As noted above, a key issue in identifying an independent prospective study with appropriate risk factor collection included the need for details of age at each pregnancy, a refinement of usual reporting of age at first birth and number of births typical of epidemiologic studies. Details on age and type of menopause were also important since this is omitted from the Gail model despite a long record of being established as a modifier of future breast cancer risk [5, 13, 14]. Other key risk factors not included in the Gail model are duration and type of postmenopausal HT used [15], BMI [16], and alcohol intake [17]. These are all in the Rosner–Colditz log-incidence model.
CTS This cohort contains the necessary data collected at baseline in 1995 for the cohort. The CTS approach to questionnaire follow-up, after 2 years, then after 3 more years, then at varying intervals each updating some exposures, together with case ascertainment ongoing annually through the California tumor registry, meant we use baseline data only. We limit the population to women who were postmenopausal at baseline. To compare incidence during common follow-up time periods we use the time frame for CTS from baseline 1995 to 2009.
NHS This cohort of women followed from 1976 has routinely updated information every 2 years on reproductive risk factors for breast cancer, family history of breast cancer, use of postmenopausal hormones, and from 1980 onwards alcohol intake. The original Rosner–Colditz model was developed in the broader NHS cohort [1, 2, 5]. For comparability with data available from the CTS, we limit the population for this analysis to women who were postmenopausal at baseline in 1994. Thus the corresponding time available for the NHS is 1994–2008. In 1994, NHS participants were 47–74. Hence, we limit the CTS participants included in the analysis to a comparable age range, excluding their older cohort members.
Model fitting issues
Limited only to baseline data from the CTS, we modified the Rosner–Colditz model to omit updating. Because this differs from our standard approach of updating exposure information every 2 years [2], we estimate the impact of this modification on overall performance.
Duration of current use of postmenopausal HT is significantly related to incidence of breast cancer [2, 18], and to type of menopause, age at menopause, and time since menopause. These factors are all importantly related to postmenopausal breast cancer incidence. We, therefore, used imputation methods to estimate future duration of use for postmenopausal HT in the CTS [19]. We used a two-step process to estimate use according to type of hormone used currently, and duration of use. We first fit a model to NHS data to estimate the duration of hormone use from 1994 to the return of the 2006 follow-up questionnaire for each type of HT (estrogen, E, alone and estrogen plus a progestin, E&P). Predictors included menopause type and time since menopause, and duration of use of HT among current users (see Tables 8 and 9). In addition to these characteristics of menopause, parity was positively related to ever use of E alone but not E&P, and positively to duration of use of estrogen alone, but inversely to duration of estrogen plus progestin. BMI was inversely related to ever use of E and E&P, but was unrelated to duration of use of either. Alcohol use was inversely related to ever use of E alone and to ever use of E&P, but not to duration of use of either formulation. We developed this model separately for use of E alone and for use of E&P. We then used this model with baseline CTS data to impute future use by type and duration for participants, taking the average of 5 imputations for each participant. (See Tables 8 and 9 for the imputation models and Appendix 2 for a summary of the imputation strategy.)
Time frame
To compare incidence of breast cancer in the two cohorts over a common time frame, we identified common subsets from the two cohorts. We use the CTS baseline in 1995 and 1994 as the start point for inclusion of NHS follow-up. We then draw on the age range of the NHS participants to define a comparable age range for CTS participants. Thus we limit NHS follow-up data to the interval 1994–2008. CTS data for the corresponding years are included with follow-up from 1995 to 2009.
During follow-up of the NHS cohort from 1994 to 2008, we identified 2,026 invasive breast cancer diagnoses among postmenopausal women during 540,617 person years. In the CTS, we identified 1,400 incident invasive breast cancer diagnoses among postmenopausal women during 288,111 person–years.
Description of the log-incidence model of breast cancer
We assume that the incidence of breast cancer at time t (I
t
) is proportional to the number of cell divisions accumulated throughout life up to age t (i.e., I
t
= kC
t
).
C
t
is obtained from
$$C_{t} = C_{0} {\text{x}}\mathop \prod \limits_{i = 0}^{t - 1} \left( {C_{i + 1} /C_{i} } \right) = C_{0} {\text{x}}\mathop \prod \limits_{i = 0}^{t - 1} \lambda_{i}$$
(1)
Thus, \(\lambda_{i} = \frac{{C_{i + 1} }}{{C_{i} }} =\) the rate of increase in \(C_{t}\) from age \(i\) to age \(i + 1\).
Log (\(\lambda_{i} )\) is assumed to be a linear function of risk factors that are relevant at age \(i.\) The set of relevant risk factors and their magnitude and/or direction may vary according to the stage of reproductive life. We fit PROC NLIN of SAS to estimate the parameters of the model with breast cancer risk factors including (1) duration of premenopause, (2) duration postmenopause, (3) type of menopause, natural or surgical (4) parity, (5) age at each birth, (6) current, past HRT use, (7) duration of HT use by type, (8) BMI, premenopause ≡ BMI1, (9) BMI, postmenopause ≡ BMI2, (10) height, (11) benign breast disease (BBD), (12) alcohol intake, (13) family history of breast cancer.
We fit the base model using baseline variables and imputed HT duration without updating exposures and assessed covariates using the CTS comparing their magnitude and direction to the variables in the NHS. We assess the performance of the model from the NHS in the CTS by fitting the NHS model and averaging five imputations of HT use. We fit the Gail model [11] using the formula from page 1880, with the caveat that in each cohort the number of previous biopsies is scored 0 or 1 and the number of relatives with family history is scored 0 or 1. We compare the c-statistic for Gail versus Rosner–Colditz log-incidence using the Wilcoxon rank sum test [20].
To assess calibration, we use the NHS model to estimate relative risks for individual women in the CTS and combine these with SEER data to estimate absolute risk. We then group the CTS participants by decile of estimated absolute risk and compare observed and expected counts of incident breast cancers and test for trend using Poisson regression approaches (for additional details, see Appendix 1).
To assess calibration, we apply the NHS risk model to the CTS population using imputed data for HRT use over 12 years. Suppose there are N subjects in the CTS population who are followed for T person–years. We divide the T person–years into L age strata and let T
l
= number of person–years in the lth age stratum. Based on the NHS risk model, we compute the relative risk for the ith person at the jth person–year given by RR
ij
compared to a hypothetical person at baseline risk where all covariate values are 0. Let h
*1
(l) be the age-specific incidence rate for the lth age group from SEER 1995–2006. We use the methods of Gail (1989) to combine the RR
ij
from the NHS model with h
*1
(l) to estimate h
1(l) = baseline incidence rate for the lth age group of CTS. An estimate of the incidence rate for the ith subject in the jth person–year is then given by
$$\hat{I}_{ij} = \mathop \sum \limits_{l = 1}^{L} h_{1} (l)\delta_{ijl} RR_{ij}$$
where \(\delta_{ijl} = { 1}\) if the ith subject is in age group l at the jth person–year, = 0 otherwise.
The corresponding estimate of cumulative incidence for the ith subject over t
i
person–years is given by
$$E_{i} = 1 - { \exp }( - \mathop \sum \limits_{j = 1}^{{t_{i} }} \hat{I}_{ij} )$$
Let O
i
= 1 if the ith subject develops breast cancer over t
i
person–years, = 0 otherwise.
If the NHS model is well calibrated in the CTS population, then O
i
should follow a Poisson distribution with mean = \(E_{i}\). To test this we let \(\mu_{i} = E(O_{i} )\) and consider the Poisson regression model
$$\ln \left( {\mu_{i} } \right) = \alpha + { \ln }(E_{i} )$$
A test of the calibration of the model at the individual level is
$$H_{0} : \alpha = 0 \, {\text{vs}}. \, H_{1} :\alpha \ne 0$$
which we can perform using a Poisson regression model with intercept only and offset given by \({ \ln }(E_{i} )\).
We also can group the subjects into deciles by cumulative incidence per year (or \(E_{i}^{*} = E_{i} /t_{i}\)) and compute the observed (O
(d)) and expected (E
(d)) number of cases in the dth decile and run a Poisson regression at the aggregate level of the form:
$$\ln \left( {\mu^{(d)} } \right) = \alpha + { \ln }(E^{(d)} )$$
where \(\mu^{(d)} = E(O^{(d)} ).\)
The individual and aggregate Poisson regression models are actually equivalent. The Poisson regression approach should be a more sensitive model of goodness of fit than the Hosmer–Lemeshow statistic given by
$$X_{HL}^{2} = \mathop \sum \limits_{d = 1}^{10} \frac{{(O^{(d)} - E^{(d)} )^{2} }}{{E^{(d)} }}$$
which is more similar to a test of hetereogeneity than the test for trend approach given by Poisson regression.
Finally, to combine inferences over several imputed data sets, multiple imputation approaches are used to obtain an overall test of calibration based on averaging estimates of \(\alpha\) over several imputations. More detail on the calibration methodology is given in Table 8.