SUMMARY Nonproportional hazards can often be expressed by extending the Cox model to include time varying coefficients; e.g., for a single covariate, the hazard function for subject i is modelled as exp { fl(t)Zi(t)}. A common example is a treatment effect that decreases with time. We show that the function /3(t) can be directly visualized by smoothing an appropriate residual plot. Also, many tests of proportional hazards, including those of Cox (1972), Gill & Schumacher (1987), Harrell (1986), Lin (1991), Moreau, O'Quigley & Mesbah (1985), Nagelkerke, Oosting & Hart (1984), O'Quigley & Pessione (1989), Schoenfeld (1980) and Wei (1984) are related to time-weighted score tests of the proportional hazards hypothesis, and can be visualized as a weighted least-squares line fitted to the residual plot.

Proportional hazards tests and diagnostics based on weighted residuals

Many survival studies record the times to two or more distinct failures on each subject. The failures may be events of different natures or may be repetitions of the same kind of event. In this article, we consider the regression analysis of such multivariate failure time observations. Each marginal distribution of the failure times is formulated by a Cox proportional hazards model. No specific structure of dependence among the distinct failure times on each subject is imposed. The regression parameters in the Cox models are estimated by maximizing the failure-specific partial likelihoods. The resulting estimators are shown to be asymptotically jointly normal with a covariance matrix that can be consistently estimated. Simultaneous inferential procedures are then proposed. Extensive Monte Carlo studies indicate that the normal approximation is adequate for practical use. The new methods allow time-dependent covariates, missing observations, and arbitrary patterns of censorship. They are illustrat...

Regression analysis of multivariate incomplete failure time data by modeling marginal distributions

The trial was stopped for efficacy of prenatal surgery after the recruitment of 183 of a planned 200 patients. This report is based on results in 158 patients whose children were evaluated at 12 months. The first primary outcome occurred in 68% of the infants in the prenatal-surgery group and in 98% of those in the postnatalsurgery group (relative risk, 0.70; 97.7% confidence interval [CI], 0.58 to 0.84; P<0.001). Actual rates of shunt placement were 40% in the prenatal-surgery group and 82% in the postnatal-surgery group (relative risk, 0.48; 97.7% CI, 0.36 to 0.64; P<0.001). Prenatal surgery also resulted in improvement in the composite score for mental development and motor function at 30 months (P = 0.007) and in improvement in several secondary outcomes, including hindbrain herniation by 12 months and ambulation by 30 months. However, prenatal surgery was associated with an increased risk of preterm delivery and uterine dehiscence at delivery. Conclusions Prenatal surgery for myelomeningocele reduced the need for shunting and improved motor outcomes at 30 months but was associated with maternal and fetal risks. (Funded by the National Institutes of Health; ClinicalTrials.gov number, NCT00060606.)

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A Randomized Trial of Prenatal versus Postnatal Repair of Myelomeningocele

Subjects often drop out of longitudinal studies prematurely, yielding unbalanced data with unequal numbers of measures for each subject. Modern software programs for handling unbalanced longitudinal data improve on methods that discard the incomplete cases by including all the data, but also yield biased inferences under plausible models for the drop-out process. This article discusses methods that simultaneously model the data and the drop-out process within a unified model-based framework. Models are classified into two broad classes—random-coefficient selection models and random-coefficient pattern-mixture models—depending on how the joint distribution of the data and drop-out mechanism is factored. Inference is likelihood-based, via maximum likelihood or Bayesian methods. A number of examples in the literature are placed in this framework, and possible extensions outlined. Data collection on the nature of the drop-out process is advocated to guide the choice of model. In cases where the drop-...

Modeling the Drop-Out Mechanism in Repeated-Measures Studies

Effect of intensive diabetes treatment on the development and progression of long-term complications in adolescents with insulin-dependent diabetes mellitus: Diabetes Control and Complications Trial☆☆☆★★★

In a sequential medical trial, a simple randomized treatment assignment rule is proposed and analyzed. On the average this rule assigns more patients to the better treatment, and it is applicable to the case where patients have delayed responses to treatments. This new assignment rule is studied for both a fixed sample size and an inverse stopping rule.

The Randomized Play-the-Winner Rule in Medical Trials

The median is a simple and meaningful measure for the center of a long-tailed survival distribution. To examine the covariate effects on survival, a natural alternative to the usual mean regression model is to regress the median of the failure time variable or a transformation thereof on the covariates. In this article we propose semiparametric procedures to make inferences for such median regression models with possibly censored observations. Our proposals can be implemented efficiently using a simulated annealing algorithm. Numerical studies are conducted to show the advantage of the new procedures over some recently developed methods for the accelerated failure time model, a special type of mean regression models in the survival analysis. The proposals discussed in the article are illustrated with a lung cancer data set.

Survival analysis with median regression models

Asymptotically distribution-free tests for equality of two multivariate distributions based on censored observations are proposed. By choosing appropriate test scores, we obtain the multivariate versions of the Gehan and log-rank tests. An important application of the proposed tests is in the area in which multiple observations are collected on study subjects, some of whom may have missing observations. The patterns of missing observations are allowed to differ for the two groups to be compared. These tests are particularly useful for longitudinal studies—for example, the analysis of repeated measurements and growth curves. Real examples are also provided for illustration.

https://www.jstor.org/stable/info/2288413

Two-Sample Asymptotically Distribution-Free Tests for Incomplete Multivariate Observations

In the comparison of K treatments, assume that patients appear singly and must be treated immediately. Suppose that patients having the same combination of prognostic factor levels are grouped into the same stratum. If the number of different strata is small, we treat each stratum as a separate independent subtrial. A treatment assignment rule is proposed that forces a small subtrial to be balanced, but tends toward the complete randomization scheme as the size of the subtrial increases. When the number of strata is large, we propose an overall assignment rule which can achieve a degree of treatment balance simultaneously across all prognostic factors.

An Application of an Urn Model to the Design of Sequential Controlled Clinical Trials

In comparing two treatments, eligible subjects come to the experiment sequentially and must be treated at once. To reduce experimental bias and to increase the precision of inference about treatment effects, the adaptive biased coin design, which offers a compromise between perfect balance and complete randomization, is proposed and analyzed. This new design has the property that it forces a small-sized experiment to be balanced, but tends toward the complete randomization scheme as the size of the experiment increases.

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L. J. Wei

Papers

The Randomized Play-the-Winner Rule in Medical Trials

Survival analysis with median regression models

Two-Sample Asymptotically Distribution-Free Tests for Incomplete Multivariate Observations

An Application of an Urn Model to the Design of Sequential Controlled Clinical Trials

The Adaptive Biased Coin Design for Sequential Experiments