1,2 ,..., x and F’= (F,,F 2, . . . , FJ. Denote by Ri(a) the rank of Xi’“’ among {X,‘“‘, j=l,..., k} for a=1,2 ,..., p and i=l,2 ,..., k. Define u!!‘(F) = P#%j].I

(1.4)

lJ

The k population profiles were defined to be parallel (1977, 1978) if F satisfies HT:

u(!)(F) = u!?‘(F) = 1, [J

. .. =

&‘(F) IJ

’

in Bhapkar and Patterson

(1.5)

for i,j=l,2 ,..., k. Under the location model we have i=1,2 ,...,

F,(x)=F(x-,M;),

k,

(1.6)

for some continuous c.d.f. F. In Bhapkar and Patterson (1977) it has been shown that if the marginal c.d.f.‘s Fca), ct= 1,2, . . . , p, of F are identical, then the condition (1.2) implies the condition (1.5). Under the scalar model, we assume that all the populations have the same location but possibly different scale parameters. Thus Fi (x) = F(B,:‘x),

(1.7)

where Bi is a diagonal p xp matrix with positive elements e!‘), 0?‘, . . . , O!p) along the diagonal. Bhapkar (1979) has shown that if the marginal c.d.f.‘s F@) of F are identical for a= 1, . . . . p, then the condition e!l)

eP)

e!p)

e,“’ -$5j=-*=+-)

(1.8)

for i=2,..., k, implies the condition (1.5). The nonparametric procedures for the homogeneity hypothesis H,* are based on

V.P. Bhapkar / Profile analysis

287

either (a) the generalized U-statistics (see, e.g. Bhapkar (1965), Puri and Sen (1971), Suguira (1965)) or (b) the linear rank statistics (see Puri and Sen (1971), Tamura (1966)). The procedures offered by Bhapkar and Patterson (1977, 1978) for the profile analysis in the location problem and by Bhapkar (1979) for the scalar problem are, for the most part, in terms of the generalized U-statistics. Recently, Chinchilli and Sen (1982) have offered linear rank statistics for profile analysis. In Section 2, first we develop another nonparametric analog of the hypothesis of parallelism of population profiles, which is particularly suited to the use of linear rank statistics. Section 3 deals with the broad regularity assumptions and, then, develops a unified treatment of profile analysis under these broad conditions. In Section 4, asymptotic power and consistency properties of a class of tests are developed. This class includes tests for the hypothesis that the profiles coincide, as well as those for the hypothesis of ‘parallelism of profiles’. Also, tests for subhypotheses are obtainable from the class discussed in Section 4.

2. Rank order analogs of parallelism The condition (1.5) appears to be one reasonable nonparametric analog of the hypothesis of parallelism of population profiles. This formulation has been seen in Bhapkar (1979), Bhapkar and Patterson (1977, 1978) to be particularly appropriate for test criteria based on generalized U-statistics in terms of ranks. In this section we consider another formulation that is appropriate for criteria based on linear rank statistics. Let all the random observations {X,l”‘, t = 1,2, . . . , ni, i = 1,2, . . . , k} on variable cx be arranged in increasing order, and suppose R:F) denotes the rank of X2) in this increasing sequence. Note that the probability of any tie is zero in view of continuity of the c.d.f.‘s Fi. Define Zi(z) = 1 if the u-th smallest in the ranking of {Xf), t = 1,2, . . . , nj, j= 1,2,..., k} is some observation from the i-th sample; otherwise Zi’,a’=O, for u=l,2 ,..., N, i=1,2 ,..., kand a=l,2 ,..., p. Let [ii”‘(F) = PF [z/t) = I].

(2.1)

We may say that the population profiles are parallel with respect to the rank order of N observations (in the combined sample) if &(;‘(l;) = &$‘(F) = -. . = &qI;),

(2.2)

forallu=l,2 ,..., Nandi=l,2 ,..., k. In contrast to this condition (2.2), the earlier condition (1.5) may now be interpreted as a definition of parallelism with respect to the rank order of a random k-tuplet. Consider now a linear rank statistic (2.3)

V.P. Bhapkar / Profile analysis

288

where {s,, u=l,2 ,..., N) is a given system of scores. Then E

F

[@)I = I =

-!_ f n. ,.,

u=

@‘(I;) rn ’

say.

(2.4)

Note that under H,*: Fi = F2= a-.= Fk,
.

(2.5)

All the profiles, for statisics S, are parallel to the axis (and hence automatically parallel to each other) if (2.5) is true for i = 1,2, . . . , k. These profiles are parallel to the axis and, furthermore, coincide when viz’= Cr=‘=,s,/N for all i and a. In the case of a generalized U-statistic

(2.6) where @(“‘(X X 2, . . . ,X,) = $(Ry)), (X9) j 11 , 2l’ , ,.., k}. Here / ’ q;;)(p) = ‘#i(a)]

RI”) being the rank of Xi’“’ in the k-tuplet

=j$, b(j)r$%,

= $“‘(I;) 9 for all n*>

(2.7)

u@) are defined in (1 . 4; . lJ Here the i-th population profile, for the statistics S defined by (2.6), is parallel to the axis if the condition (2.5) is satisfied for q’s defined by (2.7). In general, then, we shall define parallelism (to the axis) of the i-th population profile by the relation (2.5) for statistics S. Observe that for linear rank statistics S, parallelism with respect to the rank order, viz. condition (2.2), would imply parallelism for every S. Similarly, for generalized U-statistics S, parallelism for the ranks of k-tuplets, viz. condition (1.5), would imply parallelism for every S. However, the converses are not true. Nor does it appear to be possible to be able to define parallelism, in an intrinsic manner, e.g. (1.5) or (2.2), for populations F, which would work for different types of statistics S. 3. Tests for profile analysis Let X, denote k independent random samples of p-vectors, of sizes n’ = n,), respectively from continuous populations F = (F,, F2, .. . , Fk). We ($$..., assume the following regularity conditions: (A) (i) The sample sizes nj -+ 03 in such a way that

for N = ci ni ni/N + Pit

V.P. Bhapkar / Profile analysis

O
289

is a p-vector based on X, for i = 1,2, . . . , k and for each a= 1,2, .., , p, there is a linear con-

(3.1) where a’s are constants such that ain + ai, Ci a, = 1 and is a constant possibly depending on n. (iii) Assume that for F in some family 9* N”2(S, - TV,@‘))y.

NW, W’)),

(3.2)

for suitable functionals q,(F), T(F), where N denotes a normal vector, Y denotes convergence in distribution, and q,(F) -+ q(F). (iv) For F in $*={F: FEY*, F,=F,=...=F,=F say} T(F) = Z@P(F),

tl,(F) = rr,.i,

(3.3)

for qfl in (3.1), wherej’=(l,l,..., 1) and q,-+q. In (3.3), A @B denotes the Kronecker product [aijB] with A = [cQ], Z is a kx k positive semi-definite matrix [aij] of rank k- 1 such that Za= 0 for a’= (a,, a2, . . . , ak) in (ii). Also, P(F) is a matrix of functionals Q&F) of common c.d.f. F for F in %*. (v) For F in 4 C %*, P(F)is positive definite. (vi) There exists a sequence {9$,} such that, for FN in 9$,, there corresponds a c.d.f. F for a member of Z$ such that, under the sequence {FN} tl,(FN) = rl,j + N -1’2&F) + o(N-I”),

(3.4)

T(F,)

(3.5)

= Z@P(F)

+ 0(N-1’2);

there are statistics En E E(X,)

furthermore,

J% p,

P(F).

for which (3.6)

Remarks on regularity conditions (A). (1) The statistic Si(,) compares the i-th sample to the remaining k- 1 samples, for variable, CY,and the condition (3.1) implies that only k- 1, among these k, are functionally independent.

In view of conditions (3.1) and (3.2), the ‘asymptotic means’ s$’ are subject to the linear constraint k

c

qqI”)(F)

= q.

Hence, the S in (3.4) satisfies the constraint

i=l

(3.7)

290

K P. Bhapkar / Profile analysis

(2) Some distribution-free features, mostly in the asymptotic sense, are required of statistics S@‘. If the populations have common distribution F, the asymptotic mean q:$does nrt depend on either i, a or F, while the ‘asymptotic covariance’ of N”2S,z) and N”2s;/’ is o~Q~~(F), in view of (3.3). In applications we take Si@’ based only on {Xi’p’) such that for each a = I,2 ,..., p, (s~),s~) )..., SF’) is strictly distribution-free under H,*. Furthermore, this null distribution is the same for all cr. In fact, distribution of N”2(S,(a), . . . , SF’) has ‘asymptotic mean’ vj and Z as the ‘asymptotic covariance matrix’ under H,*. This is the basic commensurability feature, at least in the asymptotic sense, that makes it possible to compare Sy) with respect to samples i and/or variables a. Note that such a feature can be attained by using rank order statistics (e.g. U-statistics or linear rank statistics) even if the initial variables Xi(a), (Y= 1, . . . , p, are not commensurable. (3) qn(F) =q(F) and qn=v for all n for many rank statistics. For instance, this is the case for those based on generalized u-statistics. This is also true for statistics based on linear rank statistics for location parameters with skew-symmetric scores. (4) The regularity condition (v) may be interpreted as requiring ‘asymptotic linear independence’ of {S!:), (Y= 1,2, . . . , p} for each i, at least under H,*. In applications this condition is satisfied if the distribution F is assumed to be nonsingular; by this we mean that there is no set of p-dimensional Lebesgue measure zero which contains the whole F-probability. (5) The regularity condition (vi) appears to be the restrictive side-condition that is needed for testing parallelism (although not for Hz); one would like to relax it somehow if it were reasonably possible. For each FN outside 3c* in 9* it requires the existence of some ‘neighbor’ F in F,, such that (3.4), (3.5) and (3.6) are satisfied. Under reasonable assumptions, it is possible to show that for suitable ‘local’ alternatives qg, (see Bhapkar and Patterson (1977, 1978), Puri and Sen (1971) for the families of local translation and scalar alternatives, respectively), that the conditions (3.4) and (3.5) are meaningful and satisfied; furthermore, then, E can be appropriately defined so that the condition (3.6) holds. However, for more general F outside Fe*, the Kronecker-product structure (3.3) is not necessarily possible. Under condition A(vi) one can construct test criteria (see T,, given by (3.10)) for testing parallelism using consistent estimator E of p(p + 1)/2 ‘nuisance’ parameters P(F). However, without such a condition as in A(vi), it will be necessary to estimate practically all kp(kp + 1)/2 nuisance parameters T(F), making the construction of any reasonable test criterion too complex in order to be feasible in practice. We now proceed to describe briefly the tests used in profile analysis. To-Tests. For testing the homogeneity hypothesis H,*, some tests proposed in the literature (see Bhapkar (1965), e.g.) are based on statistics of the type

T,, =N(S,-f7nj)‘(~-OE-1)(S,-rl,j).

(3.8)

V.P. Bhapkar / Profile analysis

291

Here L’- is any g-inverse of Z, i.e. a matrix such that .Y,?Y_Z=Z. For finite samples we take ai = a,,; also Ca = 0 and from (3.1) and (3.7) $I ai,(

r&@) = 0.

T,, can then be seen to be invariant under any choice of g-inverse Z-. The upper a-th quantile of the chi-square distribution with (k - 1)~ degrees of freedom is the large-sample critical point for To at level CX. If ‘I,, f I?, an alternative statistic that has been suggested (see, e.g. Puri and Sen (1971), Tamura (1966)) is T& = IV@,: - qitj’)(Z-@E-‘)(Sn

- rlj),

(3.9)

provided qn - q = o(N-“~). However, the invariance under any choice of g-inverse is not necessarily true here in the strict sense, although it would hold asymptotically. In any case, one could define T’ for a specific g-inverse (see e.g. Section 4). T,-Tests.

For parallelism of profiles Bhapkar and Patterson (1077) have proposed

statistics (3.10)

T,, =N(S,:-~nj’)[~-O{E--gE-1JE-1}](S,-~7,1),

for the special case of generalized U-statistics; here J = [l] and g = 1/“E- ‘j. T,, is invariant for any choice of g-inverse. If )ln $ q, we could consider an alternative statistic T;,, by using q, instead of v,, in (3.10), provided v”-r,r= o(N-“~). Here too a specific g-inverse could be considered. We may write T;, = T;,, - T;,, ;

it is proposed to use T,, or (TL,J as a large-sample chi-square criterion with degrees of freedom (k - l)(p - 1) for parallelism of profiles hypothesis, while T,, (or Tin) is used as a large-sample chi-square criterion with k- 1 degrees of freedom for testing that the profiles are identical given that they are parallel. The generalized U-statistics (2.6) satisfy the regularity assumptions A($, (iii) and (iv) for ain = l/k, ‘I, = 17= Et=, $(j)/k, qJF) = q(F) defined by (2.7) and 1 au = (k

k2S,. k _LJ_-_-+i 4 Pi

k Pj

I I=I PI

,

(3.11)

were 6, = 1,O according as i =j or i #j. Then T

=

0 T

(k-O2 jTl k ni(S;-S)‘E-‘(Si-S),

7

=W-1)’ ,F, k ni(Si-S)‘[E-l-gE-‘JE-‘l(Si-S), 1 7

(3.12)

292

V.P. Bhapkar / Profile analysis

where S= C piSi and E= [e,& is given by it, F @ja)(xit,, ...,xj~,)~ja,(x~,*+,, **.,xi12k) - V2; i$1 n;(nj-1)***(n,-2k+2)

ea/3=

(3.13)

here P denotes the sum over all permutations of (2k- 1) integers (t ,, . . . , fzk), with tk+i = tit that can be chosen from (1, . . . , ni). It is shown in Bhapkar and Patterson (1977), Bhapkar (1979) that the condition A(vi) is satisfied for the sequences {FN) of location or scale alternatives. Similarly, the linear rank statistics (2.3) satisfy the assumption A(ii) for a, = n/N, n, = Cr=, s,/N. Suppose we define a function J_,,,such that J,(u/(N+ 1)) =sU, for u= 1,2,..., N, and constant-valued over intervals [u/(N+ I), (u + l)/(N + l)]. Letting J(o) = lim,,, J,(o), O< o< 1, V= j; J(o) du, it can be shown (see, e.g. Puri and Sen (1971), Tamura (1966)) that A(iii), (iv) are satisfied for q,(F) defined by (2.4), assuming qn - q = o(N-I”); q!“‘(F) = ODJ(H@‘( y)) dF.@‘( 1 I y) 9 I -m

aij =

6ij

-

L pi

- 1 .

Here H(x) = Ci pit (x), and F@) (F(n*8)) denote the marginal c.d.f.‘s variables cr (((r, fi)) respectively. The statistics are TO= i ni(sj - q,j)‘E-$9,

(3.14)

1 of F for

- qJ),

i=l

(3.15) 7’i = i ni(Sj-v,.j)‘[E-’

-gE-‘JE-‘](Sj-qJ),

i=l

and the alternative criteria are T& T;, using q instead of qn in (3.15). Here eaa is some consistent estimator (see Chinchilli and Sen (1982)) of Co

en/#) =

s 01

i --oo -co

J(F@)(y)) J(FcB’(z)) dF(“‘B,(y 3z) - q2.

It has been shown in Puri and Sen (1971) that A(vi) is satisfied for the sequences {FN) of location or scale alternatives.

4. Asymptotic

properties

of a class of tests

Let Q be a q xp matrix of rank q up and L be a 1 x k matrix of rank 15 k. The choice of Q will depend on the subset of p variables being singled out for further analysis, while that of L will be related to the subset of k populations for further investigation.

KP. Bhapkar / Profile analysis

293

Thus, we could take Q= [l,, 0] for comparing the profiles with respect to the subset of the first q variables, q5p. On the other hand, for investigating the parallelism of profiles for the first q+ 1 variables, we can take Q = [A, 01, where A is a qx(q+l) matrix of rank q such that Aj=O, qsp-1. Similarly, we can take L =Ik for comparing the profiles (or for studying parallelism) of the whole set of k populations. However, for comparing the first I + 1 populations, we can take L = [B, 01, where B is an 1 x (/+ 1) matrix of rank 1 such that Bj =0, Isk1; in such a case we assume that LZL’ is nonsingular. Moreover, a subset of variables can be considered in conjunction with a subset of populations, and this will be accomplished by considering the matrix R=L@Q,

(4.1)

and the vector V,, = RS,,.

In view of (3.2) for C,(F) = Rq,(F),

we have

N1’2(V, - C,(F)) --% where Y(F) = RT(F)R’.

N(O, Y(F)), in view of (3.4) and (3.5) we have

For F, E 9$,,

&,(F,) = q,Rj + N-“2R6

(4.2)

+ o(N-“2)

(4.3)

and p(F)

= (LZL’) 0 (QP(F)Q’)

for some F. These considerations T, = N(V,‘-q,

+ 0(N-“2),

(4.4)

lead us to the statistic

j’R’)[(LZL’)-@

(QEQ’)-‘](V,-

qnRj).

(4.5)

If q, f q, we define T’ using q instead of q, in (4.5). Here and hereafter we use simply T,, instead of more elaborate TL,an, for notational convenience. It is understood that for nonsingular LZL’, the g-inverse superscript - would be replaced by the traditional -1 for the inverse of a nonsingular matrix, 4.1. Assume the regularity condition A(i)-(vi). {F,,} in 9$,, and I < k,

Theorem

Then for the sequence

T,,+x2Uw3, where the non-centrality is given by

parameter

r = G’R’[(LZL’)-’

< of the limiting non-central chi-square variable

@ (QP(F)Q’)-‘1

Rd.

If ~,,-rl=~(N-“~), then T,‘z x2(lq, 0. For I = k, the degrees of freedom (k- 1)q and the inverse is replaced by g-inverse of LEL’ in <.

are

V.P. Bhapkar / Profile analysis

294

For 1=k, although a g-inverse is not unique, l is seen to be invariant under any choice of g-inverse for the same reason as T,,, given by (3.8).

Remark.

The proof is straight-forward and the details are omitted. The degrees of freedom are equal to the rank of the matrix (LL5’) 0 (QP(F) Q’). Corollary

4.1.

Under conditions of Theorem 4.1, for I< k, T, (or T,‘) 3

x2(lq) if

and only if R6=0. For I= k, T, (or Ti) sx2((k-

(4.6) 1)q) if and only if (4.6) holds.

It is enough to show that <=O if and only if the condition (4.6) holds. For the case Z
Nosere that taking L =Ik and Q =I., T, reduces to T,,. On the other hand, let Q be a (p - 1) xp matrix P of rank p - 1 such that Pj =0, while we let L =Ik as before. Then T, is seen to lead to T,, in view of the identity E-’ -gE-‘JE-’

= P’(PEP’)-‘P

(4.7)

9

and the relations (3.10) and (4.5). Thus we have Under regularity conditions A(i)-(vi), T,, (or Td,) -% x2((k - 1)~) if and only if S=O, and T,, (or T$zx2((kl)(p- 1)) if and only if

Corollary 4.2.

(Ik 0 P) 6 = 0,

(4.8)

for P in (4.7). For establishing the consistency properties of the tests, the condition A(vi) can be relaxed somewhat by weakening the requirement (3.6) slightly while discarding the requirements (3.4) and (3.5) altogether. First we need the lemma as in Bhapkar and Patterson (1977). Lemma 4.1. Suppose N”2(V, - g(F)) 5 N(0, Y(F)) for every Fin 3* and W, is a quadratic form N( V, - &)‘A( V, - E), where E is a constant vector and A is a positive definite matrix of constants. Then W,, 4 00, in the sense that

P,[W,>c]-,l

as ni --f 03, for all i,

and for every fixed c, if and only tf F is such that r(F) #E. Theorem 4.2. Assume the regularity conditions A(i)-(vi),

where n,, - n = o(N-“~),

V.P. Bhapkar / Profile analysis

295

and suppose that for every F in 3~ 3* the random matrix E of statistics based on X, satisfies the property E --%

G(F),

(4.9)

where G(F) is positive definite. Let T, be the statistic defined by (4.5). Then T, (or T,‘) % 00 for every F in 9 such that R[q(F) - qj ] # 0. Proof. For the case I
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