American Journal of Epidemiology
Copyright C 1996 by The Johns Hopkins University School of Hygiene and Public Health
All rights reserved
Vol. 143, No. 1
Printed In USA.
Evaluation of Birth Cohort Patterns in Population Disease Rates
Robert E. Tarone1 and Kenneth C. Chu 2
Interpretation of trends in disease rates using conventional age-period-cohort analyses is made difficult by
the lack of a unique set of parameters specifying any given model. Because of difficulties inherent in
age-period-cohort models, neither the magnitude nor the direction of a linear trend in birth cohort effects or
calendar period effects can be determined unambiguously. This leads to considerable uncertainty in making
inferences regarding disease etiology based on birth cohort or calendar period trends. In this paper, the
authors demonstrate that changes in the direction or magnitude of long term trends can be identified
unequivocally in age-period-cohort analyses, and they provide parametric methods for evaluating such
changes in trend within the usual Poisson regression framework. Such changes can have important implications for disease etiology. This is demonstrated in applications of the proposed methods to the investigation
of birth cohort trends in female breast cancer mortality rates obtained from the National Center for Health
Statistics for the United States (1970-1989) and from the World Health Organization for Japan (1955-1979).
Am J Epidemiol 1996; 143:85-91.
breast neoplasms; cohort effect; models, statistical; mortality; Poisson distribution; women
Many investigations of incidence and mortality data
involve analyses of temporal trends in rates by age and
calendar year period (1, 2). Such analyses are termed
"cross-sectional," because they cut across birth cohorts at a given point in time (3). Interpretation of
cross-sectional analyses can be misleading when longitudinal trends, such as trends with successive birth
cohorts, influence the pattern of rates over time ( 4 12). Longitudinal trends can provide critical clues
regarding disease etiology which would be missed by
strictly cross-sectional analyses. For example, crosssectional analyses of trends in US female breast cancer
mortality rates indicate a decreasing calendar period
trend in young women at the same time the calendar
period trend in older women is increasing (1, 12),
which has led to suggestions that there are different
etiologies for pre- and postmenopausal breast cancer,
differences between pre- and postmenopausal breast
cancer patients in their response to chemotherapy, or
different patterns of exposure to environmental carcinogens in young and old women (2, 13, 14). Analysis of
birth cohort trends, on the other hand, demonstrated
that the observed disparity between trends in young
and old women results from a moderation of the birth
cohort trend in risk, the timing of which suggests that
decreasing breast cancer mortality in young women
may be largely due to changes in childbearing patterns
following World War II (12). This example illustrates
that examination of longitudinal trends in rates can
have a marked impact on inferences regarding disease
etiology.
Conventional age-period-cohort analyses of variation in disease rates over time assume a log-linear
relation of age, calendar period, and birth cohort effects and invoke Poisson maximum likelihood methods to estimate the corresponding parameters. In spite
of the recognized importance of determining birth
cohort patterns of risk, the identification of such patterns has proven difficult because of the inability to
separate unequivocally linear trends in birth cohort
effects from linear trends in calendar period effects
(15-18). There are infinitely many sets of maximum
likelihood estimates of age, calendar period, and birth
cohort effects for any given data set (i.e., in statistical
terms, the parameters are not identifiable). The magnitude, or even the direction, of the linear trend in birth
cohort effects (or calendar period effects) cannot be
determined with certainty (15, 18). For example, examination of three sets of maximum likelihood estimates resulting from age-period-cohort analyses of
Japanese female breast cancer mortality rates indicates
that birth cohort parameters are increasing with advancing birth cohort in the first set of estimates but
Received for publication October 17, 1994, and in final form
October 2, 1995.
1
Biostatistics Branch, Division of Cancer Etiology, National Cancer Institute, Bethesda, MD.
2
Early Detection Branch, Division of Cancer Prevention and Control, National Cancer Institute, Bethesda, MD.
Reprint requests to Dr. Robert E. Tarone, National Cancer Institute, Executive Plaza North, Room 403, Bethesda, MD 20892.
85
86
Tarone and Chu
decreasing with advancing birth cohort in the third set
of estimates (18).
Such uncertainty has led to an understandable reluctance on the part of researchers to make strong inferences about birth cohort trends in disease rates. In this
paper, we demonstrate that, unlike the direction of a
birth cohort trend in rates, a change in the magnitude
or direction of the linear trend in birth cohorts (or
calendar periods) can be evaluated unambiguously
(i.e., such a change in magnitude is identifiable). Identifiable parameters representing local departures from
linearity have been proposed (15, 18), but the etiologic
interpretation of these parameters can be difficult.
Although such second-order effects have traditionally
been of less interest than first-order effects (e.g., the
magnitude of a slope), identification of changes in the
magnitude of long term trends can have important
etiologic implications (12). In the current paper, identifiable parameters are derived to permit unequivocal
inferences regarding such changes in the slope of birth
cohort effects in different eras using standard ageperiod-cohort analyses, and the parametric method is
applied to US white female breast cancer mortality
rates to confirm the results of the previous nonparametric analysis (12). The method is also applied to
Japanese female breast cancer mortality data for comparison of the patterns of birth cohort risk in US and
Japanese women (18).
MATERIALS AND METHODS
Sources of mortality data
US white female breast cancer mortality data for the
years 1970 through 1989 were obtained from the Division of Vital Statistics of the National Center for
Health Statistics (Hyattsville, Maryland). The breast
cancer mortality rates analyzed in the current study
differ somewhat from those used in the previous study
(12) in that 1989 mortality data are included and
population sizes have been updated on the basis of the
1990 US Census. The current study examines trends
from the period 1970-1989, as opposed to the 19691988 interval examined in the previous study (mortality data by single year of age are unavailable for all
years prior to 1969).
Construction of 2-year mortality rates
In the past, 5-year age and calendar periods have
been used routinely in age-period-cohort analyses,
presumably because rates are usually reported by
5-year age interval. The birth cohort intervals corresponding to 5-year age and calendar periods, however,
are 10 years in length, with considerable overlap be-
tween adjacent 10-year intervals. Because shorter birth
cohort intervals allow better discrimination of birth
cohort risk patterns, age-period-cohort analyses using
age and calendar period lengths of less than 5 years
should be undertaken whenever feasible. The annual
numbers of US breast cancer deaths are available by
single year of age. Annual population data are only
obtainable by 5-year age intervals, but a catalogue of
standard interpolation methods provides techniques
with which to estimate 1-year population figures from
the 5-year data (19). To estimate 1-year population
sizes, we used osculatory interpolation, which is designed to produce interpolated results that have a high
degree of smoothness. We used Beers' method based
on six-term ordinary formulas, which minimizes the
squares of the fifth-order differences and preserves the
5-year population figures (19).
The interpolated population sizes were used to create age-specific rates for 2-year age intervals. With
2-year age intervals and 2-year calendar periods, the
corresponding birth cohort intervals are 4 years in
length, with 75 percent of births occurring in the
middle 2 years of the 4-year interval. In the following
exposition, each birth cohort will be identified by the
first of the middle 2 years; for example, "the 1905
birth cohort" will refer to the years 1904-1907, with
75 percent of births occurring in the years 1905 and
1906.
Model and identrflability considerations
Table 1 shows the breast cancer mortality rates of
US white women by 2-year age interval from age 24
years to age 83 years and by 2-year calendar period
from 1970 to 1989. In general, let Ru denote the
disease rate for the /th of A age intervals and the
jth of P calendar periods. In conventional age-periodcohort notation, the birth cohort would be indicated
by an additional subscript, c = A + j — i, but to
simplify notation we will suppress the birth cohort
subscript. With A age intervals and P calendar periods,
the number of birth cohorts is C = A + P — 1.
In this notation, c — 1 corresponds to the earliest birth
cohort and c — C corresponds to the most recent birth
cohort In the usual age-period-cohort model, the
mean, EtJ, of the logarithm of the disease rate, RiJt is
modeled as
EQ
= a, +
TTJ
+
yc,
where the a / s are the age effects, the TT^'S are the
calendar period effects, and the y c 's are the birth
cohort effects.
As is shown in Appendix 1, the equality c = A +
j — i guarantees that there are infinitely many sets of
Am J Epidemiol
Vol. 143, No. 1, 1996
Analysis of Birth Cohort Trends
TABLE 1.
87
Brent cancer mortality rates amonjI US white women, 1970-1989*
Calendar period
Age
(years)
24-25
26-27
26-29
30-31
32-33
34-35
36-37
38-39
40^*1
42^*3
44-45
45-47
48-49
50-51
52-53
54-55
56-57
58-59
60-61
62-63
64-65
66-67
68-69
70-71
72-73
74-75
76-77
78-79
80-81
82-83
1970-1971
1972-1973
1974-1975
0.8311
(45)t
1.331
(66)
2.251
(103)
4.029
(172)
6.313
(256)
8.484
(334)
11.35
(442)
16.55
(661)
21.66
(905)
27.77
(1,205)
33.79
(1.501)
43.13
(1.945)
4756
(2,118)
54.48
(2,348)
63.35
(2,634)
70.41
(2,825)
73.44
(2,827)
81.35
(3,000)
77.85
(2.744).
84.29
(2,814)
88.77
(2,770)
9151
(2,649)
8857
(2,382)
95-22
(2,374)
101.41
(2,310)
11138
(2.279)
121.33
(2,169)
126.46
(1.946)
13335
(1.723)
139.77
(1,424)
0.628
(37)
1.157
(64)
2.692
(138)
4.264
(201)
5.822
(254)
8.296
(341)
11.01
(433)
16.06
(631)
19.60
(792)
26.94
(1,123)
32.84
(1,408)
41.53
(1,826)
50.86
(2,260)
56.97
(2529)
60.89
(2.654)
67.37
(2,808)
75.87
(2,994)
79.66
(3,006)
83.49
(3,025)
88.82
(3,065)
91.88
(2.996)
94.96
(2.905)
96.81
(2,737)
101.01
(2,599)
112.26
(2,614)
112.33
(2.366)
121.77
(2,290)
123.74
(2,023)
133.37
(1334)
137.72
(1539)
0.429
(27)
1.307
(78)
2.561
(143)
3.981
(206)
6.312
(301)
8.170
(359)
11.15
(455)
14.89
(587)
19.09
(750)
25.41
(1,013)
32.21
(1,328)
39.93
(1,707)
49.88
(2,192)
5437
(2,464)
60.40
(2,695)
65.82
(2,807)
73.90
(2377)
78.20
(3,015)
83.21
(3,087)
86.16
(3,056)
87.57
(2.962)
95.49
(3,059)
94.18
(2,797)
98.61
(2,662)
106.58
(2598)
113.90
(2,502)
116.38
(2.277)
131.18
(2.237)
129.63
(1384)
140.21
(1,711)
1976-1977
0.284
(19)
0.892
(57)
2.408
(145)
3.476
(195)
6.138
(317)
8.703
(410)
10.69
(460)
1336
(567)
1935
(786)
22.43
(878)
31.20
(1.251)
39.83
(1.651)
47.60
(2,038)
54.16
(2,389)
63.46
(2,823)
69.72
(3,019)
74.65
(3,111)
81.43
'(3,258)
83.16
(3,175)
91.93
(3,351)
91.46
(3.216)
95.87
(3,231)
101.34
(3,181)
10433
(2395)
10533
(2,722)
109.64
(2,513)
120.33
(2,428)
126.36
(2,208)
136.96
(2.047)
14458
(1,848)
1978-1979
1980-1981
1982-1983
1984-1985
1986-1987
1988-1989
0.412
(28)
0368
(63)
1320
(114)
3.734
(226)
5.480
(314)
8.144
(427)
11.25
(536)
13.00
(572)
18.96
(782)
25.39
(1,005)
28.85
(1,139)
36 55
0,463)
4550
(1.875)
52.82
(2,256)
60.79
(2,657)
71.31
(3,117)
7423
(3.211)
7531
(3,170)
80.63
(3,183)
88.07
(3,301)
95.84
(3,473)
9238
(3,237)
97.43
(3,177)
102.88
(3,078)
109.63
(2,975)
114.40
(2,777)
11452
(2,448)
121.94
(2,256)
12639
(1398)
14520
(1320)
0.300
(21)
0.816
(55)
1.880
(124)
3.150
(205)
6591
(410)
7.317
(412)
12.05
(608)
15.48
(713)
19.12
(824)
23.30
(950)
29.64
(1,167)
33.32
0.291)
44.89
(1,779)
5126
(2,122)
59.36
(2,545)
69.30
(3,026)
7427
(3271)
79.39
(3,407)
8428
(3,431)
88.34
(3,421)
92.62
(3,462)
99.01
(3,546)
10535
(3,562)
105.80
(3282)
11327
(3206)
117.73
(3,000)
120.10
(2,710)
122.77
(2,403)
133.86
(2219)
140.15
(1315)
0278
(20)
0.835
(59)
1.859
(128)
3.311
(220)
5.639
(358)
9.144
(549)
10.87
(612)
14.04
(729)
2028
(961)
22.61
(985)
28.10
(1,147)
35.19
(1,367)
39.36
(1,523)
49.85
(1,981)
59.13
(2,421)
64.40
(2,710)
74.84
(3221)
8125
(3,478)
86.97
(3,626)
9029
(3,637)
9338
(3,623)
98.17
(3,584)
105.94
(3,642)
10928
(3,511)
11236
(3,344)
121.60
(3258)
123.34
(2338)
131.85
(2,740)
132.41
(2.340)
14938
(2,167)
0.305
(22)
0.859
(62)
1.735
(123)
3342
(229)
5249
(346)
8.851
(562)
12.47
(756)
15.18
(856)
19.33
(995)
24.32
(1,144)
28.50
(1237)
34.94
(1,412)
4134
(1,632)
49.71
(1.913)
56.76
(2207)
68.02
(2,739)
73.00
(3.047)
80.94
(3,423)
86.86
(3,662)
94.72
(3316)
98.02
(3,866)
9923
(3,685)
111.60
(3312)
114.13
(3,766)
117.61
(3,600)
12525
(3,495)
134.76
(3368)
130.69
(2,868)
139.35
(2,610)
151.07
(2,306)
0282
(20)
0593
(43)
1307
(138)
3.192
(226)
5.028
(346)
7.449
(499)
11.35
(732)
16.40
(992)
1733
(997)
23.29
(1,184)
29.06
(1,348)
3437
(1,484)
43.35
(1,728)
47.98
(1.839)
57.65
(2,185)
62.44
(2,432)
69.68
(2310)
76.87
(3,159)
88.81
(3,676)
93.97
(3,869)
9958
(3392)
10322
(3377)
106.85
(3,888)
116.56
(3,935)
12038
(3,746)
126.30
(3,627)
128.07
(3,346)
136.81
(3,155)
145.78
(2,863)
14936
(2,419)
0219
(15)
0.875
(62)
1565
(113)
3.056
(222)
4.698
(336)
7.172
(493)
11.66
(761)
13.92
(869)
20.44
(1225)
2338
(1,351)
27.56
(1,417)
3652
(1,701)
4023
(1.728)
49.13
(1.977)
57.06
(2,195)
60.46
(2299)
7025
(2,682)
80.63
(3,147)
88.88
(3,496)
91.76
(3,750)
96.64
(3306)
101.57
(4,008)
109.52
(4,097)
116.32
(4,016)
126.55
(4,021)
132.61
(3319)
13522
(3,670)
136.82
(3294)
154.37
(3.175)
159.64
(2,732)
•
* Data ware obtained from the National Center for Health Statistics (Hyattsville, Maryland).
t Breast cancer deaths per 100,000 women.
| Numbers in parentheses, no. of breast cancer deaths.
age, period, and cohort effects which specify exactly
the same mean values for all /'s and a l l / s . Suppose,
however, that the slope of the linear trend in birth
cohort effects is /3j in one era, while the corresponding
slope in a second era is /32. That is, suppose that there
Am J Epidemiol
Vol. 143, No. 1, 1996
are cohort indices, c\ < c2, such that the linear
trends in birth cohort effects in the two eras can be
expressed as yc = 0, + ^lc for c < cl and yc = 02 +
f$2 c for c S: c2, where 0] and 02 are the intercepts of the
lines corresponding to the first and second era, respec-
88
Tarone and Chu
tively. It is shown in Appendix 1 that the difference
between the slopes, /32 ~ /3i> nas m e same value for all
of the infinitely many sets of parameters that specify
the mean values associated with a particular model
(i.e., the difference is identifiable). As a result, unequivocal inferences can be made regarding changes
in the magnitude or direction of birth cohort trends.
Contrasts in birth cohort parameters are defined in
Appendix 2 for comparison of the slopes of the linear
trends in two eras and quantification of the magnitude
of the difference between the two slopes. Each of these
contrasts is the difference between linear contrasts
defined over the two eras being compared, and is, like
the difference in slopes, identifiable. Similar contrasts
can be used to compare the slopes in different calendar
periods.
RESULTS
Figure 1 shows the maximum likelihood estimates
of the birth cohort effects for the breast cancer mortality data presented in table 1. Maximum likelihood
estimation for an age-period-cohort model requires the
specification of a parameter constraint (15-18), and
the estimates in figure 1 were obtained under the
constraint that the first and last cohort effects are zero.
It is evident that a marked change in the slope of the
birth cohort trend occurred around 1925. A different
constraint might lead to a curve with very different
first-order properties (e.g., declining cohort effects
from 1886 to 1920), but the decrease in slope around
1925 would be apparent for each constraint. The previous nonparametric analysis found a significantly increasing birth cohort trend from 1900 to 1914 and a
significantly decreasing trend from 1924 to 1938 (12).
To compare the slopes of the linear trends in birth
cohorts in these two eras using the linear contrasts
defined in Appendix 2, we let h — 8, corresponding to
the 1901 birth cohort, and k = 20, corresponding to the
1925 birth cohort. The contrast C, takes the value
—0.353 with a standard error of 0.018, and the contrast
C2 takes the value —4.157 with a standard error of
0.175. Both contrasts indicate a highly significant
change in the magnitude of the slope of the linear birth
cohort trend, with a moderation of risk for women
born from about 1925 to 1940, confirming the results
of the nonparametric analysis (12).
The trend in calendar period effects was examined
for evidence of a change in slope in the 1980s compared with the 1970s. The contrast,
TTg
7T4 -
7T2 -
27T,),
takes the value 0.168 with a standard error of 0.017,
indicating an increase in the calendar period slope in
the 1980s.
Clayton and Schifflers (18) presented age-periodcohort analyses of Japanese female breast cancer mortality using 5-year age intervals and calendar periods.
To determine whether the Japanese rates provide evi-
0.8 -i
0.6-
o
S 0.4 H
•c
.c
o
O
0.2CO
0.0-
Birth Cohort
RGURE 1. Maximum likelihood estimates of birth cohort effects In breast cancer mortality rates for US white women, 1970-1989. Data
were obtained from the National Center for Health Statistics (Hyattsville, Maryland). The maximum likelihood estimates were calculated under
the constraint that the first and last birth cohort effects were equal; the last birth cohort effect Is not plotted.
Am J Epidemiol
Vol. 143, No. 1, 1996
Analysis of Birth Cohort Trends
dence of a moderation in breast cancer risk by birth
cohort similar to that seen in the US data, the contrast
89
would mark a clear departure from the US birth cohort
trend, which gives no evidence of an increase in slope
after 1920.
yk+2
3 + Jh+2 ~ Jh+l ~
was evaluated for h = 5 and it = 10. The birth cohort
indexed by h = 5 is the birth cohort centered around
1900, while the birth cohort indexed by it = 10 is the
birth cohort centered around 1925; thus, the eras being
compared using C3 in the Japanese data are the same
as those compared in the US data using contrasts Cx
and C2. Based on maximum likelihood estimation
under the parameter constraint that the first and last
cohort effects are zero, we calculate for the Japanese
data that C3 = —0.568 with an estimated standard
error of 0.123. It can be easily verified that C3, being
an identifiable contrast, takes the same value for the
three sets of maximum likelihood estimates for the
Japanese data reported previously (18), each previous
set of estimators being based on a different parameter
constraint than that used in the current analysis. The
contrast indicates that the Japanese breast cancer mortality data, like the US breast cancer mortality data,
provide evidence of a moderation of mortality risk
beginning with women born around 1925. This moderation of the birth cohort trend in the Japanese data is
not evident from an examination of previously presented identifiable contrasts (18).
Previously proposed local curvature contrasts provided evidence in Japanese breast cancer mortality
rates of two sudden changes in the birth cohort trend
(18). One change occurred at the turn of the century.
Applying the contrast C3 to the Japanese data, taking
h = 2 (i.e., the 1885 birth cohort) and it = 6 (i.e., the
1905 birth cohort), gives C3 = 1.402 with a standard
error of 0.126, confirming a marked increase in birth
cohort slope. Applying C2 to the US data with h = 2
(i.e., the 1889 cohort) and k = 10 (i.e., the 1905
cohort) gives C2 = 0.467 with a standard error of
0.185. Thus, the birth cohort slope also increased
somewhat in US women around the turn of the century, but to a much smaller extent than in Japanese
women. A second sudden change in the Japanese birth
cohort trend which occurred around 1935 was indicated previously (18). Evaluating the contrast
7*
~
~
7A)
at h = 10 (i.e., the 1925 cohort) and k = 12 (i.e., the
1935 cohort) gives C4 = 0.180 with a standard error of
0.059. Trends in the most recent birth cohort effects
must be interpreted cautiously; however, this apparent
increase in the Japanese birth cohort slope following
1935, if confirmed with more recent mortality data,
Am J Epidemiol
Vol. 143, No. 1, 1996
DISCUSSION
We have shown that, in spite of the identifiability
problems inherent in age-period-cohort models, useful
inferences can be made regarding trends in age, calendar period, or birth cohort effects. A change in the
magnitude of the slope of a long term trend can be
identified unambiguously, and such a change can have
important interpretations regarding disease etiology.
Although first-order properties of trends (e.g., the
magnitude of the slope of the trend) are usually of
primary interest in investigations of disease rates, unequivocal inferences cannot be based on the magnitude or direction of trends in parameters in age-periodcohort models (15-18).
A recent analysis of Swedish cancer incidence rates
interpreted increasing birth cohort parameter estimates
with successive birth cohorts as evidence that cancer
risk was increasing because of recent increasing exposure to carcinogenic influences (20). As Appendix 1
shows, however, such inferences based on the direction of the birth cohort trend are tenuous at best. Only
if the parameter constraint used in the maximum likelihood estimation procedure holds exactly in the underlying true population rate structure will the direction of the trend be informative (16), and the validity
of the constraint can never be known with certainty.
Although it was not noted by the Swedish investigators (20), there is a decrease in the slope of the birth
cohort trend in the male Swedish cancer incidence data
beginning with birth cohorts of the 1940s. This diminution of the rate of increase in rates with recent
cohorts, which would seem to be at odds with the
published interpretation of the cohort trend, could be
evaluated using methods developed in the current paper.
Interpretation of the Swedish incidence data for
women is complicated by a marked moderation of the
risk of reproductive cancers beginning with cohorts
born in the 1920s (20). Interestingly, because breast
cancer would represent a major component of the
reproductive cancers, it appears that breast cancer
rates for Swedish women give evidence of the same
moderation in risk beginning with cohorts born in the
mid-1920s that is seen in US and Japanese breast
cancer mortality rates. Thus, it appears that the factor
or factors responsible for the moderation of breast
cancer mortality rates observed in US women born
from about 1925 through 1940 played a similar role in
Japan, Sweden, and possibly many other countries. It
would be useful to determine the extent to which the
90
Tarone and Chu
moderation of breast cancer risk with birth cohorts
beginning in the mid-1920s is a common feature in
international data, and to determine any factors other
than worldwide changes in childbearing patterns following World War II which could account for the
timing of the moderation. For example, a portion of
the prepubescent years of women born between 1925
and 1940 would have coincided with the Great Depression or World War II, and thus trends in energy
intake early in life (21) may have contributed to the
observed moderation of birth cohort risk.
Our analysis was based on mortality rates. Therefore, the possible impact of improved medical intervention on the survival of breast cancer patients must
be considered. Improvements in breast cancer survival
are unlikely to have had a major impact on population
breast cancer mortality rates over the period considered in this study (22). Such improvements usually
affect all or several age groups simultaneously, and
thus typically appear in an age-period-cohort model as
a decrease in the calendar period trend. The documented increase in the slope of the calendar period
trend in the 1980s rules out a major contribution of
improved medical treatment to the recent mortality
rate decreases in women under the age of 60 (12). The
fact that breast cancer mortality rate trends have been
driven predominantly by birth cohort trends over the
last three decades suggests that changes in breast cancer mortality rates have been influenced largely by
changes in etiologic factors.
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APPENDIX 1
Because of the equality i — j + c = A, it follows for any arbitrarily chosen constant, 8, that 8 (i - j + c —
A) = 0. Thus, for any such 8, any constants ax, a^, and a 3 chosen such that a, — a^ + a3 = A and any set of
parameters, a,, TTJ, and yc, which specify the underlying expected values, Etj, defining the model, the set of
parameters given by
a,* = a,
Tj* =
y * =
TTJ
i -a,)
- 8 (J ~ a
8(c-
a3)
specifies exactly the same expected values for every rate (i.e., for every i andj, a,* + TTJ* + yc* = a, + IT, +
yc). Thus, there are infinitely many sets of parameters that specify exactly the same model.
Am J Epidemiol Vol. 143, No. 1, 1996
Analysis of Birth Cohort Trends
91
For a parameter to be identifiable, it must take the same value (i.e., be invariant) for all transformations of the
above type (i.e., for all choices of 8, a,, a2, and a3). Suppose now that the slope of the linear trend in birth cohort
effects differs in different eras (that is, there are cohort indices c, < c2) such that
7c = 0i + /3ic
for c ^ Cj, and
Jc = 02 + &C
for c ^ c 2 . To see that the difference in slopes, f52 ~ /3X, is identifiable, note that under the transformation
7C* = yc + 8(c - a3),
y* = 9i + plc + 8(c-
a,)
= dl - 8a3 + (/3, + 8)c
for c ^ cl and
7c* = 02 + ftc + 8 (c - a3)
= d2 - 8a, + (ft + 8)c
for c 2: c2. The slopes for the transformed parameters, /3,* = 0! + 8 and /32* = 0 2 + 8, are not equal to the
slopes for the original parameters, /3t and /32. That is, the slopes are not invariant under the transformation (i.e.,
are not identifiable). Since 8 can take any value, it follows that different sets of maximum likelihood estimates
can indicate different magnitudes or even directions for trends in birth cohort effects, as demonstrated in the
analysis of Japanese breast cancer mortality data (18). Regardless of the value taken by 8, however, it follows
that j32* — 0!* = /32 — jBj, so the difference in slopes is identifiable. Analogous arguments demonstrate that
similar results hold for changes in the slope of age or calendar period effects.
APPENDIX 2
To obtain identifiable parameters for detection of changes in the slope of the linear trend in birth cohort risk
in different eras, we need to derive contrasts that are invariant under the transformation defined in Appendix 1,
i.e., yc* = yc + 8 (c - a3). In the previous analysis of US breast cancer mortality, we considered trends over
intervals of eight consecutive birth cohorts (12). One simple invariant contrast for comparing slopes between two
disjoint blocks of eight consecutive birth cohorts is
Ci = 7t+7 - 7* ~ (7*+7 ~ 7A)>
where h + 1 < k. This contrast is analogous to the local curvature contrast, yc+l — 27C + 7 C _!, given previously
(18). More analogous to the previously applied nonparametric method for investigating trends over eight
consecutive cohorts, in that all relevant birth cohorts contribute to the analysis, is the difference in linear
contrasts,
C2 = lyk+i + 57t+6 + 37*+5 + 7t+4 - 7*+3 - 37*+2 ~ 57*+i ~ 77*
7*+3 -
37A+2
- 57* + i -
lyh).
Any such difference in linear contrasts can be shown to be identifiable (i.e., to be invariant under the
transformation from yc to yc*) and can be expressed algebraically as a linear combination of the parameters—that
is, C = s'y, where s is the vector of coefficients defining the contrast and y is the vector of birth cohort effects.
If Vy denotes the estimated covariance matrix for the maximum likelihood estimates of the birth cohort effects,
then the variance of C can be calculated as s'Vys.
Am J Epidemiol
Vol. 143, No. 1, 1996