"
CRUDE, DI'RECT AND I.NDlRECT STANDARDIZED RATES:
WHI'CH (IF ANY) SHOULD BE USED?
by
Regina C. E1andt-Johnson
Department of Biostatistics
Univers'ity of NorthCaro1ina at Chapel Hill
..
Institute of S'tatistics Mimeo Series No. 1414
August 1982
CRUDE, DIRECT AND INDIRECT STANDARDIZED RATES:
WHICH (IF ANY) SHOULD BE USED?
Regina C. Elandt-Johnson
Department of Biostatistics, University of North Carolina
Chapel Hill, NIX. U.S.A.
SUMMARY
Direct and indirect standardizations are briefly reviewed, emphasizing
their validity under a multiplicative model without interaction of age specific
rates, in accordance with Freeman and Holford (1980).
It is shown that when the
proportionate age distributions in the study and standard populations are the
same, crude
ra~es
can be used (Section 2).
standardization (Section 5).
This is also required for internal
The paper is mainly concerned with the interpre-
tation and use of standardized rates, with special reference to hypothesis testing.
Two important situations are distinguished: (i) data represent complete popue l a t i o n s ; (ii) data represent samples.
No meaningful test is needed in situation
(i), but some possible indicators of the fit of a model are discussed (Section 3).
Their use is illustrated in searching for declining trend in mortality from
Ischemic Heart Disease in U.S. White Males over period
1968~77
(Example 1).
Testing for heterogeneity among different clinics (samples) using a multiplicative
Poisson model is discussed in Section 4 and Example 2.
Contrary to the con-
elusion of Freeman and Holford (1980), indirect standardization -- more precisely,
the use of standardized mortality ratios -- rather than direct standardization is
here favored, because of the interpretation of data being fitted to a model.
Formally, however, expected values of indirect and direct standardized rates are
the same, provided a multiplicative model without interaction is valid, . so
that either form of standardization leads to the same conlusion.
2
Keywords:
Age specific, crude and adjusted rates; Direct and indirect
standardized rates; External and internal standardization; Multiplicative
models; Interaction; Ischemic Heart Disease; Follow-up studies.
This work was supported by U.S. National Heart, Lung, and Blood
. Institute contract NIH-NHLI-7l2243 from the National Institutes of Health.
3
1.
1.1.
INTRODUCTION
Suppose we are interested
i~
comparison of incidence functions
I
of a certain event occurring in G distinct groups of individuals, each group
subclassified into I strata determined by levels of a certain characterisic.
For convenience, assume that the event under consideration is death, and we
wish to compare mortality patterns of G populations over I age strata.
A useful technique, at the initial stage of analysis, is to display
age specific rates as functions of age and compare them graphically.
survival functions can be evaluated and compared.
Further,
It is, however, customary
in epidemiological research to seek for a single summary index for each
population which includes as much as possible of the information on mortality,
and to use these indices as a kind of "test statistic" in comparative analysis.
~
The indices customarily used by epidemiologists are:
overall or crude rate;
age adjusted or directly standardized rate; and standardized mortality ratio
(SMR) or the latter multiplied by the crude rate for the standard population
often referred to as the indirectly standardized rate.
Suppose that data were collected over a fixed period T.
For the gth
population and the ith stratum, let N . denote the midperiod population
gl.
exposed to risk, Dgi - the number of deaths, and A • = D ilN . - the corgl.
g
gl.
responding age specific death rate (per person per year of life) over
period T.
are:
For a model or standard population (S), the corresponding quantities
NSi ' DSi ' ASi ' etc.
We use the customary notation for sums:
I
Ng . = I N . , D. i
i=l gl.
G
I D , etc.
g=l gi
=
4
I
Let W i = N i/N ,with L W . = 1, represent the proportionate age
g.
g
g.
g=l g~
distribution in the gth population, and similarly, let w = NSi/N • be the
Si
S
corresponding (proportionate) age distribution in the standard population.
For the gthpopulation, the following summary indices can
1.2.
be defined:
(i)
The overall or crude rate,
A(C)
g
(ii)
=
I
I
= Ng. i=l
L Ngi
= Dg. IN g.
A •
g~
(1.1)
The age-adjusted or directly standardized rate,
I
~
L
A(DS) =
g
(iii)
-1
L W ).,
i=l gi g-i
i=l
Vl
A .
Si g~
=-
I
-1
N • L~ N A· •
Si g~
S i=l·
(1.2)
The indirectly standardized rate,
I
Lvi. A •
i=l g~ g~
I
L
i=l
w i
g
A..••
I
D
w A(C)= ~ A(C)
Si S1
Eg. S '
i=l
L
(1.3)
s~
where
I
E
g.
=
L
i=l
Egi
(1.4)
Here
(1.5)
i~
the expected number of deaths in the ith stratum, under the null hypothesis
that the standard population (S) is the correct mortality model, and Eg. is
~
5
the total number of expected deaths under this model.
Dg. IE g.
=
The ratio,
(SMR) g ,
(1.6)
called the standardized mortality ratio (SMR), is, in fact, used more
commonly by epidemiologists than the indirectly standardized rate (1.3),
which may now be simply expressed in the form
>.(IS)
g
1.3.
=
(SMR) >. (C)
g
(1.3a)
S
It has been realized for a long time that the crude rates may
give invalid comparisons since their values may be affected by differences
in age compositions of the compared populations.
Adjustment by using age
distribution of a standard population (direct standardization) was introduced
to remove this effect.
This procedure, however, has some pitfalls of its
own; the values of age adjusted rates, and their comparisons depend on the
composition of the standard population used.
Earlier publications on this
topic were mainly concerned with the choice of the standard population
(e.g. Kitagawa (1964), Spiegelman and Marks (1966»,or searching for indices
other than directly standardized rates (Yerushalmy (1951), Kitagawa (1966»).
Despite criticism, age adjusted rates are still in common use,
especially in the analysis of long-term mortality trends in chronic diseases.
For example, age adjusted rates have been used recently in the analysis of
trends in cardiovascular mortality (in:
Proceedings (1979».
Also,
Doll and Peto (1981) in a recent monograph on trends in cancer mortality
in the United States, recommended the use of directly standardized rates.
6
Indirect standardization is also subject to criticism:
if the
differences between the observed and expected numbers of deaths are in
opposite directions over some strata (i.e., there is some 'interaction'
between age and population effects) then the overall difference, D - E
g.
g"
might be very small, or (SMR)
g
~
1, while, in fact, the mortality pattern
in the gth population is different than that in the standard.
These remarks indicate that the functional form of the age specific
rates plays an essential role in the validity of both types of standardization.
A very important and valuable contribution to the assessmE!Ut of conditions
for validity of standardization is the work by Freeman and Holford (1980).
These authors have emphasized that the use or misuse of standardized rates
depends on the appropriateness of implicit models for age specific rates.
Only models without 'interaction' terms are appropriate.
Two kinds of
such models are in common use:
(a)
multiplicativemodel
A • = 0
g~
g
(1. 7)
Ed
~
which assumes that age specific rates in any two populations are proportional
over all strata, and
(b)
additive model
(1.8)
Freeman and Holford (1980) have shown that· i f either model is vs:lid t direct standardized rates can be used, while indirect standardization is
justifiable only with multiplicative models.
In their conclusions, they are
not in favor of indirect standardization, especially internal, that is, when
the standard population is formed by combining all the study populations.
7
1.4.
The present paper is concerned with some further results on
the use of summary indices.
sidered here.
Only the multiplicative model (1.7) is con-
The main purpose is to develop some techniques for "testing"
whether a multiplicative model (1. 7) is justified, and discuss the role of
standardized indices in testing hypotheses about survival distributions.
Distinction is made between comparisons based on complete population data,
and those based on samples from experimental populations.
In Section 2, it is shown that if the proportionate age compositions
are the same in all populations, stratification is unnecessary; comparisons
of crude rates is valid.
In Section 3 we are concerned with comparison
of populations, using an external (standard) population as a model to which
the data are fitted.
In Section 4, on the other hand, we deal with samples
and tests of homogeneity.
Internal standardization is discussed in Section 5.
Two examples are given to illustrate the methods described, with emphasis
on interpretation of various summary indices.
The first is concerned with
a search for trend in death rates from Ischemic Heart Disease in the
u.s.
white male population, the second discusses fitting the multiplicative
model (1. 7) and tests of homogeneity in a follow-up study, in which the
samples represent different clinics.
2.
CAN THE CRUDE RATES EVER BY USED?
Consider G experimental populations (or samples) stratified into I
age strata.
Suppose that the proportionate distributions of exposed to
risk over different age strata are the same, that is
(2.1)
8
We notice that no assumption is made about the age specific rates,
It
is natural to select a standard population with the same age
distribution, that is,
i
= 1,2,. .. ,1.
(2.2)
We then have
A(C)
g
=
I
I
I
i=l
wOo A i
1. g
=
I
i=l
w . A •
S1. g1.
A(DS)
g
(2.3)
,
That is, the crude rates are identical with the directly standardized rates.
Similarly, from (1.3) and (2.2)
I
I
A(IS) =
g
i=l
Si Agi
I
I
i=l
~hat
W
I
I
i=l
W
S1.· A~.
~1.
= A(C)
(2.4)
g
wSi AS·
.1.
is, the indirectly standardized rates are identical with the crude
rates.
Also
,
[A(C)/A(C)]
= (SMR) g •
g
S
Therefore, any valid comparison of crude rates is as valid as (in fact,
is equivalent to) the comparison of standardized rates.
Since A(C) = D IN
g
g. g"
this also implies that stratification is unnecessary for valid comparisons
of population effects on mortality.
Proportionate distributions (2.1) are often attainable in clinical
trial type experiments, or retrospective or cross-sectional studies, but
are rather unlikely to occur in epidemiological follow-up studies.
In the
next two sections, we will discuss methods for some general situations.
9
3. 'EXTERNAL STANDARDIZATION: HOW TO "TEST" FOR ITS VALIDITY
3.1.
Consider G study populations and an external (standard),population
(S), stratified into I age strata.
We wish to compare mortality patterns
in the study populations with that of the standard population using
standardized rates (external standardization).
Assume first that multiplicative models for age specific rates of
the form
A. = 0 E., g = 1,2, ... ,G,
g1
g 1
Model M:
g
(3.la)
(3 .lb)
for i = 1,2,
...,
I, hold •
Here the E. 's represent the main stratum
1
effects, and the 0g 's represent the main population effects; there is no
'interaction' term, 6 , say (in statistical terminology, no ~nteraction'
8i
on a logarithmic scale) among the populations and age strata.
Under (3.1), the expected number of all deaths in the gth population
is
I
E(D ) =
,E(D g. 1M)
g =
g.
N .0 E = N 0 -E
gl. g i
g- g g
i=l
L
(3.2)
where
-E
I
-1
=
N
LN.
g. i=l gl. E.l.
g
(3.3)
and similarly,
E(DS .IMs ) = E(D S
'>
= NS.OS£S
(3.4)
10
3.2.
Indirect
~tandardization.
We wish to compare death rates in
each study population with the corresponding rates in the standard population.
Formally, for the gth population, we wish to test the null nypothesis
(3.5)
or equivalently ,(if lllodel (3.1) 'is- va1i.d},
H(g) •
o •
0
g
~
= Us
'
(3.6)
against the alternative
(g)·
HA
.
(3.?)
In other words, we wish to fit model M of the standard population
S
to each study population separately.
If H(g) is true we have,
o
(3.8)
so that
(3.9)
g
If, however, Hci ) is not true,
E(D g. )/E(Dg. IHO(g»
= 0g los = y g ,say.
(3.10)
Comparing (3.8) and (1.4), we notice that under model (3.1),
(3.11)
11
gives the expected number of deaths under indirect standardization.
This
also implies that (3.10) is approximately the expectation of the standardized
mortality ratio, that is
(3.12)
Thus, assuming model (3.1) is
valtd~
the SMR's measure the relative
(with respect to a standard population) mortality effects of the study
populations.
We also have
(3.13)
3.3.
Direct standardization.
In indirect standardization, the
mortality rates of the standard population are used as; a.-.odel f-or:;eacht of the
d
study populations.
In direct standardization, we say that the same weights,
wSi's, are used in calculating an average rate for each study population.
But this can also be interpreted in a different way.
The standard population
can be considered as an experimental population toWhich we try to fit dif(Sg)
We now test a null hypothesis, Ho . , say,
(Sg)
: ASi = Agi = <5 g E.1.
that .the·popuIati~nGS)~~;tt$.a.
.. mOdel.Mg' that is, H
O
ferent models (study populations).
for all i.
Formally, we also have
E(D
jH(Sg»
S· 0
=
I
I
i=l
Ns ·<5 E.
1. g 1.
= NS· . <5 gEs
(3.14)
so that
E(D
s. !H(Sg»/E(D
0
s· )
=
(3.15)
12
which is the same as (3.10).
(Note, however, the differences in the structure
and meanings of (3.10) and (3.15).) Also,
(3.16)
which is identical with (3.13).
Thus, when model (3.1) is valid, the expected values of indirect and
direct standardized rates are the same.
This implies that effectively, either
kind of standardization leads to the same result.
difference in interpretation.
There is, however, some
With indirect standardization we fit the
data (i.e., each study population) to the model (standard population); with
direct standardization, the model is fitted to the data, which is not quite
what we should do.
For this reason, I would prefer indirect standardization.
Freeman and Holford (1980) favored directly standardized rates, because they
also give valid comparisons under additive. model (1.8).
3 ..4.
"Testing" for multiplicative model (3.1).
The results presented
above are based on the assumption that the multiplicative model (3.1) holds.
How do we know that this assumption is correct, so that the standardized
indices would give valid comparisons?
This may be known from past experience or, perhaps, obtained from the
data.
Since we are dealing with finite populations of known sizes, we can
treat them as samples from infinite populations.
Formally, we then can
construct some tests of goodness of fit, but since the populations are
very large, any test will nearly always give highly significant results.
Therefore, we may use some other indicators which would suggest that model
(3.1) is acceptable.
~
13
(i)
A simple, but very useful method of assessment of (3.1) is to calculate
ratios, Ag/A
Si
= 1,2, ••• ,1.
' i
If these ratios are fairly consta;nt over
all strata in each population, but may differ from population to population,
then this would be an indicator that a multiplicative model is applicable.
Often it holds approximately over a restricted age range, for example, 30-75.
(c.f. Table lB).
In such cases, we may confine our analysis to this age
range.
(ii)
It was mentioned in Section 3.3, that if model (3.1) holds,
the directly and indirectly standardized rates should be approximately equal
in each population.
This would provide another check, but should be used
in conjunction with the method described in (i); close values of direct and
e
indirect standardized rates can be obtained even if (3.1) does not hold
(see Table lD).
3.5.
"Testing" for significance of population effects.
culate some kind of formal Chi-square statistics.
We can cal-
These, of course, will
be practically almost always statistically "significant" (yielding very
often huge values), but they may be used as relative measures of discrepancy
as described below.
(i)
Since the numbers of deaths, D . 's are rather small as compared
g~
to population sizes Ng~. 's, it is reasonable to. regard each Dg~. as a Poisson
variable, with mean,
are independent.
~gi
= NgiA gi ,
and conditional on Ngi's,
the Dgi'S
Assuming model (3.1) is valid, and Hag): Og = Os is true,
the statistic
(3.17)
~.
14
= NgiOSE is defined in (1.4), is approximately distributed
where E = NgiA
gi
i
Si
g
2
as X with I degrees of freedom. Also, under H6 ) ,
2
X
(3.18)
~omb,g
is approximately distributed as
X2 with 1 degree of freedom.
Finally,
J2
-"TIif , g
is approximately distributed as
= X2 _ X2
g
(3.19)
'Comb,g
X2 with I-I degrees of freedom.
All these X2 ,s will usually be highly significant.
We now introduce
a quantity
(3.20)
Assuming model (3.1) is correct, and
(a)
If H(g):
o
2
°g = °S is true, XComb,g
should be rather small as
compared to X2 , so that we should have H2 relatively large (for large I, H +1).
g
g
g
(b)
If the alternative Hig>:%s=Yg is true, then
X~Omb,g
should be of the same
2
2
2
order of magnitude as X , so that H should be relatively small (H +O). Thus,
g
g
g
H~ measures conformity with or departure from H6 g ) under th~ assumption that
(3.1) is correct.
Some theoretical justification for (a) and (b) is given
in Appendix.
We should be aware, however, that if model (3.1) does not hold, i.e.
we observe 'interaction', then we have a similar situation as in (a), but
H~
g
measures departure from model (3.1) rather than conformity with H6 )·
Therefore, it is advisable to check first the adequacy of model (3.1).
(ii)
Finally, we may fit multiplicative model (3.1) to all data
including (or excluding) the standard population.
Let
6g
and ~i be the
15
corresponding estimates of 0
and E., respectively. The estimated expected
1
I
A
A
2
value of D . is then N iO E •• Now, the quantity, L (D i- N i O E.) IN .8 ~i
g1
g g 1
i=l
g
g g 1
g1 g
no longer represents a Chi-square statistic. It is a contribution to the
g
A
A
overall Chi-square,
2
XTotal =
G
I
A
A
2
A
A
L L [(Dgi-Ng1.0 gEi ) IN g1.0 g E.]
g=l i=l
1
(3.21)
If model (3.1) is correct, (3.21) is approximately distributed as
X2 with (G-l)(I-l) degrees of freedom.
2
Similarly, X
is only a contribution to
Comb,g
G
=
L (D g. -Eg' )2 /E g.
(3.22)
g=l
2
Nevertheless, the indices, H ,s are still useful checking for conformity
g
with multiplicative model (3.1), considered jointly with indicators discussed
in Section 3.4.
(More details on fitting multiplicative models are given
in Section 4.)
EXAMPLE 1.
The data in Table lA represent the U.S. white male pop-
ulations and counts of death from Ischemic Heart Disease (IHD); as defined
in ICDA, Eighth Revision by code numbers 410-413, over the period 1968.
data are from U.S. Vital Statistics, Part IIA,1968-l977.
The
It is claimed
(in: . Proceedings (1979)} that there is a definite declining
trend in mortality from IHD; the directly standardized rates using U.S.
White Male, 1940 population as standard were used in this monograph.
We now
apply the methods discussed in this paper to search for this trend, and to
measure it, using external standardization.
TABLE lA
U.S. WHITE MN,ES - ISCHEMIC HEART DISEASE; (410-413) OVER PERIOD 1908-1977
(i)
Age
Group
1
2
3
4
5
6
7
0-1
1-5
5-10
10-15
15-20
20-25
25-30
SUM
8
9
10
30-35
35-40
40-45
45-50
50-55
55-60
60-65
65-70
70-75
11
12
13
14
15
16
SUM
17
75-80
80-85
85+
18
19
SUM
Total
Age
(i)
Group
1
2
3
4
5
6
7
0-1
1-5
5-10
10-15
15';'20
20-25
25-30
SUM
8
9
10
30-35
35-40
40-45
45-50
50-55
55-60
60-65
65-70
70-75
11
12
13
14
15
16
SUM
17
18
19
75-80
80-85
85+
SUM
--- .
_Tote
1968
N '10- 3
1969
N .10- 3
2i
D
U
11
1970 (Census)
D2i
N3i
1971
1,444
6,434
9,049
8,839
7,906
6,367
5,536
7
7
3
23
55
188
1,471
6,139
8,988
8,954
8,037
6,671
5,740
10
3
6
6
27
71
190
1,501,250
5,873,083
8,633,093
9,033,725
8,291,270
6,940,820
45,575
300
46,000
313
46,123,033
4,789
4,910
5,333
5,235
4,747
4,224
3,494
2,761
2,047
655
2,386
6,572
13,259
21,495
31,913
41,691
48,244
54,860
4,876
4,822
5,270
5,297
4,799
4,297
3,551
2,817
2,043
650
2,333
6,300
13,043
21,182
31,471
41,170
47,854
53,362
4,925,069
656
4,784,375
5,194,497
5,257,619
4,832,555
2,203
6,006
37,540
221,075
37,772
217,365
1.472
824
414
54.878
43,828
38,299
1,471
842
429
2,710
137,005
85,825
358,380
1973
N ·10-3
6i
17
7
3
7
6
21
32
223
349
5,849,792
1972
N ·10-3
4i
D3i
D4i
N5i '10-3
D51
1,543
5,851
8,339
9,140
8,551
7,572
6,051
16
5
7
14
43
74
213
1,412
5,889
8,058
9,103
8,727
7,650
6,588
4
8
22
74
212
47,047
372
47,427
330
701
2,175
5,876
12,398
20,868
30,945
40,998
47,544
51.411
5,295
4,777
5,071
5,205
5,061
4,393
3,788
2,971
2,117
634
1,980
5,652
12,192
20,944
30,162
42,052
48,494
52,006
4
6
4,310,921
31,134
3,647,243
40,874
2,807,974
2,107,552
37,867,805
52,029
5,097
4,761
5,139
5,256
.4,946
4,370
3,728
2.891
2,116
214,149
38,304
212,916
38,678
214,116
53,677
44,258
39,189
1,437,628
53,000
805,564
486,957
43,794
39,756
1,458
826
465
53,399
44,149
41,982
1,453
845
477
54,014
44,854
42.222
2,742
137,124
2,730,149
136,550
2,749
139,530
2,775
141,090
86.514
354,802
88.100
352,818
88,880
355,536
12,629
20,924
47,694
86,720,987
TABLE 1A (Continued)
1974
351,048
1975
1976
1977
N '10-3
10i
Total
D61
N .10-3
1,318
5,836
7,774
9,032
8,868
7,803
6,793
12
1
5
10
34
71
202
1,286
5,668
7,531
8,950
8,971
7,990
7,068
6
1
1
7
27
64
178
1,312
5,417
7,401
8,780
9,029
8,219
7,366
7
5
2
5
22
69
207
1,293
5,189
7,397
8,491
9.107
8,390
7,729
5
5
2
5
17
55
220
1,345
5,079
7.293
8,200
9,059
8,557
7,652
9
1
4
4
24
52
183
13,925,250
57.375,083
80.1+63,093
88,522,725
86,546,270
76,159,820
66,372,792
93
34
45
68
260
667
2,016
47,424
335
47,464
284
47,524
317
47,596
306
47,185
277
469,365,033
3,183
5,532
4,831
4,988
5,180
5,130
4,419
3,835
3,049
2,152
659
1,974
5,430
11,864
20,397
30,594
40,971
48,357
51;444
5,856
4,906
4,907
5,129
5,178
4,466
3,883
3,127
2,193
663
1,829
4,988
11,380
19,723
28,658
39,299
47,486
50,144
6,044
4,973
4,820
5,095
5,180
4,553
3,907
3,220
2,226
641
1,779
4,775
10,463
19,105
27,703
38,195
46,409
48,521
6,161
5,098
4,795
5,042
5,165
4,643
3,952
3,279
2,284
656
1,732
4,634
10,050
18,116
27,030
37,625
45,991
48,458
6,7.02
5,282
4,803
4,971
5,114
4,759
3,979
3,342
2,365
679
1,750
4,284
9,360
17,557
26,367
36,115
45,115
48,641
55,377,069
49,144,375
50,320,497
51,667,619
50,152,555
44,434,921
37,764,243
30,264,974
21,650,552
6.594
20,141
54,517
116,638
200,311
295,977
398,990
473,188
510,876
39,216
211,690
39,645
204,170
40,018
197,591
40,419
194,292
41;317
189,868
390,776,805
2,077,232
1,428
865
493
52,363
45,819
43,979
1,419
870
520
49,429
43,717
44,342
1,435
874
549
47,689
42,468
43,052
1,442
892
561
47,687
42,623
44,624
1,448
897
583
46,8(}2
42,039
44,794
14,463,628
8,540,564
4,977,957
512,938
437,549
422,239
133,209
2,895
134,934
2,928
133,635
27,982,149
1,372,726
331'-
90,910
329,532
91,430
323,780
888,123,987
AJ,141
7i
D
7i
N '10-3
8i
2,786
142,161
2,809
137,488
2,858
89,426
354,186
90,118
341,942
90,400
D8i
-3
N '10
9i
D
9i
D10i
No!
D' i
I-'
'"
-
1968
1969
1970(CenGus)
(i)
Age
Group
1
2
3
4
5
6
7
0-1
1-5
5-10
10-15
15-20
20-25
25-30
1.18
0.11
0.08
0.03
0.29
0.86
3.40
252.49
212.99
95.40
51.10
114.86
73.12
89.08
0.68
0.05
0.07
0.07
0.34
1.06
3;81
. 145.79
95.67
82.33
100.89
132.64
90.09
86.83
0.47
0.05
0.08
0.07
0.25
8
9
10
11
12
30-35
35-40
40-45
45-50
14
15
16
55-60
60-65
65-70
70-75
102.68
105.54
106.58
105.44
104.58
104.61
106.47
102.87
108.56
13.33
48.38
119.54
246.23
441.38
732.39
1159.39
1698.76
2611.94
100.08
105.07
103.39
102.51
101.94
101.41
103.45
100.01
105.80
13.32
13
13.68
48.59
123.23
253.27
452.81
755.52
1193.22
1747.34
2680.02
17
18
19
75-80
80-85
85+
3728.12
5318.93
9250.97
101.13
97.84
113.31
3649.01
5256.29
9134.97
98.98
96.69
111.89
50-5~
-
B
T:tt
DEATH RATES AND THEIR RATlOS TO DEATH RATES OF
STANDARD (EXTERNAL) POPULATION WM 1970 (CENSUS)
).
Ii
.10 5
Y .10 2
.Ii
).
Zi
YZi ·t0 2
.10 5
).3i .10 5
1972
1971
Y3i ·10 2
5
.10 2
Y4i '10 5
).4i· 102
.100
100
100
100
100
100
100
1.04
0.09
0.08
0.15
0.50
0.98
3.52
222.39
167.30
103.53
230.62
198.54
82.72
92.34
0.28
0.10
0.05
0.09
0.25
0.97
3.22
60.75
199.46
61.22
132.32
99.53
81.88
84.41
2468.69
100
100
100
100
100
100
100
100
100
13.75
45.68
114.34
235.88
421.92
708.12
1099.73
1644.55
2456.59
103.26
99.21
98.89
98:20
97.44
98.05
98.13
96.82
98.42
11.97
41.45
111.46
234.24
413.83
686.59
1110.14
1632.25
2456.59
89.89
90.02
96.40
97.52
95.58
95.07
99.06
96.10
99.51
3686.63
5436.44
8164.17
100
100
100
3662.48
5344.92
9028.89
99.35
98.32
110.59
3717.41
5308.17
8851.57
100.84
97.64
109.27
1.18
3.81
46.05
115.62
240.20
432.98
722.21
1120.68
1098.62
y
5i ·10
).
5i
TABLE 1B (Continued)
1973
1974
(i)
Age
Group
).6i·105
1
2
3
4
5
6
7
0.,.,1
1-5
5-10
10-15
15-20
20-25
25-30
0.91
0.02
0.06
0.11
0.38
0.91
2.97
195.26
33.55
79.32
166.70
151.38
77.02
78.01
8
9
10
11
12
13
14
15
16
30-35
35-40
40-45
45-50
50-55
55-60
60-65
65-70
70-75
11.70
40.86
108.86
229.03
397.60
692.33
1068.34
1586.00
2390.52
17
18
19
75-80
80-85
85+
3666.88
5296.99
8920.69
1976
1975
Y7i .10 2
).8i· 105
Y8i· 102
).9i· 105
Y .102
9i
).10i· 105
0.47
0.02
0.01
0.08
0.30
0.80
2.52
100.06
34.54
16.38
117.76
118.83
67.80
66.06
0.53
0.09
0.03
0.06
0.24
0.84
2.81
114.42
180.70
33.33
85.74
96.20
71.06
73.72
0.39
0.04
0.03
0.06
0.19
0.66
2.85
82.93
75.46
33.35
88.66
73.70
55.49
74.67
0.67
0.02
0.05
0.05
0.26
0.61
2.39
87.85
88.74
94.15
95.35
91.83
95.86
95.33
93.38
96.83
11.32
37.28
101.65
221.88
380.90
641.69
1012.08
1518.58
2286.55
85.00
80.96
87.92
92.37
97.97
88.85
90.31
89.41
92.62
10.61
35.77
99.07
205.36
368.82
608.46
977 .60
1441.27
2179.74
79.62
77.69
85.68
85.49
85.18
84.25
87.23
84.85
88.30
10.65
33.97
96.64
199.33
350.75
582.17
952.05
1402.59
2121.63
79.94
73.78
83.58
82.98
81.01
80.61
94.95
82.58
95.94
99.46
97.43
109.27
3483.37
5024.94
8527.31
94.49
92.43
104.45
3323.28
4859.04
7841.89
90.14
89.38
96.05
3307.00
4778.36
7954.37
89.70
87.90
97.43
y
6i ·10
2
).7i. 105
1968-77
1977
y
.
lOi
·102
lOi· 105
143.51
38.54
67.65
73.44
104.60
51.44
62.74
0.67
0.06
0.06
0.08
0.30
0.88
3.04
10.13
33.13
89.19
188.29
343.31
554.04
907.64
1349.94
2056.70
76.06
71.95
77.14
78.39
79.29
76.71
80.99
79.48
83.31
11.91
1563.48
2359.64
3232.18
4686.62
7683.36
87.67
86.21
94.11
5123.19
8482.17
40.98
108.34
225.75
399.40
666.09
1056.53
3546.40
f-'
-.,J
......
00
-
e
e
19
As a standard (S) population, we use here the U.S. white male,
1970 (Census) population.
Table lB exhibits observed rates,
A
i'105, and
,g
2
their ratios to the death rates of standard population multiplied by 10 ,
Y '10
g
2
for g = 1,2, ••• ,G.
ages, showing
These ratios seem to be more stable for adult
a rather excessive departure from stability for very young
ages and moderate departures for old ages.
We decided first to analyze
mortality trends in the 30-75 age range; this is also the range which
is often used in epidemiological studies.
The results are given in Table lC.
The rows 3-6 exhibit the indices discussed in this section for external
standardization.
in Section 5.J
(Internal standardization, shown in rows 7-10, is discussed
2
We assume that values of the H ,s of the order of magnitude
g
0.10 - 0.20 are sufficiently small, to reject the hypothesis
~
population 5 (1972) seems to deviate from these limits.
H~g). Here,
Generally, we can
reasonably assume that the data fit fairly well the multiplicative model (3.1).
To give some idea about the magnitudes of x
2
g
,s, we quote the results, for
populations 1, (1968), population 8 (1975), and total (1968-77).
1968
1975
(1968-77)
xg2 :
752.41
4548.54
24951.84
2
xComb,g'
•
664.04
4474.76
24203.92
88.37
73.78
747.92
0.1174
JO.0162
0.0300
2
XDif ,g:
2
H
g
We also fitted the multiplicative model to the data, including the
standard 1970 (Census) by the maximum likelihood method, subject to the
G
condition
I
g=l
Og =1. The results (for G = 10, and I = 9) are:
e
e
e
21
0
g
.
E. :
~
0.1124060,
0.1006243,
0.1124593,
0.0957242,
0.1063621,
0,0914330,
0.1039086,
·0.0886082,
0.1032941,
0.0851820 ;
0.0012023,
0.0668153,
0.0041075,
0.1060325,
0.0107985,
0.1572720,
0.0225374,
0.2368974.
0.0400394,
The values Go ·10 2 are given in the last row in Table Ie; we will discuss
g
their meaning in Section 5.
As
can be seen from Table lB, model (3.1) should not fit the data over the
whole age range.
Nevertheless, we carried out the analysis -for i,llustrativepurpose
2
(Table lD). The Hg ,s in row 5 of ,Table lD confirm our conclusion drawn from Table lB.
4.
TESTING FOR HETEROGENEITY IN SAMPLES FROM DIFFERENT POPULATIONS
4.1.
In this section, we are still concerned with multiplicative
models, but our data represent samples from finite populations.
For convenience,
we retain the same notation as in Section 3, so that N . now denotes the size
g~
of the sample from gth population, in the ith stratum.
Often instead of
sample size, the amount of person-years exposed to risk, A ., say, over a
g~
fixed period (T) of investigation is
N .'s are replaced by A i's.
g~
g
g~ven
(see Example 2).
In the analysis, the
Our interest is usually in comparison of mor-
tality patterns in different groups rather than in searching for mortality
trends, that is, in tests for heterogeneity.
conditional on the set {N
gi
As in Section 3, we assume that
}, the counts of deaths, D , are independent
gi
Poisson variables with expected values, E(D .) = N .J.. .,
g~
g~
g~
In particular, we
consider a Poisson multiplicative model with parameters
11 • = N .0
g~
g~
gE ~• •
(4.1)
22
Several authors (e.g. Osborn (1974), Breslow and Day (1975), Gail
(1978)) have discussed these models and their applications to various sets
of data.
Theoretical bases for constructing various tests are elegantly
presented by Andersen (1977), and can be used for further reference for constructing various test statistics.
We confine ourselves here to the Pearson-
type X2-tests.
4.2.
The likelihood function leads to the set of GI likelihood
(4.2)
(4.3)
(4.4)
(Andersen (1977), Section 2).
These equations
can be solved using a computer
program.
2
The Pearson-type X
goodness of fit criterion is
(4.5)
A
(cf. (3.21)).
A
If the N iO E. 's are large enough (Le. ~ 10), (4.5) is
g
g
~
approximately distributed as X2 with (G-l)(I-l) degrees of freedom, )provided the
multiplicative Poisson model (4.1) is correct.
Of special interest is the null hypothesis, H ' that all 0g'S are the
O
1
same, and in view of (4.4), HO: 0g = G for g = 1,2, ••• ,G. In this case,
23
equations (4.2) can be solved explicitly, yielding
D .
A
E.
1.
.l.
=G
N -.
(4.6)
•1.
Since the null hypothesis, H ' can also be expressed in the form
O
HO·• Agi = A
Oi for g = 1,2, ••• ,G"
the ma:x:imum likelihood estimate of A01.' is
A D • l.•
Ei
=
(4.7)
N:" '
.1.
and the estimated expected value of D . under H is
gl.
O
E[D
If H
O
t"
.IH )]
gl. o
E ••
g1.
(4.8)
is true, the statistic
(4.9)
is approximately distributed as X2with IG-I=I(G-l) degrees of freedom.
The quantity
2
2
Y = (D g i-Ng1..A Oi ) IN g1..A 1.'
gi
O
A
A
(4.10)
2
represents the contribution of the (gi)th cell to the X -statistic (4.9).
I 2
Note that each sum
Y . represents only a contribution of the gth sample
i=l gl.
2
to the overall x -statistic defined in (4.10), but is not itself distributed
L
4.3.
It appears that D.1..'s are sufficient statistics for E.'S
(see
l.
formula (4.6».
Since the distribution of D.i's does not depend on 0g'S,
24
they are also ancillary for 0 'so
(Andersen (1977), Section 5).
g
Therefore,
inferences about 0 's can be derived from the distributions of the
g
D its conditional on D .'s.
g
·1
For the ith stratum, the Poisson variates, D , conditional on D. i have
gi
a multinomial distribution with index D• i and parameters Ng1. IN ·1., when H is
O
valid.
Note that
N .
D .
A
~
·1
Dg i-D.
N
=
D
.-N
i
-N-=
D
.-N
.A
·1.
g1 g . i
g1 g1 O1·
i
Therefore,
(4.11)
under H ' the statiatic
O
(4.12)
has, indeed, an. approximate
X2 distribution with G-l degrees of freedom,
and
2
XTotal =
has a n approximate X2
identical with
I
X2
=
x~1
L
i=1
(4.13)
(G~l)I
distribution with
degrees of freedom; it is
(4.9).
Also,
G
2
A
2
L (D -E ) IEg.
g=l g. g.
XComb =
(4.14)
where
I
E
g.
=
L
i=l
I
A
E. =
g1
I.
A
D
i
INA
=IN
_.i=l gi Oi 11;;1, gi N. i
is approximately distributed as X2 with G-l degrees of freedom.
2
2
x-~if
= XTotal
_ X2
Comb
(4.15)
Also,
(
4.16)
2
is approximately distributed as X with (G....l) (1-1) degrees of freedom. It should
25
be noticed t how.ever t that
the_X~omb and~if ,are
not_ precisely independent.
The three X2-criteria should be interpreted jointly
2
XDif
2
X
Comb
2
XTotal
2
H
H : 8
O g
= -1G , g = 1,2, ••• ,G
(a)
NS
NS
NS
Large
Not rejected, model (4.1) valid,
(b)
Signif
Signif
NS
Small
Rejected, model (4.1) valid,
(c)
Signif
NS
Signif
Large
Rejected, model (4.1) invalid.
(4.16)
NotiCe that if tne multf.plicative model 'is assumed, . (c) should not arise.
EXAMPLE 2.
The data in Table 2A represent mortality from all causes
over a follow-up period of about 7 years, collected by 8 regional clinics
(LRCF, 1974). The samples were selective; each consists of 15% of random
selectio~and
of all individuals with cholesterol or triglycerides above the
corresponding 95th percentiles at the screening date.
They also represent
different occupational groups such as university employees, industrial plant
unit workers t etc.
The recruitment period was about 3 years, and the average length of
follow-up was about 5 years.
Table 2A exhibits the (exact) amounts of person-
years exposed to risk, A ., over the follow-up rather than the average sample
g1.
sizes.
Also, Dg i denotes
the number. of deaths from all causes in the gth
.
sample and the ith stratum.
The multiplicative model (4.1) was fitted yielding
the estimates of the parameters (for G
o.:
g
0.08701,
0.09861,
0.11511,
0.10375,
0.15410,
0.12107;
e:.1. :
0.12645,
0.47024,
084695.
= 8 and
I
0.20228,
= 3):
0.11807,
TABLE ZB
TESTING FOR HETEROGENEITY
CLINICS
Age
Group D
(i)
li
1
Eli
Z
Z
3
Z
.5
4
Yli
D
Zi
E
Zi
Y2i
D
31
E
31
2
Y3i
D
41
E41
2
Y4i
6
D
5i
E51
2
Y51
D
61
E
6i
2
Y6i
D71
E
7i
Total
a
7
2
Y7i
D
a1
E8i
2
Y"81
X2
1
d.f.
2
X. 95
30-45
1
0.1+8
0.5633
2
1.81
0.0199
3
1.68
1.0371
0
1.27
1.2700
1
1.60
0.2250
1
2.47
0.8749
2
2.62
0.1467
6
4.08
0.9035
5.0404
7 14.07
NS
45-55
0
1.33
1.8300 1
2.26
0.0725
7
5.03
0.7716
3
1.59
1.2504
6
4.19
0.7819
6
6.22
0.0078
5
4.94
0.0007
5
6.95
0.5471
5.8920
7
14.07
NS
0.3682
2
1.21
0.5158
3
3.37
0.0406
3
0.64 18.70251 2
3.35
0.5440
3
3.53
0.0796
6
7.49
0.2964
6
5.87
0.0029 10.5500
7
14.07
NS
2.7615
5
5.28
1.2382 13 10.08
1.8493
6
3.50111.22291 9
9.14
1.5509 10 12.22
1.4535 21.4824 21
x2
3.5902 7
32.67
14.07
NS
NS
X2
Dif
H2
23.67
NS
55-70._1_ 0.55
SUM
2
-. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -2.86
0.9623 13 15.05
0.4438 17 16.90
COmb
17.8922 14
0.8329
--.,...,------------------'--------------------------------....:----'---------N
'"
e
-
e
27
4It
2
The X -statistic" calculated from (4. ) is equal to 15.04 with 14 d.f. and
is not significant (NS) at the significance level a
= 0.05
(X
2
•
95
;14
= 23.67).
Next, we test the hypothesis, H ' that the 0g'S are all equal to
O
A
(l/G)= (1/8).
The estimated expected values, E ., and the contributions to
g~
X2 are given in Table 2B.
the H is not rejected.
O
We have the situation (4.l6(a»
It
discussed above, so
seems that the non-sign1ficantresults are mainly
due to the smallness of the data set; in some classes there are only 1 or 0
deaths.
4.4.
The tests of homogeneity discussed above can be formally applied
to population data.
countries.)
(For example, in studies of mortality patterns in different
However, if they are, the X2-values are usually highly significant,
and situations (4.l6(a)} and (4.16(c»
geneity of type (4.l6(c»
might be sometimes indistinguishable.
is detected, the populations might be arranged in
smaller subsets, for which either (4.l6(a»
or (4.l6(b»
holds.
Alternatively,
we may try to get different models allowing for geographical effects, or
including interactions.
5•
5.1.
model (3.1)
INTERNAL STANDARDIZATION
We now return to Example 1 and assume that the multiplicative
is
valid
(including the 1970 census population), for age
range 30-75.
With external standardization, we compared the effect (0 ) of each
g
study population with the effect (os) of the standard.
wish to compare" the
°
We may also
g 's with an average effect of all populations,
28
that is, when the standard population is composed of all study populations,
and the expected number of deaths in the gth population and ith stratum is
calculated from the formula
(5.1)
where D.i/N.i = A is the corresponding age specific rate in the combined
Oi
population.
This is called internal standardization.
encounter some difficulty.
Here,however, we
If the multiplicative model M given by (3.1a)
g
is valid, then in the general case, the model M given by (3.1b) is now not
S
valid, so that M and M are inconsistent.
g
S
To see this from (4.2), we have
(5.2)
where
_
0i
-1
= N.
.~
G
L N ·i·o~
g=l g ....
(5.3)
Hence
(5.4)
and
= N .e:.o i
g~
Now the effect for the standard population,
~
(5.5)
6i , depends on i (the stratum), so
that (5.4) is not a multiplicative model.
5.2.
(Section 2).
Suppose that the study population distributions are proportional
In this case, we also have
29
Ngi =
N' i
G
with
I
a
g=l og
= 1,so
a Og ' saYJ for all i,
(5.6)
that
G
I
g=l
a
og 0g
=
6,
for all i
(5.7)
Under these conditions, we have
E(Dg.
IHO)
(5.8)
= Eg.
We also have
I
E(D
g.
)
=6IN.E.,
i=l g3. 3.
(5.9)
so that
E(SMR) g.::: E(D g' )/E(D g. .IH ) = 0 g /6
O
Now, (SMR)
g
(5.10)
measures the relative effect of the gth population with
respect to the average effect of all study populations.
5.3.
As already mentioned, 'perfect' proportionate distributions of study
populations are seldom encountered even in studies of the same population
over a certain period of years as can be seen in Example 1.
Tables of wOi's
and aOg's (not given in this text) reveal such departures from proportionate
distributions.
period (e.g.
~
But in studies of the same population over a fairly short
10 years) these departures are quite commonly not so great that
they make a substantial difference to our results.
The ratios A(C) /A (C)
g
S'
30
compared with the values of (SMR)g in row 5, do not differ considerably,
indicating that the assumption of proportionate distribution can be used
approximately.
We have also computed a table of Agi!A
Oi
' analogous to Table lB
(not given in this text), which suggests that the combined population may be
reasonably used as a standard.
in rows 7-10 of Table lC.
Results of internal standardization are given
2
Except for population 6 (1973), the H ,s are
g
smaller than for external standardization, indicating a fairly good fit of
multiplicative model, and reasonably small effect of the
proportionality among
distributions.
A final check for the multiplicative model is given in the last row (11)
of Table lC.
We notice that for the
mult~plicative model
E(SMR) g ~ E(D g. )!E(D g. IH ) = 0g!
o
l
G
= Go g.
(5.11)
If internal standardization is approximately valid, (5.11) given in row 11 of
Table lC should not deviate too much from the observed (SMR)
of Table lC.
g
given in row 9
Except for population 2 (1969) the fit is rather good.
Surprisingly good agreement of row 9 and 11 in Table lD is also observed, though
not all ;::the l. :data seem to "fit. the Dlultiplica.t:i:ve model. Probably these
indices are sufficiently robust,that is, are not too sensitive to assumptions.
6.
6.1.
DISCUSSION
Though standardized rates have been widely criticized, they are
still used in epidemiological research.
The criticism, however, did not relate
to an essential feature that these indices can be regarded as playing the role
of "statistics" in statistical inference.
Thus, their appropriateness depends
on the underlying model for age specific rates as was recently emphasized by
Freeman and Holford (1980).
31
Suppose, for example, that we compare two means of a given characteristic,
using t-test (which is a kind of "index").
If the assumption that these means
are from normal populations does not hold, t-test is invalid.
to these indices:
Agi
= 0g8 i
6.2.
The same applies
they are valid, for example, if a multiplicative model,
(and ASi
= 0S8 i )
is
valid,. or equivalently> when . \/A 8i = %s.
The simplest way to analyze the data would be to fit the data to
the model (usually by the maximum likelihood method), and "test ll for its
2
appropriateness, using the H -index
:4n conjUmctionW±ththet:atioS,Ag/~i'
Which should be approximately constant 'f~r any' g;&h, if model M is val·id. In case
g
of good fit.• the GO's
the effects of differenj:
P9pulations.
g '"C(5.ll"reveal
"
11
_
6.3.
More traditionally, we may calculate standardized rates.
out in the text
6.4.
•
As pointed
indirect standardization, especially (SMR) g 1.,$ pre(erab·le •.
If the proportionate distributions of study populations are
approximately the same, crude rates are identical to external (directly and
indirectly) standardized rates.
Internal standardization can be used to evaluate
population effect as compared to the average effect of all study populations.
6.5.
It seems that the standardized rates are 'robust estimators,' in
that they are not sensitve to assumptions required for their validity.
6.6.
Multiplicative models appear to be appropriate for different types
(not necessary mortality) data, and are convenient mathematically.
It would
be of some interest to perform similar analyses for additive models, A .
g1
= ¢g +
and check for their validity for standardization.
Acknowledgement
I would like to thank Mr. G. Samsa for computing summary indices, and
Dr. N.J. Johnson for obtaining the maximum likelihood estimates.
a.,
1
32
REFERENCES
1.
Andersen, E.B. (191,7). Multiplicative Poisson models with unequal
cell rates. Second J. Statist. 4, 153-158.
2.
Breslow, N. and Day, N.E. (1975). Indirect standardization and multiplicative models for rates with reference to the age adjustment of
cancer incidence and relative frequency data. J. Chron. Dis. ~~,
289-303.
3.
Doll, R. and Peto, E. (1981). The causes of cancer: quantitative estimates
of avoidable risks of cancer in the United States today. J. Nat.
Cancer Inst. 66, 1193-1308.
4.
Freeman, D.H. and Holford, T.R. (1980).
195-205.
5.
Gail. M. (1978). The analysis of heterogeneity for indirect standardized mortality ratios. J. Roy. Statist. Soc. Ser. A, ~~~, 224-234.
6.
Internat~onal
7.
Kitagawa, E.M. (1964). Standardized comparisons in population research.
Demography ~, 296-315.
8.
Kitaga~a,
9.
Lipids Research Clinics Program (LRCP) (1974). Protocol of the Lipid
Research Clinics Prevalence Study. Central Patient Registry and
Coordinating Center, Dept. of Biostatistics, University of North
Carolina at Chapel Hill.
Summary rates.
Biometrics 36,
Classification of Diseases Adapted (ICDA) for use in the
United States, Eighth Revision. U.S. DHEW, Public Health Service
Publication No. 1693, 1968.
E.M. (1966). Theoretical considerations in the selection of
a mortality index and some empirical comparisons. Human Biology 38,
293-308.
..
10.
Osborn, J. (1975). A multiplicative model for the analysis of vital
statistics rates. App1. Statistics :~, 75-84.
11.
Proceedings of the conference on the Decline in Coronoary Heart Disease
Mortality (Eds. Havlik, R.J. and Feinlieb, M.) U. S. DEHW, Public
Health Service, NIH Publication No. 79-1610, 1979.
12.
Spiegelman, M. and Marks, H.H. (1966). Empirical testing of standards
for the age adjustment of death rates by direct method.
Human Biology
38, 280-292.
13.
Vital Statistics of the United States, 1968-17.
A. DHEW Publications, 1972-1982.
14.
Yerushalmy, J. (1951). A mortality index for use in place of the age adjusted
death rate. f\m~_-!:..1.~~. Hea!!.l:!. il' 907-922.
Vol. II.
MOrtality.
Part
e
33
APPENDIX
Expected Values of "X2-Statistics" Under Different Hypotheses
We assume that the D .'s and DS"S are approximately Poisson variates
g~
..
~
which have multiplicative models with expected values, ].lSi = Ngi(\Si and
~Si = NSiOSS i ' respective1y,and conditionally on sets {N , N } they are
gi
Si
mutually independent.
g
Consider the null hypothesis H6 ): Og/oS = 1, and the alternative
HA(g):
°los = y .
g
g
We then have
E(D
e
gi
IH(g»
0
= Var(D . IH(g)= Nos = E .
g~
0
gi S i
g~
(A.1)
=Var(D
(A.2)
and
E(D .IH(g»
g~
A
gi
IH(g»
A
= y E
g gi
Similarly,
I
=
L Eg~. = Eg'
i=l
(A.3)
and
E(D
g.
/H(g»
A
= Var(D
g.
IH(g»
A
- y E
g g'
(A.4)
We now evaluate
=
--!.-.
E •
g~
[Var (Dg-t I HA(g»
...
+
(y -1) 2E~.]
g
g~
(A.5)
34
Thus
E(X 2 IH(g»
g.
A
I (D
-E
= E ~ . gi gi
.L
~=
1
)2
E i
IH(g)
A
g
= Iy +(y -1) 2E
g g
g.
(A.6)
Similarly,
E(X 2
IH(g»
Comb,g A
(D -E
)2
IHA(g)] = ~ +(y _1)2 E
E
gg
g.
g.
= E[ g. g'
(A.7)
so that
(A.B)
Hence
=1
(y
-
_1)2 E +Y
g.
g.
g + 1-1
(y -1)2E +y
g
g' g
=0
,
(A.9)
•