Weighted Wilcoxon-Type Rank Test for Interval Censored Data

Abstract

Interval censored (IC) failure time data are often observed in medical follow-up studies and clinical trials where subjects can only be followed periodically, and the failure time can only be known to lie in an interval. In this paper, we propose a weighted Wilcoxon-type rank test for the problem of comparing two IC samples. Under a very general sampling technique developed by Fay (1999), the mean and variance of the test statistics under the null hypothesis can be derived. Through simulation studies, we find that the performance of the proposed test is better than that of the two existing Wilcoxon-type rank tests proposed by Mantel (1967) and R. Peto and J. Peto (1972). The proposed test is illustrated by means of an example involving patients in AIDS cohort studies.

1. Introduction

Interval censored (IC) failure time data often arise from medical studies such as AIDS cohort studies and leukemic blood cancer follow-up studies. In these studies, patients were divided into two groups according to different treatments. For example, in leukemic cancer studies, one group of the patients was treated with radiotherapy alone, and the other group of patients was treated with initial radiotherapy along with adjuvant chemotherapy. The two groups of patients were examined every month, and the failure time of interest is the time until the appearance of leukemia retraction; the object is to test the difference of the failure times between the two treatments. Some of the patients missed some successive scheduled examinations and came back later with a changed clinical status, and they contributed IC observations. For our convenience, we assume that in such a medical study, the underlying survival function can be either discrete or continuous, and there are only finitely many scheduled examination times. IC data only provide partial information about the lifetime of the subject, and the data is one kind of incomplete data. To deal with such incomplete data, Turnbull [1] introduced a self-consistent algorithm to compute the maximum likelihood estimate of the survival function for arbitrarily censored and truncated data. For IC data, there have been some related studies in the literature as well. For example, Mantel [2] extends Gehan’s [3, 4] generalized Wilcoxon [5] test to interval censored data, and R. Peto and J. Peto [6] also develop a different version. Sun [7] applied Turnbull’s algorithm to estimate the number of failures and risks of IC data and then propose a log-rank type test.

Fay [8], Sun [7], Zhao and Sun [9], Sun et al. [10], and Huang et al. [11] extend the log-rank test to interval censored data. Petroni and Wolfe [12] and Lim and Sun [13] generalize Pepe and Fleming’s [14] weighted Kaplan-Meier (WKM) [15] test to interval censored data.

For the purpose of comparing the power of the test statistics, Fay [8] proposed a model for generating interval censored observation. A similar selection scheme can also be seen in the Urn model of Lee [16] and mixed cased model of Schick and Yu [17]. In this paper, we propose a Wilcoxon-type weighted rank test to compare with the existing two Wilcoxon-type rank tests proposed by Mantel [2] and R. Peto and J. Peto [6]. We restrict ourselves to the Wilcoxon-type rank tests because these tests are simple to use and have the robustness property that their powers are fairly stable under different lifetime distributions.

This paper is organized as follows. In Section 2, we review the Turnbull’s [1] algorithm and introduce Fay’s [8] selection model for generating interval censored data. This selection model can be extended to a more general one, and the consistency property can be found in Schick and Yu [17]. In Section 3, we introduce Mantel’s [2] and R. Peto and J. Peto’s [6] generalized Wilcoxon-type rank tests and propose our weighted rank test. In Section 4, a simulation study is conducted to compare the performance of the three tests under different configurations. Finally, an application to AIDS cohort study is presented in Section 5.

2. Data Treatment

Assume that 𝑋 is the lifetime random variable of a survival study, measured in discrete units and taking values 0=𝑥₀<𝑥₁<𝑥₂<⋯<𝑥_𝑚. Let 𝑈={(𝑥_𝑖,𝑥_𝑗],0≤𝑖<𝑗≤𝑚} be the collection of all 𝑚(𝑚+1)/2 admissible intervals, and define 𝑝_𝑗=𝑃(𝑋=𝑥_𝑗), where ∑^𝑚_𝑗=1𝑝_𝑗=1, so that 𝐹(𝑥)=∑_{𝑥𝑗≤𝑥}𝑝_𝑗, and 𝑆(𝑥)=∑_{𝑥𝑗>𝑥}𝑝_𝑗. Note that the observed failure time data in a clinical trial can be discretized if the underlying variable is continuous.

2.1. Turnbull’s Algorithm

Suppose that there is a sample of 𝑛 i.i.d. observations (𝑋^𝑖_𝐿,𝑋^𝑖_𝑅] of 𝑋, 𝑖=1,2,…,𝑛. Here, (𝑋^𝑖_𝐿,𝑋^𝑖_𝑅] is the IC observation of the 𝑖th individual in the sample, where 𝑋^𝑖_𝐿,𝑋^𝑖_𝑅∈{𝑥₀,𝑥₁,𝑥₂,…,𝑥_𝑚}, and 𝑋^𝑖_𝐿<𝑋^𝑖_𝑅. The case 𝑋^𝑖_𝑅=𝑥_𝑚 is to denote that the failure time of the 𝑖th subject occurs after the last examination time 𝑥_𝑚−1. Turnbull [1] proposed an algorithm to estimate the unknown probabilities 𝑝=(𝑝₁,𝑝₂,…,𝑝_𝑚). The algorithm can be described by the following four steps.

Step 1. Start with initial values 𝑝⁽⁰⁾=(𝑝₁⁽⁰⁾,𝑝₂⁽⁰⁾,…,𝑝_𝑚⁽⁰⁾).

Step 2. Obtain improved estimates 𝑝_𝑗⁽¹⁾ by setting 𝑝_𝑗⁽¹⁾=1𝑛^𝑛_𝑖=1𝛼^𝑖_𝑗𝑝_𝑗⁽⁰⁾∑^𝑚_𝑙=1𝛼^𝑖_𝑙𝑝_𝑙⁽⁰⁾,𝑗=1,2,…,𝑚,where𝛼^𝑖_𝑗=𝐼𝑥_𝑗∈𝑋^𝑖_𝐿,𝑋^𝑖_𝑅. (1)

Step 3. Return to Step 1 with 𝑝⁽¹⁾ replacing 𝑝⁽⁰⁾.

Step 4. Stop when the required accuracy has been achieved.

The algorithm is simple and converges fairly rapidly. The estimate 𝑝=(𝑝₁,𝑝₂,…,𝑝_𝑚) yielded from the iteration is in fact the unique maximum likelihood estimate of 𝑝=(𝑝₁,𝑝₂,…,𝑝_𝑚) and is a self-consistent estimate.

2.2. Return Probability Model

To comply with the periodical clinical inspection, Fay [8] proposed a simulation model for generating IC data. He assumed that the probability for a patient to return to the clinic for inspection at time points 𝑥₁,𝑥₂,…,𝑥_𝑚−1 are i.i.d. Bernulli random variables 𝐴₁,𝐴₂,…,𝐴_𝑚−1; that is, 𝑃(𝐴_𝑖=1)=𝑞, 𝑃(𝐴_𝑖=0)=1−𝑞, 0<𝑞<1, 𝑖=1,2,…,𝑚−1. 𝐴_𝑖=1 means that the patient returned to the clinic at the inspection time 𝑥_𝑖, and 𝐴_𝑖=0 means that the patient missed the inspection. In our model, we always assume that 𝐴_𝑚=1. The failure time 𝑋 is independent of (𝐴₁,𝐴₂,…,𝐴_𝑚−1), and the observable random interval is 𝑋_𝐿,𝑋_𝑅=𝑥_𝑠𝑗,𝑥_𝑡𝑗,where𝑠_𝑗=max_𝑙0≤𝑙<𝑗∶𝐴_𝑙,𝑡=1_𝑗=min_𝑙𝑗≤𝑙≤𝑚∶𝐴_𝑙,𝑥=1_𝑠𝑗<𝑋≤𝑥_𝑡𝑗. (2)

2.2.1. Model Consistency

Under Fay’s [8] selection model, the consistency property has been proved. This selection model can be generalized to the case that the return probability at each examination time point may be different; say that 𝑃(𝐴_𝑖=1)=𝑎_𝑖,  𝑖=1,2,…,𝑚. To demonstrate the generalized return model, we set 𝑚=3 and 𝑥₁=1, 𝑥₂=2, and 𝑥₃=3. The selection probabilities for all admissible intervals are shown in Tables 1 and 2.

Table 1 

The probability of selected interval.

Table 2 

Selection probability 𝑄(𝐼) for all admissible intervals.

It is not difficult to see that the selection probability of the interval 𝐼=(𝑥_𝑢,𝑥_𝑣] is 𝑄{𝐼}=𝑃(𝐼)𝑎_𝑢×^𝑣−1_𝑖=𝑢+11−𝑎_𝑖×𝑎_𝑣,𝑄𝑥𝑢=0,1,2,…,𝑣−1,𝑣=1,2,…,𝑚−1,_𝑢,𝑥_𝑚=𝑃𝑥_𝑢,𝑥_𝑚𝑎_𝑢×^𝑚−1_𝑖=𝑢+11−𝑎_𝑖,0≤𝑢≤𝑚−1, (3)(4) where 𝑃(𝐼)=(𝑝_𝑢+1+𝑝_𝑢+2+⋯+𝑝_𝑣), 𝑥₀=0, and 𝑎₀=1. For instance, the interval (0,2] may be selected under two possibilities. First, the true value of 𝑋 is 𝑋=1, and the patient who missed the inspection at 𝑥₁=1 then goes to inspection at 𝑥₂=2; in this case, the interval is selected with probability 𝑝₁(1−𝑎₁)𝑎₂. Second, the true value of 𝑋 is 𝑋=2, and the patient missed the inspection at 𝑥₁=1 then goes to inspection at 𝑥₂=2; in this case, the interval is selected with probability 𝑝₂(1−𝑎₁)𝑎₂, and therefore 𝑄{(0,2]}=(𝑝₁+𝑝₂)(1−𝑎₁)𝑎₂.

The generalized return probability model can be viewed as a special case of the mixed case model in Schick and Yu [17]; under very mild conditions, the estimate of 𝑝=(𝑝₁,𝑝₂,…,𝑝_𝑚) computed by Turnbull’s algorithm is still consistent.

3. Wilcoxon-Type Rank Tests for Interval Censored Data

Two-sample Wilcoxon rank test is a well-known method to test whether two samples of exact data come from the same population. The method is constructed by ranking the pooled samples and giving an appropriate rank to each observation. However, this ranking technique is in general not admissible for intervals. In this section, we will discuss how to generalize the ranking technique and then propose a Wilcoxon-type rank test for IC data to compare with two existing rank tests proposed by Mantel [2] and R. Peto and J. Peto [6]. Suppose that two samples of IC data for 𝑋 and 𝑌 are, respectively, (𝑋^𝑖_𝐿,𝑋^𝑖_𝑅],  𝑖=1,2,…,𝑛₁ and (𝑌^𝑖_𝐿,𝑌^𝑖_𝑅],  𝑖=1,2,…,𝑛₂. To test whether these two samples come from the same population is equivalent to testing the equality of survival functions 𝑆_𝑋(𝑡) and 𝑆_𝑌(𝑡),forall𝑡≥0; that is, 𝐻₀∶𝑆_𝑋(𝑡)=𝑆_𝑌(𝑡),∀𝑡≥0. (5)

3.1. Mantel’s Test

Mantel [2] extended Gehan’s [3, 4] generalized Wilcoxon test to interval censored data by defining the score of the 𝑘th observation as the number of observations that are definitely greater than the 𝑘th observation minus the number of observations that are definitely less than the 𝑘th observation. He proposed the test statistic 𝑊=^𝑛1_𝑘=1𝑉_𝑘,where𝑉_𝑘=^𝑛1+𝑛2_ℎ=1𝑉_𝑘ℎ,𝑉_𝑘ℎ=1ifweknowforsureobs-𝑘>obs-ℎ,−1ifweknowforsureobs-𝑘<obs-ℎ,0ifnotsure. (6)

Under 𝐻₀, the test statistic is approximately normal distributed with mean 0 and variance Var(𝑊)=𝑛₁𝑛₂^𝑛1+𝑛2_𝑘=1𝑉²_𝑘𝑛₁+𝑛₂𝑛₁+𝑛₂.−1 (7)

3.2. R. Peto and J. Peto’s Test

Different from the Mantel’s generalized version, R. Peto and J. Peto [6] defined the score of the 𝑖th observation as 𝑈_𝑖=𝑓𝑆𝑋^𝑖_𝐿−𝑓𝑆𝑋^𝑖_𝑅𝑆𝑋^𝑖_𝐿−𝑆𝑋^𝑖_𝑅, (8) where 𝑆 is the estimated survival function, 𝑓(𝑦)=𝑦²−𝑦; hence, 𝑈_𝑖=𝑆(𝑋^𝑖_𝐿)+𝑆(𝑋^𝑖_𝑅)−1. They proposed the test statistic 𝑍²=𝑌²₁/𝑛₁+𝑌²₂/𝑛₂𝑠²,where𝑌₁=^𝑛1_𝑖=1𝑈_𝑖,𝑌₂=^𝑛1+𝑛2_𝑖=𝑛1+1𝑈_𝑖,𝑠²=∑^𝑛1+𝑛2_𝑖=1𝑈²_𝑖𝑛₁+𝑛₂.−1 (9) Under 𝐻₀, the test statistic 𝑍² is approximately distributed as 𝜒²₁.

3.3. Our Proposed Wilcoxon-Type Weighted Rank Test

To transform an IC data to exact, we first assign each inspection time 𝑥_𝑖 a primary rank 𝑅_𝑖; for instance, 𝑅_𝑖=𝑖. Rewrite any observation, say (𝑋^𝑗_𝐿,𝑋^𝑗_𝑅], as (𝑋^𝑗_𝐿,𝑋^𝑗_𝑅]=(𝑥_𝑢(𝑗),𝑥_𝑣(𝑗)], where 𝑥_𝑢(𝑗),𝑥_𝑣(𝑗)∈{0,𝑥₁,𝑥₂,…,𝑥_𝑚}, and 𝑥_𝑢(𝑗)<𝑥_𝑣(𝑗). Then, we associate the observation (𝑋^𝑗_𝐿,𝑋^𝑗_𝑅] with the weighted rank rank𝑋^𝑗_𝐿,𝑋^𝑗_𝑅=^𝑣(𝑗)_{𝑙=𝑢(𝑗)+1}𝑝_𝑙𝑝_𝑢(𝑗)+1+⋯+𝑝_𝑣(𝑗)𝑅_𝑙. (10) Let 𝑊₁, 𝑊₂ be, respectively, the average weighted rank of the 𝑋 and 𝑌 samples, so that 𝑊₁=1𝑛₁^𝑛1_𝑖=1rank𝑋^𝑖_𝐿,𝑋^𝑖_𝑅=1𝑛₁^𝑛1_𝑖=1^𝑣(𝑖)_{𝑙=𝑢(𝑖)+1}𝑝_𝑙𝑝_𝑢(𝑖)+1+⋯+𝑝_𝑣(𝑖)𝑅_𝑙,𝑊₂=1𝑛₂^𝑛2_𝑗=1rank𝑌^𝑗_𝐿,𝑌^𝑗_𝑅=1𝑛₂^𝑛2_𝑗=1^𝑣(𝑗)_{𝑙=𝑢(𝑗)+1}𝑝_𝑙𝑝_𝑢(𝑗)+1+⋯+𝑝_𝑣(𝑗)𝑅_𝑙. (11) To test whether two IC samples come from the same population, we propose the test statistic 𝑊W.R.T=¹−𝑊₂Var𝑊₁+Var𝑊₂. (12) Under 𝐻₀, the central limit theorem implies that W.R.T is approximately distributed as a standard normal random variable. However, the mean and variance of 𝑊₁ and 𝑊₂ may depend on the probability space where they are defined; it means, different selection probability for IC intervals in (4) leads to different mean and variance of 𝑊₁ and 𝑊₂. We therefore only consider the selection model of Fay defined in Section 2.2. In this model, the selection probability of an IC interval is in one of the following categories: (i)𝑄0,𝑥_𝑟=^𝑟_𝑗=1𝑝_𝑗𝑞1−𝑞^𝑟−1()𝑄,1≤𝑟<𝑚,ii0,𝑥_𝑚=^𝑚_𝑗=1𝑝_𝑗1−𝑞^𝑚−1,(iii)𝑄𝑥_𝑢,𝑥_𝑣=^𝑣_𝑗=𝑢+1𝑝_𝑗𝑞²1−𝑞^𝑟−1()𝑄𝑥,1≤𝑢<𝑣<𝑚,iv_𝑢,𝑥_𝑚=^𝑚_𝑗=𝑢+1𝑝_𝑗𝑞1−𝑞^{𝑚−𝑢−1},1≤𝑢<𝑚. (13)(14)(15)(16) Consider the probability space (𝑈,2^𝑈,𝑄), where the probability measure 𝑄 is defined in Section 2. To compute the variance of 𝑊₁ and 𝑊₂, we define a random variable 𝑍 on this space by assigning value 𝑍{(𝑥_𝑢,𝑥_𝑣]} to the interval (𝑥_𝑢,𝑥_𝑣] in 𝑈, where 𝑍𝑥_𝑢,𝑥_𝑣=^𝑣_𝑙=𝑢+1𝑝_𝑙𝑝_𝑢+1+⋯+𝑝_𝑣𝑅_𝑙,0≤𝑢<𝑣≤𝑚. (17) The value 𝑍{(𝑥_𝑢,𝑥_𝑣]} can be viewed as the weighted rank of (𝑥_𝑢,𝑥_𝑣]. If 𝑅_𝑙, 𝑙=1,2,…,𝑚 are chosen as in the Wilcoxon test for exact data, then our proposed test statistic W.R.T is a Wilcoxon-type weighted rank test. Under this probability space, the expectation 𝐸(𝑍) can be simplified as in the following theorem.

Theorem 1. Suppose that 𝑍 is the random variable defined on the probability space (𝑈,2^𝑈,𝑄) according to (17). Then, the expectation of 𝑍, 𝐸(𝑍), can be simplified as 𝐸(𝑍)=^𝑚_𝑙=1𝑝_𝑙𝑅_𝑙, (18) which is independent of the choice of 𝑞.

Proof. It is obvious that 𝐸(𝑍) can be written as 𝐸(𝑍)=∑^𝑚_𝑙=1𝑏_𝑙𝑝_𝑙𝑅_𝑙, where the coefficients 𝑏_𝑙,  𝑙=1,2,…,𝑚 are to be determined. The theorem is, hence, proved if we can show that all the coefficients 𝑏_𝑙 are ones.
Consider 𝑏₁ first. An interval (𝑥_𝑢,𝑥_𝑣] contributes 𝑝₁𝑅₁ in 𝐸(𝑍) if and only if it contains the point 𝑥₁. Therefore, it must be of the form (0,𝑥_𝑣],  𝑣=1,2,…,𝑚. For intervals (0,𝑥_𝑣], 1≤𝑣≤𝑚−1, the probabilities 𝑄{(0,𝑥_𝑣]} are defined in (13). For interval (0,𝑥_𝑚], the probability 𝑄{(0,𝑥_𝑚]} is defined in (14). Therefore, the coefficient 𝑏₁ is 𝑏₁=^𝑚−1_𝑣=1𝑞1−𝑞^𝑣−1+1−𝑞^𝑚−1=𝑞1−1−𝑞^𝑚−1+1−1−𝑞1−𝑞^𝑚−1=1. (19) Next, consider the coefficient 𝑏_𝑗 for 1<𝑗≤𝑚−1. An interval contributes 𝑝_𝑗𝑅_𝑗 if and only if it contains the point 𝑥_𝑗. Therefore, it must be of the form (𝑥_𝑢,𝑥_𝑣], where 0≤𝑢<𝑗≤𝑣≤𝑚. It is necessary to study the contribution of the interval (𝑥_𝑢,𝑥_𝑣] to 𝑏_𝑗 in four different categories.(i)𝑢=0,𝑣≤𝑚−1.
By (13), this category contributes ∑^𝑚−1_𝑣=𝑗𝑞(1−𝑞)^𝑣−1.(ii)𝑢=0,𝑣=𝑚.
By (14), the interval (0,𝑥_𝑚] contributes (1−𝑞)^𝑚−1.(iii)1≤𝑢<𝑣≤𝑚−1.
By (15), this category contributes ∑^𝑗−1_𝑢=1∑^𝑚−1_𝑣=𝑗𝑞²(1−𝑞)^{𝑣−𝑢−1}.(iv)𝑢≥1,𝑣=𝑚.
By (16), this category contributes ∑^𝑗−1_𝑢=1𝑞(1−𝑞)^{𝑚−𝑢−1}.
Consequently, the coefficient of 𝑏_𝑗 is 𝑏_𝑗=^𝑚−1_𝑣=𝑗𝑞1−𝑞^𝑣−1+1−𝑞^𝑚−1+^𝑗−1_𝑢=1^𝑚−1_𝑣=𝑗𝑞²1−𝑞^{𝑣−𝑢−1}+^𝑗−1_𝑢=1𝑞1−𝑞^{𝑚−𝑢−1}=𝑞1−𝑞^𝑗−11−1−𝑞^𝑚−𝑗+1−1−𝑞1−𝑞^𝑚−1+^𝑗−1_𝑢=1𝑞𝑞1−𝑞^{𝑗−𝑢−1}1−1−𝑞^𝑚−𝑗+1−1−𝑞_{𝑗−1𝑢=1}𝑞1−𝑞^{𝑚−𝑢−1}=1−𝑞^𝑗−1−1−𝑞^𝑚−1+1−𝑞^𝑚−1+^𝑗−1_𝑢=1𝑞1−𝑞^{𝑗−𝑢−1}−^𝑗−1_𝑢=1𝑞1−𝑞^{𝑚−𝑢−1}+^𝑗−1_𝑢=1𝑞1−𝑞^{𝑚−𝑢−1}=1−𝑞^𝑗−1+1−1−𝑞^𝑗−1=1. (20) Finally, the proof for the case 𝑗=𝑚 is 𝑏_𝑚=^𝑚−1_𝑢=1𝑞1−𝑞^{𝑚−𝑢−1}+1−𝑞^𝑚−1=1−1−𝑞^𝑚−1+1−𝑞^𝑚−1=1. (21)

The variance of 𝑍, Var(𝑍), is Var(𝑍)=𝐸𝑍²−𝐸²(𝑍)=^{𝑚(𝑚+1)/2}_𝑖=1𝑄𝐼_𝑖𝑅²𝐼_𝑖−𝐸²(𝑍), (22) where 𝑄(𝐼_𝑖) and 𝑅(𝐼_𝑖) are the selected probability and the weighted rank of the 𝑖th admissible interval of 𝐼_𝑖, respectively, 𝐼_𝑖∈𝑈.

Consider the formulas (13)–(16), the selection probability 𝑄(𝐼) depends on 𝑝=(𝑝₁,𝑝₂,…,𝑝_𝑚) and 𝑞; therefore, the likelihood function can be written as 𝐿𝑝₁,𝑝₂,…,𝑝_𝑚,𝑞=𝑃𝑝₁,𝑝₂,…,𝑝_𝑚𝐺𝑞, (23) where 𝐺(𝑞)=𝑞^𝑘1(1−𝑞)^𝑘2, 𝑘₁ and 𝑘₂ are positive integers determined by the sample. Since the probability 𝑝=(𝑝₁,𝑝₂,…,𝑝_𝑚) can be estimated by Turnbull’s [1] algorithm discussed in Section 2.2, and 𝑞 can also be estimated by 𝑘₁/(𝑘₁+𝑘₂) trivially.

For demonstration, we set 𝑚=6, inspection times 𝑥_𝑖=𝑖, 𝑖=1,2,…,6, and the true lifetime 𝑋 is exponentially distributed with 𝜆=1/3. For different sample sizes 𝑛=50, 100, and 150, different return probabilities of inspection 𝑞=0.8, 0.5, and 0.3, and simulation with 100 replications, Table 3 presents the estimates of 𝑞 and sample variance and sample deviation of 𝑞. To show the normality of W.R.T, we assume that the two populations (sample size 𝑛₁=𝑛₂=30) are coming from the same distribution exponential (1/5). By simulation with 10000 replications and different return probabilities of inspection 𝑞=0.8, 0.5, and 0.3, Table 4 presents the quantiles of W.R.T and 𝑁(0,1). Figure 1 shows the CDF plots of 𝑁(0,1) and W.R.T with 𝑞=0.5.

Table 3 

The mean, sample variance, and sample deviation of 𝑞.

Table 4 

The quantiles of W.R.T and 𝑁(0,1).

Figure 1 

CDF of standard normal and simulation result of W.R.T. Line: standard normal. Point: simulation result of W.R.T (𝑞=0.5).

4. Simulation Study

In this section, we carry out simulation studies to compare the performance of W.R.T test with Mantel’s [2] and Peto’s [6] tests. In the study, we assume that the failure time random variable is distributed as exponential, total sample sizes are 𝑛=100 and 200, and each sample has (𝑛/2) subjects. The interval censored data are generated by the following four steps.

Step 1. Generate a failure time 𝑡_𝑗 from some distribution.

Step 2. Create a 0, 1 sequence 𝐴={𝐴₀,𝐴₁,𝐴₂,…,𝐴_𝑚} with probabilities 𝑃(𝐴_𝑖=1)=𝑞,  𝑖=1,2,…,𝑚−1, and 𝑃(𝐴₀=1)=𝑃(𝐴_𝑚=1)=1.

Step 3. The observation is (𝑎,𝑏], if 𝑎<𝑡_𝑗≤𝑏,𝐴_𝑎=𝐴_𝑏=1, and 𝐴_𝑎+1=𝐴_𝑎+2=⋯=𝐴_𝑏−1=0.

Step 4. Repeat Step 1 to Step 3 for 𝑛 times.

We consider three return probabilities, 𝑞=0.8, 0.5, and 0.3, two sets of inspection time points, 𝑚=6,10, and 1000 replications at significance level 0.05.

In the case of 𝑚=6, 6 return points, we set the hazards 1/3 for population 1 and 1/3𝑒^𝛽 for population 2. Figure 2 shows the density plot of exponential distribution with 𝛽=−0.4, −0.2, 0, 0.2, 0.4. In the case of 𝑚=10, 10 return points, we set the hazards 1/4 for population 1 and 1/4𝑒^𝛽 for population 2. Figure 3 shows the density plot of exponential distribution with 𝛽=−0.6, −0.3, 0, 0.3, 0.6. Tables 5 and 6 present the powers of the three tests with sample size 𝑛=100 and 200. Simulation result shows that when the failure times come from the exponential distribution, our proposed test W.R.T is the most powerful.

Table 5 

Power comparison of tests under exponential distribution with sample 𝑛=100.

Table 6 

Power comparison of tests under exponential distribution with sample 𝑛=200.

Figure 2 

Density plot of exponential distribution with hazards 1/3𝑒𝛽.

Figure 3 

Density plot of exponential distribution with hazards 1/4𝑒𝛽.

5. An Application to AIDS Cohort Study

Consider the data of 262 hemophilia patients in De Gruttola and Lagakos [18], among them, 105 patients received at least 1,000 𝜇g/kg of blood factor for at least one year between 1982 and 1985, and the other 157 patients received less than 1,000 𝜇g/kg in each year. In this medical study, patients were treated between 1978 and 1988, the observations (𝑋_𝐿,𝑋_𝑅] for the 262 patients, based on a discretization of the time axis into 6-month intervals. The failure time of interest is the time of HIV seroconversion. The object is to test the difference of the failure times between the two treatments. Applying our proposed test, namely, W.R.T, Mantel’s [2] and Peto’s [6] tests to this data set, the values of the three test statistics are −7.815, −7.352, and 56.476, respectively. All the three 𝑃 values are less than 0.001 and have the same conclusion that the HIV seroconversion appeared in the two groups of patients being significantly different.