Age-period-cohort analysis of cancers not related to tobacco, screening, or HIV: sex and race differences

Abstract

Objective

To identify trends in a residual category of cancers not typically associated with tobacco, screening, or human immunodeficiency virus (HIV) infection.

Methods

For persons aged 20–84, we used sex- and race-specific age-period-cohort (APC) models to describe temporal patterns of incidence (1975–2004) and mortality (1970–2004) in the U.S. for a residual cancer category that excluded non-Hodgkin lymphoma, Kaposi sarcoma, and cancer of the oral cavity and pharynx, esophagus, pancreas, larynx, lung and bronchus, urinary bladder, kidney and renal pelvis, colon and rectum, prostate, female breast, and cervix uteri.

Results

Age-specific incidence rose (0.1–0.9% per year, on average) in every sex-race group, with factors related to both time period and birth cohort membership appearing to accelerate the increases in women. Age-specific mortality fell (0.6–0.9% per year, on average) for black and white men and women, with the declines decelerating in white women but accelerating in the other sex-race groups. Extrapolations of APC models predicted higher age-adjusted incidence rates in white women (11%), black women (5%), and white men (4%) in 2005–2009, relative to 2000–2004, and lower rates in black men (−3%), accompanied by lower age-adjusted mortality rates in every sex-race group (−8% in black men, −3% in black women, −1% in white men, and −1% in white women).

Conclusions

The possibility that increased incidence in women over time reflects changes in underlying risks, diagnostic practices, or better case ascertainment should be actively explored. Declining mortality may signify improvements in cancer care.

Appendix

Suppose cancer data (incidence or mortality) are cross-classified into N^a age groups and N^p time periods of equal width, thereby determining N^c birth cohorts. Assume cancer events (new cases or deaths) follow a Poisson distribution and consider the age-period-cohort (APC) model:

$$ {\text{ln[E(R}}_{ijk} ) ]= \mu + {\text{A}}_{i} + {\text{P}}_{j} + {\text{C}}_{k} , $$

(1)

where ln[E(R _ijk )] is the natural logarithm of the expected value of the cancer rate, R _ijk , associated with the ith age group (i = 1,…,N^a), jth time period (j = 1,…,N^p), and kth birth cohort (k = 1,…,N^c). This full APC model includes an overall mean (μ) and separate main effects for age group (A _i ), time period (P _j ), and birth cohort (C _k ). The relationship among these three temporal factors dictates that N^c = N^a + N^p − 1 and k = N^a + j − i.

Using the approach of Dinse et al. [11] as originally proposed by Holford [10], we reparameterized the APC model in terms of trend lines and deviations from the trend lines. For each temporal factor (i.e., age, period, cohort), imagine fitting a straight line to the main effects and then representing each main effect as a point along that line plus the vertical distance (or deviation) from that line. Within this framework, the main effects in model (1) can be expressed as follows:

$$ {\text{A}}_{i} = \left[ {i - {\frac{{{\text{N}}^{a} + 1}}{2}}} \right]\beta^{a} + \delta_{i}^{a} , $$

(2)

$$ {\text{P}}_{j} = \left[ {j - {\frac{{{\text{N}}^{p} + 1}}{2}}} \right]\beta^{p} + \delta_{j}^{p} , $$

(3)

$$ {\text{C}}_{k} = \left[ {k - {\frac{{{\text{N}}^{c} + 1}}{2}}} \right]\beta^{c} + \delta_{k}^{c} , $$

(4)

where $\delta_{i}^{a} = \sum_{m = 1}^{{N^{a} - 1}} {H_{im}^{a} \theta_{m}^{a} }$, $\delta_{j}^{p} = \sum_{m = 1}^{{N^{p} - 1}} {H_{jm}^{p} \theta_{m}^{p} }$, and $\delta_{k}^{c} = \sum_{m = 1}^{{N^{c} - 1}} {H_{km}^{c} \theta_{m}^{c} }$. The β and θ terms are unknown regression parameters and the H terms are elements of design matrices of orthogonal polynomials (see Holford [10] for details). The first component in each of these three equations uses the slope parameter for age (β^a), period (β^p), or cohort (β^c) to determine the ith, jth, or kth point along the appropriate trend line, respectively. Each second component represents a deviation from the appropriate trend line that is specifically associated with the ith age group ($\delta_{i}^{a}$), jth time period ($\delta_{j}^{p}$), or kth birth cohort ($\delta_{k}^{c}$), respectively.

Because of the interdependency of age, period, and cohort, some parameters in the full APC model, such as the three individual slopes (β^a, β^p, β^c), are not uniquely estimable [10]. One function of the APC parameters that is estimable, however, is the sum of the period and cohort slopes, which is known as the drift and is denoted by D = β^p + β^c. For any age group, D summarizes the long-term linear temporal trend in the log cancer rates. From the drift, we can calculate the average annual percent change (AAPC):

$$ {\text{AAPC}} = 100\,\left[ {{ \exp }\left( {D/ 5} \right) - 1} \right], $$

(5)

which is the age-independent relative change in cancer rates associated with a temporal change of 1 year. Our APC analysis used 5-year age and time period groupings. Equation 5 divides D by 5 to rescale from 5-year time units to 1-year time units.

Analogously, the main effects (A _i , P _j , C _k ) in the full APC model are not uniquely estimable, but their deviations from linearity ($\delta_{i}^{a}$, $\delta_{j}^{p}$, and $\delta_{k}^{c}$) are estimable. Deviations provide information about how well the log cancer rates are summarized by the trend lines and whether any discrepancies are attributable specifically to age, period, or cohort effects. Under model (1), the ratio of the expected cancer rates for a fixed age group i and consecutive time periods and birth cohorts is:

$$ {\text{E(R}}_{i,j + 1,k + 1} ) / {\text{E(R}}_{ijk} )= \exp [({\text{P}}_{j + 1} - {\text{P}}_{j} )+ ( {\text{C}}_{k + 1} - {\text{C}}_{k} )], $$

(6)

which does not appear to be uniquely estimable. However, substituting from Eqs. 3 and 4 for the period and cohort effects, this ratio can be rewritten in terms of estimable quantities:

$$ {\text{E(R}}_{i,j + 1,k + 1} ) / {\text{E(R}}_{ijk} ) { } = \, \exp [D \, + { (}\delta_{j + 1}^{p} \, - \, \delta_{j}^{p} ) { } + { (}\delta_{k + 1}^{c} \, - \, \delta_{k}^{c} )]. $$

(7)

Furthermore, taking the ratio of two adjacent expected rate ratios for a fixed age group i yields:

$$ {\frac{{{\text{E(R}}_{i,j + 1,k + 1} ) / {\text{E(R}}_{ijk} )}}{{{\text{E(R}}_{ijk} ) / {\text{E(R}}_{i,j - 1,k - 1} )}}} = \exp [(\delta_{j + 1}^{p} - 2\delta_{j}^{p} + \delta_{j - 1}^{p} )+ (\delta_{k + 1}^{c} - 2\delta_{k}^{c} + \delta_{k - 1}^{c} )], $$

(8)

where the exponent in (8) is the sum of two second-order differences, one involving only period deviations and the other involving only cohort deviations. The constant-curvature model holds these second-order differences constant in j and k, respectively. Hence, the ratio of adjacent rate ratios in (8) reduces to a constant, which we call the constant curvature index (CCI). In this case, we write $\delta_{j + 1}^{p} - 2\delta_{j}^{p} + \delta_{j - 1}^{p} \equiv \Updelta^{p}$ for j = 2,…,N^p − 1 and $\delta_{k + 1}^{c} - 2\delta_{k}^{c} + \delta_{k - 1}^{c} \equiv \Updelta^{c}$ for k = 2,…,N^c − 1, and we express the CCI in terms of the constants Δ^p and Δ^c:

$$ {\frac{{{\text{E(R}}_{i,j + 1,k + 1} ) / {\text{E(R}}_{ijk} )}}{{{\text{E(R}}_{ijk} ) / {\text{E(R}}_{i,j - 1,k - 1} )}}} = \exp (\Updelta^{\text{p}} + \Updelta^{\text{c}} ) = {\text{CCI}} . $$

(9)

Thus, the constant curvature index can be written as the product CCI = CCI^p · CCI^c, where the period component is CCI^p = exp(Δ^p) and the cohort component is CCI^c = exp(Δ^c). Unlike the AAPC, the CCI retains the 5-year time unit of analysis.