Introduction

To assess the effectiveness of the treatment provided to patients and the risk of death during a given period, many cancer healthcare facilities design information systems that analyze data and assess patient survival using the Kaplan-Meier curve [1]. Three nonparametric tests are usually used in the survival analysis based on the Kaplan−Meier estimator: the Cochran-Mantel-Hansel log-rank test [2], the Wilcoxon test [3], and the Tarone-Ware test [4]. To test statistical hypotheses about the identity of the survival functions, these tests mainly calculate the values of the standard normal distribution. However, these tests give reliable results only if the samples are large enough and additional conditions are met. The most popular is the log rank test, which gives the maximum power under the alternatives with proportional hazards [5]. However, these tests give reliable results only if the samples are large enough and additional conditions are met. For example, the Wilcoxon test is preferable when deaths at early time points have more weights [6], and the Tarone-Ware test also places more heavy weight on hazards at the early time [7].

The nonparametric Kaplan-Meier estimate measures the survival time of patients, i.e. the interval of time between a certain date (for example, the date of surgery) and the moment of death or censuring. It allows the construction of survival functions based on data on the life expectancy of patients and estimates the risk of death during a given time period. Similarly, it can be used to estimate the time to equipment failure or other significant event. Thus, it can be used for assessment of the risk of a specific event (death, failure, etc.) based on observations (censored and uncensored).

The aim of this paper is to describe an alternative nonparametric test that does not use any assumption excepting the most general (continuity of the distribution) and allow using small samples (size less than 50). This test use the p-statistics investigated in [8][9][10][11] and base on the A(n) Hillʼs assumption [12]. The theoretical background of the p-statistics is developed by Matveichuk and Petunin [8,9] and later by Johnson and Kotz [10], and Klyushin and Petunin [11]. The high sensitivity and specificity of the nonparametric test for homogeneity of two samples based on the p-statistics is demonstrated in [11]. Here we propose new application of this test for comparison of two survival curves.

Theoretical backgroundi P x x x j i n − ∈ = < + (1)) ( ) ( ) ( ) , , , 1 i j j

where ( ) i x and ( ) j x are the i-th and j-th order statistics. Find the relative frequency ij h of the event

( ) ( ) ( ) , m i j y x x ∈

for the elements of y and estimate the deviation of ij h from the expected probability 1

j i n − + using the Wilson confidence interval ( ) ( ) ( ) ( )1 2

,

n ij ij ij I p p = where 2 2 (1) 2 2 2 (2) 2 2(1 ) 4 , 2 (1 ) 4

.

ij ij ij ij ij ij ij ij h n g g h h n g p n g h n g g h h n g p n g + − − + = + + + − + = +(2)

The significance level of this interval is the function of g. When g = 3 the significance level of ( ) n ij I does not exceed 0.05 [11]. P-statistics, estimating the homogeneity of samples x and y, is

( ) ( ) 1 # 1 2 n ij ij n n j i h p I n  −  −   = = ∈     +     ,(3)

It is the relative frequency of the event ( )

1 n ij ij j i p I n −   = ∈   +  

. Therefore, using ( 2) and (3) we may construct the Wilson interval I for the p-statistics an formulate the following test: the null hypothesis on identity of the survival functions is accepted if the upper bound of I is greater than 0.95, else it is rejected.

For the true null hypothesis is true, the events ( )

1 n ij ij j i p I n −   = ∈   +   form a generalized

Bernoulli scheme [8,9]. For the false null hypothesis they form a modified Bernoulli scheme. If the null hypothesis may be either true or false, they form the Matveichuk-Petunin scheme [10].

If the null hypothesis is true,

( ) lim 0,1 1 n j i n →∞ − ∈ + ,and( ) lim 0,1 1 n i n →∞ ∈ +

, then the asymptotic significance level β of a sequence of confidence intervals ( )

n ij

I is less than 0.05 [11].

Experiments and results

To confirm the high sensitivity and specificity of the proposed test, we considered two groups of patients with a nondifferential diagnosis of bladder cancer of stages T2 and T3, who in 1998-2016 received special surgical care (radical and salvage cystectomy) at the urology department of the Kiev City Clinical Oncological Dispensary. For the analysis, patients were taken who had a complete history and an accurate survival result (uncensored). Characterization of the prevalence of the malignant process was carried out according to the clinical classification TNM 7th ed. (2010).

The first group (stage T2) consists of 38 patients, among them 22 patients were underwent to radical cystectomy (17 died and 5 are alive), and 16 were underwent to the salvage cystectomy (7 died and 9 are alive). The second group (stage T3) consists of 51 patients, among them 33 patients were underwent to radical cystectomy (24 died and 9 are alive), and 18 were underwent to the salvage cystectomy (10 died and 8 are alive). The survival curves for the first and second groups are demonstrated in Fig. 1 As we see, in the first group (stage T2) the survival curve of the patients who were underwent to radical cystectomy goes above the survival curve of the patients who were underwent to salvage cystectomy. Therefore, intuitively, the risk of death for the former patients is less than for the latter ones and the salvage cystectomy prolongs life of patients better than the radical cystectomy. However, this hypothesis must be rigorously tested using statistical tests. Traditionally, to estimate the significance of the deviation between to survival curves the logrank test, the Wilcoxon test, and the Tarone-Ware are used. Their p-values are the critical values of these tests. In the second group (stage T3) the survival curve of the patients who were underwent to radical cystectomy also goes above the survival curve of the patients who were underwent to salvage cystectomy. We again may suppose that the risk of death for the former patients is less than for the latter ones. Note, that since the stage T3 is harder that T2, the survival interval became mush shorter. The maximum survival time in the first group is avout 4000 days (almost 11 years) but in second group it is about 2500 days (almost 7 years). Thus, the effectiveness of the cytectomy in this group is compensated by the stage of tumors. To estimate the significance of the deviation between to survival curves we again used the log-rank test, the Wilcoxon test, and the Tarone-Ware and their p-values.

In both cases we completed the traditional analysis by computing the p-statistics as an alternative to the three above tests. Descriptive statistics of the data are provided in Tables 1-3 The hypothesis of the identity of the two survival functions (0 -the salvage cystectomy and 1 -the radical cystectomy) in the first and second groups (stages T2 and T3, respectively) was tested using four tests at a significance level of 0.05. In all the results, there were no statistically significant differences between the survival curves, since the observed values did not exceed the critical value and the upper confidence bound for the p-statistics exceeds 0.95. The log-rank test, the Wilcoxon test and the Tarone-Ware test acceps the null hypothesis is the corresponding pvalues are less than 0.05, and the test based on the p-statistics, in opposite, accepts the null hypothesis if its p-value is greater than 0.05. Noteworthy is the fact that the observed p-value (the probability of rejecting the null hypothesis, provided that it is true) in the p-statistics test is an order of magnitude less than in the three traditional nonparametric tests used in the analysis of survival. This is the evidence of high sensitivity and specificity of the proposed test.

Conclusions

Mathematical basis of modern medical information systems for assessing the effectiveness of treatment and the risk of death during a given time period must be more rigorously justified. Traditional nonparametric tests used in survival analysis (the log-rank test, the Wilcoxon test, and the Tarone−Ware test) assume conditions that not always are met in practice. These tests reduce the verification of statistical hypotheses about the equivalence of survival functions to calculating the critical value of the standard normal distribution. This is justified only when samples are large enough and additional conditions are met. Thus, to develop an effective medical information system for survival analysis, we need in nonparametric tests with minimal preliminary assumptions and minimal requirements to the size of samples.

In paper, we described a test for verification of the hypothesis of the equivalence of the survival functions and risk of death during a given time period, which does not depend on the sample size and does not use additional preconditions, except for the condition that the samples have not ties.

We have provided the strong mathematical background and demonstrated high sensitivity and specificity of testing homogeneity of two samples of random samples from continuous distributions in comparison with three traditional tests. We have shown the practical application of this test in survival analysis of the patient with bladder cancer and demonstrated its high performance. This test may be used for the development of effective medical information systems that perform survival analysis of cancer patients. Note, that the scheme described in the paper is easily expanded on much wider spectrum of problems connected with the assessment of the risk of device failure or risk of some significant event based on the censored and uncensored observations. Future work will be directed to the improvement of computational complexity of the proposed test and its expanding to the various problem of the risk assessment.