On the use and utility of the Weibull model in the analysis of survival data
Introduction
Cox's proportional hazards regression model (or equivalently in the case of two treatment groups, the log-rank test) has become the statistician's mainstay in the analysis of survival data [1], [2], [3], [4], [5], [6]. Its predominance stems from 3 decades of application and experience, together with the fact that it is distribution free; no assumption has to be made about the underlying distribution of survival times to make inferences about relative death rates. While this is a key strength of the model, it does introduce some limitations. Specifically, a direct quantification of the improvement in survival time is not possible, except in the special case of truly exponentially distributed lifetimes where the reciprocal of the hazard ratio estimates the ratio of median times to event [7]. However, lifetimes are seldom truly exponential in their distribution, so statisticians have tended to rely on Kaplan-Meier estimates of the underlying survivor function to read off estimated percentiles. The reliability and precision of these estimates depends upon the number of deaths and patients remaining at risk at any given time point on the curve. Median survival time is often used as a measure of improvement in time, though this measure is often unavailable at the earlier analyses of longer-term trials with relatively low event rates. Even when median survival can be estimated from the Kaplan-Meier curve, tests for differences in medians between treatments are generally approximate and do not directly link with tests for parity of hazard rates [8].
The Weibull model provides an alternative, fully parametric approach to the Cox model. Both these models are, in fact, closely related; both assume proportional hazards and both provide asymptotically unbiased, equally efficient estimates of the hazard ratio between two treatments. The Weibull model, in addition to being proportional, is simultaneously an accelerated failure-time model (AFT), and is the only parametric distribution to possess both properties [4], [9]. AFT models simply examine survival times via a log-linear model so that treatment effects are expressed in terms of the relative increase or decrease in survival time. The Weibull, being both accelerated and proportional, therefore allows the simultaneous description of treatment effects both in terms of hazard ratios and also in terms of the relative increase or decrease in survival time; we might conveniently refer to this latter quantification of treatment effect as an “event time ratio,” if only to illustrate the close parallel with the better known hazard (or event) rate ratio. Cox has suggested that these kinds of analyses are most favorable when a direct interpretation of the treatment effect is desired [10].
It is important to recognize that the Weibull and other AFT models are not new, having previously been described in the literature [11]. A good, accessible overview can be found in Colette [4]. Prentice and Kalbfleisch [9] and Wei [12] have discussed the potential use of AFT models in survival analyses and, more recently, Chen and Wang [13], [14] have discussed AFT models alongside a new class of models, the “accelerated hazards model,” which models how the underlying hazard changes over time.
Despite such coverage in the literature, the Weibull model is rarely used in the routine analysis and reporting of clinical trial data. Given that the Weibull allows simultaneous estimation of both the usual hazard ratio and an event time ratio, in addition to allowing a more thorough examination of proportionality and providing a means for predicting how data might mature over time, further consideration of its use and usefulness seems worthwhile.
The remainder of this paper is therefore structured as follows: the next section provides an overview of the Weibull model, including its form, estimation of hazard and event time ratios, examination of proportionality, and prediction of data maturation. After this a comparison of Cox and Weibull models in the analysis of real clinical trial data is made, followed by a brief discussion on the need for an exact distributional match when using the Weibull model. A brief summary of key results is then followed by the final section discussing the practical value and application of the Weibull and related models in the analysis of survival data in arising clinical trials.
Section snippets
The Weibull model
Before describing the Weibull model, it is helpful to consider a general distribution for lifetimes for which proportionality holds.
Let T = t denote the time to some event of interest; this could be time to death or progression-free survival in an oncology setting. If f(t) denotes the probability density function of T, S(t) the survivor function, and h(t) the hazard function, then, as is well known,Under proportionality, hA(t) = θhB(t), so that SA(t) = [SB(t)]θ, where θ is the
The hazard ratio
Based on the parameterization in Eq. (5), the hazard ratio for two treatments is given by
Hence, if αA≠αB, hazards are not proportional. When proportionality does hold, θ = λA/λB.
Based on the parameterization in Eq. (6), the log hazard ratio for an individual with covariates xi relative to an individual with covariates is
In the case of just two treatments, the log hazard ratio is therefore given by −β/σ. In SAS the variance of the estimated log hazard ratio,
Percentiles and the event time ratio
The percentiles of a Weibull are easily derived,where tp denotes the time taken to reach the pth percentile. The relative difference in the time to achieving the pth percentile between treatments A and B iswhich, under proportional hazards, simplifies to the acceleration factor or event time ratio,Again, based on the parameterization in Eq. (6), the log event time ratio for an individual with covariates relative to an individual with
Assessing proportionality in a Weibull analysis
In the analysis of survival data, graphical methods are routinely employed to assess the extent to which proportionality holds [4]. These methods may also be supplemented by a simple test for proportionality [16]. If data follow a Weibull distribution, then a direct, model-based test of proportionality can easily be achieved by comparison of shape parameters. If a Weibull is fitted separately for each treatment group, the two shape parameters, σ1 and σ2, say, together with their variances, can
Assessing treatment differences when proportionality does not hold
While some interpretation of treatment effect estimates may be possible in the presence of modest nonproportionality, some statisticians will rightly feel unease in drawing conclusions. This being the case, the Weibull allows the hazard ratio to be plotted as a function of time, via Eq. (7). From this description of the hazard ratio, it is possible to compare treatments in terms of the average or integrated hazard over some time interval (0−T). The integrated hazard is given by λTα−1 so that
Predicting data maturation
The Weibull has been used in the field of engineering to predict the proportion of future failures after having observed a failure process to a given point in time [17]. In the context of clinical trials, predicting how deaths are likely to accumulate over time is often important, especially in the many trials designed with prespecified, event-driven interim analyses. In such trials, it is of great interest to accurately predict the time course of emerging deaths so that the appropriate
An example
Analysis by both Cox's regression and the Weibull model is illustrated in the following example [18]. Patients with early prostate cancer were randomized to one of two treatments, active (bicalutamide 150 mg) or placebo. The primary endpoint was progression-free survival. The analysis took place at a minimum of 2 years and a median of 3 years follow-up. All patients were followed to disease progression or death irrespective of withdrawal of randomized therapy or addition of other, systemic
The affect of departures from the Weibull distribution
Concerns may arise when using Weibull-based analyses in that the data collected may not conform exactly to a Weibull distribution. Simple graphical checks can be used to assess the extent to which data have a Weibull distribution and residual diagnostics can be also examined to assess goodness of fit [2], [4].
Nevertheless, concerns may still be present that without a close distributional match, inferences based on a Weibull analysis may be misleading. However, for modest departures from a true
Summary
This paper has shown the Weibull model can provide a useful, parametric alternative to conventional Cox's regression modeling in the analysis of survival data. In addition to the hazard ratio, Weibull analysis provides a means of directly estimating the relative improvement in survival time, the event time ratio. This quantification of treatment effect is of some clinical relevance and is likely to be better understood by some nonstatisticians than the conventional hazard ratio. Further, it has
Discussion
The key implication of this paper is that in those very frequent instances where two or more treatments are to be compared for survival (or some other time-to-event endpoint) with adjustment for one or more baseline prognostic factors, the Weibull is at least as informative as a corresponding Cox analysis, and probably more so. Use of the Weibull provides researchers and data analysts with an estimate of treatment effect as per routine Cox analysis but, furthermore, provides a clinically
Acknowledgements
The author would like to thank Mr. Stuart Ellis, AstraZeneca Pharmaceuticals, Alderley Park, UK, and two anonymous referees for their thoughtful comments that greatly helped to improve the clarity and focus of this manuscript.
References (24)
- et al.
Estimating a treatment effect with the accelerated hazards model
Control Clin Trials
(2000) Regression models and life tables (with discussion)
Journal of the Royal Statistical Society, Series B
(1972)- et al.
The statistical analysis of failure-time data
(1980) - et al.
Analysis of survival data
(1984) Modeling survival data in medical research
(1994)- et al.
Asymptotically efficient rank invariant test procedures (with discussion)
Journal of the Royal Statistical Society, Series A
(1972) - et al.
Statistical methods in medical research
(1987) - et al.
When do statistics lie?
UroOncology
(2001) Confidence intervals for the difference of median survival times using the stratified Cox proportional hazards model
Biometrical Journal
(2001)- et al.
Hazard rate models with covariates
Biometrics
(1979)