Survival Tests for r Groups

In this paper we give the generalization of the score tests covering the case of ties and we give examples where the expressions in matrix form are completely specified for the weighted tests and the score tests for the case of r groups. It is worth mentioning that although the score tests are not generally included in the commercial software, these tests should be used if it can be assumed that the censoring mechanism is equal in the r groups or if there is no censoring (Lawless , 1982). We establish the equivalence between “numerators” of these families of tests. As result of this equivalence we define four new tests that complete the classification of score and weighted tests. The Kruskal‐Wallis test (1952) appears as a particular case of the score tests for the case of non‐censoring. A simulation study has been done in order to compare the performance of the tests described in this paper. An example is included to make the understanding of the paper easier.     

Randomization Tests for Censored Survival Distributions

This paper discusses the application of randomization tests to censored survival distributions. The three types of censoring considered are those designated by MILLER (1981) as Type 1 (fixed time termination), Type 2 (termination of experiment at r-th failure), and random censoring. Examples utilize the Gehan scoring procedure. Randomization tests for which computer programs already exist can be applied to a variety of experimental designs, regardless of the presence of censored observations.     

Generalized pairwise comparisons for censored data: An overview

The method of generalized pairwise comparisons (GPC) is an extension of the well-known nonparametric Wilcoxon–Mann–Whitney test for comparing two groups of observations. Multiple generalizations of Wilcoxon–Mann–Whitney test and other GPC methods have been proposed over the years to handle censored data. These methods apply different approaches to handling loss of information due to censoring: ignoring noninformative pairwise comparisons due to censoring (Gehan, Harrell, and Buyse); imputation using estimates of the survival distribution (Efron, Péron, and Latta); or inverse probability of censoring weighting (IPCW, Datta and Dong). Based on the GPC statistic, a measure of treatment effect, the “net benefit,” can be defined. It quantifies the difference between the probabilities that a randomly selected individual from one group is doing better than an individual from the other group. This paper aims at evaluating GPC methods for censored data, both in the context of hypothesis testing and estimation, and providing recommendations related to their choice in various situations. The methods that ignore uninformative pairs have comparable power to more complex and computationally demanding methods in situations of low censoring, and are slightly superior for high proportions (>40%) of censoring. If one is interested in estimation of the net benefit, Harrell's c index is an unbiased estimator if the proportional hazards assumption holds. Otherwise, the imputation (Efron or Peron) or IPCW (Datta, Dong) methods provide unbiased estimators in case of proportions of drop-out censoring up to 60%.     

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.     

A Bayesian Chi‐Squared Goodness‐of‐Fit Test for Censored Data Models

Summary We propose a Bayesian chi‐squared model diagnostic for analysis of data subject to censoring. The test statistic has the form of Pearson's chi‐squared test statistic and is easy to calculate from standard output of Markov chain Monte Carlo algorithms. The key innovation of this diagnostic is that it is based only on observed failure times. Because it does not rely on the imputation of failure times for observations that have been censored, we show that under heavy censoring it can have higher power for detecting model departures than a comparable test based on the complete data. In a simulation study, we show that tests based on this diagnostic exhibit comparable power and better nominal Type I error rates than a commonly used alternative test proposed by Akritas (1988, Journal of the American Statistical Association 83, 222–230). An important advantage of the proposed diagnostic is that it can be applied to a broad class of censored data models, including generalized linear models and other models with nonidentically distributed and nonadditive error structures. We illustrate the proposed model diagnostic for testing the adequacy of two parametric survival models for Space Shuttle main engine failures.     

The Relationship Between Two Statistical Tests for the Comparison of Two Exponential Distributions

To compare two exponential distributions with or without censoring, two different statistics are often used; one is the F test proposed by COX (1953) and the other is based on the efficient score procedure. In this paper, the relationship between these tests is investigated and it is shown that the efficient score test is a large-sample approximation of the F test.     

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.     

Altitudinal patterns for longevity, fecundity and senescence in Drosophila buzzatii

We tested for variation in longevity, senescence rate and early fecundity of Drosophila buzzatii along an elevational transect in Argentina, using laboratory-reared flies in laboratory tests performed to avoid extrinsic mortality. At 25 °C, females from lowland populations lived longer and had a lower demographic rate of senescence than females from highland populations. Minimal instead of maximal temperature at the sites of origin of population best predicted this cline. A very different pattern was found at higher test temperature. At 29.5 °C, longevity of males increased with altitude of origin of population. No clinal trend was apparent for longevity of females at 29.5 °C. There was evidence for a trade-off between early fecundity and longevity at non-stressful temperature (25 °C) along the altitudinal gradient. This trait association is consistent with evolutionary theories of aging. Population-by-temperature and sex-by-temperature interactions indicate that senescence patterns are expressed in environment specific ways.     

Median Tests for Censored Survival Data; a Contingency Table Approach

Summary The median failure time is often utilized to summarize survival data because it has a more straightforward interpretation for investigators in practice than the popular hazard function. However, existing methods for comparing median failure times for censored survival data either require estimation of the probability density function or involve complicated formulas to calculate the variance of the estimates. In this article, we modify a K ‐sample median test for censored survival data ( Brookmeyer and Crowley, 1982 , Journal of the American Statistical Association 77, 433–440) through a simple contingency table approach where each cell counts the number of observations in each sample that are greater than the pooled median or vice versa. Under censoring, this approach would generate noninteger entries for the cells in the contingency table. We propose to construct a weighted asymptotic test statistic that aggregates dependent χ² ‐statistics formed at the nearest integer points to the original noninteger entries. We show that this statistic follows approximately a χ² ‐distribution with k? 1 degrees of freedom. For a small sample case, we propose a test statistic based on combined p ‐values from Fisher’s exact tests, which follows a χ² ‐distribution with 2 degrees of freedom. Simulation studies are performed to show that the proposed method provides reasonable type I error probabilities and powers. The proposed method is illustrated with two real datasets from phase III breast cancer clinical trials.     

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)     

GERMINATION ECOLOGY OF DANTHONIA SERICEA POPULATIONS

Seed were collected from nine populations of Danthonia sericea on either wet or well-drained, sandy sites ranging from Georgia to New Jersey. Germination tests were conducted in the laboratory and in field plots established on contrasting habitat types. Factors considered were temperature, light, prechilling, scarification, and after-ripening. The majority of freshly harvested seed from all nine populations exhibited innate dormancy the intensity of which decreased with age. Dormancy was removed by scarification and prechilling treatments of 60 days. Seed from wet habitats showed significantly greater germination in light (alternating night-day) than in dark, while seed from well-drained habitats had a higher germination percentage in dark than in light. Both responses are considered to be physiological adaptations of the populations to their respective habitats. Alternating night-day temperatures of 15–25 C produced the greatest overall germination for the nine populations; 15–20 and 10–25 C produced the next highest responses. Field observations confirmed that peak germination occurred in the spring, at the time soil temperatures were in the same range found to be optimum in the laboratory. A comparison of the germination responses obtained from six New Jersey populations in field plots on contrasting habitat types further revealed distinctive physiological adaptations for populations from differing habitats.     

Asymptotic Distribution of Score Statistics for Spatial Cluster Detection with Censored Data

Summary Cook, Gold, and Li (2007, Biometrics 63, 540–549) extended the Kulldorff (1997, Communications in Statistics 26, 1481–1496) scan statistic for spatial cluster detection to survival‐type observations. Their approach was based on the score statistic and they proposed a permutation distribution for the maximum of score tests. The score statistic makes it possible to apply the scan statistic idea to models including explanatory variables. However, we show that the permutation distribution requires strong assumptions of independence between potential cluster and both censoring and explanatory variables. In contrast, we present an approach using the asymptotic distribution of the maximum of score statistics in a manner not requiring these assumptions.     

Regression on Quantile Residual Life

Summary A time‐specific log‐linear regression method on quantile residual lifetime is proposed. Under the proposed regression model, any quantile of a time‐to‐event distribution among survivors beyond a certain time point is associated with selected covariates under right censoring. Consistency and asymptotic normality of the regression estimator are established. An asymptotic test statistic is proposed to evaluate the covariate effects on the quantile residual lifetimes at a specific time point. Evaluation of the test statistic does not require estimation of the variance–covariance matrix of the regression estimators, which involves the probability density function of the survival distribution with censoring. Simulation studies are performed to assess finite sample properties of the regression parameter estimator and test statistic. The new regression method is applied to a breast cancer data set with long‐term follow‐up to estimate the patients' median residual lifetimes, adjusting for important prognostic factors.     

Statistical methods for describing developmental milestones with censored data: effects of birth weight status and sex in neonatal pigtailed macaques

Neurobehavioral tests are used to assess early neonatal behavioral functioning and detect effects of prenatal and perinatal events. However, common measurement and data collection methods create specific data features requiring thoughtful statistical analysis. Assessment response measurements are often ordinal scaled, not interval scaled; the magnitude of the physical response may not directly correlate with the underlying state of developmental maturity; and a subject's assessment record may be censored. Censoring occurs when the milestone is exhibited at the first test (left censoring), when the milestone is not exhibited before the end of the study (right censoring), or when the exact age of attaining the milestone is uncertain due to irregularly spaced test sessions or missing data (interval censoring). Such milestone data is best analyzed using survival analysis methods. Two methods are contrasted: the non-parametric Kaplan-Meier estimator and the fully parametric interval censored regression. The methods represent the spectrum of survival analyses in terms of parametric assumptions, ability to handle simultaneous testing of multiple predictors, and accommodation of different types of censoring. Both methods were used to assess birth weight status and sex effects on 14 separate test items from assessments on 255 healthy pigtailed macaques. The methods gave almost identical results. Compared to the normal birth weight group, the low birth weight group had significantly delayed development on all but one test item. Within the low birth weight group, males had significantly delayed development for some responses relative to females.     

Sweat loss prediction using a multi-model approach

A new multi-model approach (MMA) for sweat loss prediction is proposed to improve prediction accuracy. MMA was computed as the average of sweat loss predicted by two existing thermoregulation models: i.e., the rational model SCENARIO and the empirical model Heat Strain Decision Aid (HSDA). Three independent physiological datasets, a total of 44 trials, were used to compare predictions by MMA, SCENARIO, and HSDA. The observed sweat losses were collected under different combinations of uniform ensembles, environmental conditions (15–40°C, RH 25–75%), and exercise intensities (250–600 W). Root mean square deviation (RMSD), residual plots, and paired t tests were used to compare predictions with observations. Overall, MMA reduced RMSD by 30–39% in comparison with either SCENARIO or HSDA, and increased the prediction accuracy to 66% from 34% or 55%. Of the MMA predictions, 70% fell within the range of mean observed value ± SD, while only 43% of SCENARIO and 50% of HSDA predictions fell within the same range. Paired t tests showed that differences between observations and MMA predictions were not significant, but differences between observations and SCENARIO or HSDA predictions were significantly different for two datasets. Thus, MMA predicted sweat loss more accurately than either of the two single models for the three datasets used. Future work will be to evaluate MMA using additional physiological data to expand the scope of populations and conditions.     

Regression analysis for recurrent events data under dependent censoring

Summary Recurrent events data are commonly seen in longitudinal follow‐up studies. Dependent censoring often occurs due to death or exclusion from the study related to the disease process. In this article, we assume flexible marginal regression models on the recurrence process and the dependent censoring time without specifying their dependence structure. The proposed model generalizes the approach by Ghosh and Lin (2003, Biometrics 59, 877–885). The technique of artificial censoring provides a way to maintain the homogeneity of the hypothetical error variables under dependent censoring. Here we propose to apply this technique to two Gehan‐type statistics. One considers only order information for pairs whereas the other utilizes additional information of observed censoring times available for recurrence data. A model‐checking procedure is also proposed to assess the adequacy of the fitted model. The proposed estimators have good asymptotic properties. Their finite‐sample performances are examined via simulations. Finally, the proposed methods are applied to analyze the AIDS linked to the intravenous experiences cohort data.     

Role of IFN-γ +874 T/A single nucleotide polymorphism in the tuberculosis outcome among Brazilians subjects

Several genetic cytokine gene variants have been associated with host susceptibility to infectious diseases, including tuberculosis. Based upon the importance of IFN-γ in protective immunity against Mycobacterium tuberculosis, and the functional role of the IFN-γ + 874T/A single nucleotide polymorphism in IFN-γ production, we genotyped 93 Brazilian tuberculosis patients and 266 asymptomatic health care workers, including 150 individuals with a positive tuberculin skin test, and analyzed the possible association of the +874A low IFN-γ producer allele with tuberculosis occurrence. Using multivariable logistic regression models, genotype and allele frequencies of the mutant + 874A (low IFN-γ producer) allele were significantly associated with tuberculosis disease. Heterozygous carriers had a 25% increased chance, while individuals presenting the A/A homozygous genotype had an over two-fold risk of having active tuberculosis (95% CI, 1.16–5.91, P = 0.03). Despite the mixed ethnicity observed in Brazilian populations, the present data agree with observations reported in other populations and thus demonstrate that the functional +874T/A IFN-γ gene polymorphism is associated with tuberculosis in different populations.     

Comparing treatment-induced changes for k independent samples of paired observations

In a recent paper, MARASCUILO [19] has provided an asymptotic solution to the important question on how to test for differences in change parameters when paired observation of binary type (+, -) have been made on two or more independent samples of individuals. In this article, an alternative approach is presented implying asymptotic as well as exact tests for changes. They are based on pre-post test designs from clinical research and allow for controlled evaluation of one treatment modality as well as for comparing 2 or more than 2 treatment modalities. The rationale of the tests is based on McNEMARS [21] test for paired binary observations in one, two, or k samples.