On the efficiency of age-adjusted tests in animal carcinogenicity experiments

Based on asymptotic relative efficiency calculations, Ryan (1985, Biometrics 41, 525-531) concludes that, in the analysis of animal carcinogenicity experiments, age-adjusted tests of tumor rates should be routinely preferred to simple proportions tests for both lethal and nonlethal tumors. We recalculate the asymptotic efficiencies of the simple proportions test relative to the log-rank test for the lethal tumor case. For a simplified model it is shown that the relative efficiency may be easily computed as a function of the crude tumor rate and the survival rate at the time of terminal sacrifice. More generally, we calculate by numerical quadrature the asymptotic relative efficiency for all models considered by Ryan and, using simulations, examine the relevance of asymptotic efficiencies to typical sample sizes. Contrary to the numerical results of Ryan, we find, for experiments with good survival and typical tumor rates, that the relative efficiencies are greater than 95%, usually about 99%. In the nonlethal tumor case, similar results follow from Ryan for tumor rates and survival rates typically encountered in practice. As it is often difficult to determine whether or not a tumor is lethal, we conclude for equal interim mortality rates, that the simple proportions test is usually adequate in evaluating animal carcinogenicity experiments.     

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.     

Testing homogeneity in Weibull-regression models

In survival studies with families or geographical units it may be of interest testing whether such groups are homogeneous for given explanatory variables. In this paper we consider score type tests for group homogeneity based on a mixing model in which the group effect is modelled as a random variable. As opposed to hazard-based frailty models, this model presents survival times that conditioned on the random effect, has an accelerated failure time representation. The test statistics requires only estimation of the conventional regression model without the random effect and does not require specifying the distribution of the random effect. The tests are derived for a Weibull regression model and in the uncensored situation, a closed form is obtained for the test statistic. A simulation study is used for comparing the power of the tests. The proposed tests are applied to real data sets with censored data.     

A joint frailty model for survival and gap times between recurrent events

Therapy for patients with a recurrent disease focuses on delaying disease recurrence and prolonging survival. A common analysis approach for such data is to estimate the distribution of disease-free survival, that is, the time to the first disease recurrence or death, whichever happens first. However, treating death similarly as disease recurrence may give misleading results. Also considering only the first recurrence and ignoring subsequent ones can result in loss of statistical power. We use a joint frailty model to simultaneously analyze disease recurrences and survival. Separate parameters for disease recurrence and survival are used in the joint model to distinguish treatment effects on these two types of events. The correlation between disease recurrences and survival is taken into account by a shared frailty. The effect of disease recurrence on survival can also be estimated by this model. The EM algorithm is used to fit the model, with Markov chain Monte Carlo simulations in the E-steps. The method is evaluated by simulation studies and illustrated through a study of patients with heart failure. Sensitivity analysis for the parametric assumption of the frailty distribution is assessed by simulations.     

Comparing treatments in the presence of crossing survival curves: an application to bone marrow transplantation

Summary .   In some clinical studies comparing treatments in terms of their survival curves, researchers may anticipate that the survival curves will cross at some point, leading to interest in a long-term survival comparison. However, simple comparison of the survival curves at a fixed point may be inefficient, and use of a weighted log-rank test may be overly sensitive to early differences in survival. We formulate the problem as one of testing for differences in survival curves after a prespecified time point, and propose a variety of techniques for testing this hypothesis. We study these methods using simulation and illustrate them on a study comparing survival for autologous and allogeneic bone marrow transplants.     

Regression with frailty in survival analysis

In studies of survival, the hazard function for each individual may depend on observed risk variables but usually not all such variables are known or measurable. This unknown factor of the hazard function is usually termed the individual heterogeneity or frailty. When survival is time to the occurrence of a particular type of event and more than one such time may be obtained for each individual, frailty is a common factor among such recurrence times. A model including frailty is fitted to such repeated measures of recurrence times.     

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.     

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.     

A General Class of Semiparametric Transformation Frailty Models for Nonproportional Hazards Survival Data

Summary We propose a semiparametrically efficient estimation of a broad class of transformation regression models for nonproportional hazards data. Classical transformation models are to be viewed from a frailty model paradigm, and the proposed method provides a unified approach that is valid for both continuous and discrete frailty models. The proposed models are shown to be flexible enough to model long‐term follow‐up survival data when the treatment effect diminishes over time, a case for which the PH or proportional odds assumption is violated, or a situation in which a substantial proportion of patients remains cured after treatment. Estimation of the link parameter in frailty distribution, considered to be unknown and possibly dependent on a time‐independent covariates, is automatically included in the proposed methods. The observed information matrix is computed to evaluate the variances of all the parameter estimates. Our likelihood‐based approach provides a natural way to construct simple statistics for testing the PH and proportional odds assumptions for usual survival data or testing the short‐ and long‐term effects for survival data with a cure fraction. Simulation studies demonstrate that the proposed inference procedures perform well in realistic settings. Applications to two medical studies are provided.     

Flexible weighted log-rank tests optimal for detecting early and/or late survival differences

At the present time, many AIDS clinical trials compare drug therapies by a time-to-event primary endpoint that measures the durability of suppression of HIV replication. For such studies, survival differences tend to occur early and/or late in the follow-up period due to drug differences in initial potency and/or durability of efficacy, and detecting these differences is of primary interest. We propose a weighted log-rank statistic that emphasizes early and/or late survival differences. We also consider some versatile tests that also emphasize these differences but are sensitive to a wider range of alternatives. The performances of the new tests are evaluated in numerical studies. For the alternatives of interest, the new tests show greater power and flexibility than commonly used weighted log-rank tests and related versatile tests. When the main interest is in detecting early and/or late survival differences, these tests may be preferable to the other versatile and weighted log-rank tests that have been studied.     

Efficiency of age-adjusted tests in animal carcinogenicity experiments

Carcinogenicity experiments may be analysed by a simple comparison of the control and exposed groups with respect to the proportions of observed tumors amongst dead animals, when longevity is identical in control and exposed groups. This simple-proportions test is invalid when longevity varies between groups and age-adjusted methods such as the Hoel-Walburg or logrank test are needed. This paper stresses that there is an advantage to using age-adjusted tests, even when the simple-proportions test is valid. The argument is based on calculations of the asymptotic relative efficiency of the simple-proportions test with respect to both the Hoel-Walburg and the logrank tests.     

Accurate Computation of Survival Statistics in Genome-Wide Studies

A key challenge in genomics is to identify genetic variants that distinguish patients with different survival time following diagnosis or treatment. While the log-rank test is widely used for this purpose, nearly all implementations of the log-rank test rely on an asymptotic approximation that is not appropriate in many genomics applications. This is because: the two populations determined by a genetic variant may have very different sizes; and the evaluation of many possible variants demands highly accurate computation of very small p-values. We demonstrate this problem for cancer genomics data where the standard log-rank test leads to many false positive associations between somatic mutations and survival time. We develop and analyze a novel algorithm, Exact Log-rank Test (ExaLT), that accurately computes the p-value of the log-rank statistic under an exact distribution that is appropriate for any size populations. We demonstrate the advantages of ExaLT on data from published cancer genomics studies, finding significant differences from the reported p-values. We analyze somatic mutations in six cancer types from The Cancer Genome Atlas (TCGA), finding mutations with known association to survival as well as several novel associations. In contrast, standard implementations of the log-rank test report dozens-hundreds of likely false positive associations as more significant than these known associations.     

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.     

Analyses of two-way chronic studies

This paper proposes a method for analyzing tumor data from chronic studies when the experimental design includes combinations of two factors, for example, sex and dose. Both main effects and combined-effect (interaction) hypotheses are considered. A stratified log-rank statistic is presented for tests of no column or row (main) effects. The paper shows that when the numbers of animals in the cells are unequal and disproportional, the null distribution of the unstratified log-rank statistic does not have a chi-square distribution. Two simple models, additive and multiplicative, for representing the combined effect of row and column are considered under the proportional hazards model. A simple conservative statistic is proposed for testing the additivity of the row and column effects. A simulation experiment to examine the behavior of the null distribution of the combined-effect test statistic under the additive model and the power of the test against the multiplicative model is reported. The procedure is illustrated by analyzing mammary tumors induced by 7,12-dimethylbenz[a]anthracene (DMBA) in yellow and agouti F1 female mice from a laboratory experiment.     

A minimum version of log-rank test for testing the existence of cancer cure using relative survival data

Cancer survival is one of the most important measures to evaluate the effectiveness of treatment and early diagnosis. The ultimate goal of cancer research and patient care is the cure of cancer. As cancer treatments progress, cure becomes a reality for many cancers if patients are diagnosed early and get effective treatment. If a cure does exist for a certain type of cancer, it is useful to estimate the time of cure. For cancers that impose excess risk of mortality, it is informative to understand the difference in survival between cancer patients and the general cancer-free population. In population-based cancer survival studies, relative survival is the standard measure of excess mortality due to cancer. Cure is achieved when the survival of cancer patients is equivalent to that of the general population. This definition of cure is usually called the statistical cure, which is an important measure of burden due to cancer. In this paper, a minimum version of the log-rank test is proposed to test the equivalence of cancer patients' survival using the relative survival data. Performance of the proposed test is evaluated by simulation. Relative survival data from population-based cancer registries in SEER Program are used to examine patients' survival after diagnosis for various major cancer sites.     

The influence of competing-risks setting on the choice of hypothesis test for treatment effect

There is considerable debate regarding the choice of test for treatment difference in a randomized clinical trial in the presence of competing risks. This question arose in the study of standard and new antiepileptic drugs (SANAD) trial comparing new and standard antiepileptic drugs. This paper provides simulation results for the log-rank test comparing cause-specific hazard rates and Gray's test comparing cause-specific cumulative incidence curves. To inform the analysis of the SANAD trial, competing-risks settings were considered where both events are of interest, events may be negatively correlated, and the degree of correlation may differ in the 2 treatment groups. In settings where there are effects in opposite directions for the 2 event types, a likely situation for the SANAD trial, Gray's test has greater power to detect treatment differences than log-rank analysis. For the epilepsy application, conclusions were qualitatively similar for both log-rank and Gray's tests.     

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.     

Prognostic Relevance of Cytochrome c Oxidase in Primary Glioblastoma Multiforme

Patients with primary glioblastoma multiforme (GBM) have one of the lowest overall survival rates among cancer patients, and reliable biomarkers are necessary to predict patient outcome. Cytochrome c oxidase (CcO) promotes the switch from glycolytic to OXPHOS metabolism, and increased CcO activity in tumors has been associated with tumor progression after chemotherapy failure. Thus, we investigated the relationship between tumor CcO activity and the survival of patients diagnosed with primary GBM. A total of 84 patients with grade IV glioma were evaluated in this retrospective cohort study. Cumulative survival was calculated by the Kaplan-Meier method and analyzed by the log-rank test, and univariate and multivariate analyses were performed with the Cox regression model. Mitochondrial CcO activity was determined by spectrophotometrically measuring the oxidation of cytochrome c. High CcO activity was detected in a subset of glioma tumors (∼30%), and was an independent prognostic factor for shorter progression-free survival and overall survival [P = 0.0087 by the log-rank test, hazard ratio = 3.57 for progression-free survival; P<0.001 by the log-rank test, hazard ratio = 10.75 for overall survival]. The median survival time for patients with low tumor CcO activity was 14.3 months, compared with 6.3 months for patients with high tumor CcO activity. High CcO activity occurs in a significant subset of high-grade glioma patients and is an independent predictor of poor outcome. Thus, CcO activity may serve as a useful molecular marker for the categorization and targeted therapy of GBMs.     

Bayesian semiparametric dynamic frailty models for multiple event time data

Many biomedical studies collect data on times of occurrence for a health event that can occur repeatedly, such as infection, hospitalization, recurrence of disease, or tumor onset. To analyze such data, it is necessary to account for within-subject dependency in the multiple event times. Motivated by data from studies of palpable tumors, this article proposes a dynamic frailty model and Bayesian semiparametric approach to inference. The widely used shared frailty proportional hazards model is generalized to allow subject-specific frailties to change dynamically with age while also accommodating nonproportional hazards. Parametric assumptions on the frailty distribution are avoided by using Dirichlet process priors for a shared frailty and for multiplicative innovations on this frailty. By centering the semiparametric model on a conditionally conjugate dynamic gamma model, we facilitate posterior computation and lack-of-fit assessments of the parametric model. Our proposed method is demonstrated using data from a cancer chemoprevention study.     

Genes, Demography, and Life Span: The Contribution of Demographic Data in Genetic Studies on Aging and Longevity

In population studies on aging, the data on genetic markers are often collected for individuals from different age groups. The purpose of such studies is to identify, by comparison of the frequencies of selected genotypes, "longevity" or "frailty" genes in the oldest and in younger groups of individuals. To address questions about more-complicated aspects of genetic influence on longevity, additional information must be used. In this article, we show that the use of demographic information, together with data on genetic markers, allows us to calculate hazard rates, relative risks, and survival functions for respective genes or genotypes. New methods of combining genetic and demographic information are discussed. These methods are tested on simulated data and then are applied to the analysis of data on genetic markers for two haplogroups of human mtDNA. The approaches suggested in this article provide a powerful tool for analyzing the influence of candidate genes on longevity and survival. We also show how factors such as changes in the initial frequencies of candidate genes in subsequent cohorts, or secular trends in cohort mortality, may influence the results of an analysis.