A Bayesian approach for analyzing partly interval-censored data under the proportional hazards model

Abstract

Partly interval-censored time-to-event data often occur in biomedical studies of diseases where periodic medical examinations for symptoms of interest are necessary. Recent decades have seen blooming methods and R packages for interval-censored data, however, the research effort for partly interval-censored data is limited. We propose an efficient and easy-to-implement Bayesian semiparametric method for analyzing partly interval-censored data under the proportional hazards model. Two simulation studies are conducted to compare the performance of the proposed method with two main Bayesian methods currently available in the literature and the classic Cox proportional hazards model. The proposed method is applied to a partly interval-censored progression-free survival data from a metastatic colorectal cancer trial.

1. Introduction

Partly interval-censored data often occur in medical and health studies that include periodic examinations. With partly interval-censored data, the failure times are exactly observed for some subjects, while only known to be within certain time intervals for the rest. In cancer clinical trials, progression-free survival, defined as time from study entry to disease progression or death due to any cause, is often used as the primary endpoint. It is actually partly interval-censored as the exact date of death is normally known while the date of disease progression is only known to be between two assessment visits. The mainstream methods in pharmaceutical industry are to ignore this so-called arbitrary censoring attribute of the data and continue to treat it as right-censored by assuming that the event occurs at the study day when it is detected. This strategy can induce bias in the estimation especially when the intervals are wide and varied.^1,2 The standard error of the estimation is also underestimated since it assumes that failure times are exactly known when they are not.³

The current literature for partly interval-censored data is limited. From the frequentist perspective, Huang⁴ developed the asymptotic properties for the nonparametric maximum likelihood estimator (NPMLE) of the distribution function in Turnbull’s⁵ model. Kim³ developed the maximum likelihood estimator for the proportional hazards (PH) model. Zhao et al.⁶ developed a class of generalized log-rank tests that perform survival comparison. Gao et al.² developed semiparametric estimation of the accelerated failure time (AFT) model. From the Bayesian perspective, Zhou and Hanson⁷ developed a unified approach that fits PH, proportional odds, and AFT models to partly interval-censored and left-truncated spatial data. The two functions that implement their method for partly interval-censored data are survregbayes and survregbayes2 in their R package spBayesSurv.⁸ Komárek and Lesaffre⁹ developed a mixed-effects AFT model for partly interval-censored data. This method is implemented by the bayessurvreg1 function in their R package bayesSurv.¹⁰

It is worth mentioning that there are R packages for interval-censored data that can also be used for fitting partly interval-censored data. They include the intcox package that implements Pan’s method¹¹ which extends the iterative convex minorant algorithm to the Cox PH model for interval-censored data; the MIICD package¹² that implements multiple imputation for PH regression with interval-censored data; the coarseDataTools package¹³ that fits parametric AFT models to interval-censored data; the interval package¹⁴ that estimates the NPMLE of survival curve and performs log-rank and Wilcoxon type tests for interval-censored data; the SmoothHazard package¹⁵ that can fit semiparametric or parametric PH model to interval-censored data; the survBayes package¹⁶ that fits a PH model by a Bayesian approach to interval-censored data; the dynsurv package¹⁷ that fits Bayesian PH model to interval-censored data; and the icenReg package¹⁸ that fits Bayesian PH, proportional odds, and AFT models for interval-censored data.

However, there might be limitations with some of these packages. For instance, the intcox package does not provide standard error for an estimated regression coefficient. The interval package does not perform regression analysis. The MIICD package imputes exact times for finite interval-censored data and and then use the partial likelihood method. The survBayes package reduces the data to right-censored data by imputing an observed time for each finite interval-censored time. The dynsurv package also reduces the data to “augmented right-censored data” through sampling exact times for finite interval-censored times. The coarseDataTools package uses the survreg function in the survival package¹⁹ or uses the general optimization function optim and reduces intervals to their midpoints. For Bayesian inference, the icenReg package fits parametric models only. Further evaluation of the performance of these packages under partly interval-censored data may also be helpful.

In this paper, we introduce an efficient and easy-to-implement Bayesian approach specifically developed for analyzing partly interval-censored data under the semiparametric PH model. The main differences between the proposed method and the two Bayesian methods we compare with are: (1) Zhou and Hanson used the transformed Bernstein polynomial prior or mixtures of Polya trees prior to model the baseline survival function; while we use a mixture of basis I-splines to model the baseline cumulative hazard function. (2) Zhou and Hanson used an adaptive Metropolis sampler²⁰ to sample regression coefficients; while we use the Metropolis-Hastings algorithm²¹ to sample regression coefficients. (3) Komárek and Lesaffre fit an AFT model with the error term specified as a normal mixture with an unknown number of components; while we fit the PH model. Zhou and Hanson has pointed out that models using the mixtures of Polya trees prior can suffer from poor mixing and the transformed Bernstein polynomial prior is preferred. According to the discussions by Diaconis and Ylvisaker²² and Perron and Mengersen,²³ the approximation based on Bernstein polynomials can be poor for some nonlinear functions. On the other hand, the adaptive Metropolis algorithm samples a vector of parameters with proposal variance $\frac{{2.4}^2}{d}{C}_t+{10}^{-10}{I}_d$, where C_t is the sample variance of all previous draws, d is the dimension of the vector sampled, and I_d is the identity matrix. As a multidimensional sampler, it poses more difficulty in achieving convergence to target distribution and good mixing. Simulation II in Section 3 demonstrates one scenario where the proposed method outperforms Zhou and Hanson’s method.

The remainder of the paper is outlined as follows. Section 2 describes the proposed method including spline approximation, data augmentation, prior specification, and posterior computation. Section 3 presents two simulation studies that evaluate the performance of the method and compares it with Zhou and Hanson and Komárek and Lesaffre for partly interval-censored data, and the classic Cox PH model.²⁴ In Section 4, we derive partly interval-censored progression-free survival data based on the overall tumor responses from a phase III metastatic colorectal cancer trial and compared the analysis result by the proposed method with those from the other methods. Finally Section 5 provides conclusions and discussions.

2. Statistical method

2.1. Data structure and notation

Partly interval-censored data consist of exact event times and general interval-censored event times. Note that general interval-censored data include left-censored, interval-censored, and right-censored observations. The corresponding observed time intervals are (0, R_i], (L_i, R_i], and (L_i, ∞]. The proposed method can accommodate any of exact, left-censored, right-censored, interval-censored times, and a mixture of them. Let n₁ be the number of observations that are observed exactly and n₂ the number of general interval-censored observations. We have a total of N = n₁ + n₂ observations. Without loss of generality, for the first n₁ subjects, the failure times T_i, i = 1, …, n₁ are exactly known, but for the other n₂ subjects, the failure times are only known to be within a time interval, denoted as (L_i, R_i], i = n₁+1, …, N, where L_i can be 0 and R_i can be ∞. So the observed data are ${\left\{\left(T_i,X_i\right)\right\}}_{i=1}^{n_1}$ and ${\left\{\left(L_i,R_i,X_i\right)\right\}}_{i=n_1+1}^N$, where X_i is the ith subject’s covariate vector.

We assume that failure time T and examination times are independent given the covariate vector X.

2.2. Model

Let λ₀(t_i) denote the unspecified baseline hazard function, β the p × 1 vector of regression coefficients, x_i the p × 1 covariate vector. Under the Cox proportional hazards model, the hazard λ(t_i|x_i) of a failure time T is proportional to the baseline hazard:

$$\lambda \left(t_i\mid x_i\right)={\lambda }_0\left(t_i\right)\text{exp}\left({\beta }^{\prime }{x}_i\right).$$ (1)

For an exact observation, T_i is observed, and its likelihood function is

$$L_{1i}\left\{\beta ,{\lambda }_0(\cdot )\right\}=f\left(t_i\mid x_i\right)={\lambda }_0\left(t_i\right)\text{exp}\left({\beta }^{\prime }{x}_i\right)\text{exp}\left\{-{\Lambda }_0\left(t_i\right)\text{exp}\left({\beta }^{\prime }{x}_i\right)\right\},$$

where ${\Lambda }_0(t)={\int }_0^t{\lambda }_0(s)ds$ is the cumulative baseline hazard function.

For a general interval-censored observation, (L_i, R_i] is the observed time interval, and its likelihood function is

$$L_{2i}\left\{\beta ,{\lambda }_0(\cdot )\right\}={\left\{F\left(R_i\mid x_i\right)\right\}}^{{\delta }_{1i}}{\left\{F\left(R_i\mid x_i\right)-F\left(L_i\mid x_i\right)\right\}}^{{\delta }_{2i}}{\left\{1-F\left(L_i\mid x_i\right)\right\}}^{{\delta }_{3i}},$$ (2)

where $F(t\mid x)=1-\text{exp}(-{\int }_0^t\lambda (s\mid x)ds)$ is the cumulative distribution function given x and δ₁, δ₂, δ₃ are the left-, interval-, and right-censoring indicators. So the overall likelihood function is:

$$L\left\{\beta ,{\lambda }_0(\cdot )\right\}=\prod _{i=1}^{n_1}{L}_{1i}\left\{\beta ,{\lambda }_0(\cdot )\right\}\prod _{i=n_1+1}^N{L}_{2i}\left\{\beta ,{\lambda }_0(\cdot )\right\}.$$ (3)

2.3. Estimation of Λ₀(t) and λ₀(t)

Following Cai et al., Pan et al., Lin et al., and Pan et al.,^25–28 we model the cumulative baseline hazard function Λ₀(t) with a linear combination of a set of basis I-splines:³³

$${\Lambda }_0(t)=\sum _{l=1}^K{\gamma }_l{I}_l(t)$$ (4)

where {γ_l} is a set of non-negative coefficients and {I_l(t)} is a set of basis I-splines.

To construct the set of basis I-splines, we need to specify the degree (1=linear, 2=quadratic, 3=cubic, etc.) of each basis I-spline and an increasing sequence of knots within the data range. The set of basis I-splines are fully determined once the degree and the knots are specified. The number of basis I-splines (K) equals the degree plus the number of interior knots. In general, we recommend taking 2 or 3 as the degree value for adequate smoothness and 10–30 equally spaced knots for adequate modeling flexibility.

Note that knots and degree can be adjusted based on data. Together with the coefficients for the basis I-splines, a monotone spline created this way can provide great flexibility for approximating a curve. Furthermore, the shrinkage prior for the spline coefficients γ_l as shown in Section 2.5 serves to: (1) keep those important basis functions and leave those unnecessary ones out; (2) avoid over-fitting problems that may be caused by using too many knots for flexibility.

For the baseline hazard function λ₀(t), we model it with a linear combination of a set of basis M-splines:³³

$${\lambda }_0(t)=\sum _{l=1}^K{\gamma }_l{M}_l(t),$$

where {γ_l} is the same set of non-negative coefficients as in (4) and {M_l(t)} is a set of basis M-splines. I-splines are the integrated M-splines such that $\left\{I_l(t)={\int }_0^t{M}_l(s)ds\right\}$. In our model, they share the same knots and an I-spline of degree k corresponds to an M-spline of degree k − 1.

2.4. Data augmentation

Although one may use the Metropolis-Hastings algorithm to sample all the parameters from their posteriors based on the original data likelihood (3), it is difficult to find good proposal distributions to obtain reasonable acceptance rates and well mixed Markov chain Monte Carlo (MCMC) chains. To facilitate posterior computation, we construct the following data augmentations.

For general interval-censored data, a two-step data augmentation is constructed by taking advantage of the PH model structure and the spline modeling form of Λ₀(t) in (4). Assume that there is an underlying recurrent event E, for which the number of occurrences N(t) within time interval (0, t] is a nonhomogeneous Poisson process with cumulative intensity function Λ₀(t) exp(β′x). Define T = inf{t : N(t) > 0}, time of first occurrence in the Poisson process. Then we have P(T > t) = P(N(t) = 0) = exp{−Λ₀(t) exp(β′x)}, which is our survival function of interest. So T indeed follows the PH model in (1).

Now define two time points t₁ and t₂ such that 0 < t₁ < t₂. For left-censored observations (0, R], we set t₁ = R and t₂ undefined as long as greater than t₁. For interval-censored observations (L, R], we set t₁ = L and t₂ = R. For right-censored observations (L, ∞), we set t₂ = L and t₁ undefined as long as less than t₂. It is clear that N(t₁) denotes the number of occurrences of E until time t₁, and N(t₂) − N(t₁) denotes the number of occurrences of E during the interval (t₁, t₂]. By the properties of nonhomogeneous Poisson process, the random variable Z = N(t₁) ~ Poi(exp(Λ₀(t₁) exp(β′x))), the random variable W = N(t₂) − N(t₁) ~ Poi(exp({Λ₀(t₂) − Λ₀(t₁)} exp(β′x))), and they are independent. For left-censored data, since t₂ is some point greater than t₁ = R, W can take any value and will not contribute any information about the failure time T. For interval-censored data, Z = 0 and W > 0. For right-censored data, t₁ is some point less than t₂ = L, so Z = W = 0. The augmented data likelihood function for subject i is

$$L_{2\mathit{\text{aug}}1,i}\left(\theta \mid Z_i,W_i\right)=\text{Poi}\left(Z_i\right)\text{Poi}{\left(W_i\right)}^{{\delta }_{2i}+{\delta }_{3i}}\times {\left\{1\left(Z_i>0\right)\right\}}^{{\delta }_{1i}}{\left\{1\left(Z_i=0\right)1\left(W_i>0\right)\right\}}^{{\delta }_{2i}}{\left\{1\left(Z_i=0\right)1\left(W_i=0\right)\right\}}^{{\delta }_{3i}},$$

where θ = (β, λ₀(·)) denotes the set of parameters, 1(·) the indicator function, and 0⁰ = 1. Integrating out Z_i and W_i will lead to the original likelihood function in (2).

Furthermore, based on the additive property of Poisson distribution and the linear combination form of (4), decompose Z and W respectively into K independent Poisson latent variables {Z_l} and {W_l}, such that $Z={\sum }_{l=1}^K{Z}_l$ with Z_l ~ Poi(γ_lI_l(t₁) exp(β′x)) and $W={\sum }_{l=1}^K{W}_l$ with W_l ~ Poi({γ_lI_l(t₂) − γ_lI_l(t₁)} exp(β′x)), with constraints ${\sum }_{l=1}^K{Z}_l>0$ if δ₁ = 1, ${\sum }_{l=1}^K{Z}_l=0$ and ${\sum }_{l=1}^K{W}_l>0$ if δ₂ = 1, and ${\sum }_{l=1}^K{Z}_l={\sum }_{l=1}^K{W}_l=0$ if δ₃ = 1. Then for subject i, the further augmented data likelihood function is

$$L_{2aug2,i}\left(\mathit{\text{θ}}\mid Z_{il}^{\prime }s,W_{il}^{\prime }s\right)=\left\{\prod _{l=1}^K\text{Poi}\left(Z_{il}\right)\text{Poi}{\left(W_{il}\right)}^{{\delta }_{2i}+{\delta }_{3i}}\right\}\times {\left\{1\left(Z_i>0\right)\right\}}^{{\delta }_{1i}}{\left\{1\left(Z_i=0\right)1\left(W_i>0\right)\right\}}^{{\delta }_{2i}}{\left\{1\left(Z_i=0\right)1\left(W_i=0\right)\right\}}^{{\delta }_{3i}}.$$

The likelihood function is simply a product of Poisson probability mass functions, which leads to relatively straightforward posterior computation to be presented in Section 2.5.

For exact times, it will be challenging to sample the basis M-spline coefficients γ_l directly given the summation form in the likelihood function:

$$L_1(\theta )=\prod _{i=1}^{n_1}\left[\{\sum _{l=1}^K{\gamma }_l{M}_l\left(t_i\right)\}\text{exp}\left({\beta }^{\prime }{x}_i\right)\text{exp}\left\{-{\Lambda }_0\left(t_i\right)\text{exp}\left({\beta }^{\prime }{x}_i\right)\right\}\right].$$

We introduce latent variables $u_i=\left(u_{i1},u_{i2},\dots ,u_{iK}\right)~\text{Multinomial}\left(1;\frac{1}{K},\frac{1}{K},\dots ,\frac{1}{K}\right)$, then we can derive the augmented data likelihood function for the part of exact observations as:

$$L_{1aug}\left(\theta \mid u_i^{\prime }s\right)=\prod _{i=1}^{n_1}\left[\{K\prod _{l=1}^K{\left({\gamma }_l{M}_l\left(t_i\right)\right)}^{u_{il}}\}\text{exp}\left({\beta }^{\prime }{x}_i\right)\text{exp}\left\{-{\Lambda }_0\left(t_i\right)\text{exp}\left({\beta }^{\prime }{x}_i\right)\right\}\right].$$

Integrating out u_i’s will lead to the original likelihood function L₁(θ). Under this format, we can obtain a Gamma posterior distribution for each γ_l, l = 1, …, K.

2.5. Prior specification and posterior computation

For spline coefficients, we assign an Exponential prior Exp(η) for γ_l and a Gamma hyperpiror Ga(a_η, b_η) for η. This specification leads to conjugate posteriors for both γ_l and η. For a numeric (continuous or count) covariate, we assign a Normal prior $N\left(0,{\sigma }_0^2\right)$ for β_r. Since the corresponding posterior is not conjugate, the Metropolis-Hastings algorithm is used for sampling from the posterior. For a categorical covariate with c levels, we represent it using c − 1 dummy variables. The Metropolis-Hastings algorithm as a general sampler can be used for sampling for its β_r too. However, here we treat a categorical covariate differently. The reason is that by specifying a Gamma prior Ga(a_ϕ, b_ϕ) for ϕ_r = exp(β_r), the resulting posterior happens to be Gamma which can be directly sampled from and renders better MCMC chains. Then we transform ϕ_r back to β_r.

After initializing values for the parameters, the proposed MCMC algorithm proceeds in the following steps.

1. Let Z_i = 0 and W_i = 0 for all i, Z_il = 0 and W_il = 0 for all i and l. If δ_1i = 1, then sample

   $$\begin{gathered} Z_i~\text{Poi}\left({\Lambda }_0\left(R_i\right)\text{exp}\left({\beta }^{\prime }{x}_i\right)\right)1\left(Z_i>0\right), \\ \left(Z_{i1},\dots ,Z_{iK}\right)~\text{Multinomial}\left(Z_i;p_{i1},\dots ,p_{iK}\right), \\ \text{and }\left(p_{i1},\dots ,p_{iK}\right)\propto \left({\gamma }_1{I}_1\left(R_i\right),\dots ,{\gamma }_K{I}_K\left(R_i\right)\right). \end{gathered}$$
   If δ_2i = 1, then sample

   $$\begin{gathered} W_i~\text{Poi}\left(\left\{{\Lambda }_0\left(R_i\right)-{\Lambda }_0\left(L_i\right)\right\}\text{exp}\left({\beta }^{\prime }{x}_i\right)\right)1\left(W_i>0\right), \\ \left(W_{i1},\dots ,W_{iK}\right)~\text{Multinomial}\left(W_i;q_{i1},\dots ,q_{iK}\right), \\ \text{ and }\left(q_{i1},\dots ,q_{iK}\right)\propto \left({\gamma }_1\left\{I_1\left(R_i\right)-I_1\left(L_i\right)\right\},\dots ,{\gamma }_K\left\{I_K\left(R_i\right)-I_K\left(L_i\right)\right\}\right). \end{gathered}$$

2. Sample (u_i1, …, u_iK) ~ Multinomial(1; o_i1, …, o_iK) and (o_i1, …, o_iK) ∝ (γ₁M₁(t_i), …, γ_KM_K(t_i)).

3. For β_r corresponding to a numeric covariate, use the Metropolis-Hastings algorithm to sample from its full conditional distribution

   $$p\left({\beta }_r\mid Z_i^{\prime }s,W_i^{\prime }s,{\mathit{\beta }}_{-r}\right)\propto \text{exp}[\sum _{i=1}^{n_1}\left\{x_{ir}{\beta }_r-{\Lambda }_0\left(t_i\right)e^{\beta {\prime }_{x_i}}\right\}]\times \text{exp}[\sum _{i=n_1+1}^N\{x_{ir}{\beta }_r\left(Z_i{\delta }_{1i}+W_i{\delta }_{2i}\right)-e^{{\beta }_{x_i}^{\prime }}\left({\Lambda }_0\left(R_i\right)\left({\delta }_{1i}+{\delta }_{2i}\right)+{\Lambda }_0\left(L_i\right){\delta }_{3i}\right)\}]p\left({\beta }_r\right),$$

   where $p\left({\beta }_r\right)=N\left(0,{\sigma }_0^2\right)$ is the prior used for β_r, and β_−r denotes all the β’s except for β_r.

4. For β_r corresponding to a categorical covariate, let ϕ_r = exp(β_r), sample ϕ_r from

   $$\text{Ga}(a_{\varphi }+\sum _{i=1}^{n_1}{x}_{ir}+\sum _{i=n_1+1}^N{x}_{ir}\left(Z_i{\delta }_{1i}+W_i{\delta }_{2i}\right),b_{\varphi }+\sum _{i=1}^{n_1}{\Lambda }_0\left(t_i\right)e^{{\beta }_{-r}^{\prime }{x}_{i,-r}}{x}_{ir}+\sum _{i=n_1+1}^N{e}^{{\beta }_{-r}^{\prime }{x}_{i,-r}}\left\{{\Lambda }_0\left(R_i\right)\left({\delta }_{1i}+{\delta }_{2i}\right)+{\Lambda }_0\left(L_i\right){\delta }_{3i}\right\}{x}_{ir}),$$

   where x_{i, −r} is the covariate vector except for x_ir for subject i.

5. Sample γ_l, l = 1, …, K, from

   $$\text{Ga}(1+\sum _{i=1}^{n_1}{u}_{il}+\sum _{i=n_1+1}^N\left(Z_{il}{\delta }_{1i}+W_{il}{\delta }_{2i}\right),\eta +\sum _{i=m_1+1}^N{e}^{{\beta }^{\prime }{x}_i}\left\{I_l\left(R_i\right)\left({\delta }_{1i}+{\delta }_{2i}\right)+I_l\left(L_i\right){\delta }_{3i}\right\}).$$

6. Sample η from $\text{Ga}(a_{\eta }+K,b_{\eta }+{\sum }_{l=1}^K{\gamma }_l)$.

As we can see, latent variables and spline coefficients all can be sampled from standard distributions. Special sampling method (here Metropolis-Hastings) is only required for the regression coefficient of a numeric covariate.

3. Simulations

3.1. Simulation I

We evaluate the performance of the proposed method through a simulation study. A total of 100 data sets were generated. For each data set, the failure times were generated from the following PH model:

$$S\left(t\mid x_1,x_2\right)=\text{exp}\left\{-{\Lambda }_0(t)\text{exp}\left({\beta }_1{x}_1+{\beta }_2{x}_2\right)\right\},$$

where Λ₀(t) = log(1 + t), β₁ = 1, β₂ = 1, x₁’s ~ N(0, 0.5²), and x₂’s ~ Bernoulli(0.5). Note that x₁ and x₂ were independently sampled and a new set of covariates were generated for each data set. We assume that the random number of medical examinations performed for each person is 1 plus a Poisson random number with mean 2. The gap times between adjacent medical examinations follow an Exponential distribution with mean 1. The observed interval is formed by the consecutive examination times (including 0 and ∞) that contain the true failure time. In each data set, there are N = 460 subjects, around 20% of which are set to have exact event times observed.

To construct the basis I-splines and basis M-splines, we set the degree as 2 for the basis I-splines and chose 15 equally spaced knots within the range of observed times. For hyper-parameters, we tried ${\sigma }_0^2=10$, 10, 100, 1000, a_η = b_η = 0.01, 0.1, 1, and a_ϕ = b_ϕ = 0.01, 0.1, 1. The results were very similar and we chose to use ${\sigma }_0^2=100$, a_η = b_η = 1, and a_ϕ = b_ϕ = 1. Fast convergence and good mixing were observed for all key parameters. For each MCMC chain, we set total number of iterations = 11,000, burn-in = 1000, and thin = 1.

We fit the proposed method, and compare it survregbayes and survregbayes2 in the spBayesSurv package, and bayessurvreg1 in the bayesSurv package. We also treat finite interval-censored data as exact data by taking the right endpoints as the event times, as has conventionally done by practitioners, and then fit the Cox PH model using the coxph function in the survival package. The purpose is to demonstrate the potential bias this conventional approach might introduce.

Table 1 summarizes the simulation results. For each parameter, the point estimate is the average of the 100 posterior means, the sample standard deviation (SSD) is the sample standard deviation of the 100 posterior means, the empirical standard error (ESE) is the average of the 100 estimated standard errors, and the 95% coverage probability (95CP) is the percentage of the 100 credible intervals for each β_r that contains the true parameter value. Effective sample size (ESS) and absolute value of Geweke’s Z-score were computed based on the MCMC chains using the coda package.³⁷ Negative log-likelihood (NLLK) is the negative of log psuedo marginal likelihood from survregbayes and survregbayes2 and the negative of log-likelihood from coxph. Log-likelihood at each iteration from bayessurvreg1 was averaged to calculate negative log-likelihood. Deviance information criterion (DIC) was not calculated for bayessurvreg1 because the error variance at each iteration was not available.

Table 1:

Simulation I - Estimation of regression coefficient, effective sample size, absolute value of Geweke’s Z-score, deviance information criterion, and nagetive log-likelihood based on the proposed method, survregbayes, survregbayes2, bayessurvreg1, and coxph.

R function	True	Estimate	SSD	ESE	95CP	ESS	|Geweke’s Z|	DIC	NLLK
Proposed method	1	1.012	0.125	0.130	0.94	557	0.7775	560	297
	1	1.004	0.117	0.121	0.95	834	0.7790
survregbayes	1	0.993	0.120	0.130	0.97	1099	0.9755	688	344
	1	1.017	0.118	0.124	0.96	1081	0.9323
survregbayes2	1	0.982	0.118	0.129	0.96	1120	0.8147	689	345
	1	1.006	0.118	0.122	0.95	1109	0.8523
bayessurvregl	1	−1.379	0.169	0.184	-	1787	1.1591	-	769
	1	−1.443	0.188	0.178	-	1019	1.0351
coxph	1	0.634	0.116	0.108	0.11	-	-	-	1801
	1	0.702	0.126	0.110	0.23	-	-

As seen in Table 1, the proposed method, survregbayes, and survregbayes2 all perform very well. The survregbayes and survregbayes2 functions have relatively high effective sample size, however, the proposed method shows lower absolute Geweke’s Z-score, deviance information criterion, and negative log-likelihood which indicate better MCMC convergence to the stationary distribution and better model goodness-of-fit. Note that bayessurvreg1 fits a Bayesian AFT model (log(T_i) = β′x_i + ϵ_i), so it makes sense that the estimated regression coefficients are of negative signs and the coverage probabilities are not presented. The estimation from coxph shows large bias, low coverage probability, and large negative log-likelihood, which illustrates the bias it can induce if we treat partly interval-censored data as right-censored data.

We also estimated baseline survival function S₀(t) based on the 100 simulated data sets. The estimated baseline survival functions and the true baseline survival function are plotted in Figure 1. All four partly interval-censored methods provide good approximations to the true baseline survival. However, the estimated curve based on coxph deviates significantly from the true curve.

Figure 1:

Simulation I - Plot of estimated S₀(t) based on 100 simulated data sets using the proposed method, survregbayes, survregbayes2, bayessurvreg1, and coxph compared to true S₀(t) curve.

3.2. Simulation II

To explore more scenarios, we performed another simulation study where the true cumulative baseline hazard function is set to be Λ₀(t) = t². Compared to Simulation I where Λ₀(t) = log(1 + t), the risk of failure is much higher under the new function. Other settings are exactly the same as in Simulation I.

Table 2 summarizes the simulation results. As we can see, the proposed method provides the best estimation with small biases and coverage probabilities close to the nominal level. The effective sample size from the proposed method is low compared to the other three partly interval-censored methods, indicating relatively high autocorrelation among the MCMC samples. The point estimates and coverage probabilities from survregbayes and survregbayes2 are not very good, even though they have high effective sample sizes. The estimation from coxph deviates even further from the true values compared to Simulation I, which indicates that the conventional method might lead to more bias for analyzing partly interval-censored data from diseases that have fast failure rate.

Table 2:

Simulation II - Estimation of regression coefficient, effective sample size, absolute value of Geweke’s Z-score, deviance information criterion, and negative log-likelihood based on the proposed method, survregbayes, survregbayes2, bayessurvreg1, and coxph.

R function	True	Estimate	SSD	ESE	95CP	ESS	|Geweke’s Z|	DIC	NLLK
Proposed method	1	0.982	0.142	0.145	0.96	135	1.1637	422	210
	1	0.989	0.144	0.145	0.94	188	0.9371
survregbayes	1	0.873	0.122	0.138	0.89	1028	1.1991	431	216
	1	0.884	0.122	0.136	0.87	1022	1.1645
survregbayes2	1	0.804	0.125	0.138	0.69	997	1.3432	441	223
	1	0.813	0.128	0.135	0.72	952	1.5399
bayessurvregl	1	−0.499	0.061	0.072	-	1536	0.9370	-	275
	1	−0.504	0.075	0.072	-	784	1.1358
coxph	1	0.307	0.096	0.100	0	-	-	-	1998
	1	0.324	0.102	0.101	0	-	-

We also plotted the estimated baseline survival function versus the true in Figure 2. The proposed method provides the best approximation to the true baseline survival, followed by bayessurvreg1, survregbayes, and survregbayes2. The estimated curve from coxph is still noticeably different from the true curve.

Figure 2:

Simulation II - Plot of estimated S₀(t) based on 100 simulated data sets using the proposed method, survregbayes, survregbayes2, bayessurvreg1, and coxph compared to true S₀(t) curve.

4. An application to progression-free survival data

We apply the proposed method to a randomized phase III study that compares the efficacy of FOLFIRI versus panitumumab + FOLFIRI in patients with previously treated metastatic colorectal cancer. FOLFIRI is a combination of chemotherapy drugs: fluorouracil, leucovorin, and irinotecan. Panitumumab is a fully human monoclonal antibody specific to the epidermal growth factor receptor. The primary endpoint is progression-free survival. Two binary covariates are of interest: treatment arm (FOLFIRI vs. panitumumab + FOLFIRI) and patient tumor KRAS mutation status (wild-type vs. mutant). KRAS stands for the gene Kirsten rat sarcoma viral oncogene homolog. It is one of a group of genes involved in the epidermal growth factor receptor pathway.

Both treatments were administered every 2 weeks. The visit schedule for tumor response evaluation was every 8 weeks until documentation of disease progression. Based on the overall tumor responses (e.g., complete response, partial response, stable disease, and progressive disease) at each visit across their on-study period, we derived the progression-free survival for each patient. We set baseline assessment as Day 0. If a patient had disease progression at the first post-baseline assessment, then he is left-censored. If a patient had disease progression at a later assessment, then he is interval-censored. If a patient was alive without disease progression at the last on-study assessment, then he is right-censored. If a patient died while on-study, then his progression-free survival is exact. After excluding 30 test failures and 61 missing values for KRAS mutation status, the final data set contains N = 855 patients, among which 52 died on-study, 168 left-censored, 329 interval-censored, and 306 right-censored. The FOLFIRI arm contains 427 randomly assigned patients with 234 wild-type and 193 mutant, while the panitumumab + FOLFIRI arm contains 428 randomly assigned patients with 240 wild-type and 188 mutant.

The estimation results using the proposed method, survregbayes, survregbayes2, bayessurvreg1, and coxph are presented in Table 3. The proposed method, survregbayes, bayessurvreg1, and coxph all detect improvement of progression-free survival by adding panitumumab. The survregbayes2 function fails to detect a significant treatment effect. The four partly interval-censored methods have much lower negative log-likelihood than coxph, which indicates they fit the data better by taking into account that disease progression occurs between two visits instead of on a visit day.

Table 3:

Metastatic colorectal cancer trial (N = 855) - Estimation of regression coefficient, effective sample size, and negative log-likelihood based on the proposed method, survregbayes, survregbayes2, bayessurvreg1, and coxph.

R function		Estimate	SE	95% CI	ESS	NLLK
Proposed method	Treatment	−0.229	0.085	(−0.395, −0.062)	2657	1441
	KRAS	0.131	0.086	(−0.037, 0.298)	3165
survregbayes	Treatment	−0.186	0.085	(−0.355, −0.019)	1294	1557
	KRAS	0.149	0.085	(−0.018, 0.317)	1264
survregbayes2	Treatment	−0.171	0.088	(−0.341, 0.003)	1158	1564
	KRAS	0.140	0.089	(−0.035, 0.310)	1178
bayessurvregl	Treatment	0.239	0.094	(0.059, 0.428)	1047	1280
	KRAS	−0.169	0.101	(−0.365, 0.034)	640
coxph	Treatment	−0.215	0.086	(−0.384, −0.046)	-	3230
	KRAS	0.163	0.086	(−0.006, 0.332)	-

Figure 3 presents the estimated survival curves for the four groups formed by treatment arm and KRAS mutation status based on the five methods compared in Table 2 and the classic Kaplan-Meier method.³⁹ As indicated by the estimated regression coefficients, the survival expectation is the highest for wild-type patients receiving panitumumab + FOLFIRI, followed by mutant patients receiving panitumumab + FOLFIRI, wild-type patients receiving FOLFIRI, and finally mutant patients receiving FOLFIRI.

Figure 3:

Metastatic colorectal cancer trial (N = 855) - Estimated survival curves using the proposed method, survregbayes, survregbayes2, bayessurvreg1, coxph, and Kaplan-Meier method. Four curves are plotted for each method based on the four groups formed by treatment arm and mutation status.

Since panitumumab is an antibody targeted at the epidermal growth factor receptor and KRAS mutation status predicts the efficacy of such type of agents in metastatic colorectal cancer,^40,41 we also compared the efficacy of panitumumab + FOLFIRI vs. FOLFIRI among patients with wild-type KRAS tumors as well as among patients with mutant KRAS tumors. Peeters et al.^40,41 treated progression-free survival as right-censored and used the classic log-rank test and Cox PH model, but stratified by performance status, prior bevacizumab, and prior oxaliplatin exposure.

The results using the proposed method, survregbayes, survregbayes2, bayessurvreg1, and coxph are summarized in Table 4. For wild-type patients, when patitumumab was added to FOLFIRI, a significant improvement in progression-free survival was observed based on all of the five methods. The results are consistent with that from Peeters et al.^40,41 For mutant patients, only the proposed method detects a weak improvement in efficacy. The 95% CI from Peeters et al.⁴⁰ is (−0.386, 0.058), which indicates a non-significant trend toward increased progression-free survival. As in the first set of analysis, the proposed method has much higher effective sample size, indicating better mixing and more efficiency in generating effective samples.

Table 4:

Metastatic colorectal cancer trial (N = 855) - Estimation of regression coefficient, effective sample size, and negative log-likelihood among patients with wild-type KRAS tumors and patients with mutant KRAS tumors, based on the proposed method, survregbayes, survregbayes2, bayessurvreg1, and coxph.

R function		Estimate	SE	95% CI	ESS	NLLK
Proposed method	wild-type	−0.473	0.108	(−0.685, −0.264)	3220	903
	mutant	−0.260	0.116	(−0.489, −0.032)	3250	663
survregbayes	wild-type	−0.306	0.119	(−0.537, −0.077)	1532	896
	mutant	−0.031	0.132	(−0.285, 0.227)	1591	663
survregbayes2	wild-type	−0.291	0.120	(−0.525, −0.054)	1641	897
	mutant	−0.029	0.131	(−0.289, 0.228)	1675	659
bayessurvregl	wild-type	0.377	0.130	(0.129, 0.636)	1127	744
	mutant	0.059	0.135	(−0.200, 0.325)	675	496
coxph	wild-type	−0.336	0.117	(−0.565, −0.107)	-	1590
	mutant	−0.057	0.128	(−0.306, 0.193)	-	1265

5. Conclusion

In the past few decades, many statistical methods and R packages have been developed for interval-censored data. There have been limited research specifically developed for partly interval-censored data which also occur often in medical studies. The several methods developed from the frequentist perspective seem to be hard to implement by practitioners, or at least with no ready-to-use code available. The main Bayesian methods are the two R packages we have compared the proposed method to in this article: one fits PH, proportional odds, and AFT models to partly interval-censored data and left-truncated data and the other fits mixed effects AFT model to partly interval-censored data. We developed an efficient and easy-to-implement Bayesian semiparametric method under the PH model directly targeted at analyzing partly interval-censored data. The proposed method performs comparably well in terms of regression coefficient estimation and survival function estimation. It even outperforms the two R packages when the rate of failure is high as seen in Simulation II. Our developed method is a meaningful addition to the literature and we hope to provide pharmaceutical companies with another ready-to-use tool for analyzing partly interval-censored data that are commonly encountered in cancer clinical trials, e.g. progression-free survival and disease-free survival.

Our simulation and real data analysis show that, when there is only one covariate, the effective sample size of the proposed method is pretty high. However, it may have less ideal mixing when there are more than one covariate. This is largely due to the component-wise updating of regression coefficients in our algorithm.³⁰ A possible solution for this is to sample β simultaneously through the consideration of correlated proposals such as the Metropolis-Hastings algorithm based on the iterative weighted least squares.⁴² This could be an area to be explored in future research.