EVALUATION OF LONG-TERM SURVIVAL: USE OF DIAGNOSTICS AND ROBUST ESTIMATORS WITH COX'S PROPORTIONAL HAZARDS MODEL

We consider methodological problems in evaluating long-term survival in clinical trials. In particular we examine the use of several methods that extend the basic Cox regression analysis. In the presence of a long term observation, the proportional hazard (PH) assumption may easily be violated and a few long term survivors may have a large effect on parameter estimates. We consider both model selection and robust estimation in a data set of 474 ovarian cancer patients enrolled in a clinical trial and followed for between 7 and 12 years after randomization. Two diagnostic plots for assessing goodness-of-fit are introduced. One shows the variation in time of parameter estimates and is an alternative to PH checking based on time-dependent covariates. The other takes advantage of the martingale residual process in time to represent the lack of fit with a metric of the type ‘observed minus expected’ number of events. Robust estimation is carried out by maximizing a weighted partial likelihood which downweights the contribution to estimation of influential observations. This type of complementary analysis of long-term results of clinical studies is useful in assessing the soundness of the conclusions on treatment effect. In the example analysed here, the difference in survival between treatments was mostly confined to those individuals who survived at least two years beyond randomization     

A BIVARIATE PARAMETRIC MODEL FOR SURVIVAL AND INTERMEDIATE EVENT TIMES

In diseases such as acquired immune deficiency syndrome (AIDS), there is great interest in describing the frequency of secondary diagnoses that occur during the course of the disease and their effect on survival. Casting this situation in a more general framework, one distinguishes a terminal event (TE) and an intermediate event (IE) that may or may not occur. In epidemiologic applications the TE is usually death. Earlier studies of IE and TE times have used the latter to censor the IE time for individuals who do not present it. For such cases, we argue that more appropriately the TE removes the individual from the risk set for the IE. With this view, one distinguishes observations of four types, each with a different formula for its likelihood contribution. We propose an approach that uses separate parametric models for the marginal distribution of the survival time D and for the conditional distribution of the time R to the IE given D = d and R⩽D. A central quantity is the probability of presenting the IE given the occurrence of the TE at time d. This function of d can reveal important connections between the two events. We suggest a model derived from Weibull distributions where the parameters control the shape of this function. One can obtain inferences about other probabilities of interest such as the proportion of individuals who present the IE, P(R⩽D), the marginal distribution of R among the IE cases, P(R>r∣R⩽D) and the residual survival after the IE occurs, P(D−R>v∣R⩽D, R = r). We apply the model to the analysis of time to secondary Kaposi's sarcoma (KS) diagnosis and time to death in a large cohort study of homosexual men infected with the human immunodeficiency virus type 1 (HIV) and who had an initial non-KS AIDS diagnosis.     

A ROBUST METHOD FOR PROPORTIONAL HAZARDS REGRESSION

In this paper we give an informal introduction to a robust method for survival analysis which is based on a modification of the usual partial likelihood estimator (PLE). Large sample results lead us to expect reduced bias for this robust estimator compared with the PLE whenever there are even slight violations of the model. In this paper we investigate three types of violation: (a) varying dependency structure of survival time and covariates over the sample; (b) omission of influential covariates, and (c) errors in the covariates. The simulations presented support the above expectation. Analyses of data sets from cancer epidemiology and from a clinical trial in lung cancer illustrate that a better fit and additional insights may be gained using robust estimators.     

AN EMPIRICAL COMPARISON OF STATISTICAL TESTS FOR ASSESSING THE PROPORTIONAL HAZARDS ASSUMPTION OF COX'S MODEL

In the analysis of survival data using the Cox proportional hazard (PH) model, it is important to verify that the explanatory variables analysed satisfy the proportional hazard assumption of the model. This paper presents results of a simulation study that compares five test statistics to check the proportional hazard assumption of Cox's model. The test statistics were evaluated under proportional hazards and the following types of departures from the proportional hazard assumption: increasing relative hazards; decreasing relative hazards; crossing hazards; diverging hazards, and non-monotonic hazards. The test statistics compared include those based on partitioning of failure time and those that do not require partitioning of failure time. The simulation results demonstrate that the time-dependent covariate test, the weighted residuals score test and the linear correlation test have equally good power for detection of non-proportionality in the varieties of non-proportional hazards studied. Using illustrative data from the literature, these test statistics performed similarly. © 1997 by John Wiley & Sons, Ltd.     

EFFECTS OF COVARIATE OMISSION AND CATEGORIZATION WHEN ANALYSING RANDOMIZED TRIALS WITH THE COX MODEL

When analysing the survival of patients in comparative randomized clinical trials using the Cox proportional hazards model, important prognostic factors may be included for the adjustment of the treatment effect. In this paper we examine two of the most common misspecifications of the model: (i) an important prognostic factor is omitted from the analysis; and (ii) an important prognostic factor originally present on continuous scale is included in categorized form. Both situations may emerge from the occurrence of missing values. We investigate the properties of the maximum partial likelihood estimator of the treatment effect under this kind of misspecification. The estimate of the treatment effect is found to be asymptotically biased toward zero. For its asymptotic variance we obtain a quantity with the so-called ‘sandwich’ structure. Thus, variance estimation by the inverse of the second-order derivative of the likelihood is not consistent. The magnitude of overestimation or underestimation is evaluated numerically for specific settings. The precision of the treatment effect estimate under covariate omission or categorization is compared with the precision of the estimate in the correct and not misspecified model. It turns out that correct adjustment does not lead to a higher precision of the treatment effect estimate, but due to the resulting underestimation, covariate omission or categorization lead to loss of power of the test of no treatment effect. © 1997 by John Wiley & Sons, Ltd.     

ESTIMATING CONFIDENCE INTERVALS FOR COST-EFFECTIVENESS RATIOS: AN EXAMPLE FROM A RANDOMIZED TRIAL

Cost-effectiveness ratios usually appear as point estimates without confidence intervals, since the numerator and denominator are both stochastic and one cannot estimate the variance of the estimator exactly. The recent literature, however, stresses the importance of presenting confidence intervals for cost-effectiveness ratios in the analysis of health care programmes. This paper compares the use of several methods to obtain confidence intervals for the cost-effectiveness of a randomized intervention to increase the use of Medicaid's Early and Periodic Screening, Diagnosis and Treatment (EPSDT) programme. Comparisons of the intervals show that methods that account for skewness in the distribution of the ratio estimator may be substantially preferable in practice to methods that assume the cost-effectiveness ratio estimator is normally distributed. We show that non-parametric bootstrap methods that are mathematically less complex but computationally more rigorous result in confidence intervals that are similar to the intervals from a parametric method that adjusts for skewness in the distribution of the ratio. The analyses also show that the modest sample sizes needed to detect statistically significant effects in a randomized trial may result in confidence intervals for estimates of cost-effectiveness that are much wider than the boundaries obtained from deterministic sensitivity analyses.     

APPLICATIONS OF A MIXTURE SURVIVAL MODEL WITH COVARIATES TO THE ANALYSIS OF A DEPRESSION PREVENTION TRIAL

This paper presents a case study of model selection for survival analysis data. We use an approximate Bayesian method for model selection based on assessing the posterior probability of competing models given the data. We introduce the Schwarz criteria, an approximation to the logarithm of the Bayes factor, to provide an indication of evidence in favour of one model compared to another. Specifically, in the context of a depression prevention clinical trial we evaluate the efficacy of treatment in preventing or delaying the time to recurrence of depression, and evaluate how differences in the survival distributions between the two treatment groups depend on explanatory variables of interest. This investigation is based on a mixture survival model that explicitly incorporates the possibility of a surviving fraction.     

BIOLOGICAL EFFECTS OF CHRONIC ETHANOL CONSUMPTION: A REAPPRAISAL OF THE LIEBER-DE CARLI LIQUID-DIET MODEL WITH REFERENCE TO SKELETAL MUSCLE

This paper responds to the recent article by Rao et at. (Alcoholand Alcoholism 21, 369–373, 1986) which suggested thatthe Lieber-De Carli liquid diet for chronic ethanol-feedingstudies may not be suitable for the rat because of the changesin carbohydrate content. The viability of the Lieber-De Carlimodel was, therefore, re-examined. The content of the liquiddiet was shown to be nutritionally adequate, when compared toa solid laboratory chow. However, rats on an alcohol feedingregime had sub-optimal growth rates, because of a reductionin the amount of liquid diet consumed. When compared to pairfed controls, rats fed the ethanol-containing diet showed markedchanges in skeletal muscle. These observations are similar tothose in man and we conclude that the use of the Lieber-De Carlifeeding regime in experimental animals is a suitable means ofinvestigating the mechanism of skeletal myopathy.