On the Behaviour of Wald Statistics for the Disjunction of Two Regular Hypotheses

On the Behaviour of Wald Statistics for the Disjunction of Two Regular HypothesesAuthor(s): G. F. V. GlonekSource: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 55, No. 3(1993), pp. 749-755Published by: Wiley for the Royal Statistical SocietyStable URL: http://www.jstor.org/stable/2345885 .

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J. R. Statist. Soc. B (1993) 55, No. 3, pp. 749-755

On the Behaviour of Wald Statistics for the Disjunction of Two Regular Hypotheses

By G. F. V. GLONEKt

Flinders University of South Australia, Adelaide, Australia

[Received August 1989. Final revision May 1992]

SUMMARY Occasionally hypotheses arise which, although not themselves regular, have the logical structure of the disjunction of two, or in general several, regular hypotheses. An example of such a hypothesis is that of collapsibility in a three-way contingency table as defined by Whittemore. When such a hypothesis is to be tested, a Wald test appears, superficially, to offer a convenient solution. However, closer inspection reveals that the irregularity intrinsic to such problems precludes the regularity conditions required for the valid construction of a Wald test statistic. The purpose of this paper is to describe explicitly the nature of the irregularity and its effect on the asymptotic behaviour of the Wald test statistic.

Keywords: COLLAPSIBILITY; DISJUNCTION; DISTRIBUTION OF WALD STATISTIC

1. COLLAPSIBILITY IN CONTINGENCY TABLES

An important example of a statistical hypothesis whose structure is a disjunction is that of collapsibility in a three-way contingency table. Suppose that N= {Nijk} is a 2 x 2 x 2 table of (n, ir)-multinomial frequencies, obtained by the classification of n objects according to the three dichotomous variables A, B and C. In what follows, it is convenient to consider the probabilities yijk as functions of the log-linear parameters,

log 7r,jk = Go + 0i + OB + OAC + QIB + OAkC + qBC + ABC (1.1)

where, for uniqueness,

OA = 0B = oi elk = AB=BoAC=AC = 0BC = BC =ABC = AABC .ABC 0

Following Simpson (1951) and Whittemore (1978) the table iX is said to be collapsible over C provided that

Xl l k 722k = _ 1 + 722 + for k= 1, 2, 1r12k 721k 712 + 721 +

where + indicates summation over the subscript. Simpson (1951) showed that the table was collapsible if and only if it satisfied at least one of the two conditional independence relationships,

AllCIB or Bll CIA, and hence the hypothesis of collapsibility can be seen to be the disjunction of two regular hypotheses. In terms of the parameters (1.1) the hypothesis of collapsibility is

tAddress for correspondence: School of Information Science and Technology, Flinders University of South Australia, GPO Box 2100, Adelaide SA 5001, Australia.

? 1993 Royal Statistical Society 0035-9246/93/55749 $2.00



750 GLONEK [No. 3,

H: OAC= 0,A0BC=0 or BC2 =? BC=2 ?

If it is assumed that 0OA2BC= 0, then the hypothesis of collapsibility has the form Ho: 0AC=0 or OBC

=

or, equivalently, H0: OAfCOBC=O.

Collapsibility can be defined analogously for the general r x s x t contingency table, and as was shown in Davis (1989) the underlying structure is also a union of manifolds. In that paper, an intersection-union test of collapsibility was developed. However, in this paper we shall be concerned only with the Wald test for collapsibility in the 2 x 2 x 2 case under the assumption of no interaction. In practice, the assumption of no interaction would be checked first by using a standard hypothesis testing procedure.

To facilitate the discussion of the contingency table application, we first study the following simpler problem. Consider a two-dimensional parameter,

0 = (01, 02) E 0 = R ,

and the hypothesis Ho: 0102= 0.

In terms of the parameter space, Ho may be expressed as Ho: 0 e 0E, where

GO = {OeR2: 0102 = 0} = {(0, 02): 02eR} U {(01, 0): 01 eR},

and in this light it is clear that Ho is not regular in the sense that 00 is not a manifold but rather the union of two manifolds. Since the regularity conditions required for many standard results are local conditions, it is reasonable to expect the difficulties associated with the lack of regularity to occur at the intersection of the two manifolds which in this case is the single point 0 = (0, 0).

Suppose now that a consistent asymptotically normal estimator is available, i.e. a sequence of estimates { #,, } with

5[(Ofn - 0)VnJ + A(O, S) (1.2)

is given and for simplicity assume that E is known to be the 2 x 2 identity matrix I2. To construct a Wald statistic, a function of the parameter estimates such as 91 02 iS chosen and, by using the delta method, the asymptotic variance of 91 92-1n is found to be

u2(O) = 0j2 + 02.

The Wald test statistic is then given by

Wn = 1 02+) (1.3)

For 0 e 00 with 0 * (0, 0) so that o2(0) ? 0 standard theory may be applied to show that the asymptotic distribution of Wn is N(O, 1).

However, when the true parameter 0 satisfies 0 = (0, 0), the asymptotic variance of 092-1n vanishes so that the usual theory does not apply. In that case, the asymptotic

distribution of Wn may be seen from expressions (1.2) and (1.3) to be that of



1993] WALD STATISTICS 751

xy (1.4) ,= (X2 + y2-)'(14

where X and Y are independent and identically distributed N(O, 1) random variables. The distribution of Wmay be obtained by transforming (X, Y) to polar co-ordinates (R, 4) so that X = R cos 4 and Y= R sin X, where R and 4 are independent and 4 is distributed uniformly on (0, 2-r). Substituting into equation (1.4) yields

W = 1R sin(24)

and since 4 is distributed uniformly on (0, 2-r) it follows that sin(24) has the same distribution as sin 4. Finally, since 4 and R are independent

[Sf W = f(0,~ I). (1.5) An alternative proof of equation (1.5) may be obtained from the theory of stable distributions; see for example Feller (1972) or Shepp (1964).

To extend the preceding theory to the more practical situation of testing for collapsibility in the 2 x 2 x 2 contingency table, it is necessary to incorporate the following features.

(a) The problem involves a fixed number of nuisance parameters. (b) The asymptotic variance matrix of 0 is not known and cannot be assumed to be

the identity matrix. (c) The test statistic need not have the form previously assumed and may be a

function of the nuisance parameters. For example, under the model of no three-factor interaction, 04"IB= 0, the hypothesis of collapsibility can also be formulated as

HL AO AB g ( r11 + r22 ) 0. 7r12 + 721+/

When this quantity is of interest, it would be natural to consider the statistic

22 log( X1+22+ |var A l22 lo 2 , (1.6)

7r12 + 721+ I 12 + 721+

where

OA{_I~og (7r11+7r22+ var t22 -o 7r12 + 7r2l +

is a consistent estimate of the variance.

In the following section, the asymptotic distribution of test statistics such as expression (1.6) will be derived.

2. GENERAL RESULT

Consider a p-dimensional parameter 0 = (Of, 02, . . ., Qp) E 0 and let Ho be the hypothesis

Ho:0e00 = {OeO 0102=0},



752 GLONEK [No. 3,

where additional parameters 03, . . ., Op will play the role of nuisance parameters. As previously, it is assumed that a sequence of estimates { I, }. with

.Sf[(#n - 0)-/nj -*-{0 , (O)} , (2. 1) is given. However, the only assumptions now on E(0) are that E(0) is positive definite for all 0e0 and that E(O) is a consistent estimate. We note that these assumptions apply to the maximum likelihood estimates of 0 in equation (1.1), as can be seen in Bishop et al. (1975).

The class of test statistics considered here are those constructed by choosing a continuously differentiable constraint function,

f: 0 '-+ R, withf- '(o) = 0o (2.2)

so that the large sample variance of f(t)Vn is given by

2(0) = f '(0)2(0)f'(0)T

and the Wald statistic is then

Wn -=0)~

Provided that a(0) * 0, the null hypothesis behaviour of Wn is well known to be asymptotically N(0, 1). Unfortunately, it is not possible to choose a constraint function f satisfying expression (2.2) with a(0) * 0 when 01 = 02 = 0. This is demonstrated in the following result.

Proposition 1. Let f: 0 F-+ R be any continuously differentiable function satisfying f(0) = 0 for all 0 e 00. Then for any 00 with 0?0 = 02 = 0 we have

f'(00) = 0.

Proof. Observe from expression (2.2) that f(H) = 0 for any 0 with either 01 = 0 or 02=0. Now,

= ~o-to1 I0 =-p0 = 0. d@l|, t,0=00 Rf(al?'a3?0* '49 01=0 dO1I

Similar arguments apply to the remaining p - 1 partial derivatives when evaluated at 00. O

When 01=02=0 it is possible, under mild conditions on f, to investigate the asymptotic behaviour of W. The most important of these conditions is that a2f/A1 a02 * 0. Approximating f and f' by appropriate Taylor polynomials, and appealing to the consistency of E(d), it is not difficult to verify that

? 01 02 Wn =(1102+u220 +2u120102) {1 + op (l)},

where

/' a112 O22

and hence the asymptotic distribution of Wn is that of



1993] WALD STATISTICS 753

v=(X2 + y2 + 2pXY) where (X, Y) have the bivariate normal distribution with mean 0 and variance matrix Q U O12

1p IJ 'AlI 22) Although the distribution of Wdepends on p, the distribution of -2 does not and can be obtained as follows.

Proposition 2. Suppose that (X, Y) have the bivariate normal distribution with mean 0 and variance matrix

I(p w i t h - I<p l,,

and let XY

V(X2+ y2+2pXY)

Then

,i[4W2] = X12

Proof. Let Z1 and Z2 be given by Z= (X+ Y)/V{2(1 + p)},

Z2= (X- Y)/V/{2(l -p)} so that Z1 and Z2 are independent standard normal variates, and expressing W in terms of Z1 and Z2 yields

w2_ I { p (Z12 + Z22) + Z12 _ Z2212(23

(1 + p2)(Z2 + Z22) + 2p(Z2 _ Z22)(

Now, let R2 = Z?2 + Z and V = (-Z2 2)/RI2 and substitute in equation (2.3) to obtain

W2= R 2 (p + V)2 4I +p22p

From equation (1.5), it follows that 4W2 has the xl2-distribution when p = 0 and, since V and R are independent, the general result can be proved by showing that the distribution of

(p+ V)2 1 +p2+2pV

does not depend on p. This fact, which is closely related to the fact that expression (1) of McCullagh (1989) defines a probability density, may be established by observing that V has distribution function

F(v) = 1- -sin-' I )) 7ir2



754 GLONEK [No. 3, so that for 0 < v < 1

q(v) =Pr{1i + V) < v2}

1 + p2 + 2p V = F[p(v2_ 1)+vV{p2(v2_ 1)+ 1}]-F[p(v2 _ 1)-vJ{p2(v2_ 1)+ 1}

2-sin-1 (/[1 +p(l -v2)+vV{p2(v2_ 1)+ 1}])

sin- I I +p(l _V2)_ VVfp2(V2_ 1)+ 1}]

To complete the proof, observe that

cos {j q(v)} = 2 [/{(1 - p)2(1 -_v2)} + V{(1 + p)2(1 - v2)}J

= V(1 _ v2)

which does not depend on p. OI

3. DISCUSSION

To summarize these results, it has been shown that the usual distributional results for Wald statistics do not apply to tests of disjunctions such as collapsibility. This is true irrespective of how the constraint functionfis chosen. The reason for this is that the hypothesis space OO is irregular in that it is not a manifold. In this case the asymptotic behaviour of W,2 under Ho is described by

fxi if 0102 =0, (01 , 02) ? (00), .seY[n W2] L XX if21-02=- (3.1)

If a critical region of the form n W2 > C is used then expression (3.1) shows that the rejection probability will vary dramatically at certain points in the null hypothesis space. Although it is true that the size of the test can be guaranteed by choosing c from the x2 reference distribution, such a procedure can be shown to be unsatisfactory in terms of power against certain alternatives. For this reason we do not recommend the use of Wald statistics for testing disjunctions of hypotheses.

The results proved in this paper apply to the 2 x 2 x 2 contingency table under the assumption of no three-factor interaction, or, in terms of equation (1.1), 0 'IC =0. Proposition 1 can easily be extended to many other more complicated cases. For example, if a Wald statistic on 2 degrees of freedom is constructed to test simultaneously for no interaction and collapsibility, as was suggested by Ducharme and Lepage (1986), then the construction fails because the variance matrix of the vector-valued constraint function can be shown to exhibit a rank deficiency at certain points of the null hypothesis. Similar results also hold in the r x s x t case. However, the simple result of proposition 2 does not generalize to any of these cases and the asymptotic behaviour of the Wald statistic in those situations is at present unknown.

ACKNOWLEDGEMENTS I am grateful to J. N. Darroch, S. L. Lauritzen, S. M. Stigler and D. L. Wallace for

some very helpful discussions and also to the referees whose comments greatly



19931 WALD STATISTICS 755

improved the presentation of this paper. The manuscript was prepared by using computer facilities supported in part by National Science Foundation grants DMS 86-01732 and DMS 87-03942 to the Department of Statistics at the University of Chicago.

REFERENCES

Bishop, Y. M. M., Fienberg, S. E. and Holland, P. W. (1975) Discrete Multivariate Analysis. Cambridge: Massachusetts Institute of Technology Press.

Davis, L. J. (1989) Intersection union tests for strict collapsibility in three-dimensional contingency tables. Ann. Statist., 17, 1693-1708.

Ducharme, G. R. and Lepage, Y. (1986) Testing collapsibility in contingency tables. J. R. Statist. Soc. B, 48, 197-205.

Feller, W. (1971) An Introduction to Probability Theory and Its Applications, 2nd edn, vol. II, p. 64. New York: Wiley.

McCullagh, P. (1989) Some statistical properties of a new family of continuous univariate distributions. J. Am. Statist. Ass., 84, 125-129.

Shepp, L. (1964) Normal functions of normal random variables. SIAM Rev., 6, 459-460. Simpson, E. H. (1951) The interpretation of interaction in contingency tables. J. R. Statist. Soc. B, 13,

238-241. Whittemore, A. S. (1978) Collapsibility of multidimensional contingency tables. J. R. Statist. Soc. B, 40,

328-340.



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On the Behaviour of Wald Statistics for the Disjunction of Two Regular Hypotheses