Deep learning for the change-point Cox model with current status data

A change-point model is a statistical framework for detecting points or periods at which the properties of the data shift abruptly (Pons 2003; Deng et al. 2017, 2022). Such models are vital across many domains, including economics (Li et al. 2024), environmental science, signal processing, and medical settings (Deng et al. 2022), where identifying when changes occur is crucial for decision-making and subsequent analysis.

Many studies have examined change-point models in survival analysis (Loader 1991; Pons 2003; Dupuy 2006; Kosorok and Song 2007; Torben and Scheike 2007; Oueslati and Lopez 2013; Deng et al. 2022). Among these, the Cox proportional hazards model has attracted the most attention. Luo and Boyett (1997) first considered the model \(\Lambda (t)=\Lambda _0(t)\exp (\beta _0I_{\{X\le \zeta _0\}}+\alpha _0 Z)\), where the covariates X and Z are one-dimensional and a constant term \(\beta _0\) is added when the threshold \(\zeta _0\) is reached. Pons (2003) considered the Cox model \(\Lambda (t)=\Lambda _0(t)\exp \{{\varvec{\alpha }}_0^\top {\varvec{Z}}_1+{\varvec{\beta }}_0^\top {\varvec{Z}}_2I_{\{Z_3\le \zeta _0\}}+{\varvec{\gamma }}_0^\top {\varvec{Z}}_2I_{\{Z_3>\zeta _0\}}\}\), where the covariate jumps at a threshold \(\zeta _0\), and showed that the change-point estimator is n-consistent. Kosorok and Song (2007) and Song et al. (2009) extended the asymptotic properties of the Cox model to a class of transformation models and introduced a maximal score-based test for the existence of a change point using random shuffling. Lee et al. (2020) further studied tests for the change-point effect with right-censored data and proposed three maximum Wald or score-based procedures that are important for model identification. More recently, for right-censored data, Deng et al. (2022) extended the Cox model with a change point to one with a change hyperplane. They developed a computational method for this model, proved the convergence of the change hyperplane, and derived the asymptotic distribution of the estimators.

Most existing methods focus on right-censored data. In this type of data, some individuals have fully observed failure times, while others are right-censored, known only after the time of censoring. However, in some clinical settings, individuals scheduled for observation to detect clinically significant changes in their health or disease status may miss some scheduled appointments and return with altered status (Ma et al. 2014), such as in studies of traumatic brain injury (Vanier et al. 2020). In such cases, change-point models developed for right-censored data may fail because the event time of interest can be left-censored or right-censored. To overcome this limitation, we study a change-point model for “case-I” interval-censored data, also known as current status data.

Current status data arise when the exact event time is unobserved and only the occurrence status at a single examination is recorded (Huang 1996; Huang and Wellner 1997; Sun 2006). Compared with right-censored data, the precise value of the survival time T is unknown, making statistical inference with current status data more challenging. Huang (1996) first established the Cox model framework for current status data and derived its asymptotic properties. For regression analysis, a partially linear proportional hazards model was later proposed (Lu and McMahan 2018).

However, in clinical studies, some covariates can exert complex nonlinear interaction effects on the logarithm of the cumulative hazard of the failure time T. When interaction effects are present, simple linear or purely additive models may fail to capture them, leading to substantial bias. To address this, Wu et al. (2024) proposed a deep learning approach that flexibly models the nonparametric effects of multivariate covariates and their potential relationships. The partially linear Cox model is then formulated aswhere \(\Lambda _0\) denotes the baseline cumulative hazard function, \({\varvec{\beta }}_0\in \mathbb {R}^p\) is an unknown parameter vector, and \(g_0: \mathbb {R}^r\rightarrow \mathbb {R}\) represents an unknown smooth function.

Although many methods have been developed for failure-time data with change points, most are designed for right-censored data and are not directly applicable to interval-censored settings. We therefore study a partially linear Cox model with a change point for current status data.

Our main contributions are summarized below: The remainder of the paper is organized as follows. Section 2 presents the estimation procedure for a change point in a partially linear Cox model with current status data. Section 3 provides theoretical results for the estimators, including consistency, convergence rates for all components, and the asymptotic normality of the parametric estimates. Section 4 introduces a score-based testing procedure to assess the presence of a change point. Section 5 reports numerical experiments, and Sect. 6 illustrates the method with the Rotterdam breast cancer data. Section 7 offers a summary and further discussion. All technical lemmas, necessary proofs, and computational details are provided in the supplementary material.

This section establishes the consistency and asymptotic properties of the estimates in (3). Before discussing them, we first introduce some notation. The estimates of the change point \(\zeta _0\) and the remaining parameters \(\varvec{\xi }_0=(\Lambda _0,\varvec{\beta }_0,g_0,\varvec{\gamma }_0,h_0)\) are denoted by \(\hat{\zeta }_n\) and \(\hat{\varvec{\xi }}_n=(\hat{\Lambda }_{n}, \hat{\varvec{\beta }}_n, \hat{g}_n, \hat{\varvec{\gamma }}_n, \hat{h}_n)\), respectively. Note that \(\varvec{\eta }=(\varvec{\xi },\zeta )=(\Lambda ,\varvec{\beta },g,\varvec{\gamma },h,\zeta )\). For any q-dimensional vector \(\varvec{a}=(a_1,\dots ,a_q)^\top \in {\mathbb {R}}^q\), \(\Vert \varvec{a}\Vert =\sqrt{\sum _{j=1}^{q}a_j^2}\) and \(\Vert \varvec{a}\Vert _\infty =\max _j|a_j|\) denote the \(L^2\) norm and \(L^\infty \) norm, respectively. For any matrix \(W=(w_{ij})\in {\mathbb {R}}^{m\times n}\), let \(\Vert W\Vert _{\infty }=\max _{i,j}|w_{ij}|\). For any function h(x), \(\Vert h\Vert _\infty =\sup _x|h(x)|\). Denote \(a_n\lesssim b_n\) as \(a_n\le cb_n\) and \(a_n\gtrsim b_n\) as \(a_n\ge cb_n\) for some constant \(c>0\) respectively, and \(a_n\asymp b_n\) indicates \(a_n\lesssim b_n\) and \(a_n\gtrsim b_n\). Define the distance of \({\varvec{\eta }}\) aswhere \(|\cdot |\) represents the absolute value,andrespectively. Denote \(\hat{\varvec{\theta }}_n=(\hat{\varvec{\beta }}_n^\top ,\hat{\varvec{\gamma }}_n^\top )^\top \), \(\tilde{\textbf{X}}(\zeta )=(\textbf{X}^\top ,\textbf{X}^\top I_{\{E>\zeta \}})^\top \), and \(\tilde{K}({\textbf{Z}};\zeta )=g({\textbf{Z}})+h({\textbf{Z}})I_{\{E>\zeta \}}\). Writewhere \(\varvec{V}=(U,\textbf{X},\textbf{Z},E)\), and definethen \(\ell _n({\varvec{\eta }})={\mathbb {P}}_nm_{{\varvec{\eta }}}(\varvec{V}):=\frac{1}{n}\sum _{i=1}^n m_{{\varvec{\eta }}}(\varvec{V}_i)\) and define \(\ell ({\varvec{\eta }})={\mathbb {P}} m_{{\varvec{\eta }}}(\varvec{V}):=\int m_{{\varvec{\eta }}}(\varvec{V}) dP\).

To establish the theoretical results, the following regularity conditions are required: Condition (C1) determines the structure and complexity of the deep neural network. Condition (C2) is a mild regularity assumption on the knot placement and is similar to the conditions in Ramsay (1988). Condition (C3) is a standard assumption in semiparametric estimation The smoothness and monotonicity conditions on \(\Lambda _0(\cdot )\) in (C4) guarantee that it can be well approximated by a monotone spline basis. Conditions (C5)-(C7) are needed for the calculation in the proofs of Theorems 1–5. Conditions (C8) and (C9) are required to ensure that the model parameters are identifiable. Condition (C10) indicates that E is continuously distributed in a bounded neighborhood of \(\zeta _0\), which is crucial for establishing the asymptotic independence and asymptotic properties at \(\zeta _0\). (C11) ensures the asymptotic efficiency of the regression parameter \({\varvec{\theta }}_0\).

Theorem 1 limits the range of the change point, illustrating that this methodology is identifiable only when \(\zeta _0\in (\min _i(E_i),\max _i(E_i))\), i.e. \({\mathbb {P}}(\{E>\zeta _0\})\in (0,1)\). Another characteristic of identifiability is the event that \({\varvec{\gamma }}_0\ne {\varvec{0}}\) or \(g_0(\cdot )\ne 0\) occurs.

Before deriving the overall convergence rate, we introduce a Hölder class that contains both \(g_0\) and \(h_0\). Let \(\kappa \) and B be two positive constants, and \(\lfloor \kappa \rfloor \) is the largest integer smaller than \(\kappa \). A \((\kappa ,{{B}})\)-Hölder class of smooth functions is denoted aswhere \(\partial ^{{\varvec{\upsilon }}}=\partial ^{\upsilon _1}\cdots \partial ^{\upsilon _r}\) with \({\varvec{\upsilon }}=(\upsilon _1,\dots ,\upsilon _r)^\top \) and \(|{\varvec{\upsilon }}|=\sum _{j=1}^r\upsilon _j\). Denote \(K>0\), \(R\in {\mathbb {N}}^+\), \({\varvec{\kappa }}=(\kappa _0,\dots ,\kappa _{K})^\top \in {\mathbb {R}}_+^{K+1}\), and let \({\varvec{d}}=(d_0,\dots ,d_{K+1})\in {\mathbb {N}}_+^{K+2}\), \(\tilde{{\varvec{d}}}=(\tilde{d}_0,\dots ,\tilde{d}_{K})^\top \in {\mathbb {N}}_+^{K+1}\) with \(\tilde{d}_j\le d_j\) where \(j=0,\dots ,L\). Further, assume that the functions \(g_0\) and \(h_0\) belong to a composite smoothness function class:where \(\tilde{\varvec{d}}\) denotes the intrinsic dimension of the function in this class. Furthermore, denote \(\tilde{\upsilon }_i=\upsilon _i\prod _{j=i+1}^K(\upsilon _j\wedge 1)\) and \(\alpha _n=\max _{i=0,\dots ,K}n^{-{\tilde{\upsilon }_i}/{(2\tilde{\upsilon }_i+\tilde{d}_i)}}\), where \(a\wedge b=\min \{a,b\}\).

Theorem 2 illustrates that the rate is determined by the smoothness \(\alpha _n\), the intrinsic dimension \(\tilde{\varvec{d}}\) of the DNN, the smoothness \(\ell \), and the maximum number of nodes \(m_n\) of the spline function. As a result, when the intrinsic dimension \(\tilde{\varvec{d}}\) is relatively small, the suggested strategy can mitigate the curse of dimensionality and has a faster convergence rate. It is well known that when nonparametric smoothing techniques (such as splines or kernel functions) are used to estimate \(g_0\), the model suffers from a severe “curse of dimensionality”. A key advantage of the DNN estimator over traditional smoothing methods is its ability to alleviate the curse of dimensionality by accurately projecting data into a lower-dimensional representation space (Anthony and Bartlett 1999).

We next establish the semiparametric efficiency. Unlike the classical regression model, the unknown change point affects the procedure of estimation and projection, which makes it difficult to obtain the score and information functions (Pons 2003; Deng et al. 2022). Therefore, before establishing the semiparametric efficiency, we first show that the effect of the change point is asymptotically negligible; in other words, when n is large enough, \(\hat{\zeta }_n\) is asymptotically independent of \(\hat{\varvec{\xi }}_n\). In particular, denote two counting processesFor simplicity, letand

The unknown change point causes inconvenience in statistical inference, especially in deriving the asymptotic distribution. Based on asymptotic independence, we establish the effective score and information bound for \( \theta _{0}\).

Let \( \Omega _{0} = \left\{ \begin{gathered} \Lambda \in L^{2} \left( {\left[ {0,\tau } \right]} \right) \hfill \\ :\Lambda (\tau ) < \infty \hfill \\ \end{gathered} \right\} \), and let \( \begin{gathered} {\mathcal{H}}_{0} = \{ g \in {\mathcal{H}}\{ K,\kappa ,d,\tilde{d},B\} \hfill \\ \quad \quad :{\mathbb{E}}\{ g(Z)\} = 0\} \hfill \\ \end{gathered}\). Denote \(\Omega _{\Lambda _0}\) be the collection of all subfamilies \(\{\log \Lambda _t: t\in (-1,1)\}\subset \{\log \Lambda :\Lambda \in \Omega _0\}\) such that \(\lim _{t\rightarrow \infty }\Vert t^{-1}(\log \Lambda _t-\log \Lambda _0)-a\Vert _{L^2([0,\tau ])}=0\) where \(a\in L^2([0,\tau ])\), and letSimilarly, definewhere \(\mathcal {H}_{\tilde{K}_0}\) denotes the collection of all subfamilies of \(\{\tilde{K}_s(\cdot ;\zeta _0)\in L^2([0,1]^r):s\in (-1,1) \}\subset \mathcal {H}_0\) such that \(\lim _{s\rightarrow 0}\Vert u^{-1}(\tilde{K}_s-K_0)-b\Vert =0\) and \(b\in L^2([0,1]^r)\). Let \(\bar{\mathbb {T}}_{\Lambda _0}\) and \(\bar{\mathbb {T}}_{\tilde{K}_0}\) be the linear span of \({\mathbb {T}}_{\Lambda _0}\) and \({\mathbb {T}}_{\tilde{K}_0}\), respectively. Denote the single observation of the log-likelihood \(\ell ({\varvec{\eta }}|{\textbf{Y}})=\Delta \log (1-\exp \{-{\varvec{h}}_{{\varvec{\eta }}}(\varvec{V})\})-(1-\Delta ){\varvec{h}}_{{\varvec{\eta }}}(\varvec{V})\), where \(\textbf{Y}=(\Delta ,\varvec{V})\). The score function w.r.t. \({\varvec{\theta }}\) iswhereLet a parametric smooth model \((\Lambda _t(U),\tilde{K}_s({\textbf{Z}};\zeta _0))\) satisfy \(\Lambda _t(U)|_{t=0}=\Lambda _0(U)\) and \(\tilde{K}_s({\textbf{Z}};\zeta _0)|_{s=0}=\tilde{K}_0({\textbf{Z}};\zeta _0)\), with \(\left. \frac{\partial }{\partial t}\Lambda _t(U)\right| _{t=0}=a(U)\) and \(\left. \frac{\partial }{\partial s}\tilde{K}_s({\textbf{Z}};\zeta _0)\right| _{s=0}=b({\textbf{Z}})\), the score functions for \(\Lambda _0\) and \(\tilde{K}_0(\cdot ;\zeta _0)\) areandrespectively. For \({\varvec{a}}=(a_1,\dots ,a_{2p})^\top \in \bar{\mathbb {T}}_{\Lambda _0}\), let \(\dot{\ell }_{\Lambda }({{\varvec{\eta }}_0})[{\varvec{a}}]=(\dot{\ell }_{\Lambda }({{\varvec{\eta }}_0})[a_1],\dots ,\dot{\ell }_{\Lambda }({{\varvec{\eta }}_0})[a_{2p}])^\top \), and for \({\varvec{b}}=(b_1,\dots ,b_{2p})^\top \in \bar{\mathbb {T}}_{\tilde{K}_0}\), define \(\dot{\ell }_{\tilde{K}}({{\varvec{\eta }}_0})[{\varvec{b}}]=(\dot{\ell }_{\tilde{K}}({{\varvec{\eta }}_0})[b_1],\dots ,\dot{\ell }_{\tilde{K}}({{\varvec{\eta }}_0})[b_{2p}])^\top \). The score and information bound are given in the following theorem.

After determining the score function and the information bound of \({\varvec{\theta }}\), we present the asymptotic distribution of \({\hat{\varvec{\theta }}_n}\) and \(\hat{\zeta }_n\). Denote two variablesrespectively, and definewhere \(\xi _i^+=\xi ^+(U_i,\Delta _i,{\textbf{X}}_i,{\textbf{Z}}_i,E_i)\) and \(\xi _i^-=\xi ^-(U_i,\Delta _i,{\textbf{X}}_i,{\textbf{Z}}_i,E_i)\). Furthermore, for \(v\in {\mathbb {R}}^+\), definewhereand for \(v\in {\mathbb {R}}^-\), definewhere

Theorem 5 presents the \(\sqrt{n}\)-consistency and asymptotic normality of the estimate of the regression parameter and shows that the estimate of the change point converges distributionally to a distribution of the maximum value of a composite Poisson-type process with rate \(n^{-1}\).

In practice, the detection of change points is a crucial problem. Accurate identification improves understanding of the data structure, enhances predictive models, and supports better decision-making. We aim to test whether a change point exists, i.e., the null hypothesis \(H_0:{\varvec{\gamma }}_0={\varvec{0}},~ h_0(\cdot )=0.\) Note that if \({\varvec{\gamma }}^\top {\textbf{X}}+h({\textbf{Z}})\) is degenerate (identically zero) or the threshold diverges (\(\zeta _0=\pm \infty \)), the change-point model fails to be identifiable. If \(H_{0}\) holds, model (2) reduces to the deep partially linear Cox model (1) and one can estimate \({\varvec{\beta }}_0\), \(\Lambda _0(\cdot )\) and \(g_0(\cdot )\) by the estimation procedure proposed by Wu et al. (2024). To test \(H_0\), we propose a functional SUP statistic inspired by the supremum test (SUP) proposed by Kosorok and Song (2007).

Denote \(\varvec{L}_n(\varvec{\gamma },h,\Lambda ,\varvec{\beta },g,\zeta )=\sum _{i=1}^n\Delta _i\log (1-\exp \{-\varvec{h}_{\varvec{\eta }}(\varvec{V}_i)\})-(1-\Delta _i)\varvec{h}_{\varvec{\eta }}(\varvec{V}_i)\). The score functional evaluated at direction \(f\in \mathcal {G}\) is defined asand the covariance of \(\varvec{U}_n(\zeta ,f,\varvec{\xi })\) isThen the SUP statistic is defined aswhere \(\tilde{\varvec{\Omega }}_n=(\tilde{\Lambda }_n,\tilde{\varvec{\beta }}_n,\tilde{g}_n,\varvec{0},0)\), and where \(\tilde{\Lambda }_n\), \(\tilde{\varvec{\beta }}_n\), and \(\tilde{g}_n\) are the restricted MLE of \(\varvec{\beta }_0\), \(\Lambda _0\) and \(g_0\) under \(H_0\), respectively.

For application, the change point \(\zeta \) can be selected by grid search, and the function f can be chosen from the DNN function space \(\mathcal {G}\) to match the potentially complex nonlinear forms of the direction. However, directly selecting \(f \in \mathcal {G}\) may be infeasible due to the high dimensionality and complexity of the DNN function space. In practice, one can instead work with a simpler space that contains G, such as a finite-dimensional linear span \(\mathcal {T}\), and choose f from \(\mathcal {T}\). To reduce the computational burden of evaluating f(Z), the selection of direction functions can be further decomposed into a sequence of one-dimensional problems. Specifically, for each dimension \(j=1,\dots ,r\) of \(\textbf{Z}\), we can select f from \({\mathcal {T}}_{j}\), the projection on the jth dimension of \(\mathcal {T}\), and calculate the corresponding score-based statistics. Ultimately, we take the maximum of the statistics across all dimensions, which substantially reduces computational complexity while retaining flexibility. For example, if we take \({\mathcal {T}} = \text {span}\{I_{\{Z_{1}<z_1,\dots ,Z_r<z_r\}}\}\) to replace \(\mathcal {G}\), the projection on its jth dimension \({\mathcal {T}}_{j}=\text {span}\{I_{\{Z_j<z_j\}}\}\) can be used to choose f; we can take \(\textbf{Z}=_{\{Z_j<z_j\}}\in \mathcal {T}_j\) for \(j=1,\dots ,r\) and then take the supremum over j, rather than using \(\textbf{Z}=I_{\{Z_1<z_1,\dots ,Z_r<z_r\}}\in \mathcal {T}\) to compute the SUP statistic directly. In summary, the \(\textrm{SUP}_k\) statistic is defined aswhere \(\{\zeta _1^*,\dots ,\zeta _k^*\} \subset [\zeta _1,\zeta _2]\). We construct the permutation distribution of \(\textrm{SUP}_{k}\) by repeatedly permuting E and recomputing the statistic. At a significance level \(\alpha \), reject \(H_0\) if \(\textrm{SUP}_{k}\) exceeds the upper \(\alpha \)-quantile of its permutation distribution; otherwise, do not reject \(H_0\). The algorithm below summarizes the testing procedure.

In this section, we first present the computational procedure. The loss is the negative log-likelihood function, which is nonconcave in the parameters \({\varvec{\beta }}\), \({\varvec{\gamma }}\), \(W_j\), \({\varvec{\nu }}_j\), and \(c_j\); consequently, conventional optimization methods may perform poorly. To overcome this difficulty, we propose an iterative profiling estimation procedure, detailed in the algorithm below.

Step 1 gives the initial values, and it is recommended to let \(\breve{\varvec{\theta }}\) be a zero vector with dimension 2p to keep the proportional term from influencing the initial estimation and resulting in overfitting. In Step 2, given the initial values of \({\varvec{\theta }}\) and spline parameters \(c_j\)’s, the \(\hat{W}_j\)’s and \(\hat{\varvec{\nu }}_j\)’s can be obtained by minimizing the loss function using the Pytorch. We estimate \(g_0\) and \(h_0\) jointly by choosing \(p_{K+1}=2\), producing the two-dimensional output \((\hat{g}_n({\textbf{Z}}),\hat{h}_n({\textbf{Z}}))\) for each \({\textbf{Z}}\). Then, in Step 3, fixing the remaining parameters, we estimate \(\hat{c}_j\) for \(\hat{\Lambda }_n\) and set \(\hat{\Lambda }_n(\cdot )=\sum _{i=1}^{q_n}\hat{c}_j M_j(\cdot |\ell )\). In Step 4, we apply the sequential least squares programming to estimate \({\varvec{\theta }}_0\) while keeping the remaining parameters fixed. While a larger \(N_R\) in Step 5 enhances the precision of \(\zeta _0\), it also raises computational burden. In practice, one should balance accuracy against cost. Finally, in Step 6, if we define \(\hat{ \varvec{\theta }}_n^{(j)}\) as the estimation of regression parameters in the j iteration and \(\Vert \cdot \Vert _\infty \) as the \(L^{\infty }\) norm, our iteration procedures stop at the jth step when \(\Vert \hat{\varvec{\theta }}_n^{(j)}-\hat{\varvec{\theta }}_n^{(j-1)}\Vert _\infty <\epsilon \) for some given \(\epsilon >0\).

Some details of the neural network optimization are discussed here. The Adam optimizer estimates the \(W_j\) and \({\varvec{\nu }}_j\) of the deep neural network, which is reliable and practical in various models (Kingma and Ba 2014). The Adam optimization algorithm utilizes first-order moment estimates and second-order moment estimates of the gradients to dynamically adjust the learning rates for each parameter. The tuning parameters of the DNN contain the number of hidden layers K, the number of neurons \(p_k\) in all hidden layers, the number of epochs, the dropout rate and the learning rate. The learning rate is the step size of gradient descent, while the dropout rate refers to the random disregard of partial neurons during training.

In all simulations, based on Model (2), \({\textbf{X}}\) is generated from the Bernoulli distribution with a successful probability of 0.5. The covariate \({\textbf{Z}}\) is generated from a multivariate t-distribution \(t_5({\varvec{0}},{\varvec{\Sigma }})\) with truncation on [0, 2], where \({\varvec{\Sigma }} = (\sigma _{ij})\) with \(\sigma _{ii}=1\) and \(\sigma _{ij}=0.5\) for \(i\ne j\). The variable E is distributed as N(2, 1) truncated on the interval [1.5, 2.5]. We set the true values of the regression parameters \({\varvec{\theta }}_0=(\beta _0,\gamma _0)^\top = (-1, 2)^\top \), the change point \(\zeta _0 = 2\) and the baseline cumulative hazard function \(\Lambda _0(t)=\sqrt{t}/5\). For covariate \({\textbf{Z}}\), we design the following 3 cases with \({\textbf{z}}=(z_1,\dots ,z_5)^\top \in [0,2]^5\):

Case 1 (Linear): \(g_0({\textbf{z}})=-z_1/2-z_2/3-z_3/4-z_4/5-z_5/6+0.94\), \(h_0({\textbf{z}})=-z_1/3-z_2/4-z_3/5-z_4/6-z_5/7+0.71\),

Case 2 (Additive): \(g_0({\textbf{z}})=z_1^2/2+2\log (z_2+1)/5+3\sqrt{z_3}/10+e^{z_4}/5+z_5^3/10-1.62\), \(h_0({\textbf{z}})=\sin (2\pi z_1)+e^{z_2}/5+3\sqrt{z_3}/5+\log (z_4+1)/3+z_5^2/3-1.38\),

Case 3 (Deep): \(g_0({\textbf{z}})=\sqrt{z_1z_2}/5+z_3^2z_4/4+\log (z_4+1)/3+e^{z_5}/2-1.91\), \(h_0({{\textbf{z}}})=\{\sqrt{z_1z_2}/5+z_3^2z_4/4+\log (z_4+1)/3+e^{z_5}/2\}^2/5-1.36\).

All intercept terms are added to satisfy the identifiable conditions \({\mathbb {E}} g_0({\textbf{Z}})=0\) and \({\mathbb {E}} h_0({\textbf{Z}})=0\). In cases 1 and 2, the effects of all covariates on the log cumulative hazard function are linear and additive, respectively, while the effects in Case 3 are more complicated.

To compare with our proposed method (CPDPLCM), we also include an approach that estimates the effect of \({\textbf{Z}}\) using a linear model. This method assumes the form \( \Lambda (t|{\textbf{X}}, {\textbf{Z}}, E) = \Lambda _0(t)\exp \{{\varvec{\beta }_0}^\top {\textbf{X}} + \varvec{\varsigma }_0^\top {\textbf{Z}} + ({\varvec{\gamma }}_0^\top {\textbf{X}} + \varvec{\vartheta }_0^\top {\textbf{Z}} + \vartheta _0)I_{\{E >\zeta _0\}}\}, \) where \(\varvec{\varsigma }_0, \varvec{\vartheta }_0 \in \mathbb {R}^r\) and \(\vartheta _0 \in \mathbb {R}\). This approach, referred to as CPCPH, is a direct extension of methods of Huang (1996) and Pons (2003). For both the two methods, we chose \(\breve{\zeta }=\frac{1}{n}\sum _{i=1}^n E_i\), \(\breve{{\varvec{\theta }}}=(\breve{\beta },\breve{\gamma })^{\top }=(0,0)^\top \) and \(\breve{\varvec{c}}=(\breve{c}_0,\dots ,\breve{c}_{q_n})^\top \) with \(\breve{c}_j=0.1\) for \(j=0,\dots ,q_n\) to be the initial values. The sample size for each case was \(n = 1000, 2000\), or 4000, respectively, and 200 replications were performed.

We first describe the evaluation criteria for the estimates. The estimates of regression parameters are evaluated by the estimated biases (Bias), sampling standard errors (SSE), sampling means of the estimated standard error estimates (ESE), and the 95% empirical coverage probabilities (CP). To evaluate the performance of DNN functions, we consider the relative error (RE), defined aswith its SSE, where \(\hat{g}\) and \(g_0\) are evaluated at the test set’s covariates \(\{\textbf{Z}_i,i=1,\dots ,n_t\}\), with \(n_t=200\). We also consider the prediction error for the ith test observation, defined as \(\text {Error}_i(\hat{g})=\hat{g}({\textbf{Z}_i})-g_0({\textbf{Z}_i})\). Obtain the change point estimate \(\hat{\zeta }_n\) by searching a grid with a given gap; in this study, we choose the gap of 0.01 to balance the accuracy and the computational cost. The simulation results include the estimated bias, the length of the 95% empirical confidence interval (95% CI Length), and the SSE of the estimation. Specifically, for M replications, denote \(\varvec{\zeta }_M=(\hat{\zeta }^{(1)}_n,\dots ,\hat{\zeta }^{(M)}_n)\) be the estimation collection of \(\zeta _0\), the 95% CI Length is defined as the difference of the upper 2.5% quantile and the lower 2.5% quantile of \(\varvec{\zeta }_M\).

As reported in Table 1, the estimators \(\hat{{\beta }}_n\) and \(\hat{{\gamma }}_n\) exhibit improved performance as the sample size increases in all cases. For the proposed CPDPLCM method, the results indicate that \(\hat{{\beta }}_n\) and \(\hat{{\gamma }}_n\) are unbiased and asymptotically normal, in accordance with Theorem 5. Moreover, the procedure in Theorem 4 for estimating the asymptotic variance is validated as reliable. The 95% coverage probabilities for both \(\hat{{\beta }}_n\) and \(\hat{{\gamma }}_n\) under the proposed method are generally close to 0.95 as the sample size increases. Note that \(\hat{{\beta }}_n\) is estimated more accurately than \(\hat{{\gamma }}_n\) because the change-point effect does not occur for every observation, reducing the information available for estimating \(\hat{{\gamma }}_n\). In addition, in Cases 2 and 3, the CPCPH method exhibits substantial bias due to model misspecification. Although Case 1 is much simpler and satisfies the assumptions of CPCPH, our method remains highly competitive, with only a slight loss of efficiency. Overall, the proposed approach is flexible for estimating regression parameters and outperforms CPCPH.

Table 2 summarizes the relative errors (REs) and their standard deviations on the test set, and Fig. 1 visualizes the corresponding prediction errors. Both the RE and its SSE decrease as the sample size increases. The relative error of \(\hat{h}\) exceeds that of \(\hat{g}\), mirroring the behavior observed for \(\hat{\varvec{\beta }}\) versus \(\hat{\varvec{\gamma }}\). When the true functions correspond to Case 1, the proposed method is slightly less efficient than the perfectly specified CPCPH in estimating g0 and h0. In contrast, for Cases 2 and 3, as sample size grows, our method outperforms CPCPH, yielding smaller REs and prediction errors and exhibiting greater flexibility.

Table 3 presents the simulation results for the change-point estimates. When the identifiability conditions are satisfied, both methods produce valid estimates, corroborating the asymptotic independence established in Theorem 3.3. For both methods, performance improves with increasing sample size. The bias of the change-point estimates is smaller than that of the regression parameters, reflecting the faster convergence rate predicted by Theorem 3.5. Across settings, the accuracy of the change-point estimates degrades slightly as the covariate effects become more complex.

Overall, the CPDPLCM method performs well across all scenarios considered and demonstrates strong flexibility to varying model structures.

This study focuses on the Cox model with a change point at the covariate for the current status data, proposes a DNN-based estimation method, gives the semiparametric validity of the maximum likelihood estimation, and provides a computational scheme. This method is an effective tool for analyzing interval-censored survival data and identifying significant changes in risk factors, especially when the effects of covariates or nuisance parameters are complex.

This method can be directly extended to a class of generalized partial linear models, such as the transformation model (Kosorok and Song 2007) and the Cox model with right censoring (Pons 2003). Furthermore, the assumption that all covariates are affected by the change-point effect of a single univariate variable is restrictive. Thus, a multivariate \(\varvec{E}=(E_1,\dots ,E_q)^\top \) with threshold \(\varvec{\zeta }=(\zeta _1,\dots ,\zeta _q)^\top \) could be considered, and variable selection techniques may help detect these more complex change-point effects. In addition, heterogeneity across observations, common in precision medicine, suggests that the change point may be individualized; it may therefore be valuable to consider models with individualized change-point effects and to design personalized therapies.

Another challenging issue is how to effectively verify model identifiability. Identifiability is crucial for ensuring estimator consistency; however, current approaches (e.g., Kosorok and Song (2007)) rely heavily on random shuffling and sampling, which may be ineffective when training a DNN and consume substantial computing resources. An optimal approach is to avoid resampling and repeated DNN training to save time. We will explore advances on this issue in future work.