Predicting Individual Risk of Advanced Adenoma Based on the Interval-Censored Recurrent Adenoma Event and Informative Screening Time

Abstract

Panel count data is common in cancer screening. In the context of colorectal cancer screening, our work focuses on the prediction of the probability of advanced adenoma conditional on patient-level risk factors and/or event history. We implement the joint frailty model proposed by Huang et al. which involves a non-stationary Poisson process for recurrent adenoma events and informative screening time using semi-parametric Cox models correlated by a latent frailty variable, where coefficients and baseline intensity functions are estimated by estimating equations. Subject-specific frailty value is estimated by the borrow-strength method. In addition, marginal models for the adenoma and screening events are also applicable when average covariate effects on the population level are of interest. Predictions based on the marginal model and predictions based on the frailty models for patients with or without a screening history are compared. When a patient’s screening history is available and sufficient adenoma events are observed, the predictions based on the frailty model with estimated subject-specific frailty are superior. However, in the cases of early censoring when adenoma events are not observed for most patients and screening history is not available, the prediction based on the marginal model has better performance. Furthermore, for patients’ future screening, the individualized screening intervals derived from the dynamic predictions of advanced adenoma risks will detect adenoma events earlier with shorter lag time between the adenoma state transition and the screening compared to the current practice of regular screening intervals.

1 |. Introduction

Cancer screening aims to detect cancer early, when it is most curable, or even better, to detect cancer precursors whose removal prevents future cancer entirely. An example is colorectal cancer (CRC), where screening tests can identify both colorectal cancer and its adenoma precursors. According to the U.S. Center of Disease Control and Prevention (CDC), most people should begin colorectal cancer screening at 45 years old, then continue getting screened at regular intervals of 10 years [1]. Colonoscopy is the main colorectal cancer screening modality in the U.S. In addition to capturing the morphologies and numbers of adenomas, the colonoscopy allows for the removal of any observed adenoma during the screening process.

We analyzed data from 5768 individuals in the Study of Colonoscopy Utilization (SCU), an ancillary study nested within the Prostate, Lung, Colorectal, and Ovarian (PLCO) Cancer Screening trial, oversampling people with a prior history of adenomas with individual-level sampling weights. The SCU cohort includes participants with at least one surveillance colonoscopy more than 6 months and less than 10 years after their baseline colonoscopy. Consistent with standard clinical practice, adenomas identified at surveillance colonoscopy are presumed to have been removed [2]. Unlike the PLCO trial, in which screening is protocol-driven, surveillance colonoscopy in SCU is determined by community providers. Consequently, the intervals between screenings vary widely, and detailed information on non-compliance with adenoma removal is not collected. The mean (SD) time interval from baseline to the first surveillance colonoscopy is 4.4 (2.1) years, and from baseline to the second surveillance colonoscopy is 6.7 (2.1) years [3, 4].

In this paper, adenoma events are defined as transitions between biologically distinct states along the colorectal neoplasia continuum, progressing from benign lesions to invasive CRC [5]:

• State 0: No adenoma or presence of hyperplastic polyps—considered benign, low-risk lesions.

• State 1: Small tubular adenomas (≤1 cm) without dysplasia—classified as non-advanced adenomas.

• State 2: Multiple or larger tubular adenomas (> 1 cm) without villous features or high-grade dysplasia—representing borderline non-advanced adenomas.

• State 3: Adenomas with villous histology, size ≥1 cm, or high-grade dysplasia—standard criteria for advanced adenomas, as defined in PLCO protocol.

• State 4: Severe dysplasia or carcinoma in situ—considered precancerous lesions with imminent risk of invasion.

• States 5–10: Invasive CRC, ranging from early-stage tumors (e.g., confined to the submucosa or muscularis propria) to advanced-stage disease with regional or distant metastasis.

Advanced adenomas (State 3) carry a substantially greater risk of progressing to CRC. The optimal schedule of screening exams depends crucially on a patient’s risk of developing cancer and advanced adenoma. However, no current cancer screening program sequentially updates risks for the next screening exam with both individual risk factors and a history of screening test results. We aim to update the risks of advanced adenoma for the next screening by developing statistical models that handle the following major challenges:

1. Interval-censored panel count adenoma data, a common form of colorectal cancer screening data, where only the level of the state, but not the actual time of state transitions, is known.

2. Recurrent right-censored screening visits informative of cancer risks [6]. For instance, patients diagnosed with adenomas in the previous screening tend to schedule more frequent screenings.

3. Maximize the accuracy of advanced adenoma risk prediction when screening history is available.

Methodologies and packages for panel count data have been extensively studied [7–9]. Wang et al. proposed a non-stationary Poisson process for recurrent events [10]. A semi-parametric Cox model with a latent frailty variable was assumed for the intensity function, where the cumulative baseline intensity function was estimated nonparametrically. Huang et al. considered a joint model and assumed a latent variable to correlate the recurrent events and the failure process [11]. They proposed a borrow-strength method to estimate the subject-specific latent variable values from recurrent events and used the estimated latent variable values to further estimate coefficients in the failure process. For interval-censored recurrent event data, Huang et al. proposed estimating equations based on the self-consistency non-parametric Turnbull algorithm to estimate the regression coefficients [12]. Zeng et al. incorporated subject-specific random effects and time-dependent covariates to model the dependence of the recurrent events within the same subject [13]. As an alternative to the Cox-type formulation, Chiou et al. proposed an accelerated mean model for the recurrent event process under informative examination time [14]. Ma et al. studied panel count data with informative observation time and a left-truncated starting time, where they built estimating equations similar to Huang using nonparametric I-splines [12, 15].

Following the idea of Huang et al. [12] we implement a joint model of recurrent interval-censored adenoma events and informative screening visits to predict the risk of advanced adenoma for the next screening. A latent frailty variable with unspecified distribution is proposed to correlate the adenoma events and screening visits. On the other hand, when average covariate effects of populations are of interest, marginal models without a frailty term are also applicable. In terms of risk predictions, when a patient’s colorectal cancer screening history is available, we predict the probability of advanced adenoma in future time intervals based on the estimates of subject-specific frailty and risk factors. However, if the patient’s screening history is not available, we can either estimate the distribution of the frailty term or fit a marginal model to evaluate probabilities of advanced adenoma at future time points. A key feature of our proposed prediction models is that no distribution assumption is required for the adenoma and screening data. In simulations, our proposed prediction models perform well even when the models are misspecified. Compared with the standard regular screening interval, the individualized dynamic screening intervals based on our proposed prediction methods detect the adenoma state transitions earlier and lower the risk of colorectal cancer. In addition, we provide guidelines to choose between different prediction models of advanced adenoma.

2 |. Model

2.1 |. Notation

Let $i$ be the index of the subject, $n_i^s$ denote the total number of the recurrent screening events for subject $i$. Let $j$ be the index of the screening events $\left(1⩽j⩽n_i^s\right),n_{ij}^a$ denote the total number of the recurrent adenoma events for subject $i$ found during the $j\text{th}$ screening visit. Let $k$ be the index of the adenoma events in the $j\text{th}$ screening for subject $i\left(1⩽k⩽n_{ij}^a\right)$. Let $w_i$ denote the subject-specific sampling weight, $t_{ij}^s$ denote the occurrence time for the $j\text{th}$ screening of the $i\text{th}$ subject and $\tau$ is the maximum screening time over all subjects $\left(0⩽t_{ij}^s⩽\tau \right)$. Let $\eta$ denote the maximum gap time among all screening events from all subjects. As we assume all of the observed adenomas are removed during the screening visit, let $t_{ijk}^a$ denote the adenoma state transition time for the $k\text{th}$ adenoma events in the $jth$ screening for the $ith$ subject $\left(t_{ij-1}^s⩽t_{ijk}^a⩽t_{ij}^s\right)$. Let ${\lambda }_0^s(t)$ and ${\Lambda }_0^s(t)$ denote the baseline intensity function and the cumulative baseline intensity function for the screening events respectively, let ${\lambda }_0^a(t)$ and ${\Lambda }_0^a(t)$ denote the baseline intensity function and the cumulative baseline intensity function for the adenoma events respectively. Subject-specific cumulative intensity function for subject $i$ at the $jth$ interval for screening and adenoma events are denoted as ${\Lambda }_{ij}^s(t)$ and ${\Lambda }_{ij}^a(t)$ respectively.

2.2 |. Joint Frailty Model

In the SCU data, the adenoma events are interval-censored because only the number of state transitions is recorded in each screening visit. The actual occurrence time of the adenoma events is unobservable. During each screening visit, not only is the number of adenoma events captured, but the colonoscopies are also carried out to remove any observed adenoma. Each subject has up to fifteen screenings, with $0<t_{i1}^s<t_{i2}^s<\dots <t_{ij}^s<\dots <t_{i{n}_i^s}^s\le \tau$, where $j=1,2,\dots ,n_i^s$ and $\tau$ is the maximum screening time among all of the subjects. The observed screening times are right-censored with independent censoring times. To account for correlations between the adenoma and the screening events within each subject, we introduce a subject-specific frailty term $z_i$ into our joint model [10].

2.2.1 |. Recurrent Interval-Censored Adenoma Events

First, we assume a non-homogeneous Poisson process with baseline intensity function ${\lambda }_0^a(t)$ to model the recurrent interval-censored adenoma events. For the $i\text{th}$ subject at the $j\text{th}$ screening, we employ a semi-parametric coxmodel for the subject-specific intensity function ${\lambda }_i^a(t):{\lambda }_i^a(t)=z_i{\lambda }_0^a(t)\text{exp}\left({\mathit{X}}_{\mathit{i}}^{\mathit{T}}\mathit{\beta }\right)$, where ${\mathit{X}}_{\mathit{i}}$ is the subject-specific risk factors related to adenoma events, $\mathit{\beta }$ is the coefficient vector and $z_i$ is the subject-specific latent frailty. The distributions for ${\lambda }_0^a(t)$ and $z_i$ are not specified to ensure flexibility. Following Wang and Ma, we define a density function [10, 15]

$$f(t)=\frac{z_i{\lambda }_0^a(t)I(0\le t\le \tau )\text{exp}\left({\mathit{X}}_{\mathit{i}}^{\mathit{T}}\mathit{\beta }\right)}{z_i{\Lambda }_0^a(\tau )\text{exp}\left({\mathit{X}}_{\mathit{i}}^{\mathit{T}}\mathit{\beta }\right)}=\frac{{\lambda }_0^a(t)I(0\le t\le \tau )}{{\Lambda }_0^a(\tau )}$$

with cumulative distribution function $F(t)={\int }_0^{t}f(u)du=\frac{{\Lambda }_0^a(t)}{{\Lambda }_0^a(\tau )}$ for $0\le t\le \tau$. Then $F(\tau )=1$ by assuming that ${\Lambda }_0^a(\tau )$ is bounded. Let ${\tau }_i$ denote the maximum screening time for subject $i$. Define $t_{ij}^{gap}=t_{ij}^s-t_{ij-1}^s$ as the jth screening gap time and assume that all adenomas are removed by colonoscopy. Conditional on ($w_i,z_i,{\mathit{X}}_i,n_i^s,n_{ij}^a,t_{ij}^{\mathit{\text{gap}}}$), the likelihood function for the adenoma events is proportional to

$$\begin{gathered} L_c^a\propto \prod _{i=1}^n\prod _{j=1}^{n_i^s}{\left\{\frac{z_i{\Lambda }_0^a\left(t_{ij}^{gap}\right)\text{exp}\left(X_i^T\beta \right)}{z_i{\Lambda }_0^a\left({\tau }_i\right)\text{exp}\left(X_i^T\beta \right)}\right\}}^{n_{ij}^a{w}_i} \\ =\prod _{i=1}^n\prod _{j=1}^{n_i^s}{\left\{\frac{F\left(t_{ij}^{gap}\right)}{F\left({\tau }_i\right)}\right\}}^{n_{ij}^a{w}_i} \end{gathered}$$

which does not depend on $\left(z_i,{\mathit{X}}_i\right)$. Thus, $F(t)$ can be estimated by the Turnbull algorithm [16].

2.2.2 |. Recurrent Right-Censored Screening Events

For the $i\text{th}$ subject at the $j\text{th}$ screening, a semi-parametric Cox-model ${\lambda }_{ij}^s(t)$ is employed for the intensity function ${\lambda }_{ij}^s(t)$ based on the gap time $t_{ij}^{gap}$ of the screening events: ${\lambda }_{ij}^s(t)=z_i{\lambda }_0^s(t)\text{exp}\left({\mathit{V}}_{ij}(\mathit{t})\mathit{\theta }\right)$, where ${\mathit{V}}_{ij}(\mathit{t})$ is the vector of the time-varying screening-related subject-specific risk factors, $\mathit{\theta }$ is the coefficient vector. The subject-specific latent frailty $z_i$ in the screening events is assumed identical to the one in the adenoma events for subject $i$. The partial likelihood of the screening events is [17]:

$$L_p^s=\prod _{i=1}^n\prod _{j=1}^{n_i^s}{\left\{\frac{z_i\text{exp}\left({\mathit{V}}_{\mathit{i}\mathit{j}}^{\mathit{T}}\left(t_{ij}^{\mathit{\text{gap}}}\right)\mathit{\theta }\right)}{{\sum }_{i^{\prime }=1}^n{w}_{i^{\prime }}{\sum }_{j^{\prime }=1}^{n_{i^{\prime }}^s}{z}_{i^{\prime }}\text{exp}\left({\mathit{V}}_{{\mathit{i}}^{\prime }{\mathit{j}}^{\prime }}^{\mathit{T}}\left(t_{i^{\prime }{j}^{\prime }}^{\mathit{\text{gap}}}\right)\mathit{\theta }\right)I\left(t_{ij}^{\mathit{\text{gap}}}⩽t_{i^{\prime }{j}^{\prime }}^{\mathit{\text{gap}}}\right)}\right\}}^{w_i{\delta }_{ij}}$$

where ${\delta }_{ij}$ is the censoring indicator for subject $i.{\delta }_{ij}=0$ indicates that the screening event is censored and ${\delta }_{ij}=1$ otherwise.

2.3 |. Marginal Model

When average covariate effects of populations are of interest, instead of incorporating subject-specific frailty $z_i$ terms, marginal models are also applicable for adenoma and screening events, respectively. In that case, the subject-specific intensity function of the adenoma events is ${\lambda }_i^a(t)={\lambda }_0^a(t)\text{exp}\left({\mathit{X}}_{\mathit{i}}^{\mathit{T}}\mathit{\beta }\right)$ and the subject-specific intensity function of the screening events is ${\lambda }_{ij}^s(t)={\lambda }_0^s(t)\text{exp}\left({\mathit{V}}_{ij}(t)\mathit{\theta }\right)$. Compared with the joint frailty models, coefficients from the marginal models represent population-level average effects.

3 |. Parameter Estimation and Risk Prediction

3.1 |. Parameter Estimation of Joint Frailty Model

Our parameter estimation process for the joint frailty model closely follows the step-by-step method outlined in Huang et al. [12] The details of the estimation of $F(t),\beta ,z_i,{\Lambda }_0^a(t)$ and $\theta$ are as follows:

3.1.1 |. Estimation of $\mathit{F}(\mathit{t})$ by Turnbull Algorithm

Turnbull is a self-consistency Expectation Maximization non-parametric algorithm that can be implemented to estimate the cumulative distribution function $F(t)$ of interval-censored data [18]. Denote the ordered distinct screening time from all subjects $i=1,2,3,\dots ,n$ and all screening intervals $j=1,2,3,\dots ,n_i^s$ as $0=t_0^{*}<t_1^{*}<t_2^{*}<t_3^{*}<\dots <t_m^{*}<\dots <t_u^{*}=\tau$, where $m=1,2,3,\dots ,u$ and $u$ is the total number of distinct screening time. Denote the maximum screening time for subject $i$ as ${\tau }_i$. Define $a_{ijm}=1$ if $\left[t_{m-1}^{*},t_m^{*}\right]\subseteq \left[t_{ij-1}^s,t_{ij}^s\right]$ and $a_{ijm}=0$ otherwise. Define $b_{im}=1$ if $t_m^{*}\le {\tau }_i$ and $b_{im}=0$ otherwise. We maximize $L_c^a$ subject to ${\sum }_{m=1}^u{p}_m=1$, with $p_m=F\left(t_m^{*}\right)-F\left(t_{m-1}^{*}\right)$ [14]. In the E-step of the $l\text{th}$ iteration, we calculate the expected number of events in the $m\text{th}$ interval $d_m^{(l)}$ by

$$d_m^{(l)}=\sum _{i=1}^n\sum _{j=1}^{n_i^s}{n}_{ij}^a\left(\frac{a_{ijm}{p}_m^{(l)}}{\sum _{h=1}^u{a}_{ijh}{p}_h^{(l)}}+\frac{\left(1-b_{im}\right)p_m^{(l)}}{\sum _{h=1}^u{b}_{ih}{p}_h^{(l)}}\right)$$

Note that $d_m$ consists of two terms. For subject $i$, the first term is a conventional Turnbull equation that calculates the expected number of events in the $m\text{th}$ interval before the maximum screening time ${\tau }_i$ based on the observed adenoma events. However, the events after ${\tau }_i$ from the $i\text{th}$ subject are unobservable, then the second term calculates the projected number of events in the $m\text{th}$ interval after ${\tau }_i$. In the M-step, we update $p_m$ by $p_m^{(l+1)}=\frac{d_m^{(l)}}{\sum _{h=1}^u{d}_h^{(l)}}$. And $F^{(l+1)}\left(t_m^{*}\right)=\sum _{h=1}^m{p}_h^{(l+1)}$. The iterations will stop when the preset convergence criteria $\sum _{m=1}^u\left|p_m^{(l+1)}-p_m^{(l)}\right|\le ϵ$ is met. The survival probability is estimated as $S\left(t_m^{*}\right)=1-\sum _{h=1}^m{p}_h$.

3.1.2 |. Estimation of Parameter $\mathit{\beta }$ for Adenoma Events

The estimating equation for $\mathit{\beta }$ is derived from the score function of the Poisson distribution [15]. Denote ${\Lambda }_{ij}^a\left(t_{ij}^{gap}\right)={\Lambda }_{ij}^a\left(t_{ij}^s\right)-{\Lambda }_{ij-1}^a\left(t_{ij-1}^s\right)$ as the increment of cumulative intensity rate of the adenoma events for subject $i$ at the $j\text{th}$ screening. The Poisson score function with respect to $\mathit{\beta }$ is:

$$\begin{gathered} \frac{\partial }{\partial \mathit{\beta }}\text{log}\left\{\prod _{i=1}^n{\left(\prod _{j=1}^{n_i^s}\frac{\text{exp}\left(-{\Lambda }_{ij}^a\left(t_{ij}^{gap}\right)\right){\left({\Lambda }_{ij}^a\left(t_{ij}^{gap}\right)\right)}^{n_{ij}^a}}{n_{ij}^a!}\right)}^{w_i}\right\} \\ =\sum _{i=1}^n{w}_i\sum _{j=1}^{n_i^s}{\mathit{X}}_i\left(n_{ij}^a-z_i{\Lambda }_0^a\left(t_{ij}^{gap}\right)\text{exp}\left({\mathit{X}}_{\mathit{i}}^{\mathit{T}}\mathit{\beta }\right)\right) \end{gathered}$$ (1)

Specifically, $n_{ij}^a$ is the observed number of events and $z_i{\Lambda }_0^a\left(t_{ij}^{gap}\right)\text{exp}\left({\mathit{X}}_i^{\mathit{T}}\mathit{\beta }\right)$ is the expected number of events. Equation (1) will still hold even outside the context of the Poisson distribution. Since $F(t)=\frac{{\Lambda }_0^a(t)}{{\Lambda }_0^a(\tau )}$ and $E\left(z_i\mid {\mathit{X}}_i\right)=1$, we replace the unknown $F(t)$ by $\hat{F}(t)$ estimated from the Turnbull algorithm, then the estimating equation is [10]

$$\sum _{i=1}^n{w}_i\sum _{j=1}^{n_i^s}{\mathit{X}}_i\left(\frac{n_{ij}^a}{\hat{F}\left(t_{ij}^{gap}\right)}-{\Lambda }_0^a(\tau )\text{exp}\left({\mathit{X}}_{\mathit{i}}^{\mathit{T}}\mathit{\beta }\right)\right)=0$$ (2)

Considering the existence of very short follow-up time $t_{ij}^{\mathit{\text{gap}}}$ and nonzero $n_{ij}^a$ in the real data, in which case $\frac{n_{ij}^a}{\hat{F}\left(t_{ij}^{gap}\right)}$ becomes unstable, we use the following estimating equation for parameter $\mathit{\beta }$ to avoid numerical issues:

$$\sum _{i=1}^n{w}_i{\mathit{X}}_i\left(\frac{n_i^a}{\hat{F}\left({\tau }_i\right)+ϵ}-{\Lambda }_0^a(\tau )\text{exp}\left({\mathit{X}}_{\mathit{i}}^{\mathit{T}}\mathit{\beta }\right)\right)=0$$ (3)

where $n_i^a=\sum _{j=1}^{n_i^s}{n}_{ij}^a$ and ${\tau }_i$ is the maximum screening time for subject $i,ϵ$ is set to a small positive numbers to prevent denominators equals 0. ${\Lambda }_0^a(\tau )$ is a constant and can be estimated as the intercept [12].

3.1.3 |. Estimation of Frailty ${\mathit{z}}_{\mathit{i}}$

We use the borrow-strength method to estimate subject-specific latent frailty $z_i$ [11]. Although the screening events and adenoma events share the same $z_i$ based on our assumption, only the adenoma events with time-invariant covariates are used to estimate $z_i$. Conditioning on ${\mathit{X}}_i,z_i,t_{ij}^{\mathit{\text{gap}}}$,

$$E\left(n_{ij}^a\right)=z_i\text{exp}\left({\mathit{X}}_{\mathit{i}}^{\mathit{T}}\mathit{\beta }\right){\Lambda }_0^a\left(t_{ij}^{gap}\right)$$

From the definition of $F(t)$, we can estimate $z_i$ based on the screening-level data

$$\begin{gathered} {\hat{z}}_i=\frac{1}{n_i^s}\sum _{j=1}^{n_i^s}\frac{n_{ij}^a}{\text{exp}\left({\mathit{X}}_{\mathit{i}}\stackrel{\mathit{ˆ}}{\mathit{\beta }}\right){\hat{\Lambda }}_0^a\left(t_{ij}^{gap}\right)} \\ =\frac{1}{n_i^s}\sum _{j=1}^{n_i^s}\frac{n_{ij}^a}{\text{exp}\left({\mathit{X}}_{\mathit{i}}\stackrel{\mathit{ˆ}}{\mathit{\beta }}\right){\hat{\Lambda }}_0^a(\tau )\left(\hat{F}\left(t_{ij}^{gap}\right)\right)} \end{gathered}$$ (4)

or on the subject-level data

$$\begin{gathered} {\hat{z}}_i=\frac{n_i^a}{\text{exp}\left({\mathit{X}}_{\mathit{i}}\stackrel{\mathit{ˆ}}{\mathit{\beta }}\right){\hat{\Lambda }}_0^a\left({\tau }_i\right)} \\ =\frac{n_i^a}{\text{exp}\left({\mathit{X}}_{\mathit{i}}\stackrel{\mathit{ˆ}}{\mathit{\beta }}\right){\hat{\Lambda }}_0^a(\tau )\hat{F}\left({\tau }_i\right)} \end{gathered}$$ (5)

To avoid small values of $\hat{F}\left(t_{ij}^{\mathit{\text{gap}}}\right)$ in the real data, $z_i$ estimated through subject-level data is more stable numerically. However, one issue in estimating $z_i$ is the case of subjects with early right censoring time where no adenoma is observed. As noted by Balakrishnan et al. mixed censoring may arise when adenoma events are interval-censored while screening events are subject to right censoring [19]. If the right censoring time of the screening event is earlier than the occurrence time of the first adenoma event with a limited number of screening intervals from individual patients, the subject-level frailty $z_i$ will be estimated as 0 and the estimation of the distribution ${\hat{f}}_z(z)$ will be right-skewed because each ${\hat{z}}_i$ is based on very limited number of screening intervals.

3.1.4 |. Estimation of Baseline Cumulative Intensity Function for Adenoma Events ${𝚲}_0^{\mathit{a}}(\mathit{t})$

The distribution of the baseline cumulative intensity function ${\Lambda }_0^a(t)$ is not specified. However, as described in the previous section, ${\Lambda }_0^a(\tau )$ is estimated in the estimating equations for $\beta$ as the intercept, and $F(t)$ is estimated by the Turnbull algorithm. Consequently, ${\Lambda }_0^a(t)$ can be consistently estimated with ${\hat{\Lambda }}_0^a(t)={\hat{\Lambda }}_0^a(\tau )\hat{F}(t)$.

3.1.5 |. Estimation of Parameter $\mathit{\theta }$ for Screening Events

Based on the partial likelihood function for screening events, $\mathit{\theta }$ could be estimated directly from the score function of the partial likelihood function $U(\mathit{\theta })$:

$$U\left(\theta \right)=\sum _i^n{w}_i\sum _j^{n_i^s}{\delta }_{ij}\left\{V_{ij}^T\left(t_{ij}^{\mathit{\text{gap}}}\right)-\frac{{\sum }_{i^{\prime }}^n{w}_{i^{\prime }}{\sum }_{j^{\prime }}^{n_{i^{\prime }}^s}{z}_{i^{\prime }}{V}_{i^{\prime }{j}^{\prime }}^T\left(t_{i^{\prime }{j}^{\prime }}^{\mathit{\text{gap}}}\right)\text{exp}\left(V_{i^{\prime }{j}^{\prime }}^T\left(t_{i^{\prime }{j}^{\prime }}^{\mathit{\text{gap}}}\right)\theta \right)I\left(t_{ij}^{\mathit{\text{gap}}}⩽t_{i^{\prime }{j}^{\prime }}^{\mathit{\text{gap}}}\right)}{{\sum }_{i^{\prime }}^n{w}_{i^{\prime }}{\sum }_{j^{\prime }}^{n_{i^{\prime }}^s}{z}_{i^{\prime }}\text{exp}\left(V_{i^{\prime }{j}^{\prime }}^T\left(t_{i^{\prime }{j}^{\prime }}^{\mathit{\text{gap}}}\right)\theta \right)I\left(t_{ij}^{\mathit{\text{gap}}}⩽t_{i^{\prime }{j}^{\prime }}^{\mathit{\text{gap}}}\right)}\right\}$$ (6)

3.2 |. Parameter Estimation of Marginal Model

For marginal models, the parameter estimation process is similar to that of the joint frailty models, with the main difference that the subject-specific frailty $z_i$ is no longer needed. Specifically, the estimating equation for parameter $\beta$ is identical to equation (3) since frailty $z_i$ is not involved in the estimation. The estimating equation of parameter $\theta$ is analogous to equation (6) except that $z_i$ is omitted.

3.3 |. Prediction

Our main objective is to predict the probability of advanced adenoma at the next screening visit. In this section, we compare the predictions from both the frailty models and the marginal model.

3.3.1 |. Prediction Model With Individual Frailty ${\stackrel{\mathit{ˆ}}{\mathit{z}}}_{\mathit{i}}$

Denote the time until the next screening visit as $t_i^{\prime }$. The probability of advanced adenoma events could be estimated based on the proposed adenoma model and the estimated frailty ${\hat{z}}_i$. As mentioned in the previous section, the advanced adenoma is defined as the transition of adenoma states greater than the pre-specified criteria $N_a$. Based on the previous screening information for subject $i$ and the estimated latent frailty ${\hat{z}}_i$, the probability of advanced adenoma for the $i\text{th}$ subject $P_{1i}$ is:

$$P_{1i}\left(n_a>N_a\mid t=t_i^{\prime }\right)=1-\sum _{n_a=0}^{N_a}\frac{\text{exp}\left(-z_i{\Lambda }_i^a\left(t_i^{\prime }\right)\right){\left(z_i{\Lambda }_i^a\left(t_i^{\prime }\right)\right)}^{n_a}}{n_a!}$$

where ${\Lambda }_i^a\left(t_i^{\prime }\right)={\Lambda }_0^a\left(t_i^{\prime }\right)\text{exp}\left({\mathit{X}}_i^T\mathit{\beta }\right)$.

3.3.2 |. Prediction Model With Kernel Density ${\stackrel{\mathit{ˆ}}{\mathit{f}}}_{\mathit{z}}(\mathit{z})$

However, if subject $i$ has no previous screening information, the above equation is not applicable since subject-specific frailty $z_i$ cannot be estimated. Therefore, we fit a weighted kernel density ${\hat{f}}_z(z)$ based on all ${\hat{z}}_i$ values estimated from similar cohorts and estimate the probability of advanced adenoma averaging over ${\hat{f}}_z(z)$:

$$\begin{gathered} P_{2i}\left(n_a>N_a\mid t=t_i^{\prime }\right) \\ =1-\sum _{n_a=0}^{N_a}{\int }_z\frac{\text{exp}\left(-z{\Lambda }_i^a\left(t_i^{\prime }\right)\right){\left(z{\Lambda }_i^a\left(t_i^{\prime }\right)\right)}^{n_a}}{n_a!}{f}_z(z)dz \end{gathered}$$

3.3.3 |. Marginal Prediction Model Without Frailty

When the previous screening information for subject $i$ is not available, another option is to estimate the probability via marginal models without frailty:

$$P_{3i}\left(n_a>N_a\mid t=t_i^{\prime }\right)=1-\sum _{n_a=0}^{N_a}\frac{\text{exp}\left(-{\Lambda }_i^a\left(t_i^{\prime }\right)\right){\left({\Lambda }_i^a\left(t_i^{\prime }\right)\right)}^{n_a}}{n_a!}$$

3.4 |. Variance Estimation

We use the weighted bootstrap method to estimate the variance of parameter $\mathit{\beta },\mathit{\theta },{\mathit{z}}_i$ and predicted probability of advanced adenoma $P_1\left(n_a>N_a\right)$ [20]. In the $r\text{th}$ bootstrap sample, suppose we select $n^{*}$ from $n$ subjects with replacement, following Rao, weight $w_i^{(r)}$ for subject $i$ is formulated as [21]:

$$w_i^{(r)}=\left[1-{\left(\frac{n^{*}}{n-1}\right)}^{\frac{1}{2}}+{\left(\frac{n^{*}}{n-1}\right)}^{\frac{1}{2}}\left(\frac{n}{n^{*}}\right)n_i^{*(r)}\right]w_i$$

where $n_i^{*(r)}$ is the number of selections for subject $i$ in the $r\text{th}$ resampling step with $\sum _{i=1}^n{n}_i^{*(r)}=n^{*}$. Note that the above equation holds only if $\frac{n^{*}}{n-1}\le 1$. We replace $w_i$ with $w_i^{(r)}$ to get the weighted bootstrap estimator ${\stackrel{\mathit{ˆ}}{\mathit{\beta }}}^{*(r)},{\stackrel{\mathit{ˆ}}{\mathit{\theta }}}^{*(r)},{\hat{z}}_i^{*(r)}$ and the corresponding estimated predicted average probability of advanced adenomas. The above procedure is repeated $B$ times to estimate the variance. For example, the bootstrap variance for $\stackrel{\mathit{ˆ}}{\mathit{\beta }}$ is calculated by:

$$V_{\mathit{\text{boot}}}(\stackrel{\mathit{ˆ}}{\mathit{\beta }})=\frac{1}{\mathit{B}-1}\sum _{r=1}^B{\left({\stackrel{\mathit{ˆ}}{\mathit{\beta }}}^{*(r)}-{\overline{\mathit{\beta }}}^{*}\right)}^T\left({\stackrel{\mathit{ˆ}}{\mathit{\beta }}}^{*(r)}-{\overline{\mathit{\beta }}}^{*}\right)$$

where ${\overline{\mathit{\beta }}}^{*}=\frac{1}{B}\sum _{r=1}^B{\stackrel{\mathit{ˆ}}{\mathit{\beta }}}^{*(r)}$. The bootstrap variances for other parameters are calculated following the same procedure.

4 |. Simulation Studies

The SCU data includes the selection weight for each subject. To mimic the settings in SCU data and evaluate both the validity and robustness of our proposed method, we propose four different sets of simulations.

4.1 |. Data Generation

For each set of simulations, first, we generate 2500 subjects as the sample representative of the target population before selective sampling with fixed weights. We assume ${\mathit{X}}_i=\left(x_{i1},x_{i2}\right)$ are the covariates related to adenoma events with $x_{i1}~N(0,5)$ and $x_{i2}~Ber(0.5)$ and $\mathit{\beta }=\left({\beta }_1,{\beta }_2\right)=(0.1,0.2)$ are the corresponding coefficients. Subject-specific frailty $z_i$ is generated following a gamma distribution $\Gamma (2,0.5)$. Then, 250 high-risk subjects are selected with a selection probability of 1, and 250 low-risk subjects are randomly selected with a selection probability of 1/9. High-risk subjects are defined as $x_{i1}$ in top 20% quantile and $x_{i2}=1$. We assume that transitions across each adenoma state occur with identical transition intensity. The occurrence time of adenoma events follows an exponential distribution with ${\lambda }_0^a=0.2$, and each subject has up to 10 screenings and up to 10 adenoma events in each screening visit. That is, we generate a 10 × 10 adenoma event matrix time for each subject with $t_{ijk}^a=\frac{-\text{log}(U(\mathrm{0,1}))}{z_i{\lambda }_0^a\text{exp}\left({\mathit{X}}_i^T\mathit{\beta }\right)}(i=1,\dots ,n,j=1,\dots ,10\text{and}k=1,\dots ,10)$ [22]. In simulations, parameters of screening events are chosen to ensure most intervals have less than 10 adenoma events. Second, we generate 10 screening gap time sequentially with $t_{ij}^{\mathit{\text{gap}}}=\frac{-\text{log}(U(\mathrm{0,1}))}{z_i{\lambda }_0^5\text{exp}\left(V_{ij}^T\theta \right)}(i=1,\dots ,n\text{and}j=1,\dots ,10)$ and screening time $t_{ij}^s=\sum _{j^{\prime }=1}^j{t}_{i{j}^{\prime }}^{gap}$. Then the number of adenoma events for the $i\text{th}$ subject at the $j\text{th}$ screening. We assume the screening gap time follows exponential distribution with ${\lambda }_0^s=0.1$ and ${\mathit{V}}_{\mathit{i}\mathit{j}}=\left(v_{ij1},v_{ij2},v_{ij3}\right)$ are the covariates related to screening events, and $\mathit{\theta }=\left({\theta }_1,{\theta }_2,{\theta }_3\right)=(0.1,0.2,0.1)$ are the corresponding coefficients. Finally, we generate the independent censoring time $C_i$ from $U(60,100)$. The censoring indicator ${\delta }_{ij}=1$ if $t_{ij}^s\le C_i$ and ${\delta }_{ij}=0$ otherwise.

In simulations, Scenario 1 follows the simulation settings in Huang et al. where $v_{ij1}=x_{i1},v_{ij2}=x_{i2}$ and $v_{ij3}$ is generated from a normal distribution $N(0,5)$ independent from the adenoma event [12]. For Scenario 2, $v_{ij3}$ denotes the number of adenoma events found in the previous screening, i.e., $v_{i13}=0$ and $v_{ij3}=n_{ij-1}^a$ for $j=2,3,\dots ,10$. In Scenario 3, we consider the case when one or more relevant covariates are omitted in estimation and prediction. Specifically, recurrent adenoma data are generated with covariates ${\mathit{X}}_{\mathit{i}}=\left(x_{i1},x_{i2},x_{i3}\right)$ and the corresponding coefficients are $\mathit{\beta }=\left({\beta }_1,{\beta }_2,{\beta }_3\right)=(0.1,0.2,0.2)$. The screening data are generated with covariates ${\mathit{V}}_{ij}=\left(v_{ij1},v_{ij2},v_{ij3},v_{ij4}\right)$ and the corresponding coefficients are $\mathit{\theta }=\left({\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4\right)=(0.1,0.2,0.1,0.2)$. Similarly, we assume $v_{ij1}=x_{i1},v_{ij2}=x_{i2},v_{ij3}=x_{i3}$ and $v_{ij4}$ denotes the transitions of adenomas state found in the previous screening. However, covariates $x_{i3}$ and $v_{ij3}$ are not involved in the estimation and prediction procedures. Scenario 4 is analogous to Scenario 2, but with earlier censoring time generated from $U(4,8)$, where most subjects have only a few screening visits, and some of them do not observe any adenoma events.

4.2 |. Results

4.2.1 |. Performance of the Parameter Estimators

The parameters are estimated by the frailty model or the marginal model separately. In Table 1, specifically, ESD denotes the empirical standard deviation. BSD denotes the bootstrap standard deviation with bootstrap replication $B=200$. CP is the coverage probability using the bootstrap standard deviations. Because the estimating equations of adenoma events are the same for the marginal and the frailty model, the estimated parameters ${\hat{\beta }}_1,{\hat{\beta }}_2$ are identical, but the coefficient estimates for the screening events are different. In Scenario 1, the parameter estimates are unbiased, which is consistent with the results in Huang et al. [12] However, in Scenario 2, ${\theta }_3$ is under-estimated when we plug in the estimated ${\hat{z}}_i$, because the estimated ${\hat{z}}_i$ from the borrow-strength method is strongly correlated with the covariate $v_{ij3}=n_{ij-1}^a$. We also plug in the true $z_i$ into the estimating equation of $\mathit{\theta }$ and get unbiased estimates of ${\theta }_3$ (results omitted). Scenarios 3–4 evaluate the robustness of parameter estimation. In Scenario 3, since the unknown covariate $x_{i3}$ is involved in data generation, but not in estimation, the estimates from both the frailty and marginal models are biased. In Scenario 4, when censoring happens early, and 90% of the screening events are right-censored, both the frailty model and the marginal model lead to large variances and attenuate parameter estimates. Table 2 denotes the 25th, 50th, and 75th percentiles in each simulation scenario. The percentiles of the estimated frailty are consistent with the percentiles of the true frailty in Scenarios 1–2. Since the model is misspecified in Scenario 3, the percentiles of the estimated frailty are no longer aligned with the true frailty distribution. Frailty $z_i$ is severely underestimated in Scenario 4 when most screening events are right-censored.

TABLE 1 ∣.

Parameter estimates of adenoma and screening events with $z_i~\Gamma (2,0.5)$ : Scenario 1: Independent $v_{ij3}~N(0,5)$; Scenario 2: $v_{ij3}=n_{ij-1}^a$; Scenario 3: $v_{ij4}=n_{ij-1}^a$ with unobserved $x_{i3}~\text{Norm}(\mathrm{0,5})$; Scenario 4: $v_{ij3}=n_{ij-1}^a$ with early censoring $C_i~U(4,8)$.

Method	Parameter	True	Mean(Est)	ESD(Est)	BSD(Est)	CP(Est)
Scenario 1
Frailty	${\beta }_1$	0.100	0.095	0.010	0.008	0.893
	${\beta }_2$	0.200	0.202	0.084	0.089	0.944
	${\theta }_1$	0.100	0.095	0.010	0.009	0.883
	${\theta }_2$	0.200	0.204	0.094	0.096	0.939
	${\theta }_3$	0.100	0.098	0.005	0.005	0.919
Marginal	${\beta }_1$	0.100	0.095	0.009	0.008	0.885
	${\beta }_2$	0.200	0.194	0.085	0.087	0.935
	${\theta }_1$	0.100	0.066	0.007	0.007	0.005
	${\theta }_2$	0.200	0.133	0.072	0.068	0.850
	${\theta }_3$	0.100	0.076	0.005	0.005	0.005
Scenario 2
Frailty	${\beta }_1$	0.100	0.098	0.009	0.008	0.901
	${\beta }_2$	0.200	0.199	0.091	0.090	0.948
	${\theta }_1$	0.100	0.097	0.010	0.009	0.911
	${\theta }_2$	0.200	0.192	0.094	0.095	0.963
	${\theta }_3$	0.100	0.072	0.012	0.012	0.335
Marginal	${\beta }_1$	0.100	0.097	0.008	0.008	0.920
	${\beta }_2$	0.200	0.197	0.088	0.088	0.945
	${\theta }_1$	0.100	0.063	0.006	0.007	0.000
	${\theta }_2$	0.200	0.128	0.066	0.065	0.785
	${\theta }_3$	0.100	0.109	0.010	0.011	0.890
Scenario 3
Frailty	${\beta }_1$	0.100	0.083	0.010	0.010	0.536
	${\beta }_2$	0.200	0.163	0.107	0.109	0.939
	${\theta }_1$	0.100	0.073	0.009	0.009	0.160
	${\theta }_2$	0.200	0.146	0.084	0.090	0.923
	${\theta }_3$	0.100	0.023	0.015	0.015	0.000
Marginal	${\beta }_1$	0.100	0.085	0.011	0.010	0.622
	${\beta }_2$	0.200	0.178	0.106	0.106	0.939
	${\theta }_1$	0.100	0.049	0.006	0.006	0.000
	${\theta }_2$	0.200	0.101	0.058	0.060	0.602
	${\theta }_3$	0.100	0.095	0.011	0.011	0.908
Scenario 4
Frailty	${\beta }_1$	0.100	0.090	0.043	0.040	0.949
	${\beta }_2$	0.200	0.324	0.378	0.308	0.872
	${\theta }_1$	0.100	0.054	0.025	0.032	0.474
	${\theta }_2$	0.200	0.144	0.354	0.271	0.910
	${\theta }_3$	0.100	−0.455	0.259	0.213	0.147
Marginal	${\beta }_1$	0.100	0.094	0.048	0.035	0.943
	${\beta }_2$	0.200	0.252	0.354	0.295	0.928
	${\theta }_1$	0.100	0.049	0.012	0.011	0.021
	${\theta }_2$	0.200	0.100	0.113	0.114	0.850
	${\theta }_3$	0.100	0.086	0.072	0.069	0.943

TABLE 2 ∣.

The distribution of estimated and actual $z_i~\Gamma (2,0.5)$: Scenario 1: Independent $v_{ij3}~N(0,5)$; Scenario 2: $v_{ij3}=n_{ij-1}^a$; Scenario 3: $v_{ij4}=n_{ij-1}^a$ with unobserved $x_{i3}~\text{Norm}(\mathrm{0,5})$; Scenario 4: $v_{ij3}=n_{ij-1}^a$ with early censoring $C_i~U(4,8)$.

	25 percentile	50 percentile	75 percentile
Scenario 1
${\hat{z}}_i$	0.471	0.824	1.346
$z_i$	0.492	0.851	1.354
Scenario 2
${\hat{z}}_i$	0.458	0.814	1.345
$z_i$	0.494	0.854	1.360
Scenario 3
${\hat{z}}_i$	0.409	0.759	1.324
$z_i$	0.499	0.856	1.358
Scenario 4
${\hat{z}}_i$	0.056	0.758	1.442
$z_i$	0.682	1.072	1.595

4.2.2 |. Performance of the Risk Prediction Models

We define an advanced adenoma event in a screening visit if the transitions of the observed adenoma state exceed pre-specified criteria $n_{ij}^a>2$, which aligns with the previous definition. The non-advanced adenoma events are defined as $n_{ij}^a>0$. In Table 3, we compare the performance of proposed methods based on the frailty and marginal models with the observed percentage of advanced adenoma. ESD(Est) denotes the empirical standard deviation for the average predicted risk of advanced adenoma. BSD(Est) denotes the bootstrap standard deviation for the average predicted risk of advanced adenoma with bootstrap replication $B=200$.

TABLE 3 |.

Redicted probability of adenoma and advanced adenoma events with $z_i~\Gamma (2,0.5)$: Scenario 1: Independent $v_{ij3}~N(0,5)$; Scenario 2: $v_{ij3}=n_{ij-1}^a$; Scenario 3: $v_{ij4}=n_{ij-1}^a$ with unobserved $x_{i3}~\text{Norm}(\mathrm{0,5})$; Scenario 4: $v_{ij3}=n_{ij-1}^a$ with early censoring $C_i~U(4,8)$.

Method	# of adenoma	Mean (Est)	Mean (Perc)	ESD (Est)	BSD (Est)
Scenario 1
${\hat{z}}_i$	$n_{ij}^a>0$	0.611	0.632	0.009	0.009
	$n_{ij}^a>2$	0.263	0.270	0.008	0.008
${\hat{f}}_z(z)$	$n_{ij}^a>0$	0.608	0.632	0.007	0.008
	$n_{ij}^a>2$	0.301	0.270	0.008	0.008
Marginal	$n_{ij}^a>0$	0.616	0.632	0.009	0.008
	$n_{ij}^a>2$	0.276	0.270	0.009	0.009
Scenario 2
${\hat{z}}_i$	$n_{ij}^a>0$	0.598	0.615	0.008	0.008
	$n_{ij}^a>2$	0.232	0.236	0.007	0.007
${\hat{f}}_z(z)$	$n_{ij}^a>0$	0.593	0.615	0.007	0.007
	$n_{ij}^a>2$	0.278	0.236	0.007	0.007
Marginal	$n_{ij}^a>0$	0.605	0.615	0.007	0.007
	$n_{ij}^a>2$	0.255	0.236	0.008	0.008
Scenario 3
${\hat{z}}_i$	$n_{ij}^a>0$	0.540	0.592	0.010	0.009
	$n_{ij}^a>2$	0.222	0.232	0.008	0.008
${\hat{f}}_z(z)$	$n_{ij}^a>0$	0.551	0.592	0.009	0.008
	$n_{ij}^a>2$	0.271	0.232	0.009	0.009
Marginal	$n_{ij}^a>0$	0.557	0.592	0.010	0.009
	$n_{ij}^a>2$	0.242	0.232	0.009	0.009
Scenario 4
${\hat{z}}_i$	$n_{ij}^a>0$	0.373	0.445	0.027	0.024
	$n_{ij}^a>2$	0.108	0.090	0.013	0.012
${\hat{f}}_z(z)$	$n_{ij}^a>0$	0.528	0.445	0.052	0.039
	$n_{ij}^a>2$	0.290	0.090	0.053	0.037
Marginal	$n_{ij}^a>0$	0.454	0.445	0.056	0.040
	$n_{ij}^a>2$	0.104	0.090	0.072	0.041

In Table 3, for Scenarios 1–2, predictions based on the frailty model with estimated subject-level ${\hat{z}}_i$ always have the best performance. In the absence of prior event history, the marginal model performs slightly better than the frailty model with ${\hat{f}}_z(z)$ in terms of prediction. Scenarios 3–4 evaluate the robustness of predictions. Even though the model is misspecified in Scenario 3, the predicted probabilities estimated from the frailty and marginal models are still close to the observed percentages with small ESD for all prediction models. However, in Scenario 4, 90% percent of the screening intervals are right-censored, and no adenoma event is observed among those intervals, leading to the underestimation of frailty $z_i$ and larger ESD. In that case, the marginal model performs best because it is not affected by the underestimation of frailty $z_i$. In summary, some general guidelines for choosing appropriate prediction models for interval-censored screening data are as follows:

1. In cases where the screening history for the patients of interest is available, a prediction model based on the subject-specific frailty estimates has the best performance.

2. In cases where the screening histories of patients to be predicted are not available but the screening histories of similar cohorts are available, predictions based on either the marginal model or the estimated frailty distribution perform reasonably well.

3. In the cases of high percentages of patients with early drop-out time and no previous screening result, the marginal prediction model is preferred since frailty terms would be underestimated, leading to right-skewed estimates of the frailty distribution.

In addition, to provide reference for patients on their future screening, we also compared the screening intervals calculated by our proposed methods to the method of regular screening interval in terms of the effectiveness of detecting advanced adenoma events under Scenario 2. Specifically, the individualized screening intervals $t_i^{\prime }$ are determined by three prediction methods: The frailty prediction model with $z_i$, the frailty prediction model with $f_z(z)$, and marginal models. These intervals are obtained by setting the equation where the predicted probability of advanced adenoma is equal to the prevalence of advanced adenoma: $\hat{P}\left(n_a>2\mid t=t_i^{\prime }\right)=P\left(n_a>2\right)$. The regular screening interval is calculated as the weighted average gap time for detecting advanced adenoma ($n_a>2$) in the historic screening results across all patients. Subsequently, we simulated 10 future adenoma events for each patient, and the distribution of the detected adenoma events by different methods is summarized in Figure 1. Since adenomas are removed upon detection during the screening, a screening is considered successful if at least one adenoma event is detected. The screening interval derived from the frailty $z_i$ prediction model performs best, as it effectively detected adenoma events when the number of transitions was 1 or 2, while having a small proportion of cases detecting no adenoma. The marginal and frailty $f_z(z)$ models are more conservative, with a higher percentage of screenings detecting no adenoma. The regular screening interval performs the worst as it usually misses the opportunity to detect early-stage adenomas, significantly increasing the risk of colorectal cancer. It is not surprising that the frailty $z_i$ prediction model performs the best because the historic data provides extra information compared to the other methods without any auxiliary information besides the covariates. Also, the poor performance of a regular screening interval is expected, given its lack of flexibility.

FIGURE 1 |.

Distribution of the transitions of adenoma status detected in the next screening based on 969 patients.

5 |. Application to the Study of Colonoscopy Utilization (SCU)

We analyze the SCU cohort consisting of 5768 PLCO participants. Sampling weights are provided in the data to account for the fact that participants with baseline adenoma are oversampled.

5.1 |. Coefficients Estimate and Risk Prediction

Given the limited availability of CRC status data in the SCU cohort, we categorize individuals into three primary risk groups based on CRC progression: Non-adenoma (State 0), non-advanced adenoma (State 1), borderline non-advanced adenomas (State 2), and advanced adenoma (State 3). Because the SCU dataset lacks information for States 4–10, the analysis is restricted to States 0–3. The intensity rates for adenoma events are assumed to be identical across all state transitions. We evaluate the effects of age, gender, Body Mass Index (BMI), and adenoma state at baseline on adenoma and screening events. We categorized continuous variables age and BMI into binary variables with age ≥ 65 years old and BMI ≥ 25.

The parameter estimates of the frailty model and the marginal model for the adenoma events are summarized in Table 4. IR denotes the intensity ratio for each parameter with IR=exp(Coefficient), and SE denotes the standard error estimated from $B=200$ bootstrap replications. We get the same parameter estimates for the adenoma events from the frailty and the marginal models. Note that the estimated intensity ratios are all greater than one, which is consistent with previous clinical findings that colorectal cancer is more common in male and elderly, obese patients with baseline adenoma [23].

TABLE 4 |.

Intensity ratio (IR) estimates with 95% confidence intervals (CI) and risk prediction for adenoma events of SCU data.

Covariates (Adenoma)	Coefficient	SE	IR	95% CI
Parameter estimates
Male	0.32	0.08	1.38	(1.18, 1.63)
Age ≥ 65	0.16	0.07	1.18	(1.02, 1.35)
BMI ≥ 25	0.26	0.08	1.30	(1.10, 1.53)
Baseline Adenoma	0.31	0.03	1.36	(1.28, 1.44)

Category	Observed Percentage	Model	Predicted Probability
Risk predictions
Non adv-adenoma	0.296	frailty with ${\hat{z}}_i$	0.266	0.007
		frailty with ${\hat{f}}_z(z)$	0.261	0.008
		marginal	0.358	0.009
		Cox proportional hazard	0.231	0.006
Adv-adenoma	0.061	frailty with ${\hat{z}}_i$	0.059	0.003
		frailty with ${\hat{f}}_z(z)$	0.080	0.004
		marginal	0.037	0.003
		Cox proportional hazard	0.055	0.004

We evaluated our proposed prediction methods in SCU data using the weighted bootstrap method. In addition, a Cox proportional hazards model is fitted for comparison. This model treats the time of screening as the time of adenoma event occurrence, accounting for recurrence and incorporating subject-level weights. Table 4 also shows the performance of our proposed prediction models, with point estimates and standard errors (SE) calculated using bootstrap replication $B=200$. In terms of non-advanced adenoma events, all of the prediction model demonstrate good performance in prediction as their average estimated probability is close to the observed percentage. For advanced adenoma events, the frailty-based prediction model with ${\hat{z}}_i$ achieves the best performance, which aligns with the simulation results, as more than half of the subjects have non-negative frailty estimates. The kernel density frailty model with ${\hat{f}}_z(z)$ and the marginal model also perform reasonably well. However, the marginal model is less conservative compared to the kernel density frailty model with ${\hat{f}}_z(z)$ as it tends to underestimate the risk of advanced adenoma.

5.2 |. Advanced Adenoma Probabilities for Individual Participants

We estimate the survival probabilities of advanced adenoma for high, medium, and low-risk patients separately based on the frailty model and the marginal model in Figure 2. We define the hypothetical high-risk patient as male, age ≥ 65, BMI ≥ 25, with baseline advanced-adenoma, medium-risk patient as male, age < 65, BMI < 25, with baseline adenoma, low-risk patient as female, age < 65, BMI < 25, no adenoma. For the frailty model, the prediction is based on the coefficient estimate and kernel density estimates for the frailty term using the following equation:

$$\begin{gathered} P_{2i}\left(n_a>N_a\mid t=t_i^{\prime }\right) \\ =1-\sum _{n_a=0}^{N_a}{\int }_z\frac{\text{exp}\left(-z{\Lambda }_i^a\left(t_i^{\prime }\right)\right){\left(z{\Lambda }_i^a\left(t_i^{\prime }\right)\right)}^{n_a}}{n_a!}{f}_z(z)dz \end{gathered}$$

with ${\Lambda }_i^a\left(t_i^{\prime }\right)={\Lambda }_0^a\left(t_i^{\prime }\right)\text{exp}\left({\mathit{X}}_i^T\mathit{\beta }\right)$ and $t_i^{\prime }=0,5,10,15,20,\dots ,\tau$.

FIGURE 2 |.

Survival probability of advanced adenoma for high, medium, and low-risk patients over time.

For the marginal model, the prediction is based on the coefficient estimate from the marginal model using the following equation:

$$P_{3i}\left(n_a>N_a\mid t=t_i^{\prime }\right)=1-\sum _{n_a=0}^{N_a}\frac{\text{exp}\left(-{\Lambda }_i^a\left(t_i^{\prime }\right)\right){\left({\Lambda }_i^a\left(t_i^{\prime }\right)\right)}^{n_a}}{n_a!}$$

with $t_i^{\prime }=0,5,10,15,20,\dots ,\tau$. Our proposed prediction methods could guide different screening interval assignments for patients on different risk levels. The following survival probabilities of advanced adenoma are evaluated:

1. high-risk patient after 1 year

2. medium-risk patient after 3 years

3. low-risk patient after 10 years

While both the marginal model and the frailty kernel density model perform well in predicting advanced adenoma events, the frailty kernel density model is more conservative in risk prediction and tends to estimate higher predicted probabilities of advanced adenoma. The survival probabilities of advanced adenoma estimated from the frailty kernel density model are 0.948, 0.964, and 0.920, respectively, indicating that the risks of advanced adenoma for patients with different risk levels are managed equally based on different screening interval assignments. In addition, we also provide the reference for the next screening time for high, medium, and low-risk patients based on the frailty kernel density model. We assume the patients should be referred to the next screening when the estimated probability of advanced adenoma is equal to the prevalence of advanced adenoma among the population. The high-risk patients are recommended to take the next screening around 1.2 years. The medium-risk patients are recommended to take the next screening around 5 years. The low-risk patients are recommended to take the next screening around 8.5 years.

6 |. Discussion

In this paper, we implement a joint frailty model and a marginal model to predict risks and quantify risk factors of adenoma events in the context of colorectal cancer screening. In terms of parameter estimation for adenoma events, the marginal model and the joint frailty model yield similar results. This is because the estimating equations for adenoma events do not depend on subject-specific frailty terms ${\hat{z}}_i$. However, the two models differ in the estimation of coefficients for the screening events. In the joint frailty model, the partial likelihood incorporates the frailty term ${\hat{z}}_i$, leading to different coefficient estimates compared to the marginal model, which does not account for individual-level heterogeneity. In terms of predictions, the frailty model with ${\hat{z}}_i$ always has the best performance except for the scenario of early censoring time, since a large number of screening intervals are right-censored with zero adenoma event, leading to the underestimation of the subject-specific frailty terms. In scenarios with extensive right-censoring of screening intervals, the marginal model outperforms the frailty model in prediction as it is not influenced by the underestimation of $z_i$. To guide patients in future screenings, all of our proposed methods outperform the method of regular screening intervals by detecting adenoma events early and reducing risks. We also try the Delta method to derive the asymptotic variance of the predicted probability $P_3\left(n_a>N_a\right)$ based on the marginal model, which is a function of $\mathit{\beta },{\Lambda }_0(t)$ and $F(t)$. The difficulty is that $F(t)$ is estimated by the non-parametric Turnbull algorithm. It is hard to derive a closed form for $\hat{F}(t)-F(t)$.

In evaluating coefficients, people may choose the frailty model when they are interested in subject-level covariate effects and the marginal model when the population-average covariate effects are of interest. For the risk predictions of the advanced adenoma, when the screening history of patients and adequate recurrent screening visits are available, it is recommended to use the prediction method with ${\hat{z}}_i$. When screening history is not available, or in cases of a high percentage of early dropout time, the marginal model would perform better in prediction.

Supplementary Material

Additional supporting information can be found online in the Supporting Information section. Data S1: sim70478-sup-0001-Supinfo.pdf.

Data Availability Statement

The PLCO SCU datasets are available for delivery on CDAS. A publicly available Data Dictionary describing each dataset is provided. To obtain the actual data in SAS or CSV format, a data-only request must be submitted. Data will be delivered once the project is approved and data transfer agreements are completed. For more details, please refer to https://cdas.cancer.gov/datasets/plco/30/#dataset-100.