Semiparametric isotonic regression analysis for risk assessment under nested case-control and case-cohort designs

Abstract

Two-phase sampling designs, including nested case-control and case-cohort designs, are frequently utilized in large cohort studies involving expensive biomarkers. To analyze data from two-phase designs with a binary outcome, parametric models such as logistic regression are often adopted. However, when the model assumptions are not valid, parametric models may lead to biased estimation and risk evaluation. In this paper, we propose a robust semiparametric regression model for binary outcomes and an easy-to-implement computational procedure that combines the pool-adjacent violators algorithm with inverse probability weighting. The asymptotic properties are established, including consistency and the convergence rate. Simulation studies show that the proposed method performs well and is more robust than logistic regression methods. We demonstrate the application of the proposed method to real data from the Prostate, Lung, Colorectal, and Ovarian (PLCO) Cancer Screening Trial.

1. Introduction

Improved risk assessment is an essential step for the early-phase detection and prognosis of many diseases, which help detect diseases in a clinically manageable stage. Motivated by a pressing need to better identify high-risk populations earlier, we focus on developing optimal scoring systems by combining subject-level information, such as the expression levels of multiple biomarkers. There is substantial literature on the advantages, in terms of improved discrimination, of using combinations of biomarkers rather than a single biomarker.¹ Although combining the subjects’ characteristics with biomarker information to develop new risk scoring systems has been an active research area, most such studies have used parametric regression models with strong model assumptions. For binary outcomes, logistic regression is typically adopted to generate risk scores.² However, logistic regression poses strong model assumptions. When these model assumptions are not valid, the derived risk scores may have undesirable properties, such as biased estimation of the regression coefficients and poor discrimination between cases and controls.³

To improve the precision of risk assessment, biomarkers have been successfully included in the risk scores for many diseases. However, measuring such biomarkers can be expensive and labor intensive; hence, the collection of biomarker information for all participants in the cohort may be infeasible. As a result, two-phase sampling designs are often adopted as a cost-effective alternative.^4–6 These designs have the further advantage of addressing the inherent problem of low incidence encountered with rare diseases, providing early detection in a large cohort of disease-free subjects. In the first phase, a large cohort is sampled from the target population. The outcome variable is prospectively collected and some easy-to-obtain covariates, such as routine clinical risk factors and demographic characteristics, are recorded. In the second phase, a subcohort of all the cases and a fraction of the controls in the full cohort are selected for biomarker measurements. There are two commonly used two-phase sampling designs: the nested case-control (NCC) and the case-cohort. They differ in the approach to select controls. In the NCC design, controls are chosen without replacement from the risk set at each event time.^4,5 In the case-cohort design, controls are randomly selected at baseline.⁶ However, these cost-effective sampling strategies create challenges for statistical analysis, because of outcome-dependent data missingness for biomarker measurements.

The analysis of two-phase design data with binary outcomes has routinely been carried out using parametric models. Two popular methods are the conditional logistic regression model for the NCC design^2,7 and the logistic regression model with inverse probability weighting (IPW), hereafter termed IPW-based logistic regression, for the case-cohort design.^8,9 In application, researchers often have limited information regarding the mathematical specification of the true regression function, although a logit link between the disease probability and risk score is a convenient choice. However, the underlying relationship may differ from the logit link in many situations, leading to biased estimation of the regression coefficients and/or disease probabilities. It is more desirable to assume a semiparametric model with minimal assumptions on the link function. Isotonic regression is a least squares problem in which only monotonicity is assumed on the shape of the regression models. Pioneering work was done by Ayer et al.¹⁰ and comprehensive reviews were provided by Barlow et al.¹¹ and Robertson et al.¹² A unique solution to standard isotonic regression exists and can be obtained using the pool-adjacent violators algorithm (PAVA).^11,13,14 The computational aspects and fast implementation of PAVA in R are discussed by Mair et al.¹⁵ However, standard isotonic regression with PAVA cannot be directly applied to the data from two-phase studies due to the outcome-dependent data missingness. The goal of this paper is to handle such data under a semiparametric isotonic regression model and to develop a computationally appealing algorithm by integrating PAVA, the IPW method, and the profiling method. To the best of our knowledge, this combination, which is particularly useful to construct risk scores and assess absolute risks simultaneously under two-phase studies, has not been applied before.

1.1. A motivating study

Lung cancer is the most common cause of cancer-related mortality in the USA, with approximately 220,000 new cases and nearly 160,000 deaths each year.¹⁶ While the five-year survival rate is 50% for lung cancer patients diagnosed at early stages, it is less than 16% for those with advanced lung cancer.¹⁶ Thus, early detection of lung cancer is the key to improve survival. The Prostate, Lung, Colorectal, and Ovarian (PLCO) Cancer Screening Trial was a randomized trial that recruited approximately 155,000 participants aged 55 to 74years from 1992 to 2001 to evaluate the effects of screening on reducing cancer mortality.¹⁷ Epidemiological information, including demographic factors, smoking behavior, anthropometric measurements, medical history, and family history of cancer, were collected through baseline questionnaires. Blood samples were drawn from participants in the screening arm at baseline and annually for the subsequent five years. Using the PLCO data, some risk models have been developed for predicting lung cancer risk. For example, the PLCO_M2012 model used a diagnosis of lung cancer within six years as the binary outcome and derived the risk score by a logistic regression model.¹⁸ The PLCO_M2012 model was reported to be more sensitive in identifying high-risk subjects than the simple selection criteria accounting for only smoking information.¹⁸

The stored biospecimens in the screening arm of the PLCO trial provided invaluable resources to improve the existing risk models. Several biomarkers assayed from the blood samples have been investigated to refine risk stratification and gain better insight into lung cancer etiology. The NCC design was adopted as a cost-effective, feasible solution to collect the biomarker information.^17,19 Controls who were free of lung cancer at the time of the case’s diagnosis were individually matched to cases using baseline information: age group, sex, study year of blood draw, calendar year of blood draw, smoking status, cumulative amount of smoking for current and former smokers, and time since quitting for former smokers.¹⁷ Never smoker controls were sampled with a 3:1 ratio, whereas ever smokers were sampled with a 1:1 ratio. Several circulating inflammation markers identified by previous research, including the C-reactive protein (CRP), serum amyloid A, soluble tumor necrosis factor receptor-2, and monokine induced by gamma interferon (CXCL9/MIG), were collected from the cases and selected controls.^17,19 The associations between these biomarkers and the risk of developing lung cancer have been evaluated using conditional logistic regression.^19,20 Our purpose is to develop a robust risk assessment tool that can simultaneously identify an optimal risk score by combining these biomarkers and baseline risk factors, as well as estimate the absolute risk of lung cancer.

The remaining sections of this paper are as follows. In section 2, we introduce the semiparametric isotonic regression model. We describe the proposed computational algorithm for estimation in section 3. In section 4, we establish the asymptotic properties of the estimators. We conduct extensive simulation studies to assess the finite sample performance of the proposed algorithm and compare it with existing estimating methods in section 5. We apply the proposed method to the PLCO study in section 6 and provide a brief discussion in section 7.

2. Notation and model

2.1. General notations

Suppose that N subjects are followed prospectively in a study. Let Y_i be the binary outcome of interest, indicating whether subject i develops the disease of interest during the study. Cases and controls are respectively defined as subjects for whom Y_i = 1 and Y_i = 0. For notational simplicity, we use X_i to denote the p-dimensional vector of the covariates, including the routine variables and novel biomarkers. Note that under the two-phase sampling design, novel biomarkers are only ascertained at the second phase for the selected subcohort. Let n be the sample size of the subcohort. We consider two popular two-phase sampling designs, the case-cohort and the NCC designs, to introduce the selection probability.

In a case-cohort design, all cases are selected into the subcohort, and controls are randomly chosen at baseline from the full cohort. Accordingly, the probability of sampling the ith subject into the subcohort is

$$p_i=Y_i+\left(1-Y_i\right)\alpha$$

where α is a constant that represents the probability of being selected as a control. Under stratified case-cohort sampling, the full cohort is divided into L strata based on the baseline covariates. A subcohort is subsequently sampled from the full cohort using stratified sampling. Then p_i = Y_i + (1 − Y_i)α_l where α_l is the probability of being selected as a control for the lth stratum, and l denotes the stratum to which the ith subject belongs.

In the NCC design, cases that occur during the study are identified and for each case, a pre-specified number of controls are selected among those who have not developed the disease by the time the disease occurred for the case. Denote the risk set at time t as R(t) = {i : Z_i ≥ t}, where Z_i = min(T_i, C_i), T_i is the event time and C_i is the follow-up time. Let the number of subjects in R(t) be $n(t)={\Sigma }_{i=1}^{N}I\left(Z_i\ge t\right)$. We define n₁ to be the number of cases and t_i, i = 1,…, n₁ to be the failure times of the cases. At each failure time t_i, m controls are randomly selected without replacement from the risk set R(t_i), excluding the case. Hence, the probability of sampling the ith subject into the subcohort is

$$p_i=Y_i+\left(1-Y_i\right)\left\{1-G\left(Z_i\right)\right\}$$

where G(Z_i) denotes the probability that subject i has never been selected as a control up to the end of the study follow-up time Z_i. In stratified NCC sampling, at each case’s failure time, controls are selected randomly without replacement among those who are in the risk set and matched to the case based on some covariates.¹⁷ To accommodate stratified sampling, the G(·) in the sampling probability can be replaced by G_K(Z_i, K_i), where K defines the covariate strata.

2.2. Regression model

Our goal is to identify a scoring system S(X), where a higher score is related to a higher risk of developing the given disease, and to estimate the absolute risk given the score. We assume that the probability of Y_i =1 is related to the covariate vector through a semiparametric regression model

$$P\left(Y_i=1|{\mathit{X}}_i\right)=\pi \left\{S\left({\mathit{X}}_i;\mathit{\beta }\right)\right\}$$ (1)

where π(·) denotes an unknown monotonic nondecreasing function, S(X; β) is a pre-specified function of the subjects’ characteristics, and β is an unknown vector of the same dimension as the covariate vector X. A commonly used linear score summarizes the individual information as S(X; β) = β^TX. In this case, the model is called a single index model.²¹ Since π is left unspecified, we set the Euclidean norm of the coefficients $\Vert \beta {\Vert }_2=\sqrt{{\Sigma }_{i=1}^p{\beta }_i^2}$ to be 1, to ensure identifiability. Such a semiparametric model covers a wide range of regression models, including the logistic and the probit regression models.^22,23 It offers substantial robustness and flexibility by relaxing the assumption regarding the form of the link function. Note that the monotonic assumption on π(·) is necessary to construct the scoring system; without this assumption, the scoring system cannot be used for risk stratification.

3. Likelihood and estimation

We consider estimation procedures for both β and π(·) under model (1). After incorporating the unequal sampling probabilities, the weighted log-likelihood function of the data from the subcohort (Y_i, X_i, i = 1,…, n) is

$$l(\beta ,\pi )=\sum _{i=1}^n{\widehat{w}}_i\left(Y_i\text{log}\left[\pi \left\{S\left(X_i;\beta \right)\right\}\right]+\left(1-Y_i\right)\text{log}\left[1-\pi \left\{S\left(X_i;\beta \right)\right\}\right]\right)$$ (2)

subject to the monotonic constraint for π, where ${\widehat{w}}_i=1/{\widehat{p}}_i$ is the estimated version of w_i = 1/p_i. The weight w_i can be regarded as the contribution of the ith subject to the likelihood function.^24,25 Extensive simulations confirmed that IPW estimators, which break the matching for two-phase designs, are efficient.^26,27

For the case-cohort design, the parameter α can be straightforwardly estimated by the empirical proportion n₀/N, where n₀ is the sample size of the random samples from the full cohort at baseline. For the NCC design, the unknown function of G(·) can be consistently estimated by a Kaplan-Meier-type estimator,²⁵ where

$$\widehat{G}\left(Z_i\right)=\prod _{j:Z_j<Z_i}\left\{1-\frac{m{Y}_j}{n\left(Z_j\right)-1}\right\}$$

Thus, the sampling probability for subject i can be estimated by

$${\widehat{p}}_i=\{\begin{array}{ll}{Y}_i+\left(1-Y_i\right)\widehat{\alpha }\hfill & \text{ in the case }-\text{ cohort design,}\hfill \\ Y_i+\left(1-Y_i\right)\left\{1-\widehat{G}\left(Z_i\right)\right\}\hfill & \text{ in the NCC design }\hfill \end{array}$$

In the presence of matching or stratification, we replace $\widehat{\alpha }$ with ${\widehat{\alpha }}_l=n_{0l}/N_l$ for the case-cohort studies, where n_0l and N_l denote the sample size of the random samples at baseline on the lth stratum and the full cohort on the lth stratum, respectively. Similarly, we replace Ĝ(Z_i) with Ĝ_K(Z_i, K_i) for the NCC studies, where

$${\widehat{G}}_K\left(Z_i,K_i\right)=\prod _{j:Z_j<Z_i,K_j=K_i}\left\{1-\frac{m{Y}_j}{n_K\left(Z_j,K_j\right)-1}\right\}$$

and $n_K\left(Z_j,K_j\right)={\Sigma }_{i=1}^{N}I\left(Z_i\ge Z_j,K_i=K_j\right)$ is the size of the risk set at failure time Z_j after matching.

Note that directly maximizing the weighted likelihood in equation (2) with the monotonic constraint for π is computationally challenging. Considering that the likelihood in equation (2) belongs to the exponential family, we can apply PAVA to simplify the computational task.^13,28 Following the theory of isotonic regression,¹² maximizing the likelihood in equation (2) under the monotonic constraint is equivalent to minimizing the following sum of squares, denoted as Q(β, π), under the same constraint

$$\underset{\pi \left\{S_{(1)}(\beta )\right\}\le \cdots \le \pi \left\{S_{(n)}(\beta )\right\}}{\text{arg max}}l(\beta ,\pi )=\underset{\pi \left\{S_{(1)}(\beta )\right\}\le \cdots \le \pi \left\{S_{(n)}(\beta )\right\}}{\text{arg\hspace{0.17em}min}}\sum _{i=1}^n{\widehat{w}}_i{\left[Y_i-\pi \left\{S\left(X_i;\beta \right)\right\}\right]}^2$$ (3)

where S₍₁₎(β),…, S_(n)(β) denote the sorted S(X_i; β), i = 1,2,…, n in ascending order. To minimize the right-hand side, we design a stable and efficient algorithm based on the method of profiling. For any given β, we can apply PAVA to minimize the objective function Q(β, π), with respect to π(·) subject to the condition that if S₍₁₎(β) ≤ S₍₂₎(β) ≤ ⋯ ≤ S_(n)(β), then π{S₍₁₎(β)} ≤ π{S₍₂₎(β)} ≤ ⋯ ≤ π{S_(n)(β)}. Denote the corresponding estimate as $\widehat{\pi }(\beta )$. We then minimize $Q\left\{\beta ,\widehat{\pi }(\beta )\right\}$ with respect to β and denote the minimizer as ${\widehat{\mathit{\xi }}}_n=\left\{{\widehat{\beta }}_n,{\widehat{\pi }}_n(\cdot )\right\}$. Even though the estimation procedure involves the profiling idea, the computation is fast and can be easily implemented by existing programs. For example, the PAVA step can be accomplished using the R package isotone or Iso, and the minimization after profiling can be implemented using the R function optim.

4. Asymptotic properties

We establish the asymptotic properties of ${\widehat{\mathit{\xi }}}_n$, where true values of the parameters are denoted as ξ₀ = {β₀, π₀(·)}. Technical challenges arise due to the infinite dimension of π(·), as well as the variability due to the estimated sampling probabilities. Under the mild regularity conditions given in Appendix 1, we apply the empirical processes techniques^29,30 to prove the consistency and asymptotic normality of ${\widehat{\beta }}_n$ and the uniform convergence of ${\widehat{\pi }}_n$. We further show that ${\widehat{\pi }}_n$ converges to π₀ at a convergence rate of n^−1/3 using the technique of bracketing entropy. Let ∥ · ∥ be the Euclidean distance and define the metric ∥ · ∥ by

$$\Vert \pi \Vert ={\left\{\int \pi {(u)}^2\text{d}u\right\}}^{1/2}$$

The Hellinger distance h between the two density functions $g_{{\mathit{\xi }}_1}$ and $g_{{\mathit{\xi }}_2}$ is defined by

$$h^2\left(g_{{\mathit{\xi }}_1},g_{{\mathit{\xi }}_2}\right)=\int {\left\{\sqrt{g_{{\mathit{\xi }}_1}(u)}-\sqrt{g_{{\mathit{\xi }}_2}(u)}\right\}}^2\text{d}u$$

We summarize the theoretical results in the following theorem and provide the detailed proof in Appendix 1.

Theorem 1. Under the regularity conditions listed in Appendix 1, ${\widehat{\beta }}_n$ and ${\widehat{\pi }}_n$ are asymptotically consistent

$$\Vert {\widehat{\beta }}_n-{\beta }_0{\Vert }_2\to 0,and\Vert {\widehat{\pi }}_n-{\pi }_0\Vert \to 0$$

in probability. Furthermore, $\sqrt{n}\left({\widehat{\beta }}_n-{\beta }_0\right)$ converges to a normal distribution, while ${\widehat{\pi }}_n$ has a convergence rate of n^−1/3 in the Hellinger distance.

4.1. Variance estimation

The explicit form of the asymptotic variance relies on many unknown quantities, preventing direct estimation of the variance. Alternatively, resampling techniques can be adopted for consistent variance estimation. Note that the standard bootstrap method cannot be applied to the full cohort, since the novel biomarkers are missing for subjects outside the subcohort. For the case-cohort study, we adopt the bootstrap method by Wacholder et al.,³¹ in which the cases and controls are separately resampled from the subcohort with replacement such that the bootstrap data keep the same numbers of cases and controls.

For the NCC design, this modified bootstrap method cannot account for the complex dependence structure induced by repeatedly sampling without replacement from the risk sets. Thus, we apply the perturbation resampling method by Cai and Zheng,³² where a more delicate resampling scheme was designed to account for the dependence. Specifically, let V_i be an indicator of whether subject i has ever been selected in the second phase, V_0i be a binary variable taking the value of 1 if subject i has ever been sampled as a control, and V_0ij be a variable indicating if the jth subject has been chosen as a control for the ith subject. The resampling method perturbs these indicators with independent random numbers to mimic the Bernoulli sampling (with replacement). The sampling probabilities estimated from these perturbed indicators then recover the dependence structure in the finite population sampling (without replacement) and ensure that the corresponding perturbed IPW estimator has an appropriate limiting distribution. The formal justification of the resampling method can be found in Cai and Zheng.³² We describe the perturbation procedure below.

1. Generate non-negative random numbers $\left\{{\mathcal{I}}_{jk},j=1,\dots ,N;k=1,\dots ,N\right\}$ independently from a known distribution with $E\left({\mathcal{I}}_{jk}\right)=1$ and $\text{var}\left({\mathcal{I}}_{jk}\right)=1$, such as the unit exponential distribution.

2. Obtain perturbed weights ${\widehat{w}}_i^{*}=V_i^{*}/{\widehat{p}}_i^{*}$, where $V_i^{*}=Y_i{\mathcal{I}}_{ii}+\left(1-Y_i\right)V_{0i}^{*},{\widehat{p}}_i^{*}=Y_i+\left(1-Y_i\right){\widehat{p}}_{0i}^{*},{\widehat{p}}_{0i}^{*}=1-\text{exp}\left\{-{\widehat{\Lambda }}_{\text{marg}}^{*}\left(Z_i\right)\right\}$,

$$V_{0i}^{*}=1-\prod _{j:i\in R_j\backslash \left\{j\right\}}\left(1-Y_j{V}_{0ji}{\mathcal{I}}_{ji}\right),$$

and

$${\widehat{\Lambda }}_{\text{marg}}^{*}(t)=\sum _{j:Z_j\le t,Y_j=1}\frac{{\sum }_{k\in R_j\backslash \left\{j\right\}}{V}_{0jk}{\mathcal{I}}_{jk}}{n\left(Z_j\right)-1}$$

3. Define Q*(β, π) by replacing the ${\widehat{w}}_i$ in Q(β, π) with ${\widehat{w}}_{*}^i$ and apply the proposed algorithm. The resulting $\left({\widehat{\beta }}_n^{*},{\widehat{\pi }}_n^{*}\right)=\text{arg\hspace{0.17em}min}{\text{Q}}^{*}(\beta ,\pi )$, under the monotone constraint, is a perturbed counterpart of (${\widehat{\beta }}_n$, ${\widehat{\pi }}_n$).

Steps 1 to 3 can be repeated for B₀ times to obtain $\left\{{\left({\widehat{\beta }}_n^{*},{\widehat{\pi }}_n^{*}\right)}_{(b)},b=1,\dots ,B_0\right\}$, where B₀ is the total number of perturbations. The variance of ${\widehat{\beta }}_n$ can be estimated consistently by the empirical variance of its resampled counter-parts, which would facilitate Wald-type confidence intervals and hypothesis testing.

5. Simulation studies

We conducted simulation studies to examine the finite sample performance of the proposed method under two study designs: NCC study and case-cohort study.

5. 1. Simulation studies: NCC study

5.1.1. Data generation

We generated X₁, X₂, and X₃ independently from Beta(2, 2, 0, 2), Bernoulli(0.5), and Uniform(0, 2), respectively, such that the three covariates had similar variances. Here Beta(2, 2, 0, 2) is a four-parameter beta distribution. The binary response Y_i was simulated following a Bernoulli distribution with a success probability of π(β^TX) = π(β₁X_1i + β₂X_2i + β₃X_3i), where the true regression coefficients were (β_1, β_2, β₃)^T = (0.707,0,0.707)^T. We generated T_i from a uniform distribution Uniform(0, 5) for the cases and set the follow-up time to be 5.1 for all controls. Let Z_i = min(T_i, 5). For each case, three controls were sampled without replacement from its risk set, excluding the case.

We generated the event probabilities using different link functions. We considered four scenarios: Scenario 1 represented the case when the logistics regression model was the true model; Scenario 2 was used by other papers, such as Leitenstorfer and Tutz³³; Scenario 3 was a combination of concave and convex curves, and Scenario 4 was for the sensitivity analysis when the monotonic assumption was violated. These curves were chosen to exemplify real-life relationships between the outcome probability and covariates, which may follow various shapes and curvatures.

Scenario 1: The true curve followed the logistic curve, π(β^TX) = 1/[1 + exp{−3.1(β^TX – 2.5)}].

Scenario 2: The true curve had three curvatures, as shown in Figure 1, middle row, π(β^TX) = 0.1/[1 + exp{−20(β^TX – 1.2)}] + 0.2/[1 + exp{−16(β^TX − 2.3)}.

Figure 1.

Estimated risk functions under the nested case-control design.

Scenario 3: The true curve was concave at the left tail and convex at the right tail, as displayed in Figure 1, bottom row, π(β^TX) = tan{1.65(β^TX – 0.56) – 1.4}/95 + 0.06.

Scenario 4: The true curve was not monotone, as shown in Figure S2 of the Supplementary Materials, π(β^TX) = 0.12/[1 + exp{−20(β^TX – 1.2)}] + 0.24/[1 + exp{−16(β^TX – 2.3)}] – 0.05/[1 + exp{−5(β^TX – 2)}].

We considered two sample sizes of 2500 and 5000 for the full cohort and two sample sizes of 700 and 1400 for the subcohort. To ensure that we would locate the global minima, we implemented 25 sets of random initial values around the estimates obtained from logistic regression and identified the estimators that achieved the lowest loss. The number of simulation replicates was 1000, and the resampling number B₀ was 499.^34,35 For comparison, we implemented conditional logistic regression and IPW-based logistic regression.

5.1.2. Simulation results

The simulation results under the NCC design are summarized in Table 1 and Table S1 in the Supplementary Materials. The summary statistics are the empirical mean, empirical standard error (ESE) for β and the curve π(·) at the 25th, 50th, and 75th percentiles of the scores; average of the estimated standard error (ASE) based on resampling; and the empirical coverage probability of the 95% confidence interval. The estimate of β₁ was determined as a function of the estimates of β₂ and β₃ by the unit Euclidean norm constraint, and for completeness, all estimated coefficients were reported. The empirical means of the estimated π curves are plotted in Figure 1 and Figure S2 in the Supplementary Materials, in which the 5th and 95th percentiles of the scores β^TX are chosen as the limits of the x-axis. In Scenario 1, all three methods performed well, since the true model followed logistic regression. As shown in the top row of Figure 1, both the proposed method and IPW-based logistic regression captured the underlying π function. The empirical biases of the estimates obtained by all three methods were negligible, and the coverage probabilities were reasonably close to the nominal level. As expected, the ESEs of the two logistic regression methods were smaller than those from the proposed method, since the two logistic regression methods utilized the information on the underlying link function (logistic curve) while the proposed method did not.

Table 1.

Simulation results under the nested case-control design (ESE is the empirical standard error, ASE is the average of estimated standard error, and CP is the empirical coverage probability of the 95% confidence interval).

					Proposed method					IPW-based logistic regression					Conditional logistic regression
Scenario	N	n	PARA	True	Mean	Bias	ESE	ASE	CP	Mean	Bias	ESE	ASE	CP	Mean	Bias	ESE	ASE	CP
Scenario 1	2500	700	β1	.707	.683	−.024	.101	.104	.951	.705	−.002	.048	.048	.937	.704	−.003	.068	.068	.938
			β2	.000	−.004	−.004	.111	.103	.928	.000	.000	.065	.065	.953	−.002	−.002	.096	.093	.936
			β3	.707	.709	.001	.095	.098	.950	.703	−.004	.049	.047	.940	.702	−.005	.069	.067	.927
			π(.99)	.009	.010	.000	.006	-	-	.009	.000	.002	-	-	-	-	-	-	-
			π(1.41)	.034	.036	.002	.013	-	-	.034	.000	.006	-	-	-	-	-	-	-
			π(1.84)	.117	.124	.007	.037	-	-	.120	.003	.015	-	-	-	-	-	-	-
	5000	1400	β1	.707	.700	−.007	.072	.076	.958	.703	−.004	.036	.034	.933	.704	−.003	.049	.048	.945
			β2	.000	.000	.000	.085	.081	.942	.000	.000	.047	.046	.951	−.002	−.002	.069	.066	.933
			β3	.707	.702	−.005	.069	.075	.956	.707	.000	.036	.034	.928	.706	−.001	.049	.047	.948
			π(.99)	.009	.009	.000	.004	-	-	.009	.000	.002	-	-	-	-	-	-	-
			π(1.41)	.034	.035	.001	.010	-	-	.034	.000	.004	-	-	-	-	-	-	-
			π(1.84)	.117	.119	.002	.027	-	-	.119	.001	.011	-	-	-	-	-	-	-
Scenario 2	2500	700	β1	.707	.674	−.034	.126	.136	.932	.695	−.012	.078	.078	.943	.689	−.018	.088	.083	.922
			β2	.000	.011	.011	.130	.126	.921	.000	.000	.106	.101	.943	.009	.009	.123	.118	.928
			β3	.707	.708	.001	.111	.119	.947	.703	−.005	.076	.074	.933	.709	.002	.085	.080	.912
			π(.99)	.001	.011	.010	.018	-	-	.030	.028	.005	-	-	-	-	-	-	-
			π(1.41)	.107	.097	−.009	.017	-	-	.061	−.046	.008	-	-	-	-	-	-	-
			π(1.84)	.108	.116	.007	.017	-	-	.121	.013	.015	-	-	-	-	-	-	-
	5000	1400	β1	.707	.694	−.013	.073	.089	.970	.695	−.012	.059	.056	.934	.697	−.010	.064	.058	.927
			β2	.000	.000	.000	.094	.095	.942	.001	.001	.074	.073	.947	.000	.000	.087	.084	.933
			β3	.707	.707	.000	.069	.082	.965	.710	.003	.057	.053	.921	.709	.002	.062	.057	.923
			π(.99)	.001	.005	.004	.011	-	-	.029	.028	.004	-	-	-	-	-	-	-
			π(1.41)	.107	.100	−.007	.012	-	-	.060	−.046	.005	-	-	-	-	-	-	-
			π(1.84)	.108	.112	.004	.010	-	-	.119	.011	.010	-	-	-	-	-	-	-
Scenario 3	2500	700	β1	.707	.705	−.003	.072	.073	.921	.773	.066	.059	.058	.704	.764	.057	.098	.092	.803
			β2	.000	.003	.003	.056	.058	.934	.003	.003	.092	.092	.946	−.002	−.002	.152	.141	.910
			β3	.707	.700	−.007	.073	.074	.922	.621	−.086	.072	.070	.765	.624	−.083	.116	.108	.852
			π(.99)	.052	.049	−.003	.009	-	-	.028	−.024	.006	-	-	-	-	-	-	-
			π(1.41)	.061	.061	.000	.008	-	-	.061	.000	.008	-	-	-	-	-	-	-
			π(1.84)	.070	.074	.004	.012	-	-	.128	.058	.015	-	-	-	-	-	-	-
	5000	1400	β1	.707	.706	−.001	.049	.051	.939	.775	.068	.042	.041	.588	.772	.065	.070	.065	.758
			β2	.000	.000	.000	.033	.038	.955	.001	.001	.068	.066	.939	−.001	−.001	.108	.102	.919
			β3	.707	.704	−.003	.049	.051	.940	.625	−.082	.052	.050	.636	.625	−.082	.084	.078	.801
			π(.99)	.052	.050	−.002	.007	-	-	.028	−.024	.004	-	-	-	-	-	-	-
			π(1.41)	.061	.061	.000	.006	-	-	.061	−.001	.006	-	-	-	-	-	-	-
			π(1.84)	.070	.072	.002	.009	-	-	.126	.056	.010	-	-	-	-	-	-	-

In Scenario 2, the underlying p curve exhibited more curvature than that assumed by the logistic regression model (Figure 1, middle row). The proposed method still performed well in terms of estimating the regression coefficients β and the link function π. Using IPW-based logistic regression, the estimates of β were close to the true values, but the estimated curve deviated from the true function. As a result, the risk probability could not be reliably estimated, due to the substantial bias in the estimated π. For example, for a subject with a median risk score, IPW-based logistic regression severely underestimated the risk probability by 50%, which would give misleading information to the subject. Under this setting, although the underlying true model was not the logistic regression model, conditional logistic regression was quite robust and performed similarly to the proposed method. However, the conditional regression model could not estimate the absolute risk, which was important in our setting.

In Scenario 3, the proposed method remained robust and accurate in terms of both the regression coefficients and the π function. In comparison, IPW-based logistic regression could not estimate the underlying link well. For example, the risk estimation around the 75th quantile of the risk score overestimated the true risk by 80% (Figure 1, bottom row). Both logistic regression methods had biased estimates for the regression coefficients due to the mis-specification of the underlying link function. For IPW-based logistic regression, the biases were larger than the corresponding ESEs for some regression coefficients, leading to poor coverage probabilities as low as 59%.

In Scenario 4, the true curve was not monotonic; thus the monotonic assumption was not satisfied for the logistic methods, as well as the proposed method. Nevertheless, our proposed method still outperformed the commonly used logistic regression methods in estimating the link function (Figure S2 in the Supplementary Materials).

5.2. Simulation studies: case-cohort study

5.2.1. Data generation

We used the same sample sizes (2500 and 5000) and data generation scheme as specified in Scenario 3 of section 5.1.1 to simulate the covariates and outcomes for the full cohort. For the subcohort, we selected 550 or 1100 controls from the full cohort at baseline. We compared the performance of our proposed design with that of IPW-based logistic regression.

5.2.2. Simulation results

Simulation results under the case-cohort design are summarized in Table 2. Similar to our previous findings under the NCC design, the estimates of ${\widehat{\beta }}_n$ and ${\widehat{\pi }}_n$ by the proposed method were close to the true values, and the empirical coverage probabilities of the confidence intervals of ${\widehat{\beta }}_n$ were close to the nominal level. In contrast, IPW-based logistic regression overestimated β₁ by 10% and underestimated β₃ by 13%, which resulted in coverage probabilities as low as 53%. As shown in Figure 2, the proposed method fitted the true curve well with small biases, while logistic regression could not capture the true curve. The substantial differences between the estimated and true curves indicated that the use of logistic regression may result in misleading risk assessments when the model assumptions were not valid.

Table 2.

Simulation results under the case-cohort design (ESE is the empirical standard error, ASE is the average of estimated standard error, and CP is the empirical coverage probability of the 95% confidence interval).

					Proposed method					IPW-based logistic regression
Scenario	N	n	PARA	True	Mean	Bias	ESE	ASE	CP	Mean	Bias	ESE	ASE	CP
Scenario 3	2500	700	β1	.707	.704	−.003	.076	.084	.952	.770	.063	.058	.060	.749
			β2	.000	.000	.000	.058	.066	.958	−.003	−.003	.094	.095	.951
			β3	.707	.700	−.008	.075	.085	.957	.625	−.082	.071	.072	.792
			π(.99)	.052	.049	−.003	.010	-	-	.029	−.024	.006	-	-
			π(1.41)	.061	.061	.000	.008	-	-	.062	.001	.008	-	-
			π(1.84)	.070	.074	.004	.012	-	-	.129	.059	.015	-	-
	5000	1400	β1	.707	.706	−.002	.049	.057	.966	.781	.074	.040	.041	.532
			β2	.000	−.001	−.001	.033	.041	.973	.002	.002	.066	.067	.947
			β3	.707	.705	−.003	.048	.057	.972	.618	−.089	.050	.051	.597
			π(.99)	.052	.050	−.002	.007	-	-	.028	−.024	.004	-	-
			π(1.41)	.061	.061	.000	.006	-	-	.061	.000	.006	-	-
			π(1.84)	.070	.073	.003	.009	-	-	.127	.057	.010	-	-

Figure 2.

Estimated risk functions under the case-cohort design.

In summary, with the link function unspecified, the proposed method robustly estimated both the risk score using regression coefficients and the link function under various scenarios, given data from two-phase sampling designs. By comparison, logistic regression approaches generated severely biased estimates for both the risk score and the link function, even under the settings with large sample sizes. These results signified the advantage of relaxing the model assumptions by using the proposed method.

6. Application

We applied the proposed estimation and inferential procedures to data from the PLCO study. Following the paper by Shiels et al.,¹⁷ one aim of our study was to evaluate the association between the risk of lung cancer and biomarker measurements from the NCC subcohort. Due to the advantage of the proposed method, we also aimed to determine the risk of lung cancer by considering epidemiological risk factors, as well as biomarker measurements. We considered three epidemiological risk factors: age, smoking status (current/former/never), and cumulative smoking amount; and two biomarkers: CRP and CXCL9/MIG.^17,18 The NCC design was summarized in Section 1.1 and detailed by Shiels et al.¹⁷ We used the same data as those by Shiels et al.,¹⁷ consisting of 524 cases and 623 controls.

Considering that conditional logistic regression cannot incorporate the matching factors into the model, we first conducted analyses using only the biomarker information. To facilitate comparisons, the coefficients from the conditional logistic regression were normalized to have unit Euclidean norm. Consistent with the results by Shiels et al.,¹⁷ we found that the levels of CRP and CLCX9/MIG were significantly associated with lung cancer risk. The estimated coefficients (standard errors) by the proposed method were 0.699 (0.170) and 0.715 (0.210) for CRP and CXCL9/MIG, respectively, and 0.540 (0.187) and 0.842 (0.205) by conditional logistic regression. The two methods gave slightly different coefficient estimates, with similar standard errors and conclusions.

Next, we applied the proposed method to additionally incorporate the epidemiological risk factors, which were used for matching to select controls. Table 3 presents the estimated regression coefficients with their standard errors and Wald P-values. Figure 3 plots the estimated absolute risk curve. As seen from Table 3, the risk of lung cancer increased with increasing age and cumulative smoking amount. Also, current smokers and former smokers had a greater risk of developing lung cancer compared to nonsmokers. The coefficients for smoking-related risk factors were relatively large, suggesting that the smoking information played a key role in lung cancer prognosis. The two biomarkers were significantly associated with lung cancer risk after adjusting for the epidemiological risk factors, and they carried additional prognostic value to identify the high-risk population. To illustrate, we fixed age at 60years and cumulative smoking level at 50 pack-years. Among current smokers, subjects in the 4th quartile of CRP and CLCX9/MIG had an estimated risk of developing lung cancer of 0.037, which is more than six-fold the risk for subjects in the 1st quartile of the biomarkers (0.006). Among former smokers, the risk of developing lung cancer in subjects in the 4th quartile was almost 17-fold the risk in subjects in the 1st quartile (0.037 vs. 0.002). Similarly, among nonsmokers, the risks were 0.003 and 0.000 for subjects in the 4th and 1st quartiles, respectively. Therefore, incorporating the two biomarkers improved lung cancer risk stratification beyond that based on only the standard epidemiological risk factors. In addition, we did a graphical goodness-of-fit diagnostic of the monotonicity assumption and found that the monotonicity assumption was appropriate for the data from the PLCO study (section 1 of the Supplementary Materials).

Table 3.

Estimated coefficients and standard errors for the proposed model applied to the PLCO data.

	Coefficient	Standard Error	P-value
Age at randomization (per 10 years)	0.110	0.056	0.049
Current smoker vs. never smoker	0.805	0.030	<0.001
Former smoker vs. never smoker	0.531	0.100	<0.001
Cumulative smoking amount (per 50 pack-years)	0.152	0.056	0.007
CRP	0.144	0.061	0.02
CXCL9/MIG	0.117	0.053	0.03

Figure 3.

Absolute risk of developing lung cancer in the PLCO population estimated by the proposed model with variables of age, smoking status, cumulative smoking amount, CRP and CXCL9/MIG.

As one feature of the proposed method, the derived risk scores can be used for the risk stratification. Using the estimated median risk score as the cutoff, we divided the subcohort subjects equally into two groups (high-risk and low-risk groups). In the high-risk group, 54.2% of the subjects developed lung cancer; in the low-risk group, 37.1% developed lung cancer. For comparison purposes, we implemented the analytical method of the PLCO model, which was a logistic regression model with the following covariates: age, race (white, black, Hispanic, Asian, American Indian or Alaskan native, native Hawaiian or pacific islander), education, body-mass index, chronic obstructive pulmonary disease, personal history of cancer, family history of lung cancer, current smoker, former smoker, smoking intensity, duration of smoking, and smoking quit time, which yielded that 53.9% high-risk subjects and 38.0% low-risk subjects developed lung cancer.

7. Discussion

In this paper, we propose a semiparametric isotonic regression model for constructing risk scores and assessing absolute risks given data from two-phase studies. With these tools, we can identify high-risk patients and improve the shared decision making between at-risk patients and their physicians by providing a quantifiable personalized risk assessment. We leave the link function unspecified, other than the monotonicity assumption, which is a necessary assumption to achieve a sensible risk score. Although our model involves a nonparametric component, a profiling method and PAVA are utilized to improve computational efficiency and can be easily implemented using existing software. Thus, the proposed method offers robustness, easy implementation, and computational efficiency.

One challenge of the proposed method is model specification for the risk score: how to select the best set of risk factors and how to determine an appropriate form for combining multiple risk factors. A simple screening procedure (e.g. marginal correlation) or stepwise model selection procedures can be applied to select risk factors for the risk score; however, it may not be able to identify the optimal subset of risk factors. Next, to combine multiple factors, we focus on the linear form due to its simplicity and popularity. Standard model comparison tools, such as the likelihood ratio test, are not directly applicable and cannot handle the additional variation due to the estimated weights. Developing rigorous tools that simultaneously select the optimal set of risk factors and identify the best way to combine them is beyond the scope of this paper, though worthy of future research.

The ranked set sampling (RSS) design, as an alternative design to the two-phase sampling designs, has been proposed by McIntyre³⁶ and has received increasing attention. Recently, Zamanzade and Vock³⁷ showed that the RSS design is more efficient than the two-phase sampling designs under certain cases, and Zamanzade and Mahdizadeh³⁸ used a variation of RSS to efficiently estimate the prevalence of a rare disease in a given population. It is of interest to extend our work to RSS-based designs for constructing risk scores and assessing absolute risks simultaneously for future research.

Appendix 1. Asymptotic proofs

A.1. Regularity conditions

Denote the parameters ξ = (β, π). We summarize the regularity conditions as follows.

• (C1) The covariate vector X is in a bounded compact set $\mathcal{X}$ in R^p.

• (C2) The parameters ξ are in the space $\Theta =\mathcal{B}\times \mathcal{P}$, where $\mathcal{B}$ is a compact space in R^p and $\mathcal{P}$ is a space of monotonic nondecreasing functions of bounded variation.

• (C3) The true value β₀ is in the interior of the set $\mathcal{B}$. The true curve π₀(·) is continuously Fréchet differentiable on $\mathcal{K}$, where $\mathcal{K}=\left[{\eta }_1,{\eta }_2\right]$ is the support of β^TX, $\beta \in \mathcal{B}$, and $X\in \mathcal{X}$.

Due to identifiability constraint, we restrict that $\mathcal{B}=\left\{\beta \Vert \beta {\Vert }_2=1,\beta \in {\text{R}}^p\right\}$; particularly, we treat (β₂,…, β_p) as unknown parameters and β₁ as a function of (β₂,…,β_p). Without loss of generality, we let S(X; β) = β^TX throughout the proof, and the results can be easily extended to other forms of scoring. Let U₁,…,U_n be a random sample from a probability distribution P on a measurable space. Given a measurable function f, we define ${\mathbb{P}}_{n}f=n^{-1}{\Sigma }_{i=1}^{n}f\left(U_i\right),Pf=\int fdP$, and ${\mathbb{G}}_n=\sqrt{n}\left({\mathbb{P}}_n-P\right)f=n^{-1}{\sum }_{i=1}^n\left\{f\left(U_i\right)-Pf\right\}$.²⁹ The weighted log-likelihood function for a single subject with data D_i = (X_i, Y_i) is

$$l_{\mathit{\xi }}\left(D_i\right)={\widehat{w}}_i\left[Y_i\text{log}\left\{\pi \left({\beta }^T{X}_i\right)\right\}+\left(1-Y_i\right)\text{log}\left\{1-\pi \left({\beta }^T{X}_i\right)\right\}\right].$$

The estimator ${\widehat{\mathit{\xi }}}_n=\left({\widehat{\beta }}_n,{\widehat{\pi }}_n\right)$ maximizes the weighted log-likelihood function $n^{-1}{\sum }_{i=1}^n{l}_{\mathit{\xi }}\left(D_i\right)$.

A.2. Proof of consistency

Define a metric ∥·∥ on the parameter space Θ by $\Vert {\mathit{\xi }}_1-{\mathit{\xi }}_2{\Vert }^2=\Vert {\beta }_1-{\beta }_2{\Vert }_2^2+\Vert {\pi }_1-{\pi }_2{\Vert }^2$, where ∥·∥₂ is the Euclidean metric on a finite dimensional space, and $\Vert \pi {\Vert }^2={\int }_{{\eta }_1}^{{\eta }_2}\pi {(u)}^2\text{d}u$ The consistency of ${\widehat{\mathit{\xi }}}_n$ can be proven by verifying the conditions in Theorem 5.7 for M-estimators.³⁹

Given a known sampling weight, the likelihood function of the observed data from a single subject, denoted as d = (x, y), can be written as

$$p_{\mathit{\xi }}(d)={\left\{\pi \left({\beta }^{T}X\right)\right\}}^{wy}{\left\{1-\pi \left({\beta }^{T}X\right)\right\}}^{w(1-y)}$$

Also write ${\widehat{p}}_{\mathit{\xi }}(d)$ as the estimated counterpart of p_ξ(d), obtained by replacing the true weight w with ŵ. We consider the following class of functions that is related to Kullback-Leibler information

$$m_{\mathit{\xi }}(d)=\text{log}\left\{\frac{p_{\mathit{\xi }}(d)+p_0(d)}{2p_0(d)}\right\}$$ (4)

where $p_0(d)=p_{{\mathit{\xi }}_0}(d)$. Note that the function $m_{\mathit{\xi }}(d)-m_{{\mathit{\xi }}_0}(d)$ is bounded from below, $m_{\mathit{\xi }}(d)-m_{{\mathit{\xi }}_0}(d)\ge -\text{log}2$, which is necessary for being the Glivenko-Cantelli class. Define $M_n(\mathit{\xi })={\mathbb{P}}_n{\widehat{m}}_{\mathit{\xi }}$, ${\tilde{M}}_n(\mathit{\xi })={\mathbb{P}}_n{m}_{\mathit{\xi }}$, and $M(\mathit{\xi })=E{\tilde{M}}_n$, where ${\widehat{m}}_{\mathit{\xi }}=\text{log}\left\{\left({\widehat{p}}_{\mathit{\xi }}+{\widehat{p}}_0\right)/\left(2{\widehat{p}}_0\right)\right\}$.

We first show the uniform convergence of ${\tilde{M}}_n(\mathit{\xi })$ to M(ξ) using the permanence properties of the Glivenko-Cantelli class (section 2.6.5 of van der Vaart and Wellner).²⁹ We partition the likelihood into two parts according to the values of Y ({0, 1}). We then show that the following two classes of functions are P-Glivenko-Cantelli:

$${\mathcal{F}}_1=\left\{{\left\{\pi \left({\beta }^{T}X\right)\right\}}^w:\beta \in \mathcal{B},\pi \in \mathcal{P}\right\}$$ (5)

$${\mathcal{F}}_2=\left\{{\left\{1-\pi \left({\beta }^{T}X\right)\right\}}^w:\beta \in \mathcal{B},\pi \in \mathcal{P}\right\}$$ (6)

Given these results, the density function $\left\{p_{\mathit{\xi }}(d):\beta \in \mathcal{B},\pi \in \mathcal{P}\right\}$ is also Glivenko-Cantelli.

By regularity conditions, |β^Tx| < C_β|x| for $C_{\mathit{\beta }}={\text{sup}}_{\beta \in \mathcal{B}}\left|\beta \right|$, the class of functions x↦β^Tx is is therefore Glivenko-Cantelli. The class of functions $S\mapsto \pi (s)1(s\in \mathcal{K})$ is in the Glivenko-Cantelli class by Theorem 2.7.5 in van der Vaart and Wellner.²⁹ By the permanence properties of Glivenko-Cantelli classes, the class of functions x↦π(β^Tx)^w is Glivenko-Cantelli. It follows that the classes of functions ${\mathcal{F}}_1$, ${\mathcal{F}}_2$ are all Glivenko-Cantelli. By the1 regularity conditions, we have that p₀(d) is bounded away from zero, and $P\left\{p_0^{-1}(D)\right\}<\infty$. Finally, the following class of functions is Glivenko-Cantelli as its envelope function is bounded, {log{(p_ξ +p₀)/2p₀} : ξ ∈ Θ}. Then we can apply the property of the Glivenko-Cantelli class to show that ${\text{sup}}_{\mathit{\xi }\in \Theta }\left|{\tilde{M}}_n(\mathit{\xi })-M(\mathit{\xi })\right|={\text{sup}}_{\mathit{\xi }\in \Theta }\left|{\mathbb{P}}_n{m}_{\mathit{\xi }}-P{m}_{\mathit{\xi }}\right|\stackrel{p}{\to }0$.

Next, we verify the uniform convergence of $\left|M_n(\mathit{\xi })-{\tilde{M}}_n(\mathit{\xi })\right|$ to 0. Without loss of generality, we let ŵ_i be the weight under the NCC design. The results can be easily extended to the case-cohort design. Given the asymptotic behavior of the product limit estimator,⁴⁰ ${\text{sup}}_t\left|\widehat{G}(t)-G(t)\right|=O_p\left(N^{-\frac{1}{2}}\right)$, and we have ${\text{max}}_i\left|{\widehat{w}}_i-w_i\right|=O_p\left(N^{-\frac{1}{2}}\right)$. It follows that

$$\frac{{\widehat{p}}_0}{p_0}=\frac{{\pi }_0^{\widehat{w}y}{\left(1-{\pi }_0\right)}^{\widehat{w}(1-y)}}{{\pi }_0^{wy}{\left(1-{\pi }_0\right)}^{w(1-y)}}={\pi }_0^{(\widehat{w}-w)y}{\left(1-{\pi }_0\right)}^{(\widehat{w}-w)(1-y)}\stackrel{p}{\to }1$$ (7)

Therefore, ${\widehat{p}}_0-p_0\stackrel{p}{\to }0$ and $\text{log}\left({\widehat{p}}_0/p_0\right)\stackrel{p}{\to }0$. Similarly, ${\widehat{p}}_{\mathit{\xi }}/p_{\mathit{\xi }}\stackrel{p}{\to }1$, ${\widehat{p}}_{\mathit{\xi }}-p_{\mathit{\xi }}\stackrel{p}{\to }0$, and $\text{log}({\widehat{p}}_{\mathit{\xi }}-p_{\mathit{\xi }}+{\widehat{p}}_0-p_0+p_{\mathit{\xi }}+p_0)\stackrel{p}{\to }\text{log}\left(p_{\mathit{\xi }}+p_0\right)$. Combing these results, we have that

$$M_n(\mathit{\xi })-{\tilde{M}}_n(\mathit{\xi })={\mathbb{P}}_n{\widehat{m}}_{\mathit{\xi }}-{\mathbb{P}}_n{m}_{\mathit{\xi }}=\frac{1}{n}\sum _{i=1}^n\left\{{\widehat{m}}_{\mathit{\xi }}\left(d_i\right)-m_{\mathit{\xi }}\left(d_i\right)\right\}=\frac{1}{n}\sum _{i=1}^n\left(\text{log}\left\{{\widehat{p}}_{\mathit{\xi }}\left(d_i\right)+{\widehat{p}}_0\left(d_i\right)\right\}-\text{log}\left\{p_{\mathit{\xi }}\left(d_i\right)+p_0\left(d_i\right)\right\}-\left[\text{log}\left\{a{\widehat{p}}_0\left(d_i\right)\right\}-\text{log}\left\{a{p}_0\left(d_i\right)\right\}\right]\right)=\frac{1}{n}\sum _{i=1}^n\left\{\text{log}\left({\widehat{p}}_{\mathit{\xi }}-p_{\mathit{\xi }}+{\widehat{p}}_0-p_{\widehat{\mathit{\xi }}}+p_{\mathit{\xi }}+p_0\right)-\text{log}\left(p_{\mathit{\xi }}+p_0\right)-\text{log}\left({\widehat{p}}_0/p_0\right)\right\}\stackrel{p}{\to }0$$ (8)

Therefore, we have

$$\underset{\mathit{\xi }\in \Theta }{\text{sup}}\left|M_n(\mathit{\xi })-M(\mathit{\xi })\right|\le \underset{\mathit{\xi }\in \Theta }{\text{sup}}\left|M_n(\mathit{\xi })-{\tilde{M}}_n(\mathit{\xi })\right|+\underset{\mathit{\xi }\in \Theta }{\text{sup}}\left|{\tilde{M}}_n(\mathit{\xi })-M(\mathit{\xi })\right|\stackrel{p}{\to }0$$ (9)

Next, by the concavity of the logarithm and the definition of ${\widehat{\mathit{\xi }}}_n$, we have

$${\mathbb{P}}_n\text{log}\left(\frac{{\widehat{p}}_{{\widehat{\mathit{\xi }}}_n}+{\widehat{p}}_0}{2}\right)\ge \frac{1}{2}{\mathbb{P}}_n\left(\text{log}{\widehat{p}}_{{\widehat{\mathit{\xi }}}_n}+\text{log}{\widehat{p}}_0\right)\ge {\mathbb{P}}_n\text{log}{\widehat{p}}_0$$

Hence

$$M_n\left({\widehat{\mathit{\xi }}}_n\right)={\mathbb{P}}_n{\widehat{m}}_{{\widehat{\mathit{\xi }}}_n}\ge {\mathbb{P}}_n{\widehat{m}}_{{\mathit{\xi }}_0}=M_n\left({\mathit{\xi }}_0\right)$$ (10)

In addition, the true value ξ₀ is always the maximum point of Pm_ξ as by Jensen’s inequality

$$M(\mathit{\xi })=P\text{log}\left(\frac{p_{\mathit{\xi }}+p_0}{2p_0}\right)\le \text{log}P\left(\frac{p_0+p_0}{2p_0}\right)=M\left({\mathit{\xi }}_0\right)=0$$ (11)

and the equality sign can be achieved only when p_ξ = p₀. Thus, ${\text{sup}}_{\mathit{\xi }:d\left(\mathit{\xi },{\mathit{\xi }}_0\right)\ge ϵ}M(\mathit{\xi })<M\left({\mathit{\xi }}_0\right)$ is confirmed. Combining equations (9), (10), and (11), and Theorem 5.7 of van der Vaart,³⁹ we have $\Vert {\widehat{\mathit{\xi }}}_n-{\mathit{\xi }}_0\Vert \to 0$ in probability as n → ∞. As the model is identifiable on the parameter set Θ, we conclude that ${\widehat{\beta }}_n\stackrel{p}{\to }{\beta }_0$, and ${\widehat{\pi }}_n(s)\stackrel{p}{\to }{\pi }_0(s)$ uniformly in s.

A.3. Proof of convergence rate

We first derive the convergence rate of ${\tilde{\mathit{\xi }}}_n$, which is the counterpart of ${\tilde{\mathit{\xi }}}_n$ with known weights using Theorem 8.1 in van der Vaart³⁰ (equivalently, Theorem 3.4.4 in van der Vaart and Wellner.²⁹) Consider the class of functions defined in equation (4), ${\mathcal{M}}_{\delta }=\left\{m_{\mathit{\xi }}-m_{{\mathit{\xi }}_0}:\Vert \mathit{\xi }-{\mathit{\xi }}_0\Vert <\delta \right\}$ The first condition of Theorem 8.1 in van der Vaart³⁰ is satisfied under the Hellinger distance (Lemma 4.2 of van de Geer),⁴¹

$$E\left(m_{\mathit{\xi }}-m_{{\mathit{\xi }}_0}\right)=E\text{log}\left(\frac{p_{\mathit{\xi }}+p_0}{2p_0}\right)\le -h^2\left\{\left(p_{\mathit{\xi }}+p_0\right)/2,p_0\right\}\lesssim -h^2\left(p_{\mathit{\xi }},p_0\right)$$

where the Hellinger distance h is defined in section 4 and x ≲ y is a shorthand of x ≤ K₀y for a constant K₀. Here the first inequality is by the connection of the Kullback–Leibler divergence to the Hellinger distance. The second inequality follows from the fact that the Hellinger distance between any pair of densities f₁ and f₂ is equivalent to the Hellinger distance between f₁ and (f₁ + f₂)/2 (page 328 of van der Vaart and Wellner).²⁹

Lemma 8.6 of van der Vaart³⁰ shows that the bracketing entropy of the class of functions ${\mathcal{M}}_{\delta }$ can be shown to be of order 1/ϵ, which implies

$$J_{\left[\right]}\left(\delta ,\mathcal{F},L_2\right)={\int }_0^{\delta }\sqrt{K_1{ϵ}^{-1}}dϵ\lesssim {\delta }^{1/2}$$

where K₁ is a constant. Together with Lemma 8.2 of van der Vaart,³⁰ we have

$$E\underset{f\in {\mathcal{M}}_{\delta }}{\text{sup}}\left|{\mathbb{G}}_{n}f\right|\lesssim {\delta }^{1/2}\left(1+\frac{{\delta }^{1/2}}{{\delta }^2\sqrt{n}}\right)={\delta }^{1/2}+\frac{1}{\delta \sqrt{n}}$$

Plugging in ${\varphi }_n(\delta )=\sqrt{\delta }+1/(\delta \sqrt{n})$ and δ_n = n^−1/3 in Theorem 8.1 of van der Vaart,³⁰ we have

$$\frac{{\varphi }_n\left({\delta }_n\right)}{{\delta }_n^2}={\delta }_n^{-3/2}+\frac{{\delta }_n^{-3}}{\sqrt{n}}=2\sqrt{n}$$

Then the rest of the conditions in Theorem 8.1 of van der Vaart³⁰ are satisfied, confirming the convergence rate is n^−1/3 for ${\tilde{\mathit{\xi }}}_n$. Combined with the asymptotic normality of ŵ_i, we can show the convergence rate of ${\tilde{\mathit{\xi }}}_n$ is n^−1/3 in the Hellinger distance.

Next, we follow the efficient score method by Bickel et al.⁴² to obtain the information bound of β given known weights. Note that model (1) belongs to the type II semiparametric regression model in section 4.3 of Bickel et al..⁴² Let $\dot{\pi }(s)=d\pi (s)/ds$. By Proposition 4.3.2 of Bickel et al.,^{42$\dot{r}(X|\beta ,\pi )=\partial \pi \left({\beta }^{T}X\right)/\partial \beta =\dot{\pi }\left({\beta }^{T}X\right)X$}, and $\dot{I}(D|\beta ,\pi )=\partial l_{\mathit{\xi }}(D)/\partial \pi {|}_{\pi =\pi \left({\beta }^{T}X\right)}=w\left\{Y-\pi \left({\beta }^{T}X\right)\right\}/\left[\pi \left({\beta }^{T}X\right)\left\{1-\pi \left({\beta }^{T}X\right)\right\}\right]$. Denote ${\mathbb{B}}_0$ as the sigma field generated by π(β^TX). Then the efficient score function of β takes the form $I^{*}(D)=\tilde{r}\dot{I}(D|\beta ,\pi )$, where $\tilde{r}=\dot{r}(X|\beta ,\pi )-E_{{\mathit{\beta }}_0,{\pi }_0}\left(\dot{r}(X|\beta ,\pi ){\dot{I}}^2(d\left|\beta ,\pi )\right|{\mathbb{B}}_0\right)/I(X)$, and $I(X)=E_{{\mathit{\beta }}_0,{\pi }_0}\left\{{\dot{I}}^2(d\left|\beta ,\pi )\right|{\mathbb{B}}_0\right\}$. It follows that the information bound for β is $\Omega =E_{{\beta }_0,{\pi }_0}\left\{I^{*}{I}^{*T}\right\}$.

Last, following Liu and Qin,⁴³ we establish the asymptotic normality of ${\widehat{\beta }}_n$ by treating the monotone function π(·) as a nuisance parameter. We first establish the asymptotic normality of ${\tilde{\beta }}_n$ which is the counterpart of ${\tilde{\beta }}_n$ with known weights by verifying the conditions in Theorem 6.20 of van der Vaart.³⁰ To verify the no-bias condition, we can follow the steps in section 9.3 of van der Vaart.³⁰ Let ψ_β,π be the score function which is an approximation of the efficient score function I* by using a least favorable submodel. Then we can verify the no-bias condition,

$$|P_{{\beta }_0,{\pi }_0}{\psi }_{{\beta }_0,{\widehat{\pi }}_n}|\lesssim \underset{{\eta }_1}{\overset{{\eta }_2}{\int }}{|{\widehat{\pi }}_n-{\pi }_0|}^2(s)\text{d}s=O_p\left(n^{-2/3}\right)$$

Note that the class of functions x↦β^Tx belong to the Donsker class by Lemma 6.11 in van der Vaart,³⁰ and the functions s↦π(s) belongs to the Donsker class by Theorem 2.7.5 in van der Vaart and Wellner.²⁹ It follows that the functions x↦ψ_β,π belongs to the Donsker class by the permanence of the Donsker property (van der Vaart and Wellner, 1996, section 2.10).²⁹ Therefore, the Donsker condition is verified. Hence, by Theorem 6.20 of van der Vaart,³⁰ ${\tilde{\beta }}_n$ is asymptotically normal with a convergence rate of $\sqrt{n}$. Combining this result with the normality of the estimated weights, we have the asymptotic normality of ${\widehat{\beta }}_n$.