We applied the proposed model and estimation method to the WAS study data. We analyzed a subset of the data in which the first-degree relatives of the female probands are also first-degree relatives to each other. That is, (mother, sister), (daughter, daughter) and (sister, sister) pairs were included in the analysis. The subset was chosen because the pairwise association should be similar among these pairs of first-degree relatives. Cancer and non-cancer mortality are the two competing events considered in this example. Among these women, the majority of the cancer incidences were breast and ovarian tumors. The ages of the relatives at the time of the interview of the proband define the censoring times. The data of 12,255 subjects coming from 4,235 distinct families were used to obtain the Nelson-type nonparametric estimate of the marginal survival of time to the first failure, overall hazard and cause-specific hazard, and the probability of failure type given the individual’s failure time. The estimates of the probability of failure type being cancer is displayed in . It shows that cancer risk is higher than non-cancer death in mid-age but lower in young and old ages.
Figure 1 The non-parametric estimate of the probability of failure type being cancer given the age at onset (solid circle) and a fitted line of a fourth-degree polynomial model to this non-parametric estimate. The fitted model is used in the simulation study to (more ...)
shows the estimation results for the association parameters. The number of pairs with both members having cancer(d11), number of pairs with one member having cancer and the other member dead of non-cancer (d12), and number of pairs with both members dead of non-cancer (d22) are listed in the second column. The bootstrap standard errors were obtained from 500 bootstrap samples. In estimating the association parameters, to assure there are sufficient number of paired events in each sub-region to calculate the piecewise cross-ratios and odds-ratios, and to choose each region which is biologically meaningful, the number of knots was set at K = 3 with w1 = 50, w2 = 70, and w3 = ∞ (see ). The cut-off values 50 and 70 divide the cohort into young (< 50 years old), mid-age (50−70) and old (> 70) subgroups. The data of 13,962 pairs from 4,152 families were used to estimate the cross-ratios. All the piecewise cross-ratios are close to 1, indicating the association of times to first failure between first-degree relatives is modest and almost time invariant.
Nine hundred and fifty pairs with failures observed in both pair members were used to estimate the odds-ratios. Of these 950 pairs, there were 767 distinct paired failure times. Because at each of these distinct paired failure times, most of the time there was only one observation, the non-parametric estimate of the probability of failure type given the paired failure time,
is mostly 0 or 1. Thus, ϕij
s cannot be estimated reliably under this non-parametric estimation. Therefore, we considered the two alternative approaches for the estimation of
described in the previous Section: assuming
to be piecewise constant vs. the probability of failure type depends only on the individuals’ own failure time. The estimated odds-ratios in the six sub-regions as ordered in for the two approaches are (4.25, 1.95, 0.86, 1.35, 1.63, 1.65) and (5.22, 1.20, 0.88, 1.57, 2.07, 1.97), respectively. Both approaches yielded similar estimates of the odds-ratios. The estimates based on the latter approach are shown in . The distributions of the bootstrap estimates of θij
s were close to normal, but those of ϕij
s were skewed. Hence the estimates of ϕij
s were log-transformed. Compared to the estimates of the cross-ratios, the odds-ratios for the association of failure types between the first-degree relatives vary in magnitude over different ranges of ages at onset. Of particular note is the large odds ratio for the ages at onset younger than 50. It implies that the probability of having cancer for a woman who had an event before age 50 is more than 5 times higher when her first-degree relative had cancer before age 50 than if her first-degree relative died of non-cancer before age 50. For women older than 70 years, there is a trend that her chance of developing cancer is doubled if her first-degree relative had developed breast cancer after age 50 than if her first-degree relative died of non-cancer after age 50 (ϕ22
= 2.07, ϕ23
The estimates of cause-specific cross-ratios are displayed in . The cross-ratio for cancer is high (> 2) when the ages at onset in both members are young and slightly elevated (1.5 − 2) when both members are old. However these elevations, likely due to few cancer cases in both pair member in these age ranges, are not statistically significant.
Figure 2 Top left: cancer vs. cancer cross-ratio; top right: cancer vs. non-cancer death cross-ratio; bottom left: non-cancer death vs. non-cancer death cross-ratio. The averaged cancer-cancer cross-ratios in the six sub-regions ordered in are 1.22, 1.19, (more ...)
To see the impact of the failure time and failure type of one family member on the cumulative incidence of the other family member, we plot the conditional incidence function along with the unconditional counter part. The four plots in the left panel of display the marginal and conditional cumulative risk of a woman developing cancer, and the four plots in the right panel display the marginal and conditional cumulative risk of a woman dead of non-cancer. These plots show that a woman’s cumulative risk of cancer, compared to the marginal cumulative risk, is increased if her first-degree relative had cancer before age 70, and decreased if her first-degree relative had cancer after age 70. If the first-degree relative died of non-cancer before age 50, then the woman’s cumulative risk of cancer is still increased (top two plots in the left panel), although the magnitude of the increase is lower than if her first-degree had cancer. In contrast, if the first-degree relative died of non-cancer after age 50, the woman’s cumulative risk of cancer is decreased slightly, and her cumulative risk of non-cancer death is increased.
Left panel: marginal and conditional cumulative cancer incidence; right panel: marginal and conditional cumulative non-cancer mortality incidence.
One may be also interested in the conditional cumulative incidence given the failure type of the first-degree relative. In , the top panel shows that a woman’s conditional cumulative cancer incidence increased if her first-degree relative had cancer, and decreased if her first-degree relative had died of non-cancer. The lifetime (up to age 100) cumulative incidence of cancer increased from 40% to 46%, if a woman’s first-degree relative had cancer. Such a pattern also holds for non-cancer mortality. The lower panel of shows that the conditional cumulative non-cancer increased if the first-degree relative had died of non-cancer, and decreased if the first-degree relative had cancer. The lifetime risk of non-cancer death increased from 58% to 61%, if a woman’s first-degree relative had died of non-cancer.
Top: marginal and conditional cumulative cancer incidence given the failure type of the other family member; bottom: marginal and conditional cumulative non-cancer mortality incidence given the failure type of the other family member.