Jorjani Biomedicine Journal

 Let T be a random time and Z  be a vector of covariates in R^p . The proportional hazards model assumes that conditional on Z , the hazard rate function of T, is given by            

where β∈R^p  is an indefinite regression parameter vector, and h₀(t)  is an indefinite baseline hazard rate function (4).

Censorship represents an inherent element of medical and reliability studies, stemming from the practical challenge of acquiring comprehensive data about the complete lifespan of every individual. For instance, participants in a follow-up study might withdraw or the study might need to conclude at a predetermined time point. The 2 primary forms of censoring are type I and type II censoring (10-12). The present paper primarily delves into the realm of competing risk data within the context of progressively type II censoring. Progressive censoring is of particular significance in designing duration experiments, especially in the domain of reliability studies.
We employ competing risk data analysis within the framework of progressively type II censoring, which assumes elevated importance in shaping the design of duration experiments, particularly in the context of medical research. Both the conventional and progressive censoring structures share similar objectives, yet the latter is strategically designed to alleviate information loss by effectively managing the number of observed failures relative to the broader sample approach.
 Let ( T₁ , Z₁ ),…,( T_n , Z_n )  be n  independent of (T, Z ). A progressive type II censored sample is attained using the following method. First, suppose that we are given integers r₁ , r₂ ,…, r_m , chosen a priori, such that r₁+⋯+ r_m +m=n . Consider the lifetimes T₁ , T₂ ,…, T_n  as the failure times of n  items that are placed on the test at time zero. We denote by X ₍₁₎  (the first failure time) and i₁  (the number of the unit that failed). Immediately after the occurrence of the first failure, a certain number of items r₁  labeled i₂ ,…, i_r1+1 are randomly chosen and removed from the test. This set of removed items is denoted as I₁ =[ i₁ ,…, i _r1+1] . Then, at the second observed failure time X ₍₂₎ , the corresponding item number is i_{r1 +2} , and a new set of r₂  surviving items labeled i_r1+3,…, i _r1+r2+2 is randomly selected and removed from the test. We then denote I₂ =[ i _{r1+2 ,}…, i _r1+r2+2] . This process continues until, at the time X _(m)  of the m -the observed failure of unit number i_r1+⋯+ r_m-1+m , the surviving items i_r1+⋯+ r_m-1 +m+1 ,…, i_n  are all removed from the experiment, and we denote I_m =[ i_r1+⋯+ r_m-1 +m ,…, i_n] . Note that this censoring structure leads to a subsample X ₍₁₎ <⋯< X _(m)  of the order statistics T ₍₁₎ <⋯< T _(n)  obtained from T ₁ ,…, T _n . Note also that the sets of unit numbers I ₁ ,…, I _m  satisfy

 Note that the complete and type right-censored samples are single of the above scheme when r₁=r₂=⋯=r_m=0  and r₁=r₂=⋯=r_m-1=0,r_m=n-m , respectively. It is important to note that the order statistics that result from a progressive type II censoring scheme are a specific example of generalized order statistics, which were introduced by Kamps (13,14). For full instructions and more details on the review, the reader is referred to the book by Balakrishnan and Aggarwala (15-17) or other studies (18,19).
It is worth noting that within this structure, the variables R₁,R₂,…,R_m  are predetermined. However, practical scenarios may arise where the count of items or individuals being dropped or excluded from a study becomes a random variable. Yuen and Tse mention, for instance, that the number of patients excluded at different stages of clinical trials is subject to random selection and cannot be anticipated beforehand. Similarly, in certain reliability tests, testers might opt not to examine specific units, deeming it unnecessary or risky even if those units could potentially fail. Consequently, the patterns of exclusions become stochastic, as all instances of exclusion are subject to randomness.
Let's assume that each test unit failing the lifetime assessment is independent of other units but shares the same probability of being excluded, denoted as p. Following this, Xie et al indicate that the count of instances being excluded in each fault interval adheres to a binomial distribution (20).
This paper aims to analyze competing risk models in the presence of progressively type II censored data with binomial random removals arising from the proportional hazards model (11,21). The Cox model is a commonly used statistical technique to study the relationship between a patient's survival and various risk factors. The purpose of the model is to simultaneously delve into the effects of several variables on survival (12). 

Methods
The Model's Assumptions
The paper assumes that there are q  independent causes of failure directed toward each unit (22-24). The model studied in the paper contains the following assumptions:


To simplify the notation, we will use henceforth X_i  instead of x _(xi) (22, 23).
The Likelihood Function of the Model and Estimators
The Likelihood Function
Using the mentioned assumptions in section 2 and conditional on I₁,I₂,…,I_m , the likelihood function for the type II progressively censored model under competing risk is as follows (10):


where (10)


 Under some standard regularity conditions, V^-1(β^{^})  provides an estimate for the variance-covariance matrix of β^{^}  and so follows hypothesis tests and confidence intervals for β .
Finally, the Breslow-type estimator (24) of the baseline type-specific cumulative hazard function for the j-th failure type can be written as

where (25)


Results
Employed in a Real-World Dataset
Consider a scenario where demographic, personal, clinical, and laboratory data are gathered through interviews and physical examinations conducted on a sample of 200 individuals participating in a study on cardiovascular disease (CVD). These participants, between the ages of 50 and 79, who had no CVD at baseline, will be followed for 10 years. To further demonstrate the use of the proportional hazards model in competing risks, a subset of 68 participants from the simulated data is used. The event of interest, denoted as " t ," represents the period of time during which participants remain free from CVD. This time duration is defined in terms of years, starting from the baseline and ending at the earliest occurrence of either a participant being diagnosed with CVD or being confirmed to have passed away due to CVD-related causes. Cardiovascular disease, in this case, includes coronary heart disease (CHD) and stroke. Covariates of interest include age (AGE), sex (SEX = 1 for males and SEX = 0 for females), smoking status (SMOKE = 1 for current smokers and SMOKE = 0 otherwise), and body mass index ([BMI = weight] . in kilograms divided by height in meters squared), systolic blood pressure (SBP), logarithm of the ratio of urine albumin to creatinine (LACR), logarithm of triglycerides (LTG), hypertension status (HTN = 1 if SBP 140 mm Hg or DBP 90 mm Hg or on treatment hypertension and otherwise HTN = 0) and diabetes status (DM = 1 at fasting glucose of 126 mg/dL or on treatment for diabetes and otherwise DM = 0).
To evaluate the CVD outcome of interest, we use DG to denote the type of CVD. Specifically, DG = 0 if the CVD-free time is censored, DG = 1 if the participant had a stroke, DG = 2 if the participant had CHD, and DG = 3 if the participant had other types of CVD.
Results for Stroke
For stroke, SBP and hypertension status are the only significant variables. In contrast, gender, BMI, smoking status, body, the logarithm of urinary albumin and creatinine ratio, and diabetes status are significant variables for CHD and other CVDs. The outcomes suggest that the significant risk factors vary for different types of CVD events.
Simulation Results
Generally, in the simulated data, we can observe that the results for type II progressive right-censored sampling designs produce better results than results based on the usual type II censored sampling designs (Table 1).

Table 1. Simulation's results

Table 2. Results on cardiovascular disease event time data from the fitted competing risks model

Discussion