On testing hypotheses concerning standardized mortality ratios

THEORETICAL

POPULATIOK

BIOLOGY

On Testing

2, 290-298 (197 1)

Hypotheses

Standardized

Concerning

Mortality

Ratios*

LAWRENCE L. KUPPER AND DAVID G. KLEINBAUM Department of Biostatistics, School of Public Health University of North Carolina, Chapel Hill, North Carolina Received September

27514

15, 1970

This paper describes tests of hypotheses concerning the equality of any k standardized mortality ratios (or, equivalently, of any k indirect adjusted rates) from p( >k) populations. The distinction is made between the situation where the standard population is chosen independently of the p populations and one where the standard is formed by pooling all p populations. Different test procedures are required for these two situations. When the pooled standard is used, the appropriate test procedure is applicable only when k = p. Experimental evidence is given showing that when the pooled standard is used both test procedures lead to the same conclusion concerning the hypothesis for the case k = p. The recommendation is made, therefore, to use the “incorrect” test procedure when the standard is a pooled one even when k < p.

1. INTRODUCTION Suppose that there arep test populations rrl , nz ,..., z-2,, a standard population 7r,,, and c categories, where the term “test population” denotes a collection of individuals under study. We shall regard a test population as a sample from some universe, for only then is there any point in discussing statistical estimation and hypothesis testing(see Chiang(l961), Section I, for comments along this line). Let nij = the number of individuals in category j of vi , p, = the true probability of death in category j of rri , dij = the observed number of deaths in category j of rri , i = 0, l,.. ., p, j = 1, 2,..., c. Then, the standardized mortality ratio SMR(i) for population z-i is defined to be SMW) * Research supported

= 9$l n,iP~/~$~ nijpoj , by National

Heart Institute

290

i = 0, l,..., P. Grant No. HE-03341-13Sl.

(1.1)

ON TESTING THE EQUALITY OF SMR'S

291

(Note that SMR(0) = 1.) SMR’ s are used to compare mortality among populations with different age distributions and (1 .l) is essentially a weighted average of the proportions for the c categories (usually age groups), the weights being determined from the standard population. In this paper we shall be concerned with testing the following general hypothesis, namely Ho : SMR(i,) = SMR(i,) = ... = SMR(i,),

(1.2)

where 1 < il < is < ... < ik < p and 2 < K < p. Since the indirect adjusted death rate P(i) is equal to (Y,,SMR(i), where 06 is C,“=, n,&,& noj , the overall proportion of deaths in the standard population, it is clear that testing H,, is equivalent to testing that P(iJ = P(iz) = ... = P(zJ. Hence, in what follows we can restrict ourselves to considering tests of H,, . Finally, although we have presented the problem in a mortality-type setting, the techniques developed below can validly be used in any situation where the response is dichotomous (e.g., was or was not a migrant, did or did not have cancer).

2. CHOICE OF no AND ITS EFFECT ON TESTS OF H,

When the standard population z~sis chosen independently of the test populations 7rr , ~a ,..., n, , then it is reasonable to assume, for all practical purposes, that the quantitiesp,, = d,&.~,~are nonstochastic (cf., Chiang (1961), Hill (1956), Keyfitz (1966)). An example of such a situation would be one in which the test populations are all the different counties in England in 1963 and the standard is the U.S. population in 1963 or the 1945 population of France. In this case the best estimate of SMR(i) is StiR(i)

= j$l nij bij/?$

+ PW ,

i = 1,2,..., P,

(2.1)

where jij = d&z, is approximately normally distributed for large nij , has mean pii and variance ~~~(1- ptj)/nij , and is uncorrelated with jipj, unless both i = i’ and i = j’. The corresponding best estimate of P(i) is p(i) = %SMR(i). However, the situation is quite different when the standard population is formed by pooling the test populations together so that p, has the structure (2.2)

292

KUPPER AND KLEINBAUM

In this case not only the numerator but also the denominator of SMR(i) must be estimated using the bij’s obtained from the test populations; in particular, the best estimate of SMR(i) is then given by (2.3) where $aj = CF=, nij$&=r nij . The corresponding best estimate of P(i) is ;(i) = &,+&R(i), where 8, is obtained by substituting $aj for poj and ni = C,“=, nij for noj in the expression for 01s. The use of a “pooled standard” of the form (2.2) is not without precedent. For example, researchers associated with the Evans County Coronary Heart Disease Study(Casse1 et al., 1970) have compared the SMR of white males having low systolic blood pressure with that of white males having high systolic blood pressure, the categories being age groups and the standard population consisting of all white males in Evans County, Georgia. Similar pooled standards were used to study the effects of other physiological factors as well. Although decisions about the choice of standard are essentially arbitrary and depend somewhat upon personal philosophy, the use of a pooled standard is suggested when the response under study (e.g., incidence of coronary heart disease or cancer of the cervix) has not been the subject of a previous investigation conducted under comparable conditions (e.g., the same definition of disease, the same age groups, etc.). It is important to emphasize the distinction between the above two choices of the standard population because the different estimates (2.1) and (2.3) necessitate the use of different procedures for testing Ha . These approaches are described in the next two sections.

3. TESTING HO WHEN r,, IS CHOSEN INDEPENDENTLY OF THE TEST POPULATIONS

If we define the vector 6’ = (SI$IR(l), SfiR(2) ,..., S&&R(p)), where SfiR(E’) is given by (2.1), then it follows that E(6’) = 8’ = (SMR(l), SMR(2),..., SMR(p)) and var(6) = D = diag(cr,a,~~2,..., uPa),

ON

TESTING

THE

EQUALITY

OF SMR’S

293

where

In this framework, the hypothesis H,, can equivalently be written as f&J:03

= 0,

where the (k - 1) x p matrix C has as its (j - I)-th row the vector cipl = (0)...) 0, 1, 0 )..., 0, -1, O,..., 0) having +I in the ii-th position, -1 in the ij-th position, and zeros elsewhere, j = 2, 3,..., k. From a general criterion due to Wald (1943), it can be shown that Tl = &C’(CfX’-

C6

is approximately distributed as a central x2 variable with (K - 1) degrees of freedom when H, is true, where D = diag(e12,622,...,op2) is the matrix obtained from D by replacing pii by Pii everywhere it appears. We would then reject H,, at the 01 level of significance if Tl > x&-~, where x;-~;~, is the upper lOO(1 - a) y0 point of the central xi-i distribution. For the special case of Ha when K = p (so that we are testing the equality of all p SMR’s or all p indirect rates), C is simply the (p - 1) x p matrix 1-I 1 . . . 1

0

*.* *** .

0 0 * . .

/ -I O-l . . . 0

. . . 0

.

(3.1)

* * e-1

When K = 2, we may alternatively use as the test statistic 2 = [SfiR(i,)

- S&IR(i2)]/(6fl + 6f2)1’2,

which is approximately distributed as a standard normal random variable when H,, is true. In this case we would reject H,, at the OLlevel of significance when is the lOO(1 - 01/2)% point of the standard normal I Z I 3 -LIP, where Li2 distribution. The use of Tl to test hypotheses which involve an SMR(i) for which Pij is either 0 or 1 for every j requires special attention since the estimated variance ~~2 of Si%IR(i) is zero. Such a situation can be handled by either choosing the estimate of pij according to some criterion other than maximum likelihood (e.g., (dii + l)/(nij + 2) is the Bayes estimate for a uniform prior on Pii) or by deciding not to test hypotheses about SMR(i).

KUPPER AND KLEINBAUM

294

4. TESTING H,, WHEN rO IS FORMED FRohf THE TEST POPULATIONS When pOj is of the form (2.2) so that the estimate of SMR(i) is given by (2.3), the procedure for testing Ii, depends on the relationships expressed in the following lemmas.

LEMMA 4.1. When poj has the form (2.2), then CL1 ai SMR(i) Ui = Cj”=, nijPoj/CEl Cj”=l %jPOj . Proof.

= 1, where

From (l.l),

and the result then follows by using (2.2).

LEMMA 4.2. SMR(p)

When poi has the form (2.2), then SMR(l) = SMR(2) = em*= ij and only if there is a subset of (p - 1) SMR’s all equaE to one.

Proof. If SMR(l) = SMR(2) = ... = SMR(p) = CL, say, then, from Lemma 4.1, TVCL, ai = 1 = p since CT!, ai = 1. If there is a subset of (p - 1) SMR’s all equal to one, then, taking this subset to be the first (p - 1) SMR’s, it follows from Lemma 4.1 that C,“=;” ai + u2, SMR(p) = 1, or SMR(p) = (1 - Cici aJ/a, = a,/~, = I. This completes the proof.

LEMMA 4.3. Let Wiit = XI=, nijn,fj(pij g,=, wiip ) i, i’ = 1, 2 )...) p. Then,

- pgj)/n+

(i)

Wddj = - Witi and Wiit = 0 if i = i’,

(ii)

CE1 Wi = 0,

(iii) when poj has th e f arm (2.2), SMR(i) i = 1, 2 ,..., p.

and

define

Wi =

if

Wi = 0,

= 1 if and only

ON TESTING

(iii)

THE

OF SMR’S

EQUALITY

295

Using (1.1) and (2.2), we have

SMR(i)

-

1= (i j=l

nlj p,,/ i

nij po,) -

1

j=l

This completes the proof. So, when poj is of the form (2.2), it follows from Lemma 4.2 and from part (iii) of Lemma 4.3 that testing the hypothesis that SMR(1) = SMR(2) = . . . = SMR(p)(=l) or that P(1) = P(2) = . . . = P(p)(=ao) is equivalent to testing the hypothesis that Wi = 0 for every i, i = 1, 2,...,p. The appropriate test statistic for this latter hypothesis is

T, = @‘C’(C%‘)-1

C\jir,

where W’ and 2 are the estimates of W’ = (WI, W, ,..., W,) and C = ((cov( I@$, l@6,)))j’i,=1 obtained by putting jii for pij , and where C is given by (3.1). (From parts (ii) and (iii) of Lemma 4.3, it follows that one can alternatively use for C the matrix obtained by deleting any one row of the identity matrix of order p.) The statistic Tz is approximately distributed as a central x2 variate with (p - 1) degrees of freedom when the hypothesis that W = 0 is true. The determination of the elements of Z is somewhat tedious but straightforward. In particular,

can be evaluated directly from the following

formulae (see part (i) of Lemma 4.3):

296

KUPPER AND KLEINBAUM

cov(miL , pi,r,) = 0 if either i -= 1, i’ I= I’, or both, or if i, 1, i’ and 1’ are all different:

if i, 1 and 1’ are all different; and, for i # I,

For the special case p = 2, the test statistic takes the form @‘$&(~is), which is approximately distributed as xl2 when SMR(l) = SMR(2). This case has been studied by Cochran (1954) and Quade (unpublished). Unfortunately, the above procedure cannot be used to test the general hypothesis Ha given by (1.2) when K is to be strictly less than p. To illustrate the difficulty, consider the case where k = 2

5. SOME EXPERIMENTAL

RESULTS

Computer programs have been prepared which calculate SMR’s and indirect rates for p < 10 populations and c < 20 categories. One of these programs is written in Fortran IV for the IBM 360 computer at the Triangle University Computation Center (TUCC) and the other in Time-Sharing Fortran for the Call-A-Computer system. These programs allow the user to specify the standard population or to instruct the computer to form the standard by pooling the test populations. The user of either program may test hypotheses of the form (1.2) as described in Sections 3 and 4. These programs were used to compare the two test procedures presented in this paper in the situation where the standard population is formed by pooling

ON TESTING

THE

EQUALITY

TABLE

297

OF SMR’S

I

Values of Ti and T2 Obtained by Testing H,, for k = p When the Standard is Formed by Pooling the Test Populations

Number

of Populations 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4

p

“Incorrect” 28.2136 16.9346 21.2930 10.2976 10.4231 1.7373 3.1895 7.6659 0.1513 0.0475 1.2320 0.1386 1.4684 1.9818 0.1872 1.3005 1.4704 0.2089 0.4434 0.5522 2.2685 0.0042 0.0361 2.6843 0.0016 3.5495 8.1122 10.4558 2.5145 10.6485 17.0558 21.6299 23.9671 20.3859 16.6307 7.1490 9.7703 3.6639

T1

“Correct” 28.7157 17.4983 21.6004 10.1884 10.8275 1.7946 3.2292 7.3038 0.1533 0.0476 1.2488 0.1450 1.4437 1.9965 0.1947 1.2828 1.4462 0.2131 0.4296 0.5302 2.1455 0.0043 0.0466 2.7078 0.0016 3.5685 7.8162 10.8670 2.4496 10.5651 17.6127 21.9669 24.6090 20.9092 16.4557 7.0270 9.6688 3.6656

Ta

298

KUPPER AND KLEIPU‘BAUM

the test populations. The results obtained should be of considerable interest to those researchers who have incorrectly used the test procedure described in Section 3 when they should have employed that of Section 4, for it is conceivable that inferences drawn from using the “incorrect” test statistic could be revised when the “correct” one is used. The data used in this computer study was obtained from the Evans County Coronary Heart Disease Study mentioned earlier. There were 38 different data sets, each partitioned into the same c = 7 age groups; 25 of the sets were based on two populations, 7 on three, and 6 on four. The hypothesis (1.2) for k = p was tested and the values of the test statistics Tr and T, for each data set are presented in Table I. It can be seen from Table I that, despite the rationale for preferring the test procedure of Section 4, both test statistics give almost exactly the same numerical values in each case. This motivates us to make the conjecture (and, at present, it is only a conjecture) that a test of the hypothesis (1.2) when the pooled standard is formed from all p test populations and the “incorrect” test procedure of Section 3 is used will yield fairly reliable results not only when k = p but also for the case when k < p.

ACKNOWLEDGMENT The authors thank Professor Nathan suggestions concerning the manuscript.

Keyfitz

and the referees for a number

of helpful

REFERENCES CASSEL, J. C., HEYDEN, S., TYROLER, H. A., CORNONI, J., AND BARTEL, A. 1970. Differences

in probability of death related to physiological parameters in a biracial total community study (Evans County, Georgia), to appear. CHIANG, C. L. 1961. “Standard Error of the Age-Adjusted Death Rate,” Vital Statistics, Special Reports, No. 47, pp. 271-285, U. S. Dept. of Health, Education and Welfare, Washington, D. C. COCHRAN, W. G. 1954. Some methods of strengthening the common x2 tests, Biometrics 10, 417-451. HILL, A. B. 1956. “Principles of Medical Statistics,” Oxford University Press, London/ New York. KEYFITZ, N. 1966. Sampling variance of standardized mortality rates, Human Biology 38, 309-317. WALD, A. 1943. Tests of statistical hypotheses concerning several parameters when the number of observations is large, Trans. Amer. Math. Sot. 54, 426-482.