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.     

A class of permutation tests for stratified survival data

We propose a class of permutation tests for stratified survival data. The tests are derived using the framework of Fay and Shih (1998, Journal of the American Statistical Association 93, 387-396), which creates tests by permuting scores based on a functional of estimated distribution functions. Here the estimated distribution function for each possibly right-, left-, or interval-censored observation is based on a shrinkage estimator similar to the nonparametric empirical estimator of Ghosh, Lahiri, and Tiwari (1989, Communications in Statistics--Theory and Methods 18, 121-146), and permutation is carried out within strata. The proposed test with a weighted Mann-Whitney functional is similar to the permutation form of the stratified log-rank test when there is a large strata effect or the sample size in each stratum is large and is similar to the permutation form of the ordinary log-rank test when there is little strata effect. Thus, the proposed test unifies the advantages of both the stratified and ordinary log-rank tests. By changing the functional, we may obtain a stratified Prentice-Wilcoxon test or a difference in means test with similar unifying properties. We show through simulations the advantage of the proposed test over existing tests for uncensored and right-censored data.     

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.     

On null hypotheses in survival analysis

The conventional nonparametric tests in survival analysis, such as the log‐rank test, assess the null hypothesis that the hazards are equal at all times. However, hazards are hard to interpret causally, and other null hypotheses are more relevant in many scenarios with survival outcomes. To allow for a wider range of null hypotheses, we present a generic approach to define test statistics. This approach utilizes the fact that a wide range of common parameters in survival analysis can be expressed as solutions of differential equations. Thereby, we can test hypotheses based on survival parameters that solve differential equations driven by cumulative hazards, and it is easy to implement the tests on a computer. We present simulations, suggesting that our tests perform well for several hypotheses in a range of scenarios. As an illustration, we apply our tests to evaluate the effect of adjuvant chemotherapies in patients with colon cancer, using data from a randomized controlled trial.     

ROC-guided survival trees and ensembles

Tree-based methods are popular nonparametric tools in studying time-to-event outcomes. In this article, we introduce a novel framework for survival trees and ensembles, where the trees partition the dynamic survivor population and can handle time-dependent covariates. Using the idea of randomized tests, we develop generalized time-dependent receiver operating characteristic (ROC) curves for evaluating the performance of survival trees. The tree-building algorithm is guided by decision-theoretic criteria based on ROC, targeting specifically for prediction accuracy. To address the instability issue of a single tree, we propose a novel ensemble procedure based on averaging martingale estimating equations, which is different from existing methods that average the predicted survival or cumulative hazard functions from individual trees. Extensive simulation studies are conducted to examine the performance of the proposed methods. We apply the methods to a study on AIDS for illustration.     

Late divergence of survival curves in cancer immunotherapy trials: interpretation and implications

Late divergence of survival curves of treated patients and controls is commonly seen in successful cancer immunotherapy trials. Although late survival curve divergence may be caused by a delayed action of therapy, it may also be related to early effects of the treatment. We suggest that late survival divergence most often reflects a specific benefit of therapy for patients who suffer from a comparatively slow progression of disease. The occurrence of delayed survival curve divergence has important implications for the statistical analysis of immunotherapy trials. Thus, it leads to non-proportional hazard ratios that make commonly used statistical tests, e.g., the logrank test, suboptimal. It is therefore suggested that the statistical analysis of immunotherapy trials primarily should be based on a test that compares the survival curves at or after a prespecified, fixed, late time point.     

Analysis of competing risks survival data when some failure types are missing

We propose a method to analyse competing risks survival datawhen failure types are missing for some individuals. Our approachis based on a standard proportional hazards structure for eachof the failure types, and involves the solution to estimatingequations. We present consistent and asymptotically normal estimatorsof the regression coefficients and related score tests. An appealingfeature is that individuals with known failure types make thesame contributions as they would to a standard proportionalhazards analysis. Contributions of individuals with unknownfailure types are weighted according to the probability thatthey failed from the cause of interest. Efficiency and robustnessare discussed. Results are illustrated with data from a breastcancer trial.     

Stratified modestly weighted log-rank tests in settings with an anticipated delayed separation of survival curves

Delayed separation of survival curves is a common occurrence in confirmatory studies in immuno-oncology. Many novel statistical methods that aim to efficiently capture potential long-term survival improvements have been proposed in recent years. However, the vast majority do not consider stratification, which is a major limitation considering that most large confirmatory studies currently employ a stratified primary analysis. In this article, we combine recently proposed weighted log-rank tests that have been designed to work well under a delayed separation of survival curves, with stratification by a baseline variable. The aim is to increase the efficiency of the test when the stratifying variable is highly prognostic for survival. As there are many potential ways to combine the two techniques, we compare several possibilities in an extensive simulation study. We also apply the techniques retrospectively to two recent randomized clinical trials.     

Tests of proportional hazards and proportional odds models for grouped survival data

In this paper, we derive score test statistics to discriminate between proportional hazards and proportional odds models for grouped survival data. These models are embedded within a power family transformation in order to obtain the score tests. In simple cases, some small-sample results are obtained for the score statistics using Monte Carlo simulations. Score statistics have distributions well approximated by the chi-squared distribution. Real examples illustrate the proposed tests.     

Prediction of mean adult survival rates of southern African birds from demographic and ecological covariates

Estimates of annual survival rates of birds are valuable in a wide range of studies of population ecology and conservation. These include modelling studies to assess the impacts of climatic change or anthropogenic mortality for many species for which no reliable direct estimates of survival are available. We evaluate the performance of regression models in predicting adult survival rates of birds from values of demographic and ecological covariates available from textbooks and databases. We estimated adult survival for 67 species using dead recoveries of birds ringed in southern Africa and fitted regression models using five covariates: mean clutch size, mean body mass, mean age at first breeding, diet and migratory tendency. Models including these explanatory variables performed well in predicting adult survival in this set of species, both when phylogenetic relatedness of the species was taken into account using phylogenetic generalized least squares (51% of variation in logit survival explained) and when it was not (48%). Two independent validation tests also indicated good predictive power, as indicated by high correlations of observed with expected values in a leave‐one‐out cross validation test performed using data from the 67 species (35% of variation in logit survival explained), and when annual survival rates from independent mark–recapture studies of 38 southern African species were predicted from covariates and the regression using dead recoveries (48%). Clutch size and body mass were the most influential covariates, both with and without the inclusion of phylogenetic effects, and a regression model including only these two variables performed well in both of the validation tests (39 and 48% of variation in logit survival explained). Our regression models, including the version with only clutch size and body mass, are likely to perform well in predicting adult survival rate for southern African species for which direct survival estimates are not available.     

Changes in prostate cancer incidence,mortality and survival in relation to prostate specific antigen testing in New South Wales,Australia

BackgroundTo examine changes in prostate cancer incidence and mortality rates, and 5-year relative survival, in relation to changes in the rate of prostate specific antigen (PSA) screening tests and the use of radical prostatectomy (RP) in the Australian population.MethodsProstate cancer stage-specific incidence rates, 5-year relative survival and mortality rates were estimated using New South Wales Cancer Registry data. PSA screening test rates and RP/Incidence ratios were estimated from Medicare Benefits Schedule claims data. We used multiple imputation to impute stage for cases with “unknown” stage at diagnosis. Annual percentage changes (APC) in rates were estimated using Joinpoint regression.ResultsTrends in the age-standardized incidence rates for localized disease largely mirrored the trends in PSA screening test rates, with a substantial ‘spike’ in the rates occurring in 1994, followed by a second ‘spike’ in 2008, and then a significant decrease from 2008 to 2015 (APC −6.7, 95% CI −8.2, −5.1). Increasing trends in incidence rates were observed for regional stage from the early 2000s, while decreasing or stable trends were observed for distant stage since 1993. The overall RP/Incidence ratio increased from 1998 to 2003 (APC 9.6, 95% CI 3.8, 15.6), then remained relatively stable to 2015. The overall 5-year relative survival for prostate cancer increased from 58.4% (95% CI: 55.0–61.7%) in 1981–1985 to 91.3% (95% CI: 90.5–92.1%) in 2011–2015. Prostate cancer mortality rates decreased from 1990 onwards (1990–2006: APC −1.7, 95% CI −2.1, −1.2; 2006–2017: APC −3.8, 95% CI −4.4, −3.1).ConclusionsOverall, there was a decrease in the incidence rate of localized prostate cancer after 2008, an increase in survival over time and a decrease in the mortality rate since the 1990s. This seems to indicate that the more conservative use of PSA screening tests in clinical practice since 2008 has not had a negative impact on population-wide prostate cancer outcomes.     

Modeling survival of experimentally virus-infected laboratory animals

Although survival analysis is a well-established mathematical discipline, there seem to be almost no attempts in survival modeling for experimentally virus-infected laboratory animals. We have taken up a stochastic approach originally developed by Shortley in the sixties and have applied it to three different types of experimental data: to virus titer determination, to the dose dependence of the mean survival time and to single survival curves. Experience concerning parameter estimation is reported and new ways of working with the model parameters are proposed. A standard mean survival time is defined and suggested as a new quantitative measure of virulence. Moreover, for the comparison of two experiments for which the amount of virions inoculated is kept fixed, but for which other parameters may vary, a new scheme of systematizing survival data from experimentally virus-infected laboratory animals is proposed. It is very likely that the model can be also applied to cancer survival data or any other infectious pathogen.     

Genotype by environment interactions in winter survival in red deer

1. The extent to which environmental heterogeneity interacts with genetic hetero geneity to affect individual fitness within populations has the potential to affect the dynamics of natural populations and the amount of genetic variation maintained in natural populations, yet is a relatively poorly investigated topic in either ecology or evolutionary biology.
2. Many individual-based studies are precluded from such investigations by the practical problems of measuring heritability of traits affecting fitness and the difficulties of experimental manipulation of the study population. One way of demonstrating how commonly genotype by environmental interactions affect fitness, though not their overall importance in determining fitness, is to investigate fitness in a population subdivided by genotype at one or more marker loci.
3. We analyse data on calf winter survival from a population of red deer from the Isle of Rum, Scotland. Data on individual survival, environmental fluctuations and genotype at 13 loci were collected from 1982 to 1994.
4. We found associations between survival over the first winter of life and calf genotype at two out of three allozyme loci and five out of 10 microsatellite loci. All of the results remained significant under randomization tests. Other genotypes that initially appeared to have an association with survival were rejected when bootstrapped, usually due to insufficient data or anomalies in the data.
5. Our results suggest that associations between fitness and genotype are common. Five out of the seven associations found involved interactions with environmental variables. Four of these showed density-dependent selection with different genotypes showing high survival at high population size compared to low population size and one interacted with autumn rainfall. In a sixth case, genotype interacted with sex.     

A semiparametric approach for the two-sample comparison of survival times with long-term survivors

In the two-sample comparison of survival times with long-term survivors, the overall difference between the two distributions reflects differences occurring in early follow-up for susceptible subjects and in long-term follow-up for nonsusceptible subjects. In this setting, we propose statistics for testing (i) no overall, (ii) no short-term, and (iii) no long-term difference between the two distributions to be compared. The statistics are derived as follows. A semiparametric model is defined that characterizes a short-term effect and a long-term effect. By approximating this model about no difference in early survival, a time-dependent proportional hazards model is obtained. The statistics are obtained from this working model. The asymptotic distributions of the statistics for testing no overall or no short-term effects are ascertained, while that of the statistic for testing no long-term effect is valid only when the short-term effect is small. Simulation studies investigate the power properties of the proposed tests for different configurations. The results show the interesting behavior of the proposed tests for situations where a short-term effect is expected. An example investigating the impact of progesterone receptors status on local tumor relapse for patients with early breast cancer illustrates the use of the proposed tests.     

Strong density-dependent survival and recruitment regulate the abundance of a coral reef fish

Debate on the control of population dynamics in reef fishes has centred on whether patterns in abundance are determined by the supply of planktonic recruits, or by post-recruitment processes. Recruitment limitation implies little or no regulation of the reef-associated population, and is supported by several experimental studies that failed to detect density dependence. Previous manipulations of population density have, however, focused on juveniles, and there have been no tests for density-dependent interactions among adult reef fishes. I tested for population regulation in Coryphopterus glaucofraenum, a small, short-lived goby that is common in the Caribbean. Adult density was manipulated on artificial reefs and adults were also monitored on reefs where they varied in density naturally. Survival of adult gobies showed a strong inverse relationship with their initial density across a realistic range of densities. Individually marked gobies, however, grew at similar rates across all densities, suggesting that density-dependent survival was not associated with depressed growth, and so may result from predation or parasitism rather than from food shortage. Like adult survival, the accumulation of new recruits on reefs was also much lower at high adult densities than at low densities. Suppression of recruitment by adults may occur because adults cause either reduced larval settlement or reduced early post-settlement survival. In summary, this study has documented a previously unrecorded regulatory mechanism for reef fish populations (density-dependent adult mortality) and provided a particularly strong example of a well-established mechanism (density-dependent recruitment). In combination, these two compensatory mechanisms have the potential to strongly regulate the abundance of this species, and rule out the control of abundance by the supply of recruits.     

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.     

Associations between gene expressions in breast cancer and patient survival

We analyzed associations between gene expression in breast cancer and patient survival for 8024 genes from a previously published microarray data set. Analysis of survival, by using the logrank test, was performed automatically for each gene. After correcting for multiple testing, we identified 95 genes whose expression was significantly associated with patient survival. The independent prognostic value of the genes ranking the highest in univariate analysis, together with clinical parameters, was assessed by Cox multivariate regression analysis. The P-values from these logrank tests were also mapped to chromosomal positions and compared with previously reported amplicon regions. We used PubGene web tools to identify groups of genes that had co-occurred in the literature and whose expression patterns were associated with survival. Our analyses demonstrate the comprehensiveness of the microarray technology with respect to measuring gene expression and indicate that the technology may be used to screen for potential clinical markers.     

Analysis of survival data with cross-effects of survival functions

We present a model and semiparametric estimation procedures for analysis of survival data with cross-effects (CE) of survival functions. Finite sample properties of the estimators are analyzed by simulation. A goodness-of-fit test for the proportional hazards model against the CE model is proposed. The well known data concerning effects of chemotherapy and radiotherapy on the survival times of gastric cancer patients is analyzed as an example.     

Annual survival in female white-winged scoters and lesser scaup

Quantifying survival and understanding underlying sources of variation are important for developing population models and informing management decisions. We estimated apparent survival (i.e., true survival less permanent emigration) for adult female white-winged scoters (Melanitta fusca) and lesser scaup (Aythya affinis) breeding at a northern boreal forest site in western Canada, 2002–2008. We also evaluated variation in survival relative to indices of breeding status, individual quality, spring weather conditions, local small-mammal abundance, and overwinter climate. Breeding female scoters had higher apparent survival than did nonbreeding females, suggesting that breeders had higher survival or fidelity to the study area, or that more nonbreeders were transient birds that may have bred elsewhere in subsequent years. Apparent survival rate for breeding female scoters was unrelated to other individual and environmental covariates. Nest-trapped female scaup had higher apparent survival rates than did prenesting females captured in decoy traps, implying that more pre-nesters dispersed permanently or died after marking. Nesting female scaup with higher body condition or those in years when small mammals were more abundant had higher apparent survival; associations between survival and other environmental covariates were less certain. Overall, apparent survival rate of breeding adult female scoters was lower than reported for scoters from other North American locations or for other sea duck species, whereas estimates for nesting female scaup were comparable with those from other boreal and prairie-parkland locations. Our results indicate that for scaup in this region, factors influencing female body condition, such as maintaining high-quality habitat, areas with abundant food or low disturbance, could improve annual survival. © The Wildlife Society, 2019