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Although breast cancer incidence is higher in black women than in white women among women younger than 40 years, the reverse is true among those aged 40 years or older. This crossover in incidence rates between black and white ethnic groups has been well described, has not been completely understood, and has been viewed as an artifact.
To quantify this incidence rate crossover, we examined data for 440653 women with invasive breast cancer from the National Cancer Institute’s Surveillance, Epidemiology, and End Results database from January 1, 1975, through December 31, 2004. Data on invasive female breast cancers were stratified by race, age at diagnosis, year of diagnosis, and tumor characteristics. Standard descriptive analyses were supplemented with Poisson regression models, age–period–cohort models, and two-component mixture models. All statistical tests were two-sided.
We observed qualitative (ie, crossing or reversing) interactions between age and race. That is, age-specific incidence rates overall (expressed as number of breast cancers per 100000 woman-years) were higher among black women (15.5) than among white women (13.1) younger than 40 years (difference = 2.4, 95% confidence interval [CI] = 2.4 to 2.4), and then, age-specific rates crossed with rates higher among white women (281.3) than among black women (239.5) aged 40 years or older (difference = 41.8, 95% CI = 41.7 to 41.9). The black-to-white incidence rate crossover was observed for all tumor characteristics assessed, although the crossover occurred at earlier ages of diagnosis for low-risk tumor characteristics than for high-risk tumor characteristics. The incidence rate crossover between ethnic groups was robust (ie, reliable and reproducible) to adjustment for calendar period and birth cohort effects in age–period–cohort models (P < .001 for difference by race).
Although this ecologic study cannot determine the individual-level factors responsible for the racial crossover in vital rates, it confirms that the age-related crossover in breast cancer incidence rates between black and white ethnic groups is a robust age-specific effect that is independent of period and cohort effects.
The incidence rate for breast cancer among women younger than 40 years is higher among black women than among white women, but the incidence rate among women aged 40 years or older is higher among white women than among black women.
Population-based retrospective analysis of data for 440653 women with invasive breast cancer from the National Cancer Institute's Surveillance, Epidemiology, and End Results database from January 1, 1975, through December 31, 2004.
The age-specific ethnic crossover in incidence rates for breast cancer was confirmed.
The breast cancer incidence rate crossover between black and white ethnic groups may reflect breast cancer heterogeneity, with black women having more early-onset and less late-onset types of breast cancer than white women.
This study had the usual limitations of a descriptive epidemiologic study, including a retrospective registry assessment, missing data, nonstandardized histopathologic typing, and lack of individual-level risk factor data.
From the Editors
It is widely recognized that breast cancer incidence rates overall are higher in black women than in white women younger than 40 years. However, it is not as well known that age-specific incidence rates are higher for black women among women younger than 40 years and higher for white women among women 40 years or older. This so-called black-to-white ethnic crossover has been well described (1–4) but has not been fully understood and has sometimes been viewed as an artifact (5).
Differential screening practices and/or hormone replacement therapy use cannot explain the higher incidence rates among young black women because routine screening mammography and hormone replacement therapy begin after the age of 40 years. Differential racial profiles for age-dependent risk factors may be a more plausible explanation for the crossover (3,5–7), with black women having an elevated risk for the early-onset types of breast cancer and a reduced risk for the late-onset types of breast cancer, compared with white women. If differential risk factor profiles for different breast cancer types account for the black-to-white incidence rate crossover, then the crossover is a population-based expression of biologic and/or etiologic heterogeneity and should be a robust (ie, reliable and reproducible) feature of vital incidence rates.
To assess the reliability and reproducibility of the black-to-white breast cancer incidence rate crossover, we obtained case and population data from the National Cancer Institute's Surveillance, Epidemiology, and End Results (SEER) database. We supplemented standard descriptive epidemiology with three mathematical models. First, Poisson regression models provided statistical significance tests of age interactions between black and white race; statistically significant age interactions would be consistent with biologic interactions due to etiologic heterogeneity (8). Second, age–period–cohort models yielded fitted age-specific incidence rate curves for black and white women that were adjusted for calendar period and/or birth cohort effects. Different age-specific incidence rate patterns after adjustment for period and/or cohort effects (or artifacts) would be consistent with different age-dependent biology and/or etiology. Third, two-component mixture models identified heterogeneity in the age distributions at diagnosis among black and white women by determining whether two or more breast cancer populations fit the data better than a single breast cancer population. Different age distributions at diagnosis in black vs white women would provide further proof of biologic heterogeneity (9,10).
We obtained data for patients with invasive female breast cancer from nine registry databases in the National Cancer Institute's SEER program (the SEER 9 Registries database) from January 1, 1975, through December 31, 2004. These nine SEER registries include Connecticut, Iowa, New Mexico, Utah, Hawaii, Detroit, San Francisco–Oakland, Atlanta, and Seattle–Puget Sound, and cover approximately 10% of the US population (11). We stratified breast cancer patients by black and white race, age at diagnosis, year of diagnosis, and estrogen receptor (ER) expression status. Other racial groups were excluded from this analysis. The following four age groups were arbitrarily chosen to approximate mammography screening prevalence and/or use of hormone replacement therapy: age younger than 40 years (no routine screening or use of hormone replacement therapy), ages 40–49 years (low screening and no use of hormone replacement therapy), ages 50–69 years (likely screened and highest use of hormone replacement therapy), and age 70 years or older (low screening and very low or no use of hormone replacement therapy).
Although black and white race, age at diagnosis, and tumor grade were recorded for the entire study period in the SEER database, extent of disease codes for tumor size and axillary lymph node involvement was most consistently recorded from 1988 forward. Estrogen receptor status was collected from 1990 forward.
Tumor characteristics (tumor size, lymph node involvement, grade, and ER status) were dichotomized into favorable (low risk) vs unfavorable (high risk) groups (ie, tumor size ≤2.0 vs >2.0 cm, lymph node–negative vs lymph node–positive status, low- vs high-grade tumor, and ER-positive vs ER-negative status, respectively). Low-grade tumors included grade 1 (well differentiated) plus grade 2 (moderately differentiated). High-grade tumors included grade 3 (poorly differentiated) plus grade 4 (undifferentiated or anaplastic). Each SEER registry recorded ER status as positive, negative, missing, borderline, or unknown. For all tumor characteristics (tumor size, lymph node involvement, grade, and ER status), missing, borderline, or unknown tumor features were combined into a single group that was designated “other or unknown” for descriptive purposes but were excluded from further analyses.
Age-standardized (US standard population for the year 2000) incidence rates were obtained by use of SEER*Stat 6.3.6 statistical software and expressed as the number of breast cancers per 100000 woman-years (ie, women per year). Percent changes in incidence rates were calculated from the earliest to the latest recorded time periods; 95% confidence intervals for the percent changes were calculated by the delta method (12).
Age-standardized incidence rates were graphed on log–linear plots by six 5-year time periods (1975–1979, 1980–1984, 1985–1989, 1990–1994, 1995–1999, and 2000–2004). Age-specific incidence rates were graphed on a log–log scale by fourteen 5-year age groups (ages 20–24, 25–29, …, 80–84, and ≥85 years). Poisson regression models were used to examine the association of age and race with breast cancer incidence rates. For this analysis, we used cubic regression splines to obtain a smooth but flexible model of the age effects. We then selected a common value for the number of segments by use of the Akaike information criterion (13) and tested for an interaction between age (modeled as a regression spline) and race by use of a likelihood ratio test. Under the null hypothesis of no interaction between age and race, the age-specific incidence rate curves for black and white women would be parallel on the log–log scale or proportional on the absolute scale. Statistically significant age interactions could be quantitative (noncrossover) or qualitative (8,14,15). Quantitative interactions correspond to slope changes in incidence rates in magnitude but not in direction (eg, the black vs white incidence rate ratio [IRRBW] was always >1.0 or always <1.0). Qualitative interactions correspond to slope changes in magnitude and direction with the age lines crossing (eg, the black-to-white incidence rate ratio switched from >1.0 to <1.0 or from <1.0 to >1.0). We also compared age-specific incidence rates by ER expression status with Poisson regression models (Supplementary Figures 1 and 2, available online). As for race, statistically significant interactions between age ER status could be quantitative (ie, noncrossover) or qualitative. Even though ER status is a tumor characteristic rather than an independent predictor variable, parameterization of ER status allowed us to test for different age-related effects by ER expression by use of a likelihood ratio test.
Age–period–cohort models were used to simultaneously assess age-specific incidence rates that are adjusted for calendar period and birth cohort effects (16–19). As described by Tarone and Chu (20), we used 2-year age groups and 2-year time periods. Given that the year of birth equals the year of diagnosis minus the age at diagnosis, our analysis of fifteen 2-year time periods from 1975 through 2004 and thirty-two 2-year age groups from ages 20 through 83 years spanned forty-six 2-year birth cohorts from 1893 to 1983 (referred to by midyear of birth). Because the age, period, and cohort variables are collinear, it is not possible to completely separate the linear trend in calendar period effects from the linear trend in age effects or the linear trend in birth cohort effects from the linear trend in calendar period effects (18). This collinearity is known as the “nonidentifiability” problem of age–period–cohort models. Notwithstanding this nonidentifiability issue, certain age–period–cohort parameters can be estimated if the age, calendar period, and birth cohort trends are orthogonally decomposed into their linear and nonlinear components (19) (for additional details, see Appendix 1).
In brief, the estimable age–period–cohort parameters include intercepts, “drifts” (or linear trends) (16), “deviations” (or nonlinear departures from the linear trends) (21), “curvatures” (or second differences) (17), and “slope contrasts” (or differences between log–linear trends over adjacent blocks of ages, periods, or cohorts) (22). Another useful identifiable age–period–cohort parameter is the “fitted age of onset curve,” which is the sum of an intercept term, a drift parameter that we refer to as the “longitudinal age trend” (16,23), and an age deviation (18). We fitted the age-at-onset curves to breast cancer incidence for white and black women separately. Because the curves were obtained from separate age–period–cohort models, they also were adjusted for any period and cohort deviations that might differ between black and white women. For example, period effects could differ between black and white women if screening trends varied by race, and cohort effects could differ if the profile of breast cancer risk factors such as obesity or hormone replacement therapy changed differentially in successive cohorts of black and white women.
Kernel density estimation was used to produce smoothed age distribution curves at diagnosis (or density plots) in single years, as previously described (24). In brief, the kernel smoother estimated the underlying probability density function for breast cancer incidence by age at diagnosis in single years. The area under each density plot represented 100% of the patients with breast cancer; 95% confidence intervals were calculated with bootstrap resampling techniques.
Finally, two-component mixture models were used to determine whether the age distributions at diagnosis fit two or more breast cancer populations better than a single breast cancer population. The key model parameter was the probability of being in the earlier group, which was used to summarize changes in age distributions over time. The probability density function, g, is given by the formula: g(y) = f(y;α0)p + f(y;α1)(1 − p), where y = (xλ − 1)/λ and x = age at diagnosis with parameters (λ, p, α0, α1). The parameter λ denotes the power transformation, ρ is the mixing proportion, α0 and α1 stand for all the parameters that are used in the component densities f. Further details on the parameterization and estimations have been described (9,10). We used the log-likelihood ratio test and the Akaike information criterion to determine model fit, with the larger Akaike information criterion corresponding to the model with the better fit to the data (13). All statistical tests were two-sided.
The SEER 9 Registries database collected data on 440653 patients with invasive female breast cancer who were newly diagnosed during the period from January 1, 1975, through December 31, 2004 (Table 1). Among these 440653 patients, 34478 were black and 381122 white. Black patients had a younger mean age at diagnosis (57.6 years) than white patients (62.6 years) and a larger mean tumor size (2.8 vs 2.1 cm) (for both, P < .001 for difference). Overall breast cancer incidence was lower among black women (111.9 cases of breast cancer per 100000 woman-years) than among white women (128.5 cases of breast cancer per 100000 woman-years) (difference = 16.6 cases of breast cancer per 100000 woman-years, 95% confidence interval [CI] = 16.5 to 16.7). However, among women younger than 40 years (Table 1), incidence rates were 20% higher for black women than for white women (IRRBW = 1.2, 95% CI = 1.1 to 1.2), whereas among women aged 40 years or older, incidence rates were lower for black women than for white women (IRRBW < 1.0 for age groups 40–49, 50–69, and ≥70 years). Of note, the incidence rate ratio for black to white women was less than 1.0 for all low-risk tumor characteristics (eg, IRRBW = 0.6, 95% CI = 0.6 to 0.7, for tumor size ≤2.0 cm) and 1.0 or more for all high-risk tumor characteristics (eg, IRRBW = 1.2, 95% CI = 1.2 to 1.2, for tumor size >2.0 cm) (25). Similar patterns were observed for lymph node–negative vs lymph node–positive disease, low- vs high-grade tumor, and ER-positive vs ER-negative disease.
Temporal trends in incidence rates among women with invasive breast cancer are shown in Table 2 for the following three time periods: 1975–1979, 1990–1994, and 2000–2004. The median age at diagnosis rose from 61 to 64 years from 1975–1979 to 1990–1994, and then returned to 61 years during 2000–2004. Incidence rates overall increased 28.3% (95% CI = 27.5% to 29.0%) from 1975–1979 to 2000–2004 (ie, from 102.1 to 131.0 cases of breast cancer per 100000 woman-years, respectively). Incidence rates increased more over this period for older women (for ages 50–69 years, 38.1%, 95% CI = 35.3% to 41.1%) than for younger women (for ages <40 years, 5.8%, 95% CI = 5.6% to 6.0%). Among both black and white women, incidence rates rose substantially from 1975–1979 to 1990–1994, with little or no increase from 1990–1994 to 2000–2004 (26). Incidence rates generally rose more for tumors with favorable characteristics (eg, small tumors, lymph node–negative disease, low-grade tumor, and ER-positive status) than for tumors with unfavorable characteristics (eg, large tumors, lymph node–positive disease, high-grade tumor, and ER-negative status), although conclusions are somewhat limited by the reapportionment of unknown to known tumor characteristics over time. For example, percent unknown for ER status fell from 24.8% (or 19784 of a total of 79642 patients) during 1990–1994 to 13.6% (or 12804 of a total of 94446 patients) during 2000–2004. Additionally, given that SEER did not record tumor characteristics for all time periods, time trends for tumor characteristics also were relatively short for tumor size (from 1988 forward), lymph node status (from 1988 forward), and hormone receptor expression (from 1990 forward). Therefore, these percent changes should be interpreted with caution.
Age-adjusted incidence rates rose until the late 1990s and then began to fall (Figure 1, A), with rates greater for white women than for black women for the entire study period (1975–2004). Notwithstanding these overall secular trends, age-specific temporal trends varied “qualitatively” (ie, they switched, crossed, or reversed) by race and age (Figure 1, B). For example, among women aged 40–49, 50–69, and 70 years or older, incidence rates were greater for white women than for black women for the entire study period, whereas among women younger than 40 years, incidence rates were greater for black women. Of note, measures of interaction (effect modification) by race within age groups were statistically significantly different for women aged 40 years or older (ranging from P < .01 to P < .001) but not statistically significantly different among women younger than 40 years (P = .75).
Age-specific incidence rates also varied qualitatively (ie, they switched, crossed, or reversed) by race (Figure 2). For example, overall age-specific incidence rates rose rapidly until age 40–50 years for both white and black women and then continued to rise at a slower pace (Figure 2, A). This midlife change, pause, or inflection in age-specific incidence rates near age 50 years has been termed Clemmesen's hook and has been attributed to menopause (27). Notwithstanding overall age-specific incidence rates, age-specific incidence rates were greater among black women than among white women who were younger than 40 years, and then, incidence rates were higher for white women (P < .001 for age × race interaction). The age–period–cohort fitted age-at-onset curves (Figure 2, B; adjusted for calendar period and birth cohort effects) also were higher among black women who were younger than 40 years than among white women in the same age group and higher among white women who were aged 40 years or older than among black women in the same age group (P < .001 for difference by race). Additionally, age–period–cohort analyses detected statistically significantly different age deviations among black and white women near the age of 40 years (P < .001 for difference by race), differential calendar period effects during the 1980s (P < .001 for difference by race), and differential birth cohort effects near the 1950 birth cohort (P = .009 for difference by race).
The black-to-white incidence rate crossover was observed for all tumor characteristics (Figure 3), although at earlier ages at diagnosis for low-risk tumor characteristics than for high-risk tumor characteristics. For example, the crossing of black-to-white incidence rates occurred at ages 25–29 years for tumor sizes 2 cm or less, whereas there was no crossover for tumor sizes more than 2 cm. For lymph node–negative and lymph node–positive disease, incidence rates crossed at ages 35–39 and 50–54 years, respectively. For low- and high-grade tumors, incidence rates crossed at ages 30–34 and 80–84 years, respectively. For ER-positive tumors, the crossing of incidence rates for black and white women occurred at ages 30–34 years, but there was no crossover for ER-negative tumors (28).
The shapes of the black and white age-specific incidence rate curves (Figure 2) reflected bimodal early- and late-onset breast cancer populations, with modes (ie, peak frequencies) near ages 50 and 70 years, respectively (Figure 4). These bimodal breast cancer populations fluctuated over time. From 1975–1979 to 1985–1989, breast cancer populations shifted to predominant late-onset age distributions for black and white women. Beginning in 1990–1994, population distributions of black and white breast cancer populations began to return to dominant early-onset age distributions. By 2000 through 2004, early-onset breast cancer populations were reestablished for both black and white women. Of note, the movement of black women toward later ages at onset during the 1980s was slower than that of white women, but it was faster to return toward earlier ages at onset in the 1990s.
Two-component mixture models confirmed that two or more breast cancer populations fit the data better than a single cancer population for both black and white women (Table 3). For example, during 1975–1979, Akaike information criterion among white women was larger for a mixture than for a single cancer population. We observed similar patterns for all time periods and for both races. Additionally, the probability for being in the early-onset breast cancer population decreased in the 1980s and then increased in the late 1990s (Table 3; Figure 5), consistent with the density plots in Figure 4. For example, among white women, the probability for early age at onset fell from 77% (95% CI = 73.0% to 80.9%) in 1975 through 1979 to 25% (95% CI = 23.0% to 27.0%) in 1985–1989, and then rose from 33% (95% CI = 31.8% to 34.2%) in 1990–1994 to 64% (95% CI = 62.2% to 65.8%) in 2000–2004. Black women had a similar pattern.
Age-specific incidence rates were higher for ER-negative tumors among younger women and higher for ER-positive tumors among older women (Supplementary Figure 1, A, available online). Poisson regression models confirmed the interaction between age and ER expression (P < .001 for age × ER interaction). ER Furthermore, for all age groups and all time periods, incidence rates for ER-positive tumors were greater among white women than among black women (Supplementary Figure 1, B, available online). In contrast, for all age groups and all time periods, incidence rates for ER-negative tumors were greater among black women than among white women (Supplementary Figure 1, C, available online). For both ER-positive and ER-negative tumors, measures of interaction between age groups and race were statistically significantly different for older women (P < .001 among black and white women with ER-positive tumors and aged ≥70 years, and P < .001 among black and white women with ER-negative tumors and aged ≥70 years) but not for younger women (P = .67 among black and white women with ER-positive tumors and aged <40 years, and P = .16 among black and white women with ER-negative tumors and aged <40 years).
Age and trend interactions by ER status (Supplementary Figure 1, available online) reflected bimodal early- and late-onset breast cancer populations (Supplementary Figure 2, available online). Beginning in the 1990s, the bimodal population of ER-positive tumors began to shift from being dominant at later ages to earlier ages at onset. The shift toward earlier ages at onset was exaggerated for ER-negative breast cancers.
The crossover in incidence rates between black and white ethnic groups was a robust (ie, reliable and reproducible) finding in the SEER 9 Registries database for the entire study period of 1975 through 2004 (Figure 1). This age-related interaction (or effect modification) was statistically significant in Poisson regression models, and it remained so in age–period–cohort models, which were adjusted for calendar period and birth cohort effects (Figure 2). In women younger than 40 years, incidence rates were statistically significantly higher among black women than among white women, whereas the reverse was true among women aged 40 years or older for all time periods. The incidence rate crossover between black and white ethnic groups was also observed after stratification of women by tumor characteristics, although the crossover occurred at earlier ages for low-risk prognostic characteristics than for high-risk prognostic characteristics (Figure 3). Furthermore, age distributions at diagnosis shifted over time, as illustrated with density plots (Figure 4) and mixture models (Table 3; Figure 5), which tracked a secular rise and fall in the proportion of late-onset cases of breast cancer among white women and a similar but delayed pattern among black women.
When recognized as a qualitative (crossing or reversing) age interaction, the breast cancer incidence rate crossover between black and white ethnic groups can be seen as a manifestation of breast cancer heterogeneity, with black women having more of the early-onset breast cancer types and less of the late-onset breast cancer types than white women. Given the early-onset predominance for ER-negative breast cancers (Supplementary Figure 1, A, available online), it also is not surprising for ER-negative tumors to be more common among black women than among white women. It should be noted that although black and white women have different proportions of early- and late-onset breast cancer types, these results do not indicate that the two racial groups have different kinds of breast cancer, although black women have proportionally more of the worse types (ie, early-onset and ER-negative tumors).
Throughout this study, we have focused on the qualitative age interactions among black and white women, but similar crossing, reversing, or dual age-dependent effects have also been observed for breast cancer risk factors, such as parity and obesity (29–35). That is, for younger women, parity is associated with increased risk and obesity is associated with decreased risk, whereas for older women, parity is associated with decreased risk and obesity is associated with increased risk. Therefore, differential racial profiles for risk factors with qualitative age-related effects (eg, parity and obesity) might account for at least part of the incidence rate crossover between black and white ethnic groups (3,5–7). However, age–period–cohort models confirmed an age-specific black-to-white incidence rate crossover, even after adjustment for birth cohort (risk factor) effects. Thus, the black-to-white ethnic crossover cannot be fully explained by known risk factor variations (36).
Additionally, the black-to-white incidence rate crossover has been present despite well-documented period and cohort racial variations over time. For example, breast cancer incidence rates more than doubled from 1935 to 1998 before falling after the Women's Health Initiative report in 2002 (26,37–43), especially among older women with tumors of low malignant potential (44–46). The shift of black women toward older ages at diagnosis was slower than that of white women during the 1980s, and it was faster to return toward early ages at onset in the 1990s (Figure 4), possibly due to racial variations for calendar period (screening) and/or birth cohort (exposure) effects. That is, screening mammography initially lagged for black women but equalized by the mid-1990s (42,47), perhaps accounting for a slower initial rise in screen-derived late-onset breast cancers among black women. Additionally, although the percent changes in exposure to hormone replacement therapy before and after reports from the Women's Health Initiative were similar among racial groups (48), white women were approximately twice as likely as black women to use hormone replacement therapy (49), also potentially explaining a slower initial rise in postmenopausal breast cancers. Yet, age–period–cohort models confirmed an age-specific black-to-white incidence rate crossover, even after adjustment for these calendar period (screening) and birth cohort (exposure) effects.
Our study has the usual limitations of descriptive epidemiology (ie, retrospective registry assessment, missing data, nonstandardized histopathologic typing, and lack of individual-level risk factor data). A major strength was the supplementation of standard descriptive epidemiology with mathematical models. Therefore, although this ecologic study cannot determine the factors responsible for the incidence rate crossover, as far as we know, it is the first validation of this phenomenon in vital rates by use of a structured quantitative approach.
In summary, breast cancer–age interactions by black and white race suggest biologic heterogeneity of the qualitative variety (ie, they switched, crossed, or reversed). Although the black-to-white ethnic crossover has been attributed to a one-time (early) adverse effect and long-term risk reduction for reproductive risk factors such as parity (5), we suggest that the crossover reflects qualitative age interactions (effect modification), whereby risk factors such as parity have different effects on different types of breast cancer. Indeed, recent studies show differential racial and risk factor profiles for different breast cancer phenotypes (50–54). Empirical studies should attempt to verify this ecologic phenomenon with individual risk factor data. Future analytic studies also should carefully and probably routinely be powered to assess the interaction between age and race because if the incidence rate crossover from black to white ethnic groups is a proxy for an age-specific biologic heterogeneity, then early- and late-onset types of breast cancer are distinctly different diseases. Ignoring such heterogeneity may obscure important stratum-specific effects that result from subgroup averaging of nonpoolable and/or dissimilar breast cancer populations (14,55).
This research was supported in part by the Intramural Research Program of the National Institutes of Health, National Cancer Institute.
As discussed by Rosenberg and Anderson (unpublished material), let ρij denote the logarithm of the expected incidence rates for age group j and calendar period i, which are defined by using equally spaced age and period intervals. In age–period–cohort analysis, we assume log–linear effects for age (αj), period (πi), and cohort (γk) over j = 1, …, J age groups, i = 1, …, I calendar periods, and k = 1, …, I + J − 1 birth cohorts:
It is well recognized that this model is not identifiable; that is, the linear trends in age, period, and cohort effects, αL, πL, and γL, respectively, cannot be separated because of the collinear relation of the cohort, period, and age indices. That is, the year of birth equals the calendar period or year of diagnosis minus the age or age at diagnosis, or k = i −j + J, where i, j, and k index the observed age, period, and cohort effects. However, as described previously by Holford (19), it can be shown with the theory of estimable functions that the parameters in the restricted model with πL = 0 yield fitted log–linear rates equal to:
Each of the terms in Equation 1 is identifiable, and the fitted rates are the same as all other solutions that maximize the likelihood. In this equation, and are the central or referent indices of age and cohort, respectively; the coefficient of the age trend (j− provides an estimate of (αL + πL), and the coefficient of the cohort trend (k−) provides an estimate of (πL + γL). We designate (αL + πL) as the “longitudinal age trend” and (πL + γL) as the “net drift” (16,17,23,56). The parameter μ is the intercept term, and the parameters and are the orthogonal (and identifiable) deviations of Holford (19).
We also considered the function , which we term the “fitted age-at-onset curve.” This function provides a summary estimate of the longitudinal age-at-onset curve, which represents an extrapolation of the experience of the midcohort (k= and assume that that is based on the experience of all other cohorts included in the analysis. By construction, the fitted age-at-onset curve is adjusted for both period and cohort effects.
We fitted Equation 1 separately to incidence data for black and white women and used Wald tests to compare age–period–cohort parameters by race for the fitted age-at-onset curves; drift parameters; and age, period, and cohort deviations. For example, the Wald test statistic that was used to compare the net drifts has a chi-square distribution with 1 df:
where SD1 and SD0 are the estimated standard deviations of the net drift parameter (πL + γL) for black and white women, respectively. Similarly, Wald tests were developed to test for parallel age-at-onset curves (I − 1 df) and for equal age deviations (I − 2 df), period deviations (J − 2 df), and cohort deviations (I + J − 3 df).
None of the coauthors have a financial conflict of interest that would have affected this research. The authors had full access to all of the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis.
We thank the reviewers for helpful comments, which have greatly improved the content of this manuscript.