Inference for cumulative incidence quantiles via parametric and nonparametric approaches

In survival analysis, a point estimate and confidence interval for median survival time have been frequently used to summarize the survival curve. However, such quantile analyses on competing risks data have not been widely investigated. In this paper, we propose parametric inferences for quantiles from the cumulative incidence function and develop parametric confidence intervals for quantiles. In addition, we study a simplified method of inference for the nonparametric approach. We compare the parametric and nonparametric inferences in empirical studies. Simulation studies show that the procedures perform well, with parametric analyses yielding smaller mean square error when the model is not too badly misspecified. We illustrate the methods with data from a breast cancer clinical trial.     

CONFIDENCE INTERVALS FOR MEDIAN SURVIVAL TIMES UNDER A PIECEWISE EXPONENTIAL MODEL WITH PROPORTIONAL HAZARDS COVARIATE EFFECTS

Brookmeyer and Crowley derived a non-parametric confidence interval for the median survival time of a homogeneous population by inverting a generalization of the sign test for censored data. The 1−α confidence interval for the median is essentially the set of all values t such that the Kaplan–Meier estimate of the survival function at time t does not differ significantly from one-half at significance level α. Here I extend the method to incorporate covariates into the analysis by assuming an underlying piecewise exponential model with proportional hazards covariate effects. Maximum likelihood estimates of the model parameters are obtained via iterative techniques, from which the estimated (log) survival curve is easily constructed. The delta method provides asymptotic standard errors. Following Brookmeyer and Crowley, I find the confidence interval for the median survival time at a specified value of the covariate vector by inverting the sign test. I illustrate the methods using data from a clinical trial conducted by the Radiation Therapy Oncology Group in cancer of the mouth and throat. It is seen that the piecewise exponential model provides considerable flexibility in accommodating to the shape of the underlying survival curve and thus offers advantages to other, more restrictive, parametric models. Simulation studies indicate that the method provides reasonably accurate coverage probabilities.     

Bootstrap-type confidence intervals for quantiles of the survival distribution

In this paper we outline and illustrate an easy to program method for analytically calculating both parametric and non-parametric bootstrap-type confidence intervals for quantiles of the survival distribution based on right censored data. This new approach allows for the incorporation of covariates within the framework of parametric models. The procedure is based upon the notion of fractional order statistics and is carried forth using a simple beta transformation of the estimated survival function (parametric or non-parametric). It is the only direct method currently available in the sense that all other methods are based on inverting test statistics or employing confidence intervals for other survival quantities. We illustrate that the new method has favourable coverage probabilities for median confidence intervals as compared to six other competing methods.     

线性模型中自变量相对重要性优势分析法估计及其应用

目的在多元线性回归模型中,估计各自变量的相对重要性,并探索区间估计方法。方法在自变量间存在相关时,运用Budescu(1993),Azen(2003)提出的优势分析法估计肝手术病例预计存活时间的影响因素重要性,并运用Bootstrap法探索区间估计方法以此来评价估计结果的变异性。结果血凝素、预后指数、酶功能对预计存活时间的相对贡献分别为0.1415、0.3408和0.490,其Bootstrap法95%可信区间分别为(0.0573,0.2744)、(0.2359,0.4545)和(0.3411,0.6090)。结论酶功能对肝手术病例预计存活时间的影响最大,预后指数次之,血凝素最小。当自变量间存在相关时,优势分析法估计的自变量相对重要性结果更精确稳定,值得推广应用。     

Effective sample sizes for confidence intervals for survival probabilities

We examine various methods to estimate the effective sample size for construction of confidence intervals for survival probabilities. We compare the effective sample sizes of Cutler and Ederer and Peto et al., as well as a modified Cutler-Ederer effective sample size. We investigate the use of these effective sample sizes in the common situation of many censored observations that intervene between the time point of interest and the last death before this time. We note that there is no a priori reason to treat upper and lower confidence intervals in a symmetric fashion since censored survival data are by nature asymmetric. We recommend the use of the Cutler-Ederer effective sample size in construction of upper confidence intervals and the Peto effective sample size in construction of lower confidence intervals. Two examples with real data demonstrate the differences between confidence intervals formed with different effective sample sizes. This study also illustrates the need for caution in the application of simulation studies to real problems.     

On the analysis of life tables for dependent observations

Applying methods assuming independence when observations are positively correlated means that confidence intervals become too short and significance levels of statistical tests are less extreme. This paper discusses elements of life-table analysis based on the standard product-limit estimator and a modified Greenwood formula for its variance to be used for dependent observations. An application from oral surgery is given. Erroneously assuming independence in the analysis of a life table could have serious consequences. It is demonstrated in a simulation study that the confidence levels can be much too low. The proposed modification of the Greenwood formula for the variance of the estimated survival function most often results in confidence levels not too much below the required level. Using the upper bound for the variance will give conservative confidence intervals but also larger standard errors. Averaging individual group survival curves should only be considered for situations with large groups.     

Survival attributable to an exposure

In the survival analysis context, when an intervention either reduces a harmful exposure or introduces a beneficial treatment, it seems useful to quantify the gain in survival attributable to the intervention as an alternative to the reduction in risk. To accomplish this we introduce two new concepts, the attributable survival and attributable survival time, and study their properties. Our analysis includes comparison with the attributable risk function as well as hazard‐based alternatives. We also extend the setting to the case where the intervention takes place at discrete points in time, and may either eliminate exposure or introduce a beneficial treatment in only a proportion of the available group. This generalization accommodates the more realistic situation where the treatment or exposure is dynamic. We apply these methods to assess the effect of introducing highly active antiretroviral therapy for the treatment of clinical AIDS at the population level. Copyright © 2009 John Wiley & Sons, Ltd.     

Sample size planning of two‐arm superiority and noninferiority survival studies with discrete follow‐up

In clinical trials using lifetime as primary outcome variable, it is more the rule than the exception that even for patients who are failing in the course of the study, survival time does not become known exactly since follow‐up takes place according to a restricted schedule with fixed, possibly long intervals between successive visits. In practice, the discreteness of the data obtained under such circumstances is plainly ignored both in data analysis and in sample size planning of survival time studies. As a framework for analyzing the impact of making no difference between continuous and discrete recording of failure times, we use a scenario in which the partially observed times are assigned to the points of the grid of inspection times in the natural way. Evaluating the treatment effect in a two‐arm trial fitting into this framework by means of ordinary methods based on Cox's relative risk model is shown to produce biased estimates and/or confidence bounds whose actual coverage exhibits marked discrepancies from the nominal confidence level. Not surprisingly, the amount of these distorting effects turns out to be the larger the coarser the grid of inspection times has been chosen. As a promising approach to correctly analyzing and planning studies generating discretely recorded failure times, we use large‐sample likelihood theory for parametric models accommodating the key features of the scenario under consideration. The main result is an easily implementable representation of the expected information and hence of the asymptotic covariance matrix of the maximum likelihood estimators of all parameters contained in such a model. In two real examples of large‐scale clinical trials, sample size calculation based on this result is contrasted with the traditional approach, which consists of applying the usual methods for exactly observed failure times. Copyright © 2017 John Wiley & Sons, Ltd.     

Integrating health profile with survival for quality of life assessment

In cohort studies or clinical trials, measurements of quality of life (QoL) were averaged across available individuals for each group at given points in time to produce single measures for comparisons. However, estimates of these single measures may be severely biased if substantial mortality occurs over time. The objective of this study is to develop a method that integrates QoL measurement and survival for long-term evaluation of health services. We defined a mean QoL score function over time for an index population as the average QoL score of all individuals both alive and dead at each time point in the population. While a living subject's QoL can be assessed by asking one's subjective preference, the score of a decedent can be assigned a fixed value depending on the specific facet on health profile. The mean QoL score function over time is reduced to a single measure of expected cumulative QoL score, which is the area under the curve of mean QoL score function over a given time interval and can be estimated by taking a random sample from a cross-sectional survey. For the QoL score function to be extrapolated to life-long, it requires the assumption that the disease causes premature death or a long-term moderate impairment of QoL. We provided methods and computer programs for estimating mean QoL score functions and the reduced single measures for use in comparisons. A cohort of 779 breast cancer patients from Chiangmai, Thailand were followed for 12 years to demonstrate the proposed methods. The data included the 12-year complete survival records and QoL scores on 233 patients collected from a cross-sectional survey using WHOQOL questionnaire and standard gamble method. The expected cumulative QoL scores using utility and psychometric scales were compared among patients in four groups of clinical stages in this cohort for time from onset up to 12 years and life-long. We conclude that such an integration of QoL measurement with survival can be useful for the evaluation of health service and clinical decision.     

6 min步行试验评估百草枯中毒幸存者心肺代偿功能的研究

目的 通过6 min步行试验评估百草枯中毒幸存者心肺代偿功能情况。 方法 选择56例病程1年以上的百草枯中毒幸存者为研究组,56例同龄、同性别健康人为对照组,分别进行标准6 min步行试验,比较两组个体步行距离、步行后血氧饱和度下降分数和Borg呼吸困难评分,评估百草枯中毒幸存者心肺代偿功能情况及劳动耐受情况。 结果 百草枯中毒幸存者与健康人的6 min步行距离分别为(604.43 ±101.22) m和(640.16 ±106.57) m,步行后血氧饱和度下降分数分别为(3.01 ±1.02)%、(2.79 ±0.97)%,步行后Borg呼吸困难评分分别为(0.70 ±0.66)分和(0.57 ±0.44)分,以上数据在两组间的差异均无统计学意义(P>0.05)。 结论 百草枯中毒幸存者一年后心肺代偿功能恢复良好,能够像健康人一样胜任日常活动。此类患者应重树信心,以饱满的热情重新融入社会。     

SEQUENTIAL MONITORING OF CLINICAL TRIALS WITH MULTIPLE SURVIVAL ENDPOINTS

Many clinical trials follow patients for several different types of survival endpoints, such as mortality, disease progression, and time until dose-limiting toxicity. Conduct of such trials often requires that the accumulating data be reviewed periodically to protect the safety of participating patients and possibly identify early treatment differences. This paper proposes a group sequential method for assessing multiple survival endpoints using repeated confidence intervals. Counting processes for each survival endpoint are used to estimate both the correlation between outcomes and between times of interim analysis. The methods are illustrated using a clinical trial comparing two treatments for PCP prevention in AIDS patients. The operating characteristics of three strategies for constructing confidence intervals are assessed and compared in a simulation study.     

Optimal survival time‐related cut‐point with censored data

In biomedical research and practice, continuous biomarkers are often used for diagnosis and prognosis, with a cut‐point being established on the measurement to aid binary classification. When survival time is examined for the purposes of disease prognostication and is found to be related to the baseline measure of a biomarker, employing a single cut‐point on the biomarker may not be very informative. Using survival time‐dependent sensitivity and specificity, we extend a concordance probability‐based objective function to select survival time‐related cut‐points. To estimate the objective function with censored survival data, we adopt a non‐parametric procedure for time‐dependent receiver operational characteristics curves, which uses nearest neighbor estimation techniques. In a simulation study, the proposed method, when used to select a cut‐point to optimally predict survival at a given time within a specified range, yields satisfactory results. We apply the procedure to estimate survival time‐dependent cut‐point on the prognostic biomarker of serum bilirubin among patients with primary biliary cirrhosis. Copyright © 2014 John Wiley & Sons, Ltd.     

Finite sample pointwise confidence intervals for a survival distribution with right‐censored data

We review and develop pointwise confidence intervals for a survival distribution with right‐censored data for small samples, assuming only independence of censoring and survival. When there is no censoring, at each fixed time point, the problem reduces to making inferences about a binomial parameter. In this case, the recently developed beta product confidence procedure (BPCP) gives the standard exact central binomial confidence intervals of Clopper and Pearson. Additionally, the BPCP has been shown to be exact (gives guaranteed coverage at the nominal level) for progressive type II censoring and has been shown by simulation to be exact for general independent right censoring. In this paper, we modify the BPCP to create a ‘mid‐p’ version, which reduces to the mid‐p confidence interval for a binomial parameter when there is no censoring. We perform extensive simulations on both the standard and mid‐p BPCP using a method of moments implementation that enforces monotonicity over time. All simulated scenarios suggest that the standard BPCP is exact. The mid‐p BPCP, like other mid‐p confidence intervals, has simulated coverage closer to the nominal level but may not be exact for all survival times, especially in very low censoring scenarios. In contrast, the two asymptotically‐based approximations have lower than nominal coverage in many scenarios. This poor coverage is due to the extreme inflation of the lower error rates, although the upper limits are very conservative. Both the standard and the mid‐p BPCP methods are available in our bpcp R package. Published 2016. This article is US Government work and is in the public domain in the USA.     

The restricted cubic spline as baseline hazard in the proportional hazards model with step function time-dependent covariables

We incorporate a cubic spline function where the tails are linearly constrained, as the baseline hazard, into the proportional hazards model. We show estimation of covariable coefficients and survival probabilities with this model to be as efficient statistically as with the Cox proportional hazards model when covariables are fixed. Examples show that the inclusion of time-dependent covariables defined as step functions into the restricted cubic spline proportional hazards model reduces computation time by a factor of 213 over the Cox model. Advantages of the spline model also include flexibility of the hazard, smooth survival curves, and confidence limits for the survival and hazard estimates when there are time-dependent covariables present.     

Comparison of median survival times with adjustment for covariates

Brookmeyer and Crowley derived a non-parametric confidence interval for the median survival time of a homogeneous population by inverting a generalization of the sign test for censored data. The 1 – α confidence interval for the median is essentially the set of all values t such that the Kaplan—Meier estimate of the survival curve at time t does not differ significantly from one-half at the two-sided α level. Su and Wei extended this approach to the two-sample problem and derived a confidence interval for the difference in median survival times based on the Kaplan-Meier estimates of the individual survival curves and a ‘minimum dispersion’ test statistic. Here, I incorporate covariates into the analysis by assuming a proportional hazards model for the covariate effects, while leaving the two underlying survival curves virtually unconstrained. I generate a simultaneous confidence region for the two median survival times, adjusted to any selected value, z , of the covariate vector using a test-based approach analogous to Brookmeyer and Crowley's for the one-sample case. This region is, in turn, used to derive a confidence interval for the difference in median survival times between the two treatment groups at the selected value of z . Employment of a procedure suggested by Aitchison sets the level of the simultaneous region to a value that should yield, at least approximately, the desired confidence coefficient for the difference in medians. Simulation studies indicate that the method provides reasonably accurate coverage probabilities.     

生存率置信区间的五种估计方法

生存率是医学随访研究资料分析中常用的指标，例如适用于小样本资料的Ｋａｐｌａｎ－Ｍｅｉｅｒ乘积限估计和大样本资料的寿命表法生存率估计。本文对生存率置信区间的估计方法进行了讨论。主要介绍了五种置信区间的估计方法：经典法（基于Ｇｒｅｅｎｗｏｏｄ方差公式）、校正法、反正旋转换法、ｌｏｇ（－ｌｏｇ）转换法及ｌｏｇｉｔ转换法。文中给出了两个实例，并就生存率９５％置信区间的估计做了详细介绍，还进一步讨论了它们在     

The association between change in cognitive function and longevity in Dutch elderly

The association between rate of change in cognitive function and longevity was investigated with data from the Dutch Longitudinal Study Among the Elderly. A group of 211 Dutch persons aged 65-84 years at baseline (1955-1957) was reexamined twice during an 8-year follow-up period, after which mortality was ascertained through 1983. Cognitive function was assessed based on an adaptation of the Wechsler Memory Scale. Rate of change in cognitive function during the 8 years of follow-up was determined by regression on time for each individual. Cognitive function declined significantly over the 8-year period (mean yearly change, -0.28 units; 95% confidence interval -0.34 to -0.22). The rate of decline in cognitive function was strongly associated with subsequent survival time in the ages 70 years and over, with those with a large decline having a short survival time. No association could be demonstrated in the age group 65-69 years. Adjustment for potential confounders did not affect the magnitude of the association. These findings suggest that the rate of decline of cognitive function is an independent predictor of longevity in older persons.     

Estimation of the survival function for Gray's piecewise-constant time-varying coefficients model

Gray's extension of Cox's proportional hazards (PH) model for right-censored survival data allows for a departure from the PH assumption via introduction of time-varying regression coefficients (TVC). For this model estimation of the conditional hazard rate relies on the inclusion of penalized splines. Cubic penalized splines tend to be unstable in the right tail of the distribution and thus quadratic, linear and piecewise-constant penalized splines may be a favourable choice. We derive a survival function estimator for one important member of the class of TVC models--a piecewise-constant time-varying coefficients (PC-TVC) model. Using the first-order Taylor series approximation we also derive an estimate for the variance of the log-transformed and log(-log)-transformed survival function, which in turn leads to estimated confidence limits on the corresponding scales of the survival function. Accuracy in estimating underlying survival times and survival quantiles is assessed for both Cox's and Gray's PC-TVC model using a simulation study featuring scenarios violating the PH assumption. Finally, an example of the estimated survival functions and the corresponding confidence limits derived from Cox's PH and Gray's PC-TVC model, respectively, is presented for a liver transplant data set.     

Evaluating center-specific long-term outcomes through differences in mean survival time: Analysis of national kidney transplant data

Center-specific survival outcomes of kidney transplant recipients are an important quality measure, with several challenges. Existing methods based on restricted mean lifetime tend to focus on short- and medium-term clinical outcomes and may fail to capture long-term effects associated with quality of follow-up care. In this report, we propose methods that combine a lognormal frailty model and piecewise exponential baseline rates to compare the mean survival time across centers. The proposed methods allow for the consistent estimation of mean survival time as opposed to restricted mean lifetime and, within this context, permits more accurate profiling of long-term center-specific outcomes. Asymptotic properties of the proposed estimators are derived, and finite-sample properties are examined through simulation. The proposed methods are then applied to national kidney transplant data. The novelty of the proposed techniques arises from several angles. We utilize mean survival, in contrast to the most previous works that considered the restricted mean. Few previous studies have used the integrated survival function as a basis for center effects. Few provider profiling methods use a random effects model to estimate fixed center effects.     

Influence of breastfeeding on cognitive outcomes at age 6-8 years: follow-up of very low birth weight infants

The relation between breastfeeding and childhood cognitive development was examined in 1991-1993 among 439 school-age children weighing <1,500 g when born in the United States between 1991 and 1993. Measures of cognitive function included overall intellectual function, verbal ability, visual-spatial and visual-motor skill, and memory. Higher test scores for each domain of cognitive function except memory were observed among children who were breastfed directly. After covariate adjustment for home environment, maternal verbal ability, a composite measure of parental education and occupation, and length of hospitalization, the authors found that breastfed children evidenced an advantage only for measures specific to visual-motor integration (5.1 intelligence quotient (IQ) points, 95% confidence interval: 1.0, 9.2). Differences in test scores between breastfed children and those who did not receive any breast milk feedings were 3.6 IQ points (95% confidence interval: -0.3, 7.5) for overall intellectual functioning and 2.3 IQ points (95% confidence interval: -3.0, 7.6) for verbal ability. Indicators of social advantage confound the association between breastfeeding and cognitive function, but careful measurement can reduce residual confounding and may clarify causal relations.