Four hypothetical lifetime colonoscopy screening trajectories and the observed information (unshaded region) available from the SEER-Medicare data set. In the figure, a circle denotes the time an individual becomes due for a screening (occurs either at age 50 or 10 years after the previous screening), and an ‘X’ denotes the time of a colonoscopy screening. The solid lines represent periods of time where the subject is overdue for a colonoscopy (called a ‘lag time’), and the dashed lines represent the 10-year period during which average-risk subjects are not due for a screening. Among the hypothetical trajectories, subject A is screened twice, once before 1991 and then again in the observation window. Subject B has only one screening, which is observed. However, based on information provided in the observation window, it is not possible to tell if the lifetime trajectory for subject A is different than it is for subject B. Similarly, neither subject C or D has an observed screening, but subject C does get a colonoscopy screening after the study ends, while subject D does not. In this example, all four hypothetical subjects are both left-censored and right-censored.

The figure shows two possible trajectories for a left-censored subject who enters the the observation period at time t_L and has one observed screening at time t₁. Lag times are denoted with solid lines, and represent periods of time when the subject is due for a screening, and the 10-year post-screening period is denoted with a dashed line. Screenings are marked with ‘X’, and time points when the subject becomes due for a screening are marked with a circle. In this example, it is possible that (1) the one observed screening is the only lifetime screening or (2) the subject may have had two lifetime screenings, the first one occurring before the left-censoring time t_L and the second being observed. In the first case, the length of the first lag time, y₁, is equal to the time to the first screening, t₁, and can be written as . In the second case, we can only determine that the time to the first screening, y₁, is less than the left-censoring time, and the second lag time, y₂, is the remaining period of time between the observed screening and the previous screening (t₁ − 10 − y₁). This can be written as . This subject is not right-censored because he or she reaches the maximum screening age before the observation period ends.

Approximated hazard rate of the time to the first screening between years 1991 and 1996 (before Medicare insurance coverage changes or guidelines were set). The hazard rate is very flat, providing evidence that an exponential parametric distribution is appropriate. The hazard rate is approximated dividing the number of observed failures by the number of subjects at risk (provided by survfit() in R) and then smoothed using smooth.spline() in R.

Univariate model results showing estimated survival curves (i.e., the probability of not being screened via colonoscopy) for the time to the first screening, comparing subjects with no colonoscopy insurance coverage to those with at least some colonoscopy insurance coverage. In the colonoscopy screening context, a high survival curve indicates a poor rate of screening. It can be observed that the subjects with the higher survival curve are those without colonoscopy coverage.

Univariate model results showing the densities for the probability of not being screened for colorectal cancer via colonoscopy by age 55, 60, and 70 years, comparing those with no coverage (solid line) and those with at least some coverage (dashed line). Note that while the x-axes on all three graphs cover a different range of probability values, the size of the range is 15% for all three graphs. We can observe that the two densities are closest together at 55 years and are farthest apart at 70 years. Note that in all three figures, the densities do not overlap, providing evidence that the probability of screening under some and no coverage is statistically significantly different across the different time points. The densities are calculated using the density() function in R on the MCMC chain of survival probabilities, calculated at each iteration of the thinned and burned chains. Results are shown for the model with a maximum screening age of 75 years.

Left: Multivariate model results showing the marginal estimated survival curves (i.e., probability of not being screened for colorectal cancer via colonoscopy) for the only lifetime screening (solid line) or the first and second of two screenings (dashed and dotted lines). As expected, more people get only one screening in a lifetime rather than two lifetime screenings, and therefore that survival curve is lowest. Right: The estimated survival curves conditional on the number of lifetime screenings. These curves show that among subjects who will receive two screenings, the first screening happens quickly when compared with the second screening. Subjects who only get one colonoscopy take the longer than those who get two colonoscopies. As with the univariate models, a high survival curve indicates poor screening rates. Estimates were calculated from the multivariate model that assumes 75 is the maximum eligible age for screening.

Left: Multivariate model results showing the densities of the probability of getting at least one screening before the age of 55, 65, or 75 years. (Densities calculated using the density() function in R.) The densities are narrow, and the probability of at least one screening for 55-year-olds is far from the other two age groups, indicating that there may be a statistically significant difference in the probability of screening for individuals who are 55 years old when compared with 65- or 75-year-olds. The probability does not increase as rapidly between age 65 and 75 years. Estimates were calculated from the multivariate model that assumes 75 is the maximum eligible age for screening.

A contour plot of the multivariate model results showing the joint survival curve (i.e., the probability of not being screened via colonoscopy), conditional on subjects who get two screenings (i.e., the probability of two screenings, θ₂, is not used in the calculation of the survival probabilities) for the first 5 years. The figure shows minor differences between the time to the first screening and time to the second screening, as the contour plot is fairly symmetric. However, the gray shading extends slightly higher up the y-axis (which represents the time to the second screening), meaning that the probability of not being screened is higher for a longer time period before the second screening. As with the univariate models, a high survival curve indicates poor screening rates. Years 5 through 10 are omitted from the figure as the probabilities are very small and difficult to discern in the figure. Estimates were calculated from the multivariate model that assumes 75 is the maximum eligible age for screening.