Censoring Balancing Functions for Undetected Probably Significant Effects in Cox Regression

. Weighted Cox regression models were proposed as an alternative to the standard Cox proportional hazards models where consistent estimators can be obtained with more relative strength compared to unweighted cases. We proposed censoring balancing functions which can be built in a way that allows us to obtain the maximum possible signifcant treatment efects that may have gone undetected due to censoring. Te harm caused by this is compensated and new weighted parameter estimates are found. Tese functions are constructed to be monotonic because even the hazard ratios are not exactly constant as in the ideal case, but are violated by monotonic deviations in time. For more than one covariate, even the interaction between covariates in addition to censoring can lead to the loss of signifcance for some covariates’ efects. Undetected signifcant efects of one covariate can be obtained, still keeping the signifcance and approximate size of the remaining one(s). Tis is performed by keeping the consistency of the parameter estimates. Te results from both the simulated datasets and their application to real datasets supported the importance of the suggested censoring balancing functions in both one covariate and more than one covariate cases.


Introduction
1.1.Background.Survival data analysis has many applications in the real world such as engineering like testing the lifetime of life bulbs and medicine like testing the efciency of diferent treatments and it even found its role in social sciences.Cox proportional hazard models have been hugely exploited since their introduction by Cox [1] and many modifcations have been made with the aim of improving the models' accuracy.In its nature, a Cox model is built based on the hazard (rate) of a given event of interest in a specifc group and the two hazard models corresponding to the two groups under comparison are used to fnd the hazard ratio.Tis is also referred to as the risk ratio, and in Cox proportional hazards models, it is assumed to be proportional over the whole time of the study.
Te hazard rate is sometimes assumed to be associated with covariates which can be continuous or categorical and their efects are deducted from the computed regression parameters.Recently, weighted Cox models attracted the attention of many, and the weights used are the same as those employed in weighted logrank tests.We even recall that the weighted logrank tests are derived from the partial likelihood of the weighted Cox models [2].In our recent work, censoring balancing functions were introduced to balance the negative efects of censoring on the power of such tests and they were found to be of some remarkable improvement.In this work, such functions will be employed to investigate covariate efects.Yu et al. [3] used what they referred to as "censoring correction" in weighted Cox regression by using the inverse probability distribution of censoring, G(t) − 1 combined with the selected weight where G(t) is the Kaplan-Meier estimate of censoring.In their work, they also explored the average hazards ratio and the same technique had been used by Schemper et al. [4].

Parameter Estimation in Cox Regression. Te general form of a Cox proportional hazard model as introduced by
Cox [1] and widely used in many works such as [5], [6], and [7] is expressed as where X is a vector of covariates and β is the Cox regression parameter for the covariates and λ 0 (t) is the baseline hazard function which can be estimated from any parametric distribution.If the vector of covariates X is made of k covariates, such that X � (X 1 , X 2 , . . ., X k ), then β will also be of the same size since for each component X i , there must be the corresponding regression coefcient β i .
If one is interested in the hazard ratio, it can be immediately deducted from the two hazards with the same baseline hazard without knowing or estimating this since it may be canceled out in both quantities.Te covariates can be constant or time-dependent, and in the latter case, they are written as X(t).Te regression coefcient β can also be timedependent.We highlight that when exploring the Cox proportional hazards models, the main interest is in the failure time, let us say, T. Tus, the hazard λ(t|X) can be understood as the rate of failure for the objects that still survive till then.
If t is a failure time, then the likelihood that it is the specifc subject i that fails at that time is given by where R i is the set of all subjects at risk at time t.Te resulting partial likelihood is obtained by multiplying these individual probabilities.Tat is, When we take a more general case where there may be ties in occurrences of events, M event times which were observed, will be reduced to m tied event times, where we will denote by S i , i.e., the sum of covariates' X i of all individuals who underwent the event of interest at time t i and equation (3) will then be rewritten as In our next deductions and computations, we will refer to the notation in equation (3) since it is the one generally used in other research works.Survival data are practically characterized by censoring and the likelihood expression can be rewritten otherwise once the times considered are not specifed as the times of occurrence for the event of interest.
Tis is so because they may observe either the event of interest (such as death, failure, or any other) or the censoring.So, if the censoring times are also recorded, being known that the likelihood is considered and estimated only at time points of occurrence of the event of interest, Lin [5] and Fisher and Lin [8], among others, rewrote equation (3) as where δ i is the indicator function and it is such that δ i � 1 if it is the event of interest that occurred at time t i and 0 in the case of censoring.
Te aim is to fnd the regression parameter β which maximizes the partial log-likelihood function and this is computed as follows: Note that, M is the number of distinct times of occurrence for the event of interest [9] while N is the total number of times including the censoring ones.Any logic from either equation ( 6) or (7) is understandable because at the censoring time, the corresponding quantity in equation ( 5) is raised to power 0 and it gives 1, which, once multiplied by other quantities, brings no change.In the same way, for the log partial likelihood, the sum of logarithms will remain the same since the quantities corresponding to censoring are multiplied by 0.
In the next transformations, we will continue with the form of partial log-likelihood which does not contain δ i and we get To maximize l(t, β) in equation (7), we need to fnd  β which is a solution to the following equation: Equation ( 8) is called the score statistic from the partial log-likelihood, sometimes referred to as U(t, β) and is written as From equation ( 9), the concept of using weight functions to obtain weighted covariate efects was introduced and worked on by many authors including Fisher and Lin [8], León et al. [10], Lin and Wei [11], Lin [5], Murphy et al. [12], and Sasieni [7].

International Journal of Mathematics and Mathematical Sciences
Te weighted score function is obtained from equation (9) where the same weight functions as those used in weighted logrank tests are used.Moreover, the weight function which gives more power to the weighted logrank test is the one also preferred to obtain the regression parameter  β max once U w (t, β) � 0 is solved for β.Te weighted score function is expressed as follows: In Cox regression, hazard rates and survival times of diferent (two or more groups) groups are considered.As stated by Persson and Khamis [13], censoring distribution plays a big role in the negative efects being observed.In their work, the types of censoring under consideration were early, late, or random censoring.All types of censoring heavily afect the estimates when their proportions are relatively higher and the worst case is observed for early censoring when the proportionality of hazards is observed [13].Sasieni [7] highlighted that allocating lower weights to latter failures through artifcial censoring enhances the relative infuence of the outlier and this leads to a higher bias.In our present work, our analysis was performed under the consideration of random censoring.Dunkler et al. [9] stated that the unweighted Cox estimates are relatively more efcient than the weighted ones even though they overtake when the proportional hazards scenario is not met.Moreover, in contrast to the unweighted estimates, the weighted ones remain fairly constant irrespective of the size and type of censoring.Tis relative strength is still kept while calculating the concordance probability because the unweighted estimate tends to underestimate it.Again, even though the unweighted estimate performs well in the PH case, the weighted one remains acceptable since it does not introduce bias.
Te main problem being addressed in this work is the following: "Aren't there cases where negative or positive prognoses are found not signifcant because of censoring while they probably or really are?"If so, what might they be?If they were reported as signifcant, are they the real ones?What may be their possible maximum?Can't low efects be interfered by other covariates in addition to censoring?If yes, can their real strength be optimally detected without afecting other covariates?

Inverse Probability of Censoring Weighting (IPCW)
Method.When other conditions are fxed, the higher the censoring rate, the higher the loss of power of the analysis, or simply the higher the bias of the estimates.Schemper et al. [4] used the Fleming-Harrington (FH (ρ, c)) family of weights and the weight which gave higher power in the weighted logrank tests was the one to be used for the computation of β in Cox regression.To compensate the harm from censoring, Schemper et al. [4] attached on the weight the inverse of the censoring distribution function, often denoted by G(t), and the resulting weight was w(t)G(t) − 1 .G(t) is found the same way as the usual Kaplan-Meier estimates S(t) are found but with the reversed status variable.It is understood that G(t) is decreasing as censoring is observed and hence G(t) − 1 is increasing and is at least equal to 1.
In Yu et al.'s study [3], just as in Schemper et al.'s study [4], the same quantity G(t) − 1 was investigated and used while referred to as "censoring correction."It was still applied in León et al. [10] and Dunkler et al.'s study [9] where it was annexed to diferent weights that were explored to fnd the maximum of treatment efects.It had also been employed separately as an independent weight function by Xu and O'Quigley [14].All those transformations were performed with the aim of maximizing the eforts in restoring the ability to capture the treatment efects, which in most cases is referred to as the hazard ratio.Te concept of weighting in Cox regression led to the name "average hazards ratio" or "average regression efects" where e β is like the weighted mean.Weighting by G(t) − 1 was also explored in logrank tests where it is referred to as the inverse of the probability of censoring weight (IPCW) and it is used to address some biases due to censoring in diferent estimations such as survival probabilities.In general, IPCW has been and is still employed as a response to noncompliance from censoring as found in the studies of Kvamme and Borgan [15], Matsouaka and Atem [16], Satten and Datta [17], Wakounig et al. [18], and Robins and Finkelstein [19].Dong et al. [20] found that an estimate of treatment efects obtained through the IPCW-adjusted win ratio statistic is unbiased.However, Howe et al. [21] demonstrated its limitations in the presence of a strong selection bias or in the case of unmeasured common predictors, sample size, or model misspecifcation.

Censoring-Balanced Weight Function.
Censoring balancing functions were proposed to balance the negative efects of censoring on the respective results.In this work, they will be implied in the computation of the Cox regression coefcients which help to measure the covariates' efects or even hazards ratio.Te proposed censoring balancing function is defned as International Journal of Mathematics and Mathematical Sciences where c(t) indicates the censoring rate until event time t.
Te parameter "r" can vary and take on any value in Z depending on the researcher's judgment and aim.Te aim of this study as mentioned before is fnding the ideal maximum possible value (signifcance) of the treatment efects for the univariate (multivariate) case.Negative values reduce the size of the original weight and hence will act like penalties whereas the positive ones improve it which makes them act like compensations.
As the IPCW through G(t) − 1 on the existing weights brings some improvement, the newly proposed censoring balancing functions are built in a more fexible way which gives them the ability to be modifed diferently to have the desired nature and size.Tis will make their use more friendly and adaptive.
Te general censoring-balanced weight function will be of the following form: Te index CB is used to indicate that the weight under consideration is censoring-balanced.w(t) is the initial weight function and in our case, we will focus on w(t) � 1 as the initial weight and hence will apply diferent levels of compensations by manipulating the censoring balancing function as given above through varying the parameter r.Our main focus is on the average hazard ratio obtained under a specifc weight function.Te negative impact of censoring will be dealt with to see if there is any improvement brought by the censoring balancing function (CBFs) on the quantity exp(  β) which stands for the average hazard ratio.Under random censoring, it is more likely that the treatment efects are underestimated.Te proposed form of censoring balancing weight function will help to compensate for that loss of sensitivity.However, the loss can be overcompensated since in real applications, the true treatment efects are not known.So, we will compute the expected maximum possible treatment efects which may even exceed the true treatment efects.To understand this, a simulated dataset will be used to describe the situation.Since the monotone deviations from the proportionality are the most encountered, the proposed functions will be combined with w(t) � 1 because they themselves will act like weights.So, we do not need to combine them with another weight now, but it can be explored afterward.Terefore, the weighted Cox regression that will be dealt with will be from the weight function of the following form:

Application of the Goodness-of-Fit Test.
Te assumption of proportional hazards can be tested against linear monotone departures where the regression coefcient β will be replaced by a linear monotone function of time, let us say, a(t).Hence, the hypothesis which will be tested is a(t) is assumed to be monotone because the monotone departures from the proportional hazards assumptions are the most generally encountered ones in real scenarios of model misspecifcation.From this, monotone weight functions are the most appropriate since they are more sensitive to such monotone departures.Te weighted Cox regression was found to be efcient when some deviation from the proportionality assumption was observed.Lin [5] proposed a test for goodness-of-ft analysis which remains consistent even under model misspecifcation and helps to detect the diference between the two Cox regression parameters which are β and β w .
So, the same method can also be employed to test the following hypothesis: Tey stated that when the proportionality assumption is held, the two parameters do not difer signifcantly in terms of magnitude or absolute value.We recall that in this work, we will consider diferent levels of censoring balancing functions.Te proposed test is of the following form: where r � 0, 1, 2 and X i (t) is the covariates vector for the individual i whose event time is t (it does not mean that the covariates are time-dependent).
Te following quantities are deduced from S (r) (β, t) according to r: With E(β, t) defned as E(β, t) � S (1) (β, t)/S (0) (β, t), the score function defned in equation ( 10) can be rewritten as 4 International Journal of Mathematics and Mathematical Sciences Te resulting information matrix from which we will get the standard errors for the respective components of β is obtained by the second derivative of the partial loglikelihood function and in terms of S (r) 's, we have By using the E(β, t) defned previously, V(β, t) can be written as To make the notation more logically understandable to the reader, we will write V(β, X i ) instead of V(β, t).Tis is because the summation is made over event times and the subscript i will make the summation more logical.Te variance of the weighted regression estimator is computed as with It is easily noticed that A w (  β) and B w (  β) are diferent only in the sense that they are computed with weights w(X i ) and w(X i ) 2 , respectively.10) will be solved for many options to obtain the optimum parameter.For a single covariate, the efects can be found signifcant or not and, in that case, we will fnd their probable maximum value since censoring is expected to have afected it.For two or more covariates, we try to maintain the parameter component which was found to be signifcant and we try to fnd the solution for equation (10) under diferent values of r where the optimal solution will be the one with a small deviation from the already signifcant component (β r [j] − β[j]) or with nearly the same norm as the initial parameter (|β r | � |β|) but with the signifcance of efects for targeted covariates, β[i].In short, the optimization is performed as follows.

Optimization of the Parameter. Equation (
For a single covariate, the optimization is performed as For two or more covariates, the optimization is performed as or (25) Note that, we avoid the notation β max but used β opt for the case of more than one covariate because the aim is not the maximum in size but the signifcance of efects for as many as possible covariates.León et al. [10] computed β max from a set of weights of the Fleming-Harrington (G ρ,c ) family, but in this study, it will be found by varying r.For each dataset or case, the optimal r is reported together with the censoring level where it gave the intended optimal solution which was also reported and compared with the initially obtained parameter.

Data Description and Analysis
We dealt with two simulated datasets from a Weibull distribution and showed the proportional hazards nature and two real ones.For each of the simulated datasets, the total sample size is 200 with a 1 : 1 allocation ratio.We hence assumed three levels of censoring which are 20%, 40%, and 60%.Te censoring times were assumed randomly from the previously simulated time points.For each censoring level, the regression parameter was estimated.In the case of one covariate, the level of censoring balancing functions which yields the absolute maximum value of the parameter was obtained and this was compared with the one obtained under censoring.For two covariate cases, the alternative regression parameter was estimated with the aim of small deviation from the signifcant component targeting the parameter whose all components are signifcant.Once the targeted weighted parameter was obtained, it was compared with the one previously obtained under censoring using the stated test to see if the obtained parameter is still consistent.
For real data applications, two datasets were used.Tose are the colon cancer dataset and the lung cancer dataset which are freely accessible online and can be imported to diferent data analysis software.Te colon dataset has a total sample size of 1858 with an overall censoring rate of 50.48%.Te covariate of interest was sex where there were 968 males (sex � 1) and 890 females (sex � 0) who were subjected to three treatments and the variable of interest was the time of the event of interest (death) or recurrence.Since the two groups (from the sex variable) show the PH nature, we applied the Cox regression to investigate the sex covariate efects.Te lung cancer dataset is made of an overall sample size of 228 with an overall censoring rate of 27.63% where 90 of them are females (sex � 1 in our analysis) and 138 are males.After performing the analysis with this single covariate, we also analyzed the same dataset with two covariates: "sex" and "age." For each dataset (simulated or real one),  β and  β w were computed, respectively.Te parameter from weighted Cox regression  β w was then compared with the initial parameter from unweighted Cox regression.Tis is why in the following tables, D w (  β), Q w , and the corresponding p values are in the row of weighted regression where r ≠ 0 in the column (c, r).c represents the censoring rate and r is the level of the censoring balancing functions that resulted in the maximum or optimum covariates' efects.Te test statistic International Journal of Mathematics and Mathematical Sciences Q w has a central chi-square distribution with degrees of freedom equal to the dimension of the regression parameter.Te data analysis task was performed in Python.

Simulation Results. 4.2. Application to Real Data. 4.3.
Discussion of the Results.For the simulated datasets, the estimates in the noncensoring cases are obtained.As censoring occurs, the deviation from the true coefcient increases and the smaller components in terms of absolute value can even change the sign.For more than one covariate case, while investigating the efects' signifcance for one covariate, the magnitude of efects for the remaining one(s) may be slightly afected.Low censoring ( ≤ 20%) gives estimates still close to the real ones as seen in Table 1 and the application of censoring balancing functions may mislead since they tend to overcompensate the small harm from the low censoring.Tis can be seen in estimates in the cases of 20 and 16 with 0.54012 vs 0.6829 where obtaining a signifcant component for the failing one will cause a noticeable decrease in the preexisting signifcant component, and again, on 20 and 136, we have 1.2296 vs 0.7411 to show a higher deviation in the case of one covariate (Table 1).Tis is supported by the real data application where the lung dataset with sex covariate is treated separately as shown in Table 2, and the coefcient increase was approximately 66% (from −0.5311 to −0.8816), while for the simulated dataset 1, it was nearly 77.5% (from 0.7411 to 1.2296).
Te censoring level of 40% gives estimates which are relatively far from the real ones.However, when searching for the possible maximum, it does not go as far as the 20% censoring level.Tis means that it is the level where the negative impacts of censoring are unavoidable and the censoring balancing functions will play their role more appropriately (refer to rows 40 and 64 with 0.8503 vs 0.7411 and 40 and 13 with 0.675 vs 0.6829 in Table 1).
Te covariates with small coefcients are more likely to change the sign when it comes to searching for their signifcance when the censoring is too high ( ≥ 50%).Tis can be checked on row 60 and 20 with 0.1312 vs −0.1095 in Table 1 and is supported by the real application on row 50.48 and 7 with 0.1642 vs −0.0336 in Table 2. Te change of sign is logical and can be understood from the fact that nonsignifcant efects are represented by a parameter estimate whose confdence interval includes 0 and hence, it can be either positive or negative while for hypothetical signifcance, it must be on one side and statistically diferent from 0.
So, for colon cancer, if covariate "sex" has undetected signifcant efects, they are more likely positive and they may be of β w � 0.1642 giving the hazard rate of e 0.1642 ≈ 1.18 ∈ (1.05, 1.32).In the same way, for the lung dataset, if the efects of the covariate "sex" treated separately were afected by censoring, whatever they might be, they are likely less than β max � −0.8816 corresponding to the hazards ratio of 0.41 from 0.60.Still, if the covariate age treated together with covariate "sex" has undetected signifcant As we highlighted that r can be positive or negative, the optimal weight obtained in a row (27.63, 1) in Table 2 was also obtained with r � −409 and the diference observed was only in the corresponding 95% CIs for the respective components where the 95% CIs at r � −409 are [−0.5086,−0.4778] and [0.0164,0.0181]which are too much narrower and hence better.Note that, this level of r � −409 was the one which yielded the maximum efects when covariate "sex" was treated separately, but when treated together with age, the same level yielded the signifcance of age covariate efects with a minimum deviation from the existing sex covariate efects.In addition, by referring to the magnitudes of the two parameters, we obtain |β w | � 0.4935 and |β| � 0.5136 with a difference of 0.02 which is relatively small.
In general, by looking at the obtained test statistic in the two tables of the results either from the simulation or real data application, we fail to reject the null hypothesis, and since the p values are relatively higher, it categorically discredits the assumed model.Te same results were obtained on the three datasets analyzed in our reference work of [5].So, there is no signifcant diference between the efects of the covariates under censoring (normally obtained) and those obtained under the use of censoring balancing functions.

Conclusion
Censoring generally negatively afects the results of the Cox regression.Te signifcance of treatment efects can be detected if lost due to censoring but their magnitude (or size) must be controlled with due attention for them to remain realistic.For one covariate case, the possible maximum efects can be estimated.If censoring is very high, for example up to 40 % or above, some covariates can be judged as nonsignifcant while they really are and in some cases, they may even be in the opposite direction.Tis means that the efects were found to be positive (negative) but not significant under censoring while they were negative (positive) and signifcant.Te employed chi-square statistic test helps to know if the newly obtained regression parameter estimate is still consistent by comparing it to the initial one.When the diference is detected, Lin [5] proposed to perform a component-wise comparison of the two parameters.Tis is why the detection of the probably undetected signifcant efects is performed while controlling the deviation in the components which is already signifcant.In this work, the significance of some covariates' efects was detected through the employment of censoring balancing functions without experiencing a signifcant diference between the parameters' magnitudes.
For low censoring (for example, below 20%), the censoring does not have stronger negative efects that the probable existing signifcance might be lost unless probably worsened by smaller sample sizes.For this, censoring balancing functions can mislead the researcher by overestimating the coefcient while the harm from censoring is not really that high.Te reliance on compensation depends on the harm caused.However, the general remark is that the covariates' efects obtained under the use of censoring balancing functions are also consistent as proved by Lin [5] for the general weighted Cox regression.Te use of weight functions defned in the same context as censoring balancing functions (proposed in this work) will help researchers in clinical trials and pharmaceutical studies to maximize the strength of their statistical tests and make sure that no covariates with really signifcant efects are reported as nonsignifcant due to censoring.Te proposed censoring balancing functions can, therefore, be recommended for use while aiming at the investigation of signifcant covariates' efects which were undetected in the presence of censoring.

Data Availability
Te real datasets used in this study are available and freely accessible online and especially in R where both the lung cancer data and colon cancer data are accessed through the "survival" package.

Table 1 :
Results from simulation.International Journal of Mathematics and Mathematical Sciences efects due to censoring, they might be just around 0.01728 meaning that the person is at a higher risk with a hazard ratio of e 0.01728 ≈ 1.02 but it is still less than 1.04 compared to the one with the same sex to whom they are one year older.In that case, females are at a lower risk than males of the same age with a hazard ratio of e − 0.4932 ≈ 0.61.

Table 2 :
Results from real data application.