Parametric Models in Survival Analysis

Transcription

1 arametric Models in Survival Analysis Survival analysis in biostatistical applications involves the analysis of times to events (see Survival Analysis, Overview); for conciseness we refer to these times as lifetimes. Sometimes the objective is to model or describe the distribution of lifetimes in a single homogeneous population of individuals. More generally, we wish to compare distributions or to assess the relationship of explanatory variables to lifetimes. In some instances different types of events may occur to an individual, and the joint distribution of several lifetimes may be of interest. arametric survival models are ones in which the distribution of lifetimes is specified up to a parameter θ that is of finite, and usually rather small, dimension. A major advantage of parametric models is the availability of straightforward methods of estimation and inference basedonthelikelihood function. In addition, parametric representations facilitate the accumulation of scientific evidence across different but similar studies. However, fully parametric models involve stronger assumptions than semi- or nonparametric models. The choice of a parametric model is more often based on a combination of tractability and ability to fit the data than on any deep physical motivation, and it is therefore important to check the adequacy of models and to consider the sensitivity of inferences and conclusions to plausible variations in them. The next two sections describe the main families of models used for parametric survival analysis, and how such analysis is carried out. arametric Models Following standard terminology, let T 0bearandom variable representing lifetime and assume that T has a continuous distribution with cumulative distribution function (cdf) F(t) = r(t t), probability density function (pdf) f(t)= F (t), survivor function S(t) = r(t t), hazard function h(t) = f(t)/s(t), and cumulative hazard function H(t) = t 0 h(u) du. Any one of these five functions specifies the distribution of T (see Survival Distributions and Their Characteristics). It is useful to note that S(t) = exp[ H(t)]. arametric models are specified in terms of a parameter θ, and in this case we write F(t; θ),f (t; θ), and so on. A wide variety of models has been used for univariate lifetime distributions. The merits of specific families of distributions are often discussed both in terms of their ability to fit existing data and the shapes of their hazard functions, which specify the instantaneous probability of death or failure at time t, given survival up to t. In the following subsection we describe some important parametric families. Two additional subsections discuss models involving covariates and models that are multivariate or involve random effects. Some Important arametric Families Below we summarize the most widely used parametric lifetime distribution models. Unless specified otherwise, the range of the lifetime variable t is t 0. Exponential distribution. The exponential distribution has survivor, density, and hazard functions of the form S(t) = exp( λt), f (t) = λ exp( λt) h(t) = λ, where λ>0 is a parameter; it is easily seen that E(T ) = λ 1. The exponential family has constant hazard functions and the associated lack of memory property: r(t t + x T t) = r(t x). These characteristics are obviously restrictive, and limit the applicability of the model. Approximate samplesize calculations for clinical trials (see Sample Size Determination in Survival Analysis) are sometimes based on it, but require careful application [22]. It is a useful fact that if T is an arbitrary random variable with cumulative hazard function H(t),then the transformed variable H(T)has a standard (λ = 1) exponential distribution. Weibull distribution. The Weibull distribution with inverse scale parameter λ>0 and shape parameter δ>0has survivor, density, and hazard functions S(t) = exp[ (λt) δ ], f(t)= δλ(λt) δ 1 exp[ (λt) δ ], h(t) = δλ(λt) δ 1.

2 2 arametric Models in Survival Analysis The two parameters allow the Weibull density to take a variety of shapes, and the hazard function is either monotone increasing, decreasing, or constant according to whether δ>1,δ <1, or δ = 1. The case δ = 1 gives the exponential distribution, and, more generally, T δ has an exponential distribution with hazard function λ δ. The Weibull distribution often fits biostatistical survival data well, and in some applications there is physical justification for a Weibull model through weakest link or multistage waiting time arguments (e.g. Armitage & Doll [3]) (see Multistage Carcinogenesis Models). Lognormal distribution. The lognormal distribution has the property that log lifetime, Y = log T,is normally distributed with mean µ and variance σ 2. This gives a two-parameter family with survivor and density functions ( ) log t µ S(t) = 1 Φ, σ [ 1 f(t)= (2π) 1/2 σt exp 1 ( ) ] log t µ 2, 2 σ where Φ(x) is the standard normal cdf. The hazard functions are nonmonotonic: they rise from 0 at t = 0 to a maximum, and then decrease to 0 as t. This shape is implausible for some, but by no means all, survival analysis applications. Moreover, the proportion of the population with lifetimes greater than the hazard function mode can vary widely according to the values of (µ, σ ). The lognormal model has the advantage that when there is no censoring of lifetimes we can apply simple normal distribution inference methods to the log lifetimes (see the section Model Fitting and Inference below). However, censoring is more the rule than the exception in survival data, and so this is a relatively minor advantage. Log-logistic distribution. The log-logistic family has survivor, density, and hazard functions of the form S(t) = 1 δλ(λt)δ 1, f(t) = 1 + (λt) δ [1 + (λt) δ ], 2 h(t) = δλ(λt)δ (λt) δ, where λ>0andδ>0 are inverse scale and shape parameters. It derives its name from the fact that Y = log T has a logistic distribution. This family is similar to the lognormal family, and its hazard functions also increase from 0 at t = 0toamaximum, and then decrease to 0 as t becomes large. It is slightly more convenient than the lognormal family for survival analysis, since its survivor and hazard functions have simple closed-form expressions. Location-scale models for Y = log T. The Weibull, lognormal, and log-logistic distributions share a property: log lifetime Y has a location-scale distribution with survivor and density functions of the form ( ) y µ S Y (y) = G, f Y (y) = 1 ( ) y µ σ σ g, σ (1) where µ( <µ< ) is a location parameter, σ>0 is a scale parameter, and G(z) and g(z) = G (z) are the survivor function and pdf for the standardized variable Z = (Y µ)/σ. When T is Weibull the distribution of Y = log T is called the extreme-value distribution. When T is lognormal, Y is normal, and when T is log-logistic, Y is logistic. The three respective survivor functions for Z are: 1. extreme-value G(z) = exp( e z ), 2. normal G(z) = 1 Φ(z), 3. logistic G(z) = (1 + e z ) 1, where <z< in each case. Other lifetime models can be formed by choosing other distributions for Z or Y. Sometimes the distribution of Z is allowed to depend upon one or two additional parameters, thus giving very flexible three- or four-parameter lifetime distributions. Two such families are the generalized log gamma [16; 23, Section 5.3] and the Burr areto [6, 25]. The latter, for example, gives a three-parameter distribution with survivor function for T of the form S(t) = [1 + α 1 (λt) δ ] α, (2) where α>0,λ>0, and δ>0. This is obtained from (1) by taking G(z) = (1 + α 1 e z ) α and defining λ = e µ and δ = σ 1. It may also be obtained from a Weibull frailty model with gamma-distributed

3 arametric Models in Survival Analysis 3 random effects (e.g. [12]). The special case α = 1 gives the log-logistic distribution, and as α the Weibull distribution is obtained, so the Burr areto model may be used to discriminate between these two models. The generalized log gamma model is similar, and includes the Weibull and lognormal distributions as special cases. An even more comprehensive fourparameter family is obtained by choosing Z to have a standardized log F distribution [20, pp. 28, 63; 7]. This includes as special cases all of the distributions considered thus far. Some other models. A number of other parametric models are occasionally used as lifetime distributions; two of the more common models are the gamma and the inverse Gaussian. In some circumstances it is convenient to formulate a model so as to give certain shapes for the hazard function. For example, the Gompertz distribution has loglinear hazard, log h(t) = α 1 + α 2 t. Another example arises in contexts where the hazard function is thought to exhibit the so-called bathtub shape, in which it decreases from a local maximum at t = 0toalocal minimum, and then increases thereafter. one of the models discussed so far allows hazard functions of this shape. Models that do have been proposed by various authors; Hjorth [18], for example, considers a three-parameter family with h(t) = λt + α/(1 + βt), where λ>0,α >0, and β>0. If λ αβ, then the hazard function has the bathtub shape. Finally, lifetimes are by convention usually taken to be nonnegative, and the models discussed so far all assume T 0. In some applications it is argued that there exists an unknown minimum or threshold time γ>0before which the event in question cannot occur. In that case we can model the lifetime distribution by replacing t by t γ in previous expressions for survivor functions, hazard functions, and so on. For example, in the case of the exponential distribution this yields a two-parameter model with pdf f(t)= λ exp[ λ(t γ)],t γ. A practical difficulty when γ is unknown is that threshold parameters are difficult to estimate precisely unless there is a considerable amount of data. Regression Models In most survival analysis applications there are groups of individuals to be compared, or explanatory variables, such as individual characteristics or environmental conditions, whose relationship to lifetime is to be examined. A small number of homogeneous groups may be compared by fitting separate lifetime distributions for each group, but more generally, regression models that employ covariates to model the effects of explanatory variables are used. For example, in studying the survival time from diagnosis of patients with multiple myeloma, Krall et al. [21] considered 16 covariates representing factors such as the white blood-cell count at diagnosis, the presence or absence of infection at diagnosis, and the sex and age of the subject. Survival analysis may also involve covariates that vary over time. Time-varying covariates are usually handled through hazard-based semiparametric models, especially Cox s proportional or multiplicative hazards model [1, Chapter VI; 20, Chapters 4 and 5] (see Cox Regression Model). This is to a large extent because partial likelihood methods of analysis for such models [9] tend to be simpler than methods based on fully parametric models. Most of the discussion below deals with cases where the covariates are fixed, i.e. constant over time, since that is the main domain of application of parametric models. We return to time-varying covariates later in the article. Let T denote a lifetime and x a p 1 vector of covariates associated with each individual. Fully parametric regression models are in principle obtained by taking any parametric lifetime distribution and allowing its parameters to depend upon x in some specified way. For example, Weibull regression models are obtained by taking S(t x) = exp{ [λ(x)t] δ(x) }, (3) where we write S(t x) to denote the survivor function of T given x. To complete the model we must specify how λ(x) and δ(x) depend on x. Inmany applications it happens that δ(x) does not vary much with x, andsoδ(x) = δ is assumed fixed. Models with λ(x) = exp(x β), where β is a p 1 vector of regression coefficients, are often used; this form is flexible and automatically constrains λ(x) to be nonnegative. Although a very wide range of parametric regression models is possible, two types of models dominate. These are termed accelerated failure time (AFT) and proportional hazards (H) models; each

4 4 arametric Models in Survival Analysis has both fully parametric and semiparametric versions. They are described in turn. Accelerated failure time models. It was noted previously that for several common parametric survival models the distribution of log lifetime Y is of location-scale form (1). An important class of regression models is obtained by allowing the location parameter µ in (1) to depend on x, so that the survivor function of Y,givenx, is [ ] y µ(x) S Y (y x) = G, (4) σ where G(z) is a specified survivor function on < z<. We may also write (4) as Y = µ(x) + σε, (5) where the error ε has a distribution with survivor function G(z). In the case where µ(x) = x β, we have a linear model, although the errors are not necessarily normal. The name accelerated failure time derives from the corresponding model for T given x, which has survivor function [ ] log t µ(x) S(t x) = G σ = S 0 [ t α(x) ; δ ], (6) where α(x) = exp[µ(x)],δ = σ 1, and S 0 (t; δ) = G(δ log t). The effect of the covariates is to alter the time scale for t multiplicatively, i.e. either to accelerate or decelerate time. Other models that alter the time scale are also possible; in particular, nonlinear transformations of t may be used. In this sense, (6) is actually a special type of accelerated failure time model. The most widely used parametric AFT models are those for which ε in (5) has either a standard extreme value, logistic, or normal distribution, corresponding to T being Weibull, log-logistic and lognormal, respectively. However, other distributions may be used, as described in the preceding subsection. It may be noted that the Weibull model, (3), is an AFT model only if δ(x) = δ. Semiparametric versions of the AFT family are also based on (4), but do not assume any specific form for G(z) (see Semiparametric Regression). roportional hazards models. A proportional hazards (H) family of regression models is one for which the hazard function of T given x is of the form h(t x) = h 0 (t)r(x), (7) where r(x) is a positive-valued function and h 0 (t) is a baseline hazard function. The hazard function for any individual is proportional to h 0 (t), hence the name of the family. Fully parametric H models specify parametric forms h 0 (t; α) and r(x; β) for the two components of (7); in its semiparametric version [9], h 0 (t) is left unspecified. The specification r(x; β) = exp(x β) is often used. From (7) and the relationship S(t) = exp[ H(t)] it follows that the survivor function of T given x is of the form S(t x) = S 0 (t) r(x), (8) where S 0 (t) = exp[ H 0 (t)] is the baseline survivor function. A feature of H models is that if S 0 (t; α) is in a family of parametric models, then S(t x) is not always in the same family. It is, however, if h 0 (t) is of the form α 1 h 1 (t; α 2 ); this includes the Weibull family, (3), with δ(x) = δ. It is also easily checked that the family of Weibull regression models, (3), with δ(x) = δ, is both a H and an AFT model, and that it is the only set of distributions with this property. In other words, the class of H and the class of AFT models are distinct aside from models (3) with δ(x) = δ. The AFT and H models make fairly strong assumptions about the relationship between T and x. Among other things, they imply that the survivor functions for individuals with different covariate vectors never cross. Two extensions that relax these assumptions are often useful. The first retains a location-scale model, (4), for log T, but allows the scale parameter σ to depend on x; the parametric form σ(x) = exp(γ x) is convenient (e.g. [28]). The second is a relaxation of the H assumption that allows the regression coefficients in r(x) of (7) to change over time, giving h(t x) = h 0 (t)r[x; β(t)]. A simple example is the two-step model in which there is a specified value τ such that β(t) = β 1 for 0 t τ and β(t) = β 2 for t>τ. Covariates may be linked to lifetimes in a variety of other ways. For example, additive hazards models with h(t x) = h 0 (t; α) + r(x; β) (9)

5 arametric Models in Survival Analysis 5 are in some circumstances more plausible than H models. In another direction, so-called proportionalodds regression models [5, 25], in which [ ] [ ] S(t x) S0 (t; α) log = log + α(x; β), 1 S(t x) 1 S 0 (t; α) are sometimes useful; there is an obvious connection with logistic regression models for binary responses, and such models are especially useful in situations where T is a discrete variable. Let us now consider briefly time-varying covariates, in which case we write x(t) instead of x. Situations in which time-varying covariates can arise include (i) clinical studies where the treatment assigned to an individual may change during the course of the study, (ii) observational studies in which time-dependent environmental variables or exposures affect individuals, and (iii) studies where covariates internal to an individual are prognostic with respect to survival; for example, CD4 lymphocyte counts for subjects infected with the human immunodeficiency virus (HIV) are prognostic for timetodeath(see AIDS and HIV). In addition, synthetic time-dependent covariates may be used to test a H assumption [9]. A general treatment of time-varying covariates is delicate if we allow internal covariates; see [20, Chapter 5] or [1, Section III.5]. We restrict the discussion here to external covariates whose values are determined independently of individuals who are under study, and assume that an individual s hazard function at time t depends only on covariates x(t) whose value can be determined at time t. Such covariates are readily incorporated into models via the hazard function. For H models, (7), or additive hazards models, (8), for example, we merely replace x with x(t). The same thing can be done for AFT models by writing down the hazard function h(t x) that corresponds to (4), but the resulting models are less appealing than in the case of multiplicative or additive hazards. Even when the hazard function depends in a simple way on time-varying covariates, calculation of survival probabilities is more involved than for fixed covariates, as is the comparison of survival distributions for individuals with different covariate values. In particular, if we condition on the external covariate history x = [x(s), s 0], then { t } r(t t x ) = exp h[s x(s)]ds. (10) 0 This requires knowledge of all covariate values over (0,t). Other Models Survival models that involve more elaborate structure are sometimes needed. For example, there may be several modes of death or failure, such as in carcinogenicity studies in which animals are determined at autopsy to have died from one of several causes (see Tumor Incidence Experiments). This is often referred to as a competing modes of death or competing risks problem; the observable data consist of a lifetime T 0 and mode of death C, which is in some set (1,...,k). Models with covariates are conveniently expressed in terms of mode-specific hazard functions [23, Chapter 9]. h j (t x) = lim t 0 r(t < t + t, C = j T t,x), t j = 1,...,k. arametric representations h j (t x; θ) similar to those used for the hazard functions of parametric lifetime distributions, discussed above, can be employed here. Multivariate lifetime models in which m lifetimes (T i1,...,t im ) are associated with an individual i are sometimes needed; for example, T i1 and T i2 might represent the ages at death of identical twins [19]. It is also occasionally useful to associate individual latent failure times T i1,...,t ik with multiple modes of failure, as described in the preceding paragraph (see Competing Risks). Various parametric families have been proposed for multivariate survival analysis. Discrete or continuous mixture models are also employed frequently. For example, discrete mixtures have been used in situations where a fraction 1 p of individuals have an effectively infinite lifetime whereas the remaining fraction p have lifetimes that follow a distribution with survivor function S 1 (t; θ). A main area of application is in connection with the survival times of cancer patients, when a fraction 1 p of patients are cured (see Cure Models). The long-term survivors cannot be distinguished a

6 6 arametric Models in Survival Analysis priori, so a randomly selected individual has survivor function S(t; θ,p)= 1 p + ps 1 (t; θ). Models in which both p and S 1 (t) depend upon covariates may be considered (e.g. [15]). More general mixture models are obtained when individuals or groups ( clusters ) of individuals have unobservable random effects associated with them. Frailty models, for example, assume that there is a positive-valued random effect α associated with an individual and that, conditional on α, the individual s lifetime distribution has cumulative hazard function αh(t). The random effect is assumed to have a cdf G, in which case the unconditional survivor function for T is S(t) = 0 exp[ αh(t)]dg(α), (11) which is a Laplace transform. Multivariate lifetime models may similarly be obtained by assuming that there is a common random effect α associated with a group of lifetimes T 1,...,T m,which,givenα, are independent (e.g. [11]). Different parametric choices for H(t) and G(α) in (11) provide a wide variety of models. Model Fitting and Inference Model fitting involves estimation of the unknown parameters in the models on the basis of observed data. More generally, we may wish to estimate, or test hypotheses about, certain features of the model. Maximum likelihood methods are favored for most applications, and software is widely available for the most common lifetime models. Standard procedures are described below, followed by a discussion of model checking and an example. Likelihood-based Estimation and Inference Suppose that individuals i = 1,...,n have independent lifetimes T i with density and survivor functions f i (t; θ), and S i (t; θ), respectively. The index i is used with f and S to indicate that they may depend upon covariates associated with individual i. Data on lifetimes are frequently right-censored, so we will assume that for some individuals (i D) the exact lifetime t i is known, and for others (i C) only the fact that t i exceeds the observed censoring time c i is known. The likelihood function for θ is based on the joint pdf of the observed data. Under the assumption that the censoring of an individual at a point in time cannot be related to future events, the likelihood function is proportional to L(θ) = f i (t i ; θ) S i (c i ; θ). (12) i D i C The maximum likelihood estimate ˆθ is obtained by maximizing L(θ) with respect to θ, or equivalently, the log likelihood function l(θ) = log f i (t i ; θ) + log S i (c i ; θ). (13) i D i C With the models considered here, l(θ) can typically be maximized by solving the maximum likelihood or score equations l(θ)/ θ = 0. Maximum likelihood large-sample theory provides several ways of constructing tests or confidence intervals for model parameters [23, Appendix C]. The two which are most readily available in software are termed the Wald and likelihood ratio test procedures, and are described briefly. Suppose the p 1 parameter vector θ is partitioned as θ = (φ, λ ), where the r 1(r p) parameter φ is the parameter of interest and λ is a nuisance parameter. Thep p observed information matrix I(θ) and its inverse, I(θ) = 2 l θ θ = ( Iφφ I φλ I λφ I λλ ( V(θ) = I(θ) 1 Vφφ V φλ = V λφ V λλ ), ), play a key role in the Wald method. Under the hypothesis that φ = φ 0, the quantity ˆV 1/2 φφ ( ˆφ φ 0 ) is approximately standard r-variate normal in large samples, where ˆV stands for V( ˆθ). Equivalently, the Wald statistic (see Likelihood), W 1 (φ 0 ) = ( ˆφ φ 0 ) T ˆV 1 φφ ( ˆφ φ 0 ), (14) is approximately distributed as the chi-square distribution with r degrees of freedom, χ 2 (r). The statistic (14) can be used to test H: φ = φ 0 ; large values of W 1 (φ 0 ) provide evidence against the hypothesis. Confidence regions with approximate confidence

7 arametric Models in Survival Analysis 7 coefficient q are obtained as the set of parameter values φ that satisfy W 1 (φ) χq 2(r), whereχ q 2 (r) is the qth quantile of χ 2 (r). The likelihood ratio method utilizes the statistic W 2 (φ 0 ) = 2l( ˆφ, ˆλ) 2l[φ 0, ˆλ(φ 0 )], (15) where ˆθ = ( ˆφ, ˆλ ) ; and ˆλ(φ 0 ) is the value of λ that maximizes l(φ 0, λ). Under H: φ = φ 0,W 2 (φ 0 ) is approximately χ 2 (r) in large samples, and tests and confidence regions are obtained in the same way as with the Wald statistic. The methods based on (14) and (15) produce twosided confidence regions consisting in the case r = 1 of parameter values on either side of ˆφ. If onesided intervals are wanted, they may be based on sign ( ˆφ φ 0 )W 1 (φ 0 ) 1/2 or sign ( ˆφ φ 0 )W 2 (φ 0 ) 1/2, assumed to be approximately standard normal. The asymptotic theory upon which these methods rely requires that the models satisfy some mild regularity conditions. Of the models described earlier, the only cases which are problematic concern threshold parameters. For censored data a large sample is essentially one for which n is large and not too high a proportion of lifetimes is censored. For practical purposes what we desire is that significance levels (see Value) or confidence coefficients calculated from the asymptotic χ 2 or normal distributions for W 1 or W 2 are sufficiently close to their true values. If there is doubt about the adequacy of the approximations, they may be checked by simulation. A warning about accuracy is signaled by any large differences in confidence intervals or significance levels based on W 1 (φ 0 ) and W 2 (φ 0 ); in such cases intervals based on W 2 (φ 0 ) are likely to be more accurate. Analytic adjustments to improve accuracy have been developed (e.g. [4]), but are not yet widely available in software. arametric bootstrap procedures [13] provide another possibility for improving accuracy. Most survival analysis software relies solely on Wald or likelihood ratio procedures. Major packages with parametric survival analysis capabilities include SAS (see ROC LIFEREG and ROC RELIABIL- ITY), SAS JM, S-LUS (see especially the function Censor Reg), SYSTAT (see the Survival module), and LIMDE. All handle estimation for accelerated failure time regression models based on the Weibull, lognormal, and log-logistic distributions. In addition to right censoring, they handle interval censoring: in this case the ith lifetime t i is known to lie in an interval (a i,b i ), yielding the likelihood function n L(θ) = [S i (a i ; θ) S i (b i ; θ)]. i=1 The likelihood (12) is a special case of this. More complicated parametric models involving multivariate lifetimes, competing failure modes, or random effects may also be handled by maximum likelihood. Although software is not widely available for specific models, general purpose optimization software allows one to deal quite easily with most situations. roblems that cause difficulty tend to be ones in which the observed data are uninformative about certain aspects of the model, thus leading to flat regions in the likelihood function (see, for example, [15]). Model Checking arametric survival models assume a specific form for lifetime distributions and, in addition, the dependence upon any covariates must be specified. At preliminary stages of analysis it is useful to group individuals so that within groups they have similar values of important covariates. onparametric (Kaplan Meier) estimates Ŝ(t) of the survivor function for each group may be compared graphically to detect prominent features of the data, to assess the shape of the lifetime distributions, and to see whether a fairly simple regression model will suffice. A particularly useful procedure is to plot log[ log Ŝ(t)] (vertical axis) vs. log t (horizontal axis) for each group. If a H model is reasonable, then the curves should be roughly parallel in the vertical direction [see (8)]; if the curves are roughly parallel in the horizontal direction, an AFT model is suggested. If Weibull distributions are reasonable, then the curves should be roughly linear [see (3)]; differences in slope indicate a shape parameter that depends upon x. This is often termed a Weibull probability plot and is an example of probability or hazard plotting, which can also be used to explore other models. Graphical tools such as scatter plots or box plots are also valuable (see Graphical Displays), but must be modified to deal with censored observations. For box plots the empirical quantiles for a group of individuals may be determined from the Kaplan Meier estimate, provided they are not beyond the largest observation. For plots that involve the lifetimes

8 8 arametric Models in Survival Analysis directly it is important to plot lifetimes and censoring times with different symbols. robability plots of empirical survivor functions can provide checks on parametric models, as suggested above. We may also compare fitted parametric models S 0 (t; ˆθ) for baseline distributions, or models without covariates, with nonparametric estimates. Formal goodness-of-fit tests based on this idea exist for certain parametric models when there are strict limitations on censoring and the presence of covariates (e.g. [23, Chapter 10]), and in some cases for more complex situations [1, Sections VI.3 and VII.6]. However, most regression model checking is based on model expansion or on graphical assessment of residuals. Model expansion involves fitting models with additional parameters that represent specific types of departures from the current model; the need for the extra parameters may be assessed via likelihood ratio or Wald tests. Score tests for which only the current model has to be fitted are also useful on occasion (e.g. [23, Section ]). Some examples of model expansion are: (i) adding covariates representing interactions or other effects, as a check on a specified regression model; (ii) allowing σ in a location-scale model (4) to depend on x, as a check on the AFT assumption; and (iii) using the Burr areto distribution (2) in order to test the assumption of a baseline Weibull distribution (α = ). Location-scale models (4) are the most widely used parametric regression models, and for them residuals are naturally defined as ẑ i = y i µ(x i ; ˆβ), (16) ˆσ where yi = min(y i, log c i ) is either a log lifetime or log censoring time, depending on what was observed. If the model is appropriate, the ẑ i s should look roughly like a censored random sample from the distribution with survivor function G. robability or hazard plots of the ẑ i s may be used to assess the baseline distribution G, and plots of the ẑ i svs. covariates or other factors can be used to check on the constancy of σ or to look for systematic departures from the assumed specification µ(x; β). Departures from the location-scale form itself are harder to detect and are best examined by model expansion or the graphical methods mentioned at the beginning of this subsection. lots should designate censored and uncensored residuals with different symbols. Sometimes it may be useful to adjust censored ẑ i s upwards to give imputed uncensored residuals; a way of doing this is described below. In general, residuals for an arbitrary parametric survival model may be defined as ˆr i = H(t i x i; ˆθ), (17) where ti = min(t i,c i ) and H(t x) = log S(t x) is the cumulative hazard function. The motivation for (17) is that the variables H(T i x i ; θ 0 ) have a standard exponential distribution if the model is correct and θ = θ 0. If the model is appropriate, the ˆr i s, i = 1,...,n, should look roughly like a censored sample of exponential variates. These residuals can be plotted in ways described above to check on the model. It will be noted that the AFT residuals, (16), are related to the residuals (17) by ˆr i = log G(ẑ i ). Residuals based on score or log likelihood contributions by each individual are also used. Sometimes censored residuals ˆr i are adjusted upwards to give imputed values for corresponding uncensored residuals. This is usually done by adding either1orlog2tor i sfori C. The motivation for doing this is that if r i is a standard exponential random variable then E(r i r i >r)= r + 1and r(r i >r+ log 2 r i >r)= 0.5. If such imputed values are shown on plots, then it is also advisable to show the censored residuals from which these were calculated. Conclusions about explanatory variables are generally not affected much by mild misspecification of the baseline distribution in an AFT or H model, or by mild departures from the AFT or H frameworks themselves. Moreover, unless there is a considerable amount of data it is difficult to discriminate among parametric distributions with rather similar shapes, such as the Weibull, log-logistic, and lognormal. One should, however, check carefully for significant departures from assumed models. Observations that are unusual or highly influential should also be scrutinized; in some cases it is a good idea to refit models with certain observations omitted. The fitting of expanded models is encouraged both as a model checking device and as a way to assess the sensitivity of inferences to variations in the model that are plausible on the basis of the observed data.

9 arametric Models in Survival Analysis 9 An Example As a brief example we consider data on survival times for leukemia patients given by Feigl & Zelen [17] in an early paper on regression analysis of lifetime data. Survival times are in weeks from diagnosis, and there are two covariates: white blood-cell count (WBC) at diagnosis and a binary covariate AG that indicates a positive or negative (positive = 1, negative = 0) test related to white blood-cell characteristics. The data are given in Table 1, where wbc denotes white bloodcell count divided by The original data had no censored lifetimes, but for illustrative purposes three of the lifetimes have here been replaced with censoring times. Exploratory analysis suggests an AFT regression model in which covariates x 1 = AG and x 2 = log(wbc) are included. Figure 1 shows a plot of log (survival time) vs. log (wbc), with AG positive- and negative-valued observations denoted by and, respectively. Small letter ps denote the three censoring times. There are 17 AG-positive and 16 AGnegative subjects; two of the AG-positive subjects have wbc = 100 and t = 1, and their symbols are overlaid in the figure. The plot suggests that lifetimes tend to be shorter for subjects with higher WBC and longer for AG-positive subjects. Accelerated failure time models (5) with extreme value and logistic error distributions were fitted, corresponding to Weibull and log-logistic lifetime Table 1 Leukemia survival data Time AG wbc Time AG wbc a a a a Denotes a censoring time; wbc = WBC Log(time) Figure 1 p p Log (0.001 WBC) Leukemia data: log lifetime vs. log wbc distributions. Figure 1 suggests possibly different effects of WBC for subjects with AG = 1 and AG = 0, so models with and without a log (wbc) AG interaction term were fitted in each case. The two distributions gave very similar results and in neither case was the interaction term significant. The maximum log-likelihood values l( ˆβ 0, ˆβ 1, ˆβ 2, ˆσ) for the models without the interaction terms (i.e. with µ(x) in (5) given by β 0 + β 1 x 1 + β 2 x 2 ) were 52.9 (Weibull) and 53.0 (log-logistic), and so provide no evidence to favor one distribution over the other. Model checks described below also showed the two models to be comparable. For convenience we consider only the Weibull model in the remainder of the discussion. For the Weibull model the estimates (with standard errors obtained from the observed information matrix given in parentheses) are ˆβ 0 = 3.841(0.534), ˆβ 1 = 1.177(0.427), ˆβ 2 = 0.366(0.150), ˆσ = 1.119(0.164). There is clearly no evidence against the hypothesis that σ = 1, suggesting that the exponential model could be used. The effects of WBC and AG are both significant, and are in the directions suggested by Figure 1. Model checks may be based on residuals ẑ i of the form (16). We use the extreme value residuals with ˆσ = 1.119, but residuals based on the exponential model give essentially the same picture. lots of the ẑ i sagainstx 1i,x 2i, or fitted values µ(x i ; ˆβ) do not suggest any major problems with the model. Figure 2 shows a diagnostic probability plot designed to check on the assumed baseline extreme

10 10 arametric Models in Survival Analysis Exp EV residual Figure Obs EV residual Leukemia data: probability plot of residuals value distribution. This is obtained by treating the ẑ i s as a censored sample of 33 log lifetimes, and computing the Kaplan Meier estimate Ŝ(z) of the survivor function of Z based on them. Figure 2 is an extreme value probability plot of the points ẑ i,w i = 0.5Ŝ(ẑ i 0) + 0.5Ŝ(ẑ i + 0), based on plotting ẑ i vs. log( log w i ) for the uncensored residuals. The plot is roughly linear, and provides no evidence against the extreme value assumption. One feature of the data is worth noting: for the AG-positive group there is an individual with a high WBC along with a reasonably large lifetime. For the AG-negative group there are two such individuals. It is clear from Figure 1 that these observations are very influential. If they were omitted, then the effects of WBC and AG would be increased substantially. We have no reason to isolate these observations and so they are retained, but their influence is noted. Bibliographic otes arametric models for lifetime data have been in existence for a long time, but received increasing attention from about The books by Lawless [23] and Cox & Oakes [10] reference many models. Early work on parametric methods for the regression analysis of survival data emphasized exponential, Weibull, and lognormal models, and may especially be found in the biostatistics and reliability literature. apers with biostatistics applications include those by Sampford & Taylor [27], Feigl & Zelen [17], ike [26], and Zippin & Armitage [29]. The focus for parametric analysis quickly became the accelerated failure time and location-scale models. The books by Kalbfleisch & rentice [20, Chapter 3] and Lawless [23, Chapter 6] provide detailed treatments. Other good sources include Cox & Oakes [10], Andersen et al. [1], and Collett [8]. Some papers that study specific families of models include Farewell & rentice [16], Bennett [5], Ciampi et al. [7], and Anderson [2]. Escobar & Meeker [14] consider diagnostics and influence analysis. Lawless [24] discusses general classes of lifetime regression models. The proportional or multiplicative hazards model gained popularity quickly following the landmark paper by Cox [9]. The emphasis has been very much on semiparametric methods, however, and relatively little fully parametric inference for these models is found in the literature. References [1] Andersen,.K., Borgan, Ø., Gill, R.D. & Keiding,. (1993). Statistical Models Based on Counting rocesses. Springer-Verlag, ew York. [2] Anderson, K.M. (1991). A nonproportional hazards Weibull accelerated failure time regression model, Biometrics 47, [3] Armitage,. & Doll, R. (1954). The age distribution of cancer and a multistage theory of carcinogenesis, British Journal of Cancer 8, [4] Barndorff-ielsen, O.E. & Cox, D.R. (1994). Inference and Asymptotics. Chapman & Hall, London. [5] Bennett, S. (1983). Log-logistic regression models for survival data, Applied Statistics 32, [6] Burr, I.W. (1942). Cumulative frequency distributions, Annals of Mathematical Statistics 13, [7] Ciampi, A., Hogg, S.A. & Kates, L. (1986). Regression analysis of censored survival data with the generalized F family an alternative to the proportional hazards model, Statistics in Medicine 5, [8] Collett, D. (1994). Modelling Survival Data in Medical Research. Chapman & Hall, London. [9] Cox, D.R. (1972). Regression models and life tables (with discussion), Journal of the Royal Statistical Society, Series B 34, [10] Cox, D.R. & Oakes, D. (1984). Analysis of Survival Data. Chapman & Hall, London. [11] Crowder, M.C. (1989). A multivariate distribution with Weibull connections, Journal of the Royal Statistical Society, Series B 47, [12] Dubey, S.D. (1968). A compound Weibull distribution, aval Research Logistics Quarterly 15, [13] Efron, B. & Tibshirani, R.J. (1993). An Introduction to the Bootstrap. Chapman & Hall, London. [14] Escobar, L. & Meeker, W.Q. (1992). Assessing influence in regression analysis with censored data, Biometrics 48,

11 arametric Models in Survival Analysis 11 [15] Farewell, V.T. (1982). The use of mixture models for the analysis of survival data with long-term survivors, Biometrics 38, [16] Farewell, V.T. & rentice, R.L. (1977). A study of distributional shape in life testing, Technometrics 19, [17] Feigl,. & Zelen, M. (1965). Estimation of exponential survival probabilities with concomitant information, Biometrics 21, [18] Hjorth, U. (1980). A reliability distribution with increasing, decreasing, constant, and bathtub-shaped failure rates, Technometrics 22, [19] Hougaard,., Harvald, B. & Holm,.V. (1992). Measuring the similarities between the lifetimes of adult Danish twins born between , Journal of the American Statistical Association 87, [20] Kalbfleisch, J.D. & rentice, R.L. (2002). The Statistical Analysis of Failure Time Data, 2nd Ed. Wiley, ew York. [21] Krall, J., Uthoff, V. & Harley, J. (1975). A stepup procedure for selecting variables associated with survival, Biometrics 31, [22] Lachin, J.M. & Foulkes, M.A. (1986). Evaluation of sample size and power for analyses of survival with allowance for nonuniform patient entry, losses to follow-up, noncompliance, and stratification, Biometrics 42, [23] Lawless, J.F. (2002). Statistical Models and Methods for Lifetime Data, 2nd Ed.Wiley, ew York. [24] Lawless, J.F. (1986). A note on lifetime regression models, Biometrika 73, [25] ettitt, A.. (1984). roportional odds models for survival data and estimates using ranks, Applied Statistics 33, [26] ike, M.C. (1966). A method of analysis of a certain class of experiments in carcinogenesis, Biometrics 22, [27] Sampford, M.R. & Taylor, J. (1959). Censored observations in randomized block experiments, Journal of the Royal Statistical Society, Series B 21, [28] Smyth, G.K. (1989). Generalized linear models with varying dispersion, Journal of the Royal Statistical Society, Series B 51, [29] Zippin, C. & Armitage,. (1966). Use of concomitant variables and incomplete survival information in the estimation of an exponential survival parameter, Biometrics 22, J.F. LAWLESS

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