IPCW approach for testing independence

Here we present a novel inverse probability of censoring weighted (IPCW) adaptation of the Kochar–Gupta (KG) test of independence, in the case of bivariate randomly censored data. Three different censoring schemes are considered: one of the target variables is censored, both targeted variables are censored with the same censoring variable, and both target variables are censored with different censoring variables. The limiting properties of test statistics are explored. In order to compare the tests with a few well-known competitors in terms of powers, several resampling procedures have been utilised to approximate the null distribution. Special attention is dedicated to the comparison with a classical adaptation of the KG test related to the IPCW adaptation of U-statistics.


Introduction
Identifying and modelling of the dependence between two-lifetime variables are commonly met goals of biomedical studies.Therefore, a proposal of different association measures is still an active research topic.For an overview, we refer to Kotz, Balakrishnan, and Johnson (2004).Moreover, many independence tests have been proposed thus far (see e.g.De Wet 1980;Csörgő 1985;Hoeffding 1994;Genest and Rémillard 2004;Genest, Nešlehová, Rémillard, and Murphy 2019).Some of the proposed tests were specifically designed to test against special notions of stochastic dependence (Kochar and Gupta 1990;Shetty and Pandit 2003, against positive quadrant dependence, Amini-Seresht and Milošević 2020 against positive regression dependence etc.).
In this paper we focus on independence tests related to the well known Kendall concordance coefficient which is one of the most popular association measures between a pair of lifetime random variables (X, Y).It is defined by where (X 1 , Y 1 ) and (X 2 , Y 2 ) are independent copies of (X, Y).Based on the i.i.d.sample (X 1 , Y 1 ), . . ., (X n , Y n ) it can be naturally estimated as the difference between the average numbers of pairs that are in concordance and disconcordance, i.e.
Unlike the well-known Pearson correlation coefficient, Kendall coefficient does not require knowledge about the parametric form of the marginal distributions, which is rather important in practical applications.
The estimator (1) can also be expressed as I{argmax l∈{i,j} X l = argmax l∈{i,j} Y l } − 1. (2) This form may be regarded as a starting point for potential further generalisations.One such generalisation is the class of statistics m ∈ {2, 3, . . ., n − 1}, proposed by Kochar and Gupta (1990).They showed that statistic (3) is more efficient for m > 2 against several commonly used alternatives.For other tests based on the modification or the generalisation of the Kendall coefficient we refer to e.g.Bergsma and Dassios (2014) and Shi, Hallin, Drton, and Han (2022).
All previously mentioned tests were designed for complete data.However, in biomedical studies, a sample is often limited due to some censoring mechanisms.Hence, adapting tests for such data is of great importance and there are many adaptations of the Kendall coefficient known so far.The first few modifications were proposed by Brown, Hollander, and Korwar (1974), Weier and Basu (1980) and Oakes (1982), but none of the estimators was consistent when the true value of the Kendall coefficient is not equal to zero, i.e. when the marginals are dependent.The improvement of their estimators is considered in Wang and Wells (2000) in which authors proposed an estimator for Kendall coefficient expressed as an integral of an estimate of the bivariate survival function.Another improvement has been proposed in Lakhal, Rivest, and Beaudoin (2009).These authors exploited the fact that the estimator for complete data can be represented as a U-statistic, and determined when its kernel function can be calculated for censored data.Next, they used Inverse Probability Censoring Weighted (IPCW) approach (see Robins and Rotnitzky 1992), to propose an estimator.
In this work we focus on the adaptations of statistic V n,m for randomly censored bivariate data.The focus is on the case m = 3, while the general expressions of adapted test statistics for m > 3 are given in Appendix 2. In Section 2, we propose novel adaptations of statistic (5) for three different models of randomly right-censored data, while their limiting proprieties are studied in Section 3.All proofs are provided in Appendix 1. Section 4 contains a wide empirical study, while the tables with empirical sizes and powers are given in Appendix 3. Additional simulation results are presented in the Supplementary material.Section 5 contains two real data examples which demonstrate the application of the proposed methodology.

Adaptations of the Kochar-Gupta test for independence
In what follows we focus on the case when m = 3, while the case of arbitrary m ≥ 3 is discussed in Appendix 2. Let us introduce the following notation Here argmed l∈{i,j,k} X l is equal to the index of the median value of subsample X i , X j , X k , while the operator argmed l∈{i,j,k} Y l is defined analogously.
Then the statistics V n,3 , in the complete data case, can be represented as Let (X 1 , Y 1 ), . . ., (X n , Y n ) be the sample of independent and identically distributed (i.i.d.) copies of (X, Y) and (C X 1 , C Y 1 ), . . ., (C X n , C Y n ) the i.i.d.sequence of censoring random variables.In addition, let us assume that the mentioned sequences of survival times {(X i , Y i )} and censoring times {(C X i , C Y i )} are mutually independent.Let F X , F Y be the marginal distribution functions of X and Y respectively, and F X , F Y associated survival functions, and K X and K Y survival functions of the censoring random variables C X and C Y .Assume also that K X (x−) > 0 and K Y (x−) > 0 for all x ∈ R + . Let , the corresponding censoring indicators.Therefore, the starting point for the adaptation of tests is the sequence ( Datta, Bandyopadhyay, and Satten (2010), we may start with classical inverse probability censoring weighted (IPCW) estimator ijk , t Y ijk are defined as in (4) for X i , X j , X k and Y i , Y j , Y k .The expression for ω ijk and its nonparametric estimator (denoted by ωijk ) depend on censoring models introduced in the sequel.
By noticing that we can observe the value I{f X ijk = f Y ijk } and I{t X ijk = t Y ijk } not just in the case when all ith, jth, and kth observations are registered, following the approach from Lakhal et al. (2009), we propose the following estimator where while the D ijk is defined differently for different censorship schemes.One can notice that if the value of the kernel function can be calculated, then X f X ijk and X s X ijk are available observations.
In the following text, we present the expressions for ω ijk and D ijk in three different censoring scenarios.
• Censoring scheme 1 -only the first coordinate is randomly censored with C X : where and the weights are given by and • Censoring scheme 2 -both coordinates are randomly censored with different independent censors C X and C Y : where and the weights are equal to and • Censoring scheme 3 -both coordinates are randomly censored with the same censor C X = C Y = C, and its survival function is denoted by K: The weights p ijk and q ijk are equal to and Since the censoring distributions are unknown, weights ω ijk , p ijk , and q ijk are estimated by replacing K X , K Y , and K with their Kaplan-Meier estimators.The weights estimators are labelled with ωijk , pijk , and qijk , respectively, and the resulting test statistics are Remark 2.1: Following the reasoning from Cuparić and Milošević (2022), and the properties of Kaplan-Meier estimator presented in Zhou (1991), we have that Therefore, the stochastic limit in probability of Vc n,3 coincides with the one of V c n,3 .The same holds for statistics Tc n and T c n .Applying the Law of large numbers for U-statistics as well as the mean preserving property of their kernels (since they are obtained using IPCW method, see e.g.Datta et al. 2010), we have that both V c n,3 and T c n converge in probability to As a consequence, we immediately get that the tests | Vc n,3 | and | Tc n | will be consistent against alternatives with positive and negative quadrant dependence properties (see Kochar and Gupta 1990).

Limiting properties under the null hypothesis
In this section, we present limiting null distributions of introduced test statistics under the null hypothesis of independence in the first two censoring scenarios, while the third scenario is briefly discussed.All proofs are given in Appendix 1.
Let N X i (t) = I(X i ≤ t, δ X i = 0) be the right-censoring counting process for the ith individual and is the associated martingale defined with respect to the filtration and X (t) is the cumulative censoring hazard rate function.Finally, , and {M Y i (t)} are defined analogously.For more details about introduced processes, we refer to e.g.Fleming and Harrington (2011).Let us also introduce the following sets of assumptions that are important for deriving limiting properties of the test statistics.
Remark 3.1: Assumptions A and B are not too restrictive.For example, in Koziol-Green model (see Koziol and Green 1976), that assumes the following relation between the null and censoring distribution where β > 0 is an unknown parameter, the Assumptions will be satisfied if there is no more than 50% of censored observations (i.e.β < 1).
In the next two theorems, we present the limiting null distributions of the considered test statistics in the case of Censoring schemes 1 and 2, respectively.

sample and the null hypothesis and Assumption
, where where Remark 3.2: Using the same reasoning as in e.g.Cuparić and Milošević (2022), it might be shown that the asymptotic variances σ 2 V and σ 2 T can be consistently estimated with , and F X , and K X are Kaplan-Meier estimators of F X and K X , respectively.
, where where and The consistent estimators of σ 2 T and σ 2 V can be constructed analogously to those presented in Remark 3.2.
Remark 3.3: The limiting properties of test statistics in the case of Censoring scheme 3 can be obtained analogously (by changing Assumptions A and B into dK(z) < ∞, respectively).However, the expressions for limiting variances are too cumbersome to display.Thereforet, we omit them here.

Empirical study
In this section, we compare the empirical powers of novel test statistics, in three mentioned censoring schemes, with the following two adaptations of the original Kendall statistic given in (2): where • modified IPCW adaptation (see Lakhal et al. 2009): where Since the null distributions of all considered test statistics depend on unknown distributions, in small and moderate sample size cases, there is a need for a suitable resampling procedure.In the first two censoring schemes, we use the bootstrap Algorithm 1.When the second coordinate is not censored the algorithm can be used by changing δ Y i = 1 for i = 1, 2 . . ., n.

Remark 4.1:
The smoothing of the Kaplan-Meier estimator, mentioned in Step 3 of bootstrap Algorithm 1, is done via the monotone cubic spline.
In the third censoring scheme, the previous algorithm is modified in Step 2. Then the estimator of K is obtained using joined sample (max Other steps are straightforwardly modified.Besides the introduced bootstrap algorithm, we also explore the feasibility of the limiting normal distribution of the tests determine the value of the test statistic for application in practice.For the limiting variance, the empirical jackknife procedure is utilised (see e.g.Wasserman 2006).
In order to achieve a certain censoring rate p, we use the Koziol-Green model given in (12) for β = p 1−p .We also consider the case with the Lindley censor.Additional details are provided in the Supplementary material.
The empirical sizes of test statistics, under various null models with exponential marginals (denoted by E(λ)), obtained using Monte Carlo procedure with N = 2000 replicates and B = 1000 bootstrap cycles, are presented in Tables A1, A3, and A6 in Appendix 3. The empirical sizes obtained using the jackknife procedure, with N = 2000 Monte Carlo replicates, are given in brackets.
We can see from the tables that the modified IPCW approach resulted in well calibrated tests, which is not always the case when the traditional IPCW approach is used.It is also important to note that the use of the jackknife procedure is justifiable for a small censoring rate (although the tests are more liberal than expected), while for the censoring rates larger (or equal) than 30%, this approach requires larger sample sizes.
For alternative distributions, we consider bivariate distributions with exponential marginals generated with: • Farlie-Gumbel-Morgenstern copula (labeled by FGM(α)), i.e. with survival function • Clayton copula (labeled by CLAYTON(α)), i.e. with survival function The distributional parameters chosen in the study correspond to the values of Kendall's τ equal to 0.1 and 0.2.
The results are presented in Tables A2, A4, A5, and A7 in Appendix 3. In general, the modified IPCW approach resulted in much more powerful tests than a the classical approach.The reason for that presumably lies in the fact that with the modified approach we are using a larger effective sample size, and consequently extracting more information from the available data.Also, when comparing the jackknife and bootstrap approaches (for small censoring rate and n = 100, since for others the comparison is not justifiable), we observe rather similar behaviour (sometimes in favour of bootstrap sometimes not).When comparing Tc n and Ŝc n , the first one is shown to be a better solution in the case of Censoring schemes 1 and 3, while when both coordinates are censored with independent censors (Censoring scheme 2), Ŝc n is performing better.

Real data applications
In this section, we present two applications on real data sets.The first data set is taken from the study of the antianginal effect of oral isosorbide dinitrate (ISDN) provided by Danahy, Burwell, Aronow, and Prakash (1977).The researchers considered the length of exercise time required to induce angina pectoris in 21 heart disease patients.Let T 1 be the exercise time to angina pectoris in the control period, and T 2 , T 3 , and T 4 the exercise times to angina pectoris 1, 3, and 5 hours after taking oral isosorbide dinitrate, respectively.Censoring may occur due to fatigue-limited exercise rather than angina.Variable T 2 has 6 censored observations (approximately 28%), T 3 has 4 censored observations (approximately 19%), and T 4 has two censored observations (approximately 10%), while T 1 has no censored observations.In order to investigate the impact of the undertaken drugs, we test the independence of T 1 and T 2 , T 1 and T 3 , and T 1 and T 4 .The p-values of considered tests obtained using the bootstrap procedure are presented in Table 1.Since the sample size is small, we excluded the results obtained using the jackknife approach.The results indicate dependence of T 1 and T 3 , as well as of T 1 and T 4 , while the same conclusion cannot be made for T 1 and T 2 .These results can be interpreted by looking into the period of drug effectivenes after intake.The effect of the drug is significant after 1 hour (which reflects on the independence of considered exercise times), while after 3 hours the drug effects weaken and the dependence among considered variables becomes notable.As expected, variable dependence seems to be even more significant after 5 hours.
The second data set describes the time from insertion of a catheter into dialysis patients until it has to be removed due to infection, denoted by T 1 , and time after it is reinserted, denoted by T 2 (for details see McGilchrist and Aisbett 1991).Two successive recurrence times, measured from insertion until the next infection, were recorded for 38 patients.Censoring may be because of the removal for other reasons or the end-of-study effect (for the second infection).The variable T 1 has 6 censored observations, and T 2 has 12 censored observations.All tests did not reject null hypotheses (p-values are greater than 0.1) which is in concordance with the results presented in Wang and Wells (2000).

Concluding remarks
In this paper, we propose two modifications of the Kochar-Gupta test for independence and explore their limiting and finite sample properties under three different censoring scenarios.The powers of novel tests are compared to that of adaptations of Kendall's τ for considered data scenarios, in a wide empirical study.In general, the tests constructed via the modified IPCW approach were shown to be uniformly more powerful than those constructed via the classical IPCW approach for U-statistics.In addition, when they are inter compared, the novel tests show rather competitive behaviour in most of the considered censoring scenarios.
We conclude this paper by highlighting some potential directions for further research.First, it would be of interest to consider the generalisation of Censoring scheme 2 when two censors are not necessarily independent.Although the limiting null distributions might be determined using similar arguments as before, the bootstrap procedure would require substantial changes.We also found it interesting to obtain results when the censoring depends on covariates or to consider the more general case with double censoring.Other possible directions include the use of independence tests for marginal screening of high dimensional data from practical and theoretical perspectives (see e.g.Edelmann, Hummel, Hielscher, Saadati, and Benner 2020).

Appendices Appendix 1. Proofs
Proof of Theorem 3.1: First we prove the statement about the limiting null distribution of statistic √ n Vc n .Since, under H 0 , it can be easily shown that The idea is to approximate √ nV c n with the sum of i.i.d.random variables, and then, using martingale representation of Kaplan-Meier estimator of K X , to do so with n is a U-statistic.The first projection of its kernel is equal to By arranging the expressions above, we get II part: Using martingale representation (see Lakhal et al. 2009) and that (see Zhou 1991) as well as the law of large numbers for U-statistics (see e.g.Korolyuk and Borovskikh 1994) and applying the central limit theorem we finish the proof of the II part, from which follows the limiting distribution of √ n Vc n .Now, we focus on statistic like before, we split the proof into two parts.In part A we approximate √ nT c n with the sum of i.i.d.random variables, and then in part B, using martingale representation of Kaplan-Meier estimator of K X , we do so with n is a U-statistic.The first projection of its kernel is equal to Using independence of (X, δ X ) and Y, and as well as (A2) we get Taking into account (A2), and the independence of (X, δ X ) and Y, using similar reasoning as before we have Finally, we obtain that the first projection is equal to Using (A3) and the law of large numbers for U-statistics, we obtain Similarly, and, finally, Applying CLT, we obtain that under H 0 , √ n Ûn has N (0, σ 2 ).

Proof of Theorem 3.2:
The proof is analogous to the proof of Theorem 3.1.Therefore we present just the main steps.For both statistics Vc n and Tc n the representations (A1) and (A4) holds.V c n and T c n are nondegenerate U-statistics with the first projections given by and . Combining the expansions above and Hoeffding representation of U-statistics, we finish the proof.
Appendix 2. Generalization of statistics Vc n,3 and Tc n In this section, we introduce the adaptation of the Kochar-Gupta test statistic V n,m for arbitrary m ≥ 3, using both presented approaches (classical IPCW approach and its modification).
. sample following Censoring scheme 2 and the null hypothesis and Assumptions A.1.and A.2 hold.
the test statistic ϒ n ; 2: Obtain Kaplan--Meier estimators KX , and KY ; 3: Obtain smooth Kaplan--Meier estimators F X , and F Y of F X , and F Y , respectively.