Letters to trie Editor
TABLE 1.
797
Differences of linear contrasts and their variances
Description of difference
ol linear contrasts
Difference of two consecutive-term
linear contrasts involving kand
k + 1, compared with another two
consecutive terms, h and h + 1
Difference of three consecutive-term
linear contrasts involving k, k + 1,
and k + 2, compared with another
three consecutive terms, h, h + 1,
and h + 2
Difference of two non-consecutive-term
linear contrasts involving kand k+ i,
compared with two nonconsecutive
terms, h and h + /'
Dfferenca In linear contrasts using coefficients
obtained dractty from age-pertod-cohort model*
Variance tor difference
In linear contrasts!
V
+ V
V
+ V
-2V
—
ck, where h> k+ 1.
2
V.
(Note: If h - k + 1, then c^, - 2cA + c,_v)
- 4 V ^ , - 4 VM
.- 2V,.,
<THI " c » - c M + c», where /) > fr + 2.
(Note: For h = fr + 3, then c M - c^j 01/
ck, where h> k+ i.
.1.91/
4.91/
2V
—9\/
J
- 2V
2V
* Linear contrasts are in parentheses.
t Vu - variance/covariance of age-period-cohort coefficients c, and c;.
I encourage the use of the differences of two linear
contrasts with no common coefficient in the two contrasts.
These differences measure the change in slope from one
linear contrast to another. If the difference has a positive
value, then change in risk is increasing, and if the difference
is negative, the change in risk is declining. For data involving 5-year groupings, a difference of contrasts that may be
useful is (cjt+5 — ck+3) — (ck+1 — c j . This difference allows
the comparison of slopes from two linear contrasts that each
have three consecutive terms (k + 5, k + 4, and k + 3
vs. k + 2, k + 1, and it) but are adjacent to each other.
All such differences in contrasts are identifiable and
unique (2).
The birth cohort effects for diabetes mortality among
Hispanics have been plotted in panel a of figure 1 (next
page). The Clayton-Schifflers contrast is plotted in panel b,
and the (c t + J — c t+3 ) — (c t+2 - c*) difference of contrasts
is shown in panel c. It can be seen that the four-term
difference in contrasts is more informative than the local
curvature contrast in terms of changes in trends in the
Hispanics' cohort effects curve (panel a). The initial positive values in the four-term difference reflect the increased
slope in the cohort effects curve after 1902, and the negative
values for the difference from 1917 to 1932 reflect the
steady decline in the rate of change for the cohort effects.
The birth cohort effects among American Indians are
displayed in panel d of figure 1. The positive values in the
four-term difference of contrasts (panel f) initially indicate
the increase in the slope for the American Indian cohort
effects curve (panel d) in the early 1900s, and negative
values for the difference subsequently indicate the steady
decrease in the rate of change of cohort effects, which is
similar to that for the Hispanics' cohort effects curve. Thus,
there are two important components in both birth cohort
effects curves: 1) an increase in the slope of the long-term
trend in risk occurring early in the 20th century (possibly
occurring somewhat earlier for Hispanics) and 2) a period
of gradual and steady moderation of risk in the 1920s
and 1930s. Of course, the significance of the difference of
Am J Epidemiol
Vol. 147, No. 8, 1998
contrasts still needs to be determined with variance, a'Va,
where a represents the coefficients of the difference of the
contrasts and V is the covariance matrix from the ageperiod-cohort analysis.
REFERENCES
1. Gilliland FD, Owen C, Gilliland SS, Carter JS. Temporal
trends in diabetes mortality among American Indians and
Hispanics in New Mexico: birth cohort and period effects.
Am J Epidemiol 1997;145:422-31.
2. Tarone RE, Chu KC. Evaluation of birth cohort patterns in
population disease rates. Am J Epidemiol 1996;143:85-91.
3. Clayton D, Schifflers E. Models for temporal variation in
cancer rates, n. Age-period-cohort models. Stat Med 1987;6:
469-81.
4. Kotz S, Johnson NL, Read CB, eds. Encyclopedia of statistical
science. Vol 2. New York, NY: Wiley-Interscience, 1982.
Kenneth C. Chu
Office of Special Populations
National Cancer Institute
Executive Plaza North
Bethesda, MD 20892
THE AUTHORS REPLY
Identifying calendar period and cohort effects in incidence and mortality data can be useful in determining
etiologic factors for disease occurrence in populations. The
use of generalized linear models facilitates such analyses by
providing estimates of effects that are amenable to statistical
evaluation. Although the effects themselves cannot be
uniquely defined (1), several authors have shown that there
are testable contrasts which are both identifiable and interpretable and which can be used to draw reasonable infer-
798
Letters to the Editor
Cohort Effects
Of Hispanics
1892 1897 1902 1907 1912 1917 1922 1927 1S32 1937 1942 1947 1962
Clayton/Schifflers
Contrast
c(k+i)-2c(k) + c(k-i
1692 1607 1902 1007 1912 1917 1922 1927 1932 1937 1942 1947 1952
(1b)
4-Term Difference of
Contrasts:
[c(k+5)-c<k+3)Hc<k+2)-c(k)]
1692 1697 1902 1907 1912 1917 1922 1927 1932 1937 1942 1947 1952
(1c)
Cohort Effects
of American Indians
1.5
1
0.5
(id)
Clayton/Schifflers
Contrast:
c{k+1)-2c(k) + c(k-1
•
•I
i—•
••
1 i—i
'—i
1
1
i—i
1892 1897 1902 1907 1912 1917 1922 1927 1932 1937 1942 1947 1952
A
I\
/ \
0.5
0
-0.5
•4
/
I
\
(
I i—i
i—i
s
}
r
\ /
1892 1897 1902 1907 1912 1917 1922 1927 1932 1937 1942 1947 1952
4-Term Difference of
Contrasts:
[c{k+5)-c(k+3)Hc(k+2)-c(k)]
df)
1892 1897 1902 1907 1912 1917 1922 1927 1932 1937 1942 1947 1952
FIGURE 1. Birth cohort effects for diabetes mortality among Hispanics and American Indians and differences in contrasts.
ences. One class of such contrasts—namely, differences in
linear polynomial contrasts—was first introduced by
Clayton and Schifflers (2, 3) and then generalized by Tarone
and Chu (4).
How to choose among the many possible differences in
linear contrasts and how to report the results of age-periodcohort analyses remain open questions. Choices include
differences in linear contrasts that overlap on a central
coefficient (i.e., which compare the change in slope in
adjacent eras) and those that do not overlap (i.e., which
compare nonadjacent eras). In either case, the contrasts that
are compared may cover one or more time intervals of equal
length.
We and odiers have used this approach to investigate
temporal trends in mortality rates (4, 5). In our analysis
of period and cohort effects for diabetes mortality in
New Mexico (5), we reported results using differences
in overlapping linear contrasts. In the above letter, Chu (6)
Am J Epidemiol
Vol. 147, No. 8, 1998
Letters to the Editor
799
0.5-
0.0-
-0.5-H-
—e—
s
c(k+4>-2c<k+2)+c(k)
[c(k+5)-c(k+3)Hc(k+2)-c(k)]
c(k+1)-2c(k)+c(k-1)
-1.01887
1892
1897
1902
1907
1912
1917
1922
1927
1932
1937
1942
1947
1952
Cohort
FIGURE 1. Three different types of linear contrasts depicting birth cohort effects for diabetes mortality among Hispanlcs. H, the four-term
difference contrasts supplied by Chu; G, a difference of linear contrasts that compares two adjacent intervals spanning 10 years each; J, the
Clayton/Schifflers contrasts.
suggests that the use of these contrasts limits the analysis,
and he proposes the use of differences of linear contrasts
that share no common coefficient.
In a reanalysis of our data, Chu presents a Clayton/
Schifflers-type contrast comparing two adjacent 5-year intervals and a difference in two linear contrasts comparing
10-year nonabutting eras (6, p. 798). He suggests that using
the latter type of contrast uncovers a moderation of the
cohort effect in the 1917-1932 cohorts of Hispanics and
American Indians. He also appears to infer that it is more
informative to compare nonoverlapping eras.
We commend Chu for suggesting a graphical approach
and for comprehensively reporting the coefficients and contrasts; however, following his line of argument, we suggest
an alternative interpretation using the comparison shown in
figure 1 above. We plotted the Clayton/Schifflers contrasts
(J), the four-term difference contrasts supplied by Chu (H),
and a difference of linear contrasts that compares two adjacent intervals spanning 10 years each (G). The latter
contrast more closely approximates the cohorts being compared by Chu's four-term difference contrast. The overlapping and nonoverlapping contrasts that span approximately
the same years result in similar estimates; those in the
central portion of the plot are small in magnitude, and none
are statistically significant. All three types of contrasts indicate a significant increase in slope in the earliest years of
the 20th century.
We conclude that the evidence is weak for a substantial
moderating rate of change in the cohort effect in the 1917—
Am J Epidemiol
Vol. 147, No. 8, 1998
1932 period. Moreover, in this example the general distinction between overlapping and nonoverlapping differences in
linear contrasts does not appear to be important. Rather, the
span of years in the contrasts under consideration has a
greater influence on contrast values. The generalizability of
this conclusion is uncertain, because the relative importance
of the type of contrast may vary by the shape of the
coefficient function. Further methodological consideration
is needed to identify a valid generalized approach to the
choice of contrasts.
We have two additional comments regarding Chu's suggested approach (6). First, the use of graphical summaries in
contrasts which compare nonadjacent eras may be problematic. For example, the point plotted above the year 1902
using Chu's contrast is, by our reckoning, the estimated
difference in slope between 1887-1897 and 1902-1912.
What if the gap between the periods being compared were
much larger? The inferences drawn from such a graph may
not be clear. Second, when investigating changes in cohort
or period effects—for example, in the earliest calendar
period effects—the span of years available for use in contrasts is limited, and the one-interval overlapping contrasts
are the only ones available.
We suggest that reports of age-period-cohort analyses
include two types of graphical displays: graphs of estimated
cohort and period effects and graphs of differences of contrasts sharing a common coefficient but with varying interval lengths. Tests of significance are necessary for final
interpretation of results. It also may be instructive to test
800
Letters to the Editor
several contrasts covering nonadjacent eras if this is suggested by the plot of estimated effects.
REFERENCES
1. Kupper LL, Janis JM, Karmous A, ct al. Statistical age-periodcohort analysis: a review and critique. J Chronic Dis 1985;38:
811-30.
2. Clayton D, Schifflers E. Models for temporal variation in
cancer rates. I. Age-period and age-cohort models. Stat Med
1987;6:449-67.
3. Clayton D, Schifflers E. Models for temporal variation in cancer
rates. D. Age-period-cohort models. Stat Med 1987;6:469-81.
4. Tarone RE, Chu KC. Evaluation of birth cohort patterns in
population disease rates. Am J Epidemiol 1996;143:85—91.
5. Gilliland FD, Owen C, Gilliland SS, et al. Temporal trends in
diabetes mortality among American Indians and Hispanics in
New Mexico: birth cohort and period effects. Am J Epidemiol
1997;145:422-31.
6. Chu KC. Re: "Temporal trends in diabetes mortality among
American Indians and Hispanics in New Mexico: birth cohort
and period effects." (Letter). Am J Epidemiol 1998;147:796-8.
Frank D. Gilliland
Department of Preventive Medicine
School of Medicine
University of Southern California
Los Angeles, CA 90007
Charles Owen
Center for Health Promotion and
Disease Prevention
School of Medicine
University of New Mexico
Albuquerque, NM 87131
Am J Epidemiol
Vol. 147, No. 8, 1998