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ON TESTING HYPOTHESES CONCERNING STANDARDIZED
MORTALITY RATIOS AND/OR INDIRECT ADJUSTED RATES*
By
Lawrence L. Kupper
and
David G. K1einbaum
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 709
September 1970
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ON TESTING HYPOTHESES CONCERNING STANDARDIZED
MORTALITY RATIOS AND/OR INDIRECT ADJUSTED RATES*
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Lawrence L. Kupper and David G. Kleinbaum
University of North Carolina at Chapel Hill
1.
~.
Suppose that there are p test populations
~,
ulation
o
~l'~2' ••• '~p'
a standard pop-
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 Section I of [2] for comments along this line).
Let
nij = the number of individuals in category j of
~
.,
1
Pij = the true probability of death in category j of
d ..
1J
= the
observed number of deaths in category j of
i=O,l, ••• ,p, j=l,2, ••• ,c.
for population
~.
1
~
.,
1
~.
1
,
Then, the standardized mortality ratio SMR(i)
is defined to be
*Work supported by National Institutes of Health Grant No. GM-l2868.
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(1.1)
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C
SMR(i} = L _ nJ..jP . .
j-1
J.J
(Note that SMR(O) = 1.)
IL7J=1
n .. p j' i=O,l, ••• ,p.
J.J 0
In this paper we shall be concerned with testing
the following general hypothesis, namely
(1.2)
where 1<i1<i2< ••• <i~ and
2<k~.
The indirect adjusted death rate P(i) for
~.
J.
\
is
where
Wij =
(n.
- J. j
/L7J= 1
.IL7J= 1
n J.J
.. p0 j)(L7J= 1 nOJ.p OJ
n OJ.).
~c
Since P ( i ) = a o SMR ( i ) , where a o = Lj=l
nojPoj I~c
Lj=l noj is independent of i,
it is clear that testing the hypothesis H given by (1.2) is equivalent to
o
testing the hypothesis
Hence, in what follows we can, without loss of generality, restrict ourselves
to considering tests of H •
o
Finally, although we have presented the problem in a mortality-type
setting; the techniques developed. below can validly be used in any situation
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where the response is dichotomous (e.g., was or was not a migrant, did or
did not have cancer).
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2.
Cboice.of~
.and.itseffect.on.tests.ofH.
When the standard population TI
o
is chosen independently of the test pop-
ulations TI ,TI , ••• ,TIp ' then it is reasonable to assume, for all practical
l 2
purposes, that the quantities p = d ./n j are non-stochastic (c.f.,[2],[4],[5]).
oj
oJ 0
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 England.
In this case the best
estimate of SMR(i) is
(2.1)
where P1.'j = di./n .. is approximately normally distributed for large n .. , has
J 1.J
.
1.J
A
mean Pi' and variance p .. (l-p . . )/n .. , and is uncorrelated with p.,., unless
J
1.J
1.J
1.J
1. J
\,C
A
both i=i' and j=j'. The corresponding best estimate of P(i) is P(i) = [.. lW"P ..•
J= 1.J 1.J
~
A
However, the situation is quite different when the standard population is
formed by pooling the test populations together so that p . has the structure
oJ
(2.2)
In this case not only the numerator but also the denominator of SMR(i) must
A
be estimated using the Pij'S obtained from the test populations; in particular,
the best estimate of SMR(i) is then given by
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(2.3)
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A
_
~p
~p
A
where Poj - Li=l nijPij/Li=l n ij •
-
given by
A,
P(~)
~c'
A
'"
'"
= L' 1 wi,Pi"
J=
J
The corresponding best estimate of P(i) is
J
'"
where wi' is obtained by substituting p , for
J
. oJ
Poj and n. j = Ii=l n ij for noj in the expression for wij •
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 [1] 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.
In general, the use of such
a standard is dictated when the response under study (e.g., -the incidence of
coronary heart disease) has not been the subject of a previous investigation
conducted under comparable conditions.
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 H.
o
These approaches
are described in the next two sections.
3.
TestingHwhen~
is cbosenindependently of the test populations.
If we define the vector
A
a'
A
A
A
= (SMR(l),SMR(2), ••• ,SMR(p»,
'"
where SMR(i)
is given by (2.1), then it follows that
A
'"
E(a) = a and Var(~) = diag(cr~,cr~, ••• ,cr~) = D,
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where
a'
= (SMR(1),SMR(2), ••• ,SMR(p»
In this framework, the hypothesis H can equivalently be written as
o
H:
o
.e
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a=0
'.
(j~l)-th
row the vector
c'
-j-l = (0, .•• ,0,1,0, ••• ,0,-1,0, ••• ,0) having +1 in the il-th position, -1
in the i,-th position, and zeros elsewhere, j=2,3, ••• ,k.
From the fact that
J
T1
'"
"'-1
= a'
c' (CDC')
is approximately distributed as a central X2 variable with (k-l) degrees of
'"
freedom when H is true, where D
=
o
•
"'2 "'2
d~ag(crl,cr2'
.'
"'2
.
••• 'cr ) is the matrix obtained
p
from D by replacing p .. by p'" .. everywhere it appears, we would
reject H0 at
,
~J
~J
the a level of significance if T > X2
where X2
is the upper
1 - k-l;l-a'
k-l;l-a
100 (I-a) % point of the central X~_l distribution.
For the special case of H when k=p (so that we are testing the equality
o
of all p SMR's or all p indirect
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C
where the (k-l) x p matrix C has as its
aI
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and
rates~
C is simply the (p-l) x p matrix
1
-1
. 0
o
1
o
-1
o
o
o
1
o
o
0
o
o
o
.(3.1)
-1
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When k=2, we may alternatively use as the test
which is approximately distributed as a standard normal random variable when
H is true.
In this case we would reject H at the a level of significance
0 0
·
when Izi ~ zl-a/2' where zl-a/2 is the 100(1-a/2)% point of the standard
•
normal distribution.
A
The use of T to test hypotheses which involve an SMR(i) for which Pij
1
is either 0 or 1 for every j requires special attention since the estimated
A
A
variance cr.2 of SMR(i) is zero.
~
~
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sta~istic
choo~ing
Such a situation can be handled by either
the estimate of Pij according to some criterion other than maximum
likelihood (e.g., (d .. +1)/(n. +2) is the Bayes estimate for a uniform prior
~j
~J
on Pij) or by deciding not to test hypotheses about SMR(i).
4.
Testing.H whenTI.isformedfrom 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 H depends on the relationships expressed
o
in the following lemmas.
Lemma 4.1.
When P
has the form (2.2), then Ii=l a i SMR(i) = 1, where
oj
a i = I;=l nijPoj/li=l
Proof.
I~=l
nijPoj·
From (1.1), ~~
1 a.~ SMR(i) = L~=
~~ 1(~:
1 n ~J
.. p~J
.. /~~
1 ~: 1 n ~J
.. p OJ.)
L~=
LJ =
L~= L J =
= r;=l(Ii=l nijPij)/I~=l Ii=l
Lemma 4.2.
When
POj
nijPoj' and the result then follows by using (2.2) •
has the form (2.2), then SMR(1)=SMR(2)= ••• =SMR(p) if and
only if there is a subset of (p-1) SMR's all equal to one.
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Proof.
If SMR(1)=SMR(2)= ••• =SMR(p)=~, say, then, from Lemma ·4.1, ~~~ 1 a.=l=~
t..~=
since Ii=l ai=l.
~
If there is a subset of (p-l) SMR's all equal to one, then,
taking this subset to be the first (p-l) SMR's, it follows from Lemma 4.1
that Ii:i a i + ap SMR(p)=l, or SMR(p) = (l-li:i ai)jap=ap!ap=l.
This completes
the proof.
Lemma 4.3.
Let Wii ' = I;=l nijni'j(Pij-Pi'j)!n.j and define Wi = Ii'=l Wii "
i'=1,2, ••• ,p.
Then,
(i)
(ii)
(iii)
W.. , = -W.,. and W.. , = 0 if i=i',
~~
~~ 1 W.~
t..~=
~
~
~~
= 0,
when p . has the form (2.2), SMR(i)=l if and
OJ
only if W. = 0, i=1,2, ••• ,p.
~
Proof.
(i) Trivial.
(iii)
Using (1.1) and (2.2), we have
i,
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~
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SMR{i)-l =
So, when p . is of the form (Z.Z), it .fo110ws from Lemma 4.2 and from
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part (iii) of Lemma 4.3 that testing the hypothesis that SMR(1)=SMR{2)= ••• =SMR{p)
(=1) or that P(1)=P{2)= ••• =P{p){=a ) is equivalent to testing the hypothesis
o
that W.=O for every i, i=1,2, ••• ,p.
1
~
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(I:J=1 n 1J.. p1J. . /I:J=1 n1J.. pOJ.)--jL
The appropriate test statistic for. this
latter hypothesis is
A
where ~' and
A
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are the estimates of ~' = (W1~W2'· •• 'Wp) and
L=
A
A
P
{(cov(Wi ,Wi '»)i,i'=l
A
obtained by putting p .. for p .. , and where C is given,by (3.1). (From part
1J
1J
(ii) of Lemma 4.3, it follows that one can alternatively use for C the matrix
obtained by deleting anyone row of the identity matrix of order p.)
The
statistic T is approximately distributed as a central X2 variate with (p-1)
Z
degrees of freedom when the hypothesis that W=O is true.
The determination of the elements of
forward.
I
is somewhat tedious but straight-
In particular,
A
A
A
A
cov{Wi,wi ,) = Li=l Li'=l cov{Wii,Wi'i')
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can be evaluated directly from the following formulae (see part (i) of
Lemma 4.3):
A
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= 0
if either
i=~, i'=~',
or both, or if i,
~,
i' and
~,
are all different;
A
A
"o• W
" o ,.,);
= cov(W
cov(Wi~,Wi'~') = -
""
.
N~,
~
N
c
cov(Wi~,Wi~') = L
j =l nijn~jn~'jPij(l-Pij)!n~j if i, ~ and ~, are all different;
and, for
i:/:~,
"
var(Wi~)
II
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COV(Wi~,Wi'~')
,,2
,For the special case p=2, the test statistic takes the formw
which is approximately distributed as X~ when SMR(1)=SMR(2).
"
"
!var(W ),
12
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This case has
been studied by Cochran [3] and Quade (unpublished).
Unfortunately, ·the above procedure cannot be used to test the general
hypothesis H given by (1.2) when k is to be strictly 1ess.than p.
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the difficulty, consider the case where k=2 < p and
=SMR(i ).
2
w~
To illustrate
wish to test H :
o
SMR(i )
1
Then, because p exceeds k, we cannot claim that this hypothesis is
equivalent to the hypothesis that W. =W. =0, which is necessary for the test
~l
statistic T to be appropriate.
2
~2
A reasonable solution to this problem is
suggested by the findings presented in the next section.
5.
Computer programs have been prepared which calculate SMR's and indirect
rates for p
2
10 populations and c
2
20 categories.
One of these programs is
written in Fortran IV for the IBM 360 computer at the Triangle University
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Computation Center (TUCC) and the other in Time-Sharing Fortran for the
Ca11-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
the test populations.
stand~rd
population is formed by pooling
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 reversed when the "correct" one is used.
'The data used in this computer study was obtained from -the Evans County
Coronary Heart Disease Study mentioned in [1].
sets, each
pa~titioned
There were 38 different data
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 T and T ,for each data set
1
2
are presented in Table I.
None of the data sets gave. estimated SMR's of zero
(see the last paragraph of Section 3).
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 leads us to feel that a test of
the hypothesis (1.2) when the standard is pooled from the test populations
and the "incorrect" test procedure of Section 3 is used will yield fairly
reliable results not only for the case k=p but also when k < p •
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6.
~.
This paper describes· tests of hypot?eses of the form H :
o
=••• =SMR(i ) or
k
and 2
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k
~
population.
~
P(i )=P(i )= ••• =P(i ), where 1
1
2
k
I
p, when there are p test
popu1ation~
i
1
< i
2
< ••• <
~ ~
P
and c catagories for each
The distinction is made between a standard population which is
chosen independently of the test populations and one which is formed by pooling
the test populations.
two situations.
Different test procedures are appropriate for these
When the pooled standard is used, the appropriate test.pro-
cedure is applicable only when k=p.
•
Experimental evidence is givell showing
that when the pooled standard is used both the "correct" and "incorrect" test
procedures lead to the same conclusion concerning H
o
k=p.
Ie
~
H~:
SMR(i )=SMR(i )
1
2
(or H') for the case
0
The recommendation is made, therefore, to use the "incorrect" test pro-
cedure when the standard is formed by pooling the test populations even whe.
k < p.
•
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TABLE I
VALUES OF T AND T OBTAINED BY TESTING Ho
1
2
FOR k=p WHEN THE STANDARD IS FORMED BY
POOLING THE TEST POPULATIONS
NUMBER OF POPULATIONS P
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
"INCORRECT" T1
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
"CORRECT" T
2
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
7.816.2
10.8670·
2.4496
10.5651
17.6127
21. 9669
24.6090
20.9092
16.4557
7.0270
9.6688
3.6656
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REFERENCES
[1]
CASSEL, J. C., HEYDEN, S., TYROLER,'H. A., CORNONI J., BARTEL, A.
and HAMES, C. G. (1970). Differences in probability of death related
to physiological parameters in a biracial total community study
(Evans County, Georgia). To be published.
[2]
CHIANG, C. L. (1961). Standard error of the age-adjusted death rate.
U. S. Department of Health, Education and Welfare, Vital Statistics Special Reports, i[, 271-285.
'
[3]
COCHRAN, W. G. (1954). Some methods of strengthening the common X2
tests. Biometrics, 10, 417-451.
[4]
HILL, A. B. (1956).
University Press.
[5]
KEYFITZ, N. (1966). Sampling variance of standardized mortality rates.
Human Biology, 38, 309-317.
Principles of Medical Statistics.
New York, Oxford