3.2.1 Leadtime bias adjustment model
In this leadtime bias adjustment model, the total survival time for screen-detected cases is decomposed into leadtime and post-leadtime. We use a Markov model, based on the natural history part of Fig. 3A, to estimate the expected leadtime using screen-detected cases and normal subjects. Recall that λ0(·) and λ1(·) represent the pre-clinical incidence rate (from state 0 to state 1) and the instantaneous transition rate from the PCDP (state 1) to the CP (state 2), respectively. Additionally, since only screen-detected cases have leadtime gains, the data we use here to estimate the leadtime comprise no clinically detected cases, i.e. interval cancers. Using the method proposed by Chen et al. (2000) and Wu et al. (2004), λ0(·) and λ1(·) can be estimated without relying on interval cancers. After obtaining the estimate of the annual transition rate from the PCDP to the CP, λ2(·), the prognosis part was incorporated into the Markov model (the gray part in Fig. 3A) to estimate post-leadtime survival based on screen-detected cases only. A naïve approach is to approximate post-leadtime survival (PST, the dotted arrow in Fig. 3A) by subtracting the estimated leadtime (LT) from observed survival time (OST, the white arrow in Fig. 3A), which starts from the PCDP until the end of follow-up, including death from prostate cancer, other causes, and censored status, whichever came first.
To accommodate the possibility of a non-constant hazard for the risk of prostate cancer death, a Weibull distribution was used to model post-leadtime survival. The transition rate, which is dependent on age after correction for leadtime, can be expressed as
where λ20 is the scale parameter, and γ2 is the shape parameter of the Weibull distribution. To estimate the parameters of the Markov model for the natural history and prognosis parts shown in Fig. 3A, data on the follow-up of the screen-detected cases are required. When the distributions of sojourn time and the CP are specified by the exponential and Weibull distributions, respectively, at the end of follow-up at age at from a subject detected by screen at age ak the probability of being alive is
the density function of being dead from prostate cancer is
and the density function of being dead from other causes is
where u1(·) is the hazard rate from the PCDP to other causes of death, P11(ak, at) is the probability of staying in state 1 (PCDP) from ak to at, P12(ak, at) is the probability of transition from state 1 (PCDP) at ak to state 2 (CP) and staying in state 2 up to at, and f1(·) and f2(·) are the probability density functions for death from prostate cancer and death from other causes at time at, respectively.
3.2.2 Length-bias adjustment model
Different types of prostate cancer may have different sojourn times. As the distribution of leadtime depends on the distribution of sojourn time, the estimate of λ1(·) in the above leadtime adjustment model is biased (because screen-detected cancers tend to have longer sojourn times). In order to get an unbiased estimate of λ1(·) as well as an unbiased estimate of the hazard rate of λ2(·), we incorporate interval cancers, which tend to have shorter sojourn times.
A five-state stochastic process (Fig. 3A) is proposed to estimate transition parameters related to the leadtime, making allowance for different estimates for the sojourn time by combining screen-detected cases (long sojourn times) and interval cancers (short sojourn times). The post-leadtime estimate can be obtained from prognosis part of the model based on screen-detected cases only.
Without assuming any particular distribution for the transition rates, λ's, in the second (k=2) screen at a2 the probability of remaining in the normal state since the first screen until the second screen is denoted by
and the probability of being diagnosed as a screen-detected case at a2 is
where P00 and P11 are, respectively, the probabilities of staying in state 0 (normal) and state 1 (PCDP).
The density function that incorporates the probability of interval cases (short sojourn time) diagnosed at age akc can be expressed as
where fc (akc) is the derivative of the probability of being negative at ak (age at the k-th screen) and being diagnosed clinically due to symptoms and signs at akc.
For the first screen, the probability of being screened as normal or asymptomatic prostate cancer has to be conditional on time until the disease is clinically recognized being longer than the time to screen. Hence the conditional probability of being screened as normal or as an asymptomatic case at the first screen can be expressed as
respectively, recalling that a1 is the age at the first screen.
We assume that the rates of entering the PCDP, of the transition from the PCDP to the CP, and of the transition from the CP until prostate cancer death follow three Weibull distributions (Weibull (λ00, γ0), Weibull (λ10, γ1), and Weibull (λ20, γ2) as functions of age, and that three rates from normal, PCDP, and CP to other causes of death follow three exponential distributions. Weibull distribution, which allows for the transition rate changing with time, can accommodate non-homogeneous Markov model. Then Eqs. (4)–(6) can be rewritten as
The probabilities of having three outcomes for the follow-up of screen-detected cases, include
for the subject still alive
for prostate cancer death and
for other causes of death.
Note that if γ equals 1, Eqs. (9)–(14) are reduced to exponential distributions. Here, we also assume that u0(·), u1(·), and u2(·) follow exponential distribution. Although the proposed non-homogeneous model can be parameterized by the application of a Weibull distribution, estimation of parameters is time consuming and the estimates might have a large variance. For these reasons, we also adopted a piecewise method. Because the derivation of piecewise model is straightforward but laborious, the formulae used for the piecewise method are given in Appendix A.
In our first analysis only the last phase followed a Weibull distribution whereas the times from normal to entering the PCDP, and from the PCDP to the CP, were modeled as exponential distributions. Following expressions (1)–(8) for the model with two exponential distributions, Eqs. (9)–(14) for the Weibull model, or equations for the piecewise method (see Appendix A), the likelihood function for the model in Fig. 3A is
where and are the ages at the first and k-th screen for the i-th subject, , and are the ages at diagnosis of interval cancer, and the age at the end of follow-up for the i-th subject, respectively. and are indicator variables if the i-th subject was normal or screen-detected at the first screen. The counterparts for the k-th screen are and . The indicator of denotes interval cancer. , , and are indicator variables for screen-detected cases who are, respectively, alive, dead from prostate cancer, and dead from other causes, after the k-th screen for the i-th subject.
Based on the estimated transition parameters of λ20 and λ2, the prostate cancer survival function after correcting for leadtime and length bias, simultaneously making allowance for other causes of death, can be calculated as
where ak is age at the k-th screen, and at is age at the end of follow-up.