Interval censored (IC) failure time data are often observed in medical followup 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 Wilcoxontype 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 Wilcoxontype 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.
Interval censored (IC) failure time data often arise from medical studies such as AIDS cohort studies and leukemic blood cancer followup 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 [
Fay [
For the purpose of comparing the power of the test statistics, Fay [
This paper is organized as follows. In Section
Assume that
Suppose that there is a sample of
Start with initial values
Obtain improved estimates
Return to Step
Stop when the required accuracy has been achieved.
The algorithm is simple and converges fairly rapidly. The estimate
To comply with the periodical clinical inspection, Fay [
Under Fay’s [
The probability of selected interval.
True value of 
Selected interval  Probability 

(0,1] 


1  (0,2] 

(0,3] 


 
(1,2] 


2 
(0,2] 

(1,3] 


(0,3] 


 
(2,3] 


3  (1,3] 

(0,3] 

Selection probability
Interval 
Probability 

(0,1] 

(1,2] 

(2,3] 

(0,2] 

(1,3] 

(0,3] 

It is not difficult to see that the selection probability of the interval
The generalized return probability model can be viewed as a special case of the mixed case model in Schick and Yu [
Twosample Wilcoxon rank test is a wellknown 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 Wilcoxontype rank test for IC data to compare with two existing rank tests proposed by Mantel [
Mantel [
Under
Different from the Mantel’s generalized version, R. Peto and J. Peto [
To transform an IC data to exact, we first assign each inspection time
Suppose that
It is obvious that
Consider
By (
By (
By (
By (
Consequently, the coefficient of
The variance of
Consider the formulas (
For demonstration, we set
The mean, sample variance, and sample deviation of


0.8  0.5  0.3 

Estimate  0.8001  0.5029  0.3024  
50  Variance  0.0020  0.0021  0.0012 
Std.  0.0448  0.0461  0.0341  
 
Estimate  0.8039  0.5036  0.3012  
100  Variance  0.0010  0.001  0.0005 
Std.  0.0320  0.0312  0.0233  
 
Estimate  0.8009  0.4977  0.3033  
150  Variance  0.0005  0.0008  0.0004 
Std.  0.0225  0.0277  0.0207 
The quantiles of W.R.T and
Quantile 
Normal (0,1) 
W.R.T  

0.8  0.5  0.3  
0.05 




0.10 




0.15 




0.20 




0.25 




0.30 




0.35 




0.40 




0.45 




0.50 




0.55 




0.60 




0.65 




0.70 




0.75 




0.80 




0.85 




0.90 




0.95 




CDF of standard normal and simulation result of W.R.T. Line: standard normal. Point: simulation result of W.R.T (
In this section, we carry out simulation studies to compare the performance of W.R.T test with Mantel’s [
Generate a failure time
Create a 0, 1 sequence
The observation is
Repeat Step
We consider three return probabilities,
In the case of
Power comparison of tests under exponential distribution with sample


Test 
 



0  0.2  0.4  
W.R.T  0.419  0.131  0.050  0.150  0.371  
0.8  Mantel  0.391  0.120  0.047  0.143  0.362  
Peto  0.385  0.122  0.050  0.140  0.361  
W.R.T  0.383  0.123  0.045  0.132  0.345  
6  0.5  Mantel  0.360  0.121  0.041  0.124  0.344 
Peto  0.345  0.109  0.045  0.124  0.336  
W.R.T  0.313  0.102  0.042  0.103  0.254  
0.3  Mantel  0.307  0.101  0.040  0.096  0.255  
Peto  0.294  0.099  0.040  0.101  0.248  
 


Test 
 


0  0.3  0.6  
 
W.R.T  0.801  0.289  0.047  0.264  0.779  
0.8  Mantel  0.736  0.246  0.051  0.236  0.737  
Peto  0.717  0.242  0.050  0.237  0.740  
W.R.T  0.793  0.278  0.048  0.275  0.712  
10  0.5  Mantel  0.754  0.247  0.045  0.262  0.678 
Peto  0.718  0.240  0.052  0.256  0.663  
W.R.T  0.680  0.238  0.052  0.239  0.662  
0.3  Mantel  0.667  0.215  0.048  0.223  0.640  
Peto  0.624  0.216  0.049  0.224  0.632 
Power comparison of tests under exponential distribution with sample


Test 
 



0  0.2  0.4  
W.R.T  0.710  0.268  0.049  0.196  0.642  
0.8  Mantel  0.678  0.251  0.053  0.192  0.632  
Peto  0.667  0.253  0.054  0.190  0.630  
W.R.T  0.656  0.201  0.050  0.193  0.573  
6  0.5  Mantel  0.636  0.193  0.046  0.184  0.561 
Peto  0.621  0.188  0.047  0.188  0.558  
W.R.T  0.549  0.182  0.058  0.171  0.523  
0.3  Mantel  0.537  0.182  0.057  0.168  0.506  
Peto  0.523  0.181  0.052  0.164  0.501  
 


Test 
 


0  0.3  0.6  
 
W.R.T  0.984  0.520  0.049  0.473  0.945  
0.8  Mantel  0.964  0.472  0.050  0.441  0.930  
Peto  0.957  0.460  0.050  0.439  0.927  
W.R.T  0.971  0.484  0.046  0.448  0.957  
10  0.5  Mantel  0.961  0.458  0.045  0.424  0.946 
Peto  0.948  0.434  0.039  0.415  0.944  
W.R.T  0.942  0.429  0.053  0.402  0.901  
0.3  Mantel  0.927  0.413  0.050  0.387  0.892  
Peto  0.908  0.385  0.060  0.368  0.889 
Density plot of exponential distribution with hazards
Density plot of exponential distribution with hazards
Consider the data of 262 hemophilia patients in De Gruttola and Lagakos [