A Test of Equality of Survival Distributions for Two Sample Censored Grouped Survival Data

A common testing problem for a life table or survival data is to test the equality of two survival distributions when the data is both grouped and censored. Several tests have been proposed in the literature which require various assumptions about the censoring distributions. It is shown that if these conditions are relaxed then the tests may no longer have the stated properties. The maximum likelihood test of equality when no assumptions are made about the censoring marginal distributions is derived. The properties of the test are found and it is compared to the existing tests. The fact that no assumptions are required about the censoring distributions make the test a useful initial testing procedure.     

A linear rank test for use when the main interest is in differences in cure rates

For diseases with a positive probability of being cured, a family of alternatives to the null hypothesis of equality of survival distributions is introduced, which is designed to focus power against alternatives with differences in cure rates. The optimal linear rank test for this alternative is derived, and found to be substantially more efficient than the log-rank test for this alternative when cure rates are less than 50%, while there is little difference between the tests if the cure rates are 50% or greater. The simple test based on the difference of Kaplan-Meier estimates of the proportion cured is also examined, and found to be fully efficient for this alternative with no censoring, while its efficiency rapidly drops as censoring is increased. The new test is not a pure test of equality of cure rates when the data are censored, but rather is a test of equality of survival distributions that focuses power against late differences in the survival curves.     

Sample-size formula for the proportional-hazards regression model

A formula is derived for determining the number of observations necessary to test the equality of two survival distributions when concomitant information is incorporated. This formula should be useful in designing clinical trials with a heterogeneous patient population. Schoenfeld (1981, Biometrika 68, 316-319) derived the asymptotic power of a class of statistics used to test the equality of two survival distributions. That result is extended to the case where concomitant information is available for each individual and where the proportional-hazards model holds. The loss of efficiency caused by ignoring concomitant variables is also computed.     

A model-based statistic for detecting molecular markers associated with complex survival patterns in early-stage cancer

ABSTRACT: BACKGROUND: : In early-stage of cancer, primary treatment can be considered as effective at eliminating the tumor for a non-negligible proportion of patients whereas for the others it leads to a lower tumor burden and thereby potentially prolonged survival. In this mixed population of patients, it is of great interest to detect complex differences in survival distributions associated with molecular markers that potentially activate latent downstream pathways implicated in tumor progression. METHOD: : We propose a novel model-based score test designed for identifying molecular markers with complex effects on survival in early-stage cancer. From a biological point of view, the proposed score test allows to detect complex changes in the survival distributions linked to either the tumor burden or its dynamic growth. RESULTS: : Simulation results show that the proposed statistic is powerful at identifying departure from the null hypothesis of no survival difference. The practical use of the proposed statistic is exemplified by analyzing the prognostic impact of Kras mutation in early-stage of lung adenocarcinomas. This analysis leads to the conclusion that Kras mutation has a significant negative prognostic impact on survival. Moreover, it emphasizes that the complex role of Kras mutation on survival would have been overlooked by considering results from the classical logrank test. CONCLUSION: With the growing number of biological markers to be tested in early-stage cancer, the proposed score test statistic is a powerful tool for detecting molecular markers associated with complex survival patterns.     

Testing equality of survival functions based on both paired and unpaired censored data

We introduce two test procedures for comparing two survival distributions on the basis of randomly right-censored data consisting of both paired and unpaired observations. Our procedures are based on generalizations of a pooled rank test statistic previously proposed for uncensored data. One generalization adapts the Prentice-Wilcoxon score, while the other adapts the Akritas score. The use of these particular scoring systems in pooled rank tests with randomly right-censored paired data has been advocated by several researchers. Our test procedures utilize the permutation distributions of the test statistics based on a novel manner of permuting the scores. Permutation versions of tests for right-censored paired data and for two independent right-censored samples that use the proposed scoring systems are obtained as special cases of our test procedures. Simulation results show that our test procedures have high power for detecting scale and location shifts in exponential and log-logistic distributions for the survival times. We also demonstrate the advantages of our test procedures in terms of utilizing randomly occurring unpaired observations that are discarded in test procedures for paired data. The tests are applied to skin graft data previously reported elsewhere.     

The efficiency of a weighed log-rank test under a percent error misspecification model for the log hazard ratio.

For comparison of two survival distributions, it is natural to use a weighted log-rank test with weight function given by the log hazard ratio function that is anticipated a priori. This paper investigates the efficiency of this test when the a priori estimate of the log hazard ratio is subject to a specified percentage error. The test is shown to be the maximum efficiency robust test over the class of alternatives in question and a simple expression for the maximum efficiency is established.     

Identification and efficacy of longitudinal markers for survival

Methods for the combined analysis of survival time and longitudinal biomarker data have been developed in recent years, with most emphasis on modelling and estimation. This paper focuses on the use of longitudinal marker trajectories as individual-level surrogates for survival. A score test for association which requires only standard methods for implementation is derived for the initial identification of candidate biomarkers. Methods for assessing efficacy of markers are discussed and a measure contrasting conditional and marginal distributions is proposed. An application using prothrombin index as biomarker for survival of liver cirrhosis patients is included.     

Statistical Inference Methods for Two Crossing Survival Curves: A Comparison of Methods

A common problem that is encountered in medical applications is the overall homogeneity of survival distributions when two survival curves cross each other. A survey demonstrated that under this condition, which was an obvious violation of the assumption of proportional hazard rates, the log-rank test was still used in 70% of studies. Several statistical methods have been proposed to solve this problem. However, in many applications, it is difficult to specify the types of survival differences and choose an appropriate method prior to analysis. Thus, we conducted an extensive series of Monte Carlo simulations to investigate the power and type I error rate of these procedures under various patterns of crossing survival curves with different censoring rates and distribution parameters. Our objective was to evaluate the strengths and weaknesses of tests in different situations and for various censoring rates and to recommend an appropriate test that will not fail for a wide range of applications. Simulation studies demonstrated that adaptive Neyman’s smooth tests and the two-stage procedure offer higher power and greater stability than other methods when the survival distributions cross at early, middle or late times. Even for proportional hazards, both methods maintain acceptable power compared with the log-rank test. In terms of the type I error rate, Renyi and Cramér—von Mises tests are relatively conservative, whereas the statistics of the Lin-Xu test exhibit apparent inflation as the censoring rate increases. Other tests produce results close to the nominal 0.05 level. In conclusion, adaptive Neyman’s smooth tests and the two-stage procedure are found to be the most stable and feasible approaches for a variety of situations and censoring rates. Therefore, they are applicable to a wider spectrum of alternatives compared with other tests.     

Nonparametric comparison of two survival-time distributions in the presence of dependent censoring

When testing the null hypothesis that treatment arm-specific survival-time distributions are equal, the log-rank test is asymptotically valid when the distribution of time to censoring is conditionally independent of randomized treatment group given survival time. We introduce a test of the null hypothesis for use when the distribution of time to censoring depends on treatment group and survival time. This test does not make any assumptions regarding independence of censoring time and survival time. Asymptotic validity of this test only requires a consistent estimate of the conditional probability that the survival event is observed given both treatment group and that the survival event occurred before the time of analysis. However, by not making unverifiable assumptions about the data-generating mechanism, there exists a set of possible values of corresponding sample-mean estimates of these probabilities that are consistent with the observed data. Over this subset of the unit square, the proposed test can be calculated and a rejection region identified. A decision on the null that considers uncertainty because of censoring that may depend on treatment group and survival time can then be directly made. We also present a generalized log-rank test that enables us to provide conditions under which the ordinary log-rank test is asymptotically valid. This generalized test can also be used for testing the null hypothesis when the distribution of censoring depends on treatment group and survival time. However, use of this test requires semiparametric modeling assumptions. A simulation study and an example using a recent AIDS clinical trial are provided.     

Sample size and power for the weighted log-rank test and Kaplan-Meier based tests with allowance for nonproportional hazards

Asymptotic distributions under alternative hypotheses and their corresponding sample size and power equations are derived for nonparametric test statistics commonly used to compare two survival curves. Test statistics include the weighted log-rank test and the Wald test for difference in (or ratio of) Kaplan-Meier survival probability, percentile survival, and restricted mean survival time. Accrual, survival, and loss to follow-up are allowed to follow any arbitrary continuous distribution. We show that Schoenfeld's equation—often used by practitioners to calculate the required number of events for the unweighted log-rank test—can be inaccurate even when the proportional hazards (PH) assumption holds. In fact, it can mislead one to believe that 1:1 is the optimal randomization ratio (RR), when actually power can be gained by assigning more patients to the active arm. Meaningful improvements to Schoenfeld's equation are made. The present theory should be useful in designing clinical trials, particularly in immuno-oncology where nonproportional hazards are frequently encountered. We illustrate the application of our theory with an example exploring optimal RR under PH and a second example examining the impact of delayed treatment effect. A companion R package npsurvSS is available for download on CRAN.     

Correlation Goodness‐of‐fit Test for the Logarithmically‐Decreasing Survival Distribution

In this paper, we use the correlation‐type goodness‐of‐fit test for the logarithmically‐decreasing survival distribution. This model was intoduced by Sultan , Balakrishnan and Childs (2001) as a special case of Type‐I truncated logistic distribution. The power of the test based on normal, Weibull and gamma distributions is also calculated. We also give application to real example.     

Separating Components of Variation in Survival of Mule Deer in Colorado

ABSTRACT Survival is an important parameter for understanding population dynamics of mule deer (Odocoileus hemionus) and other large herbivores. To understand long-term dynamics it is important to separate sampling and biological process variation in survival. Moreover, knowledge of correlations in survival across space and between young and adults can provide more informed predictions of survival in unsampled areas. We estimated survival of fawn, yearling, and adult mule deer from 4 spatially separated regions of Colorado, USA, from 1997 to 2008. We also estimated process variance in survival across time for each age and site using Markov chain Monte Carlo (MCMC) methods. Finally, we estimated correlations in survival among sites and ages with MCMC methods. Average winter fawn survival was 0.721 (SD = 0.024) for the 4 regions. Average winter adult female survival was 0.935 (SD = 0.007). Annual adult female survival ranged from 0.803 (SD = 0.017) to 0.900 (SD = 0.028) for the 4 regions, excluding hunting mortality. The correlation between fawn and adult female survival was high, 0.563 (SD = 0.253). Correlations in winter fawn survival were higher between populations at the same latitude than they were for populations to the north and south. We used survival estimates from our analysis to inform prior distributions for a Bayesian population dynamics model from one population in Colorado and compared that model to one with noninformative prior distributions. Population models including informative prior distributions based on our results performed better than those noninformative prior distributions on survival, providing more biologically defensible results when data were sparse. Knowledge of process distributions of survival can help wildlife managers better predict future population status and understand the likely range of survival rates.     

Exact log-rank tests for unequal follow-up

The asymptotic log-rank and generalized Wilcoxon tests are the standard procedures for comparing samples of possibly censored survival times. For comparison of samples of very different sizes, an exact test is available that is based on a complete permutation of log-rank or Wilcoxon scores. While the asymptotic tests do not keep their nominal sizes if sample sizes differ substantially, the exact complete permutation test requires equal follow-up of the samples. Therefore, we have developed and present two new exact tests also suitable for unequal follow-up. The first of these is an exact analogue of the asymptotic log-rank test and conditions on observed risk sets, whereas the second approach permutes survival times while conditioning on the realized follow-up in each group. In an empirical study, we compare the new procedures with the asymptotic log-rank test, the exact complete permutation test, and an earlier proposed approach that equalizes the follow-up distributions using artificial censoring. Results confirm highly satisfactory performance of the exact procedure conditioning on realized follow-up, particularly in case of unequal follow-up. The advantage of this test over other options of analysis is finally exemplified in the analysis of a breast cancer study.     

A Simple Two‐Sample Rank Test for Multivariate Survival Outcomes with Left Truncation and Right Censoring

A distribution‐free two‐sample rank test is proposed for testing for differences between survival distributions in the analysis of biomedical studies in which two groups of subjects are followed over time for a particular outcome, which may recur. This method is motivated by an observational HIV (human immunodeficiency virus) study in which a group of HIV‐seropositive women and a comparable group of HIV‐seronegative women were examined every 6 months for the presence of cervical intraepithelial neoplasia (CIN), the cervical cancer precursor. Women entered the study serially and were subject to random loss to follow‐up. Only women free of CIN at study entry were followed resulting in left‐truncated survival times. If a woman is found to be CIN infected at a later examination, she is treated and then followed until CIN recurs. The two groups of women were compared at both occurrences of CIN on the basis of rank statistics. For the first occurrence of CIN, survival times since the beginning of the study (based on calendar time) are compared. For a recurrence of CIN, survival times since the first development of CIN are compared. The proposed test statistic for an overall difference between the two groups follows a chi‐square distribution with two degrees of freedom. Simulation results demonstrate the usefulness of the proposed test proposed test statistic, which reduces to the Gehan statistic if each person is followed only to the first failure and there is no serial enrollment.     

Proposal of k Sample Tests for Bivariate Censored Data for Nondecreasing Ordered Alternatives

Tests are introduced which are designed to test for a nondecreasing ordered alternative among the survival functions of k populations consisting of multiple observations on each subject. Some of the observations could be right censored. A simulation study is conducted comparing the proposed tests on the basis of estimated power when the underlying distributions are multivariate normal. Equal sample sizes of 20 with 25% censoring, and 40 with both 25% and 50% censoring are considered for 3 and 4 populations. All of the tests hold their α‐values well. A recommendation is made as to the best overall test for the situations considered.     

Exact,Distribution Free Confidence Intervals for Late Effects in Censored Matched Pairs

When comparing censored survival times for matched treated and control subjects, a late effect on survival is one that does not begin to appear until some time has passed. In a study of provider specialty in the treatment of ovarian cancer, a late divergence in the Kaplan–Meier survival curves hinted at superior survival among patients of gynecological oncologists, who employ chemotherapy less intensively, when compared to patients of medical oncologists, who employ chemotherapy more intensively; we ask whether this late divergence should be taken seriously. Specifically, we develop exact, permutation tests, and exact confidence intervals formed by inverting the tests, for late effects in matched pairs subject to random but heterogeneous censoring. Unlike other exact confidence intervals with censored data, the proposed intervals do not require knowledge of censoring times for patients who die. Exact distributions are consequences of two results about signs, signed ranks, and their conditional independence properties. One test, the late effects sign test, has the binomial distribution; the other, the late effects signed rank test, uses nonstandard ranks but nonetheless has the same exact distribution as Wilcoxon's signed rank test. A simulation shows that the late effects signed rank test has substantially more power to detect late effects than do conventional tests. The confidence statement provides information about both the timing and magnitude of late effects (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Using cross-validation to evaluate predictive accuracy of survival risk classifiers based on high-dimensional data

Developments in whole genome biotechnology have stimulated statistical focus on prediction methods. We review here methodology for classifying patients into survival risk groups and for using cross-validation to evaluate such classifications. Measures of discrimination for survival risk models include separation of survival curves, time-dependent ROC curves and Harrell's concordance index. For high-dimensional data applications, however, computing these measures as re-substitution statistics on the same data used for model development results in highly biased estimates. Most developments in methodology for survival risk modeling with high-dimensional data have utilized separate test data sets for model evaluation. Cross-validation has sometimes been used for optimization of tuning parameters. In many applications, however, the data available are too limited for effective division into training and test sets and consequently authors have often either reported re-substitution statistics or analyzed their data using binary classification methods in order to utilize familiar cross-validation. In this article we have tried to indicate how to utilize cross-validation for the evaluation of survival risk models; specifically how to compute cross-validated estimates of survival distributions for predicted risk groups and how to compute cross-validated time-dependent ROC curves. We have also discussed evaluation of the statistical significance of a survival risk model and evaluation of whether high-dimensional genomic data adds predictive accuracy to a model based on standard covariates alone.     

Statistical optimization of parametric accelerated failure time model for mapping survival trait loci

Most existing statistical methods for mapping quantitative trait loci (QTL) are not suitable for analyzing survival traits with a skewed distribution and censoring mechanism. As a result, researchers incorporate parametric and semi-parametric models of survival analysis into the framework of the interval mapping for QTL controlling survival traits. In survival analysis, accelerated failure time (AFT) model is considered as a de facto standard and fundamental model for data analysis. Based on AFT model, we propose a parametric approach for mapping survival traits using the EM algorithm to obtain the maximum likelihood estimates of the parameters. Also, with Bayesian information criterion (BIC) as a model selection criterion, an optimal mapping model is constructed by choosing specific error distributions with maximum likelihood and parsimonious parameters. Two real datasets were analyzed by our proposed method for illustration. The results show that among the five commonly used survival distributions, Weibull distribution is the optimal survival function for mapping of heading time in rice, while Log-logistic distribution is the optimal one for hyperoxic acute lung injury.     

An information ratio-based goodness-of-fit test for copula models on censored data

Copula is a popular method for modeling the dependence among marginal distributions in multivariate censored data. As many copula models are available, it is essential to check if the chosen copula model fits the data well for analysis. Existing approaches to testing the fitness of copula models are mainly for complete or right-censored data. No formal goodness-of-fit (GOF) test exists for interval-censored or recurrent events data. We develop a general GOF test for copula-based survival models using the information ratio (IR) to address this research gap. It can be applied to any copula family with a parametric form, such as the frequently used Archimedean, Gaussian, and D-vine families. The test statistic is easy to calculate, and the test procedure is straightforward to implement. We establish the asymptotic properties of the test statistic. The simulation results show that the proposed test controls the type-I error well and achieves adequate power when the dependence strength is moderate to high. Finally, we apply our method to test various copula models in analyzing multiple real datasets. Our method consistently separates different copula models for all these datasets in terms of model fitness.     

Evaluation of sample size and power for analyses of survival with allowance for nonuniform patient entry, losses to follow-up, noncompliance, and stratification

When designing a clinical trial to test the equality of survival distributions for two treatment groups, the usual assumptions are exponential survival, uniform patient entry, full compliance, and censoring only administratively at the end of the trial. Various authors have presented methods for estimation of sample size or power under these assumptions, some of which allow for an R-year accrual period with T total years of study, T greater than R. The method of Lachin (1981, Controlled Clinical Trials 2, 93-113) is extended to allow for cases where patients enter the trial in a nonuniform manner over time, patients may exit from the trial due to loss to follow-up (other than administrative), other patients may continue follow-up although failing to comply with the treatment regimen, and a stratified analysis may be planned according to one or more prognostic covariates.