Analysis of Clinical Trials with Time-to-Event Endpoints
5.2 Statistical Models
5.2.1 Primary Functions and Definitions
Prior to proceeding with analyses of these two data sets, the reader may find the following review of definitions of key functions in survival analysis helpful.
The primary functions in depicting survival are the hazard function or force of mortality, the survival function, the death density function and the cumulative death distribution function. These functions are symbolized in this chapter as h(t), S(t), f (t) and F (t) respectively, where t denotes survival time.
We will briefly define and explain these functions; greater detail may be found in Peace (2009).
Analysis of Clinical Trials with Time-to-Event Endpoints 89
TABLE 5.2: Breast cancer data
Radiation Only Radiation+Chemotherapy tL tU Status tL tU Status tL tU Status tL tU Status
0 7 1 32 NA 0 0 22 1 16 24 1
0 8 1 33 NA 0 0 5 1 16 60 1
0 5 1 34 NA 0 4 9 1 17 27 1
4 11 1 36 44 1 4 8 1 17 23 1
5 12 1 36 48 1 5 8 1 17 26 1
5 11 1 36 NA 0 8 12 1 18 25 1
6 10 1 36 NA 0 8 21 1 18 24 1
7 16 1 37 44 1 10 35 1 19 32 1
7 14 1 37 NA 0 10 17 1 21 NA 0
11 15 1 37 NA 0 11 13 1 22 32 1
11 18 1 37 NA 0 11 NA 0 23 NA 0
15 NA 0 38 NA 0 11 17 1 24 31 1
17 NA 0 40 NA 0 11 NA 0 24 30 1
17 25 1 45 NA 0 11 20 1 30 34 1
17 25 1 46 NA 0 12 20 1 30 36 1
18 NA 0 46 NA 0 13 NA 0 31 NA 0
19 35 1 46 NA 0 13 39 1 32 NA 0
18 26 1 46 NA 0 13 NA 0 32 40 1
22 NA 0 46 NA 0 13 NA 0 34 NA 0
24 NA 0 46 NA 0 14 17 1 34 NA 0
24 NA 0 46 NA 0 14 19 1 35 NA 0
25 37 1 46 NA 0 15 22 1 35 39 1
26 40 1 16 24 1 44 48 1
27 34 1 16 20 1 48 NA 0
90 Clinical Trial Data Analysis Using R 5.2.1.1 The Hazard Function
The mathematical definition of the hazard function h(t) is h(t) = lim
∆t→0
P r (t ≤ T ≤ t + ∆t|T ≥ t)
∆t (5.1)
where T denotes the random variable survival time. Ignoring limit considera-tions, h(t) is the proportion of a population expiring in the interval [t; t + ∆t]
among those at risk of expiring at the beginning of the interval, averaged over the length of the interval. So h(t) may be thought of as the probability of dying at time = t, given that death did not occur prior to t; or h(t) is the instantaneous risk of death at time t given that death did not occur prior to t.
A hazard function may remain constant with respect to time (correspond-ing to an exponential density); may increase as a function of time accord(correspond-ing to some power function (corresponding to a Weibull density); may increase linearly with time (corresponding to a Rayleigh density); or may increase ex-ponentially with time (corresponding to a Gompertz density). In addition, there may be intervals of time where the hazard may alternatingly decrease or increase according to some power, linear, or exponential function of time.
Associated with this hazard function, h(t), is the integrated or cumulative hazard function H(t), which is widely used in survival modeling and is defined as
Besides the hazard function, the survival function is another fundamen-tal function of interest in the analysis of survival data. The mathematical definition of the survival function S(t) is
S(t) = P r (T ≥ t) (5.3)
where T denotes the random variable survival time. The survival function S(t) may be interpreted as the probability of surviving to at least time t, the t-year survival rate, the cumulative proportion surviving to at least time t, or the proportion of a population surviving to at least time t.
5.2.1.3 The Death Density Function
For modeling survival data using a likelihood approach, the death density function is usually employed. The mathematical definition of the death density function f (t) is
f (t) = lim
∆t→0
P r (t ≤ T ≤ t + ∆t)
∆t (5.4)
Analysis of Clinical Trials with Time-to-Event Endpoints 91 where T denotes the random variable survival time. Ignoring limit consider-ations, f (t) is the proportion of all individuals in a population expiring in the interval [t; t + ∆t] divided by the length of the interval. So f (t) may be thought of as the unconditional probability of dying at time = t; or f (t) is the instantaneous unconditional risk of death at time t.
Associated with this death density function, f (t), the cumulative death distribution function F (t), is the complement of the survival function; i.e.,
F (t) = 1 − S(t) (5.5)
5.2.1.4 Relationships among These Functions
The hazard, density, survival, and cumulative death distribution functions are interrelated. If one knows the survival function, its complement is the cumulative death distribution function, and the death density may be obtained by differentiating the negative of the survival function with respect to t; i.e.,
f (t) = −d[S(t)]
dt (5.6)
The hazard function may then be obtained by dividing the death density by the survival function; i.e.,
h(t) = f (t)
S(t) = −d[log{S(t)}]
dt (5.7)
Also, if the hazard function is specified, the survival function may be ob-tained by exponentiating the negative integral of the hazard; i.e.,
S(t) = exp
− Z t
0
h(u)du
= exp [−H(t)] (5.8)
Thus knowledge of any one of the four enables one to determine the other three.
5.2.2 Parametric Models
For a particular set of survival data, the functions defined in Section 5.2.1 may be parametrically estimated by assuming survival follows some para-metric distribution or model, ‘fitting’ the data to the model and estimating model parameters by some valid estimation method (such as maximum likeli-hood). Five commonly used parametric models are the exponential, Weibull, Rayleigh, Gompertz, and lognormal. We describe different settings to which the models may apply.
92 Clinical Trial Data Analysis Using R 5.2.2.1 The Exponential Model
The exponential model is used to model a constant hazard function : h(t) = λ0> 0. The exponential death density function is
f (t) = λ0exp(−λ0t) (5.9)
the exponential survival function is
S(t) = exp(−λ0t) (5.10)
and the cumulative death distribution function is
F (t) = 1 − exp(−λ0t) (5.11)
5.2.2.2 The Weibull Model
The Weibull model derives from a power law hazard function: h(t) = λ0λ1tλ1−1, where λ0 > 0 and λ1 > 0. It may be noted that λ1 > 1 guar-antees that h(t) is monotone increasing.
The Weibull death density function is
f (t) = λ0λ1tλ1−1exp(−λ0tλ1) (5.12) the Weibull survival function is
S(t) = exp(−λ0tλ1) (5.13) and the Weibull cumulative death distribution function is
F (t) = 1 − exp(−λ0tλ1) (5.14)
5.2.2.3 The Rayleigh Model
The Rayleigh model is used to model a linear hazard function: h(t) = λ0+ 2λ1t, where λ0> 0 and λ1≥ 0. It may be noted that if λ1> 0 then h(t) is monotone increasing. The Rayleigh death density function is
f (t) = (λ0+ 2λ1t)exp[−(λ0t + λ1t2)] (5.15) the Rayleigh survival function is
S(t) = exp[−(λ0t + λ1t2)] (5.16) and the Rayleigh cumulative death distribution function is
F (t) = 1 − exp[−(λ0t + λ1t2)] (5.17)
Analysis of Clinical Trials with Time-to-Event Endpoints 93 5.2.2.4 The Gompertz Model
The Gompertz model derives from an exponential hazard function; i.e., h(t) = exp(λ0+ λ1t). Note that if λ1 > 0 then h(t) is monotone increasing.
The Gompertz death density function is f (t) = exp(λ0+ λ1t) × exp 1
λ1
[exp(λ0) − exp(λ0+ λ1t)]
(5.18) the Gompertz survival function is
S(t) = exp 1 λ1
[exp(λ0) − exp(λ0+ λ1t)]
(5.19) and the cumulative death distribution function is
F (t) = 1 − exp 1
The lognormal death density function is given by f (t) = 1 The lognormal survival function is S(t) = 1 − Φhlog(t)−µ
σ
i
, where Φ is the standard cumulative normal distribution.
The hazard function is h(t) = S(t)f (t). The cumulative death distribution function is F (t) = Φhlog(t)−µ
σ
i .
5.3 Statistical Methods for Right-Censored Data
5.3.1 Nonparametric Models: Kaplan–Meier EstimatorNonparametric approaches to estimate the functions in Section 5.2.1 make no distributional or specific model assumptions about the observed sur-vival times. Therefore these nonparametric methods are sometimes called distribution-free methods. The Kaplan–Meier estimator is the most commonly used non-parametric method to estimate the survival function.
Based on the definition of survival function, S(t), we can estimate S(t) from a sample of n observations as
S(t) =ˆ number of patients with observed times ≥ t
n (5.22)
94 Clinical Trial Data Analysis Using R
when there are no censored survival times in the sample. In this case, the estimate in Equation (5.22) is simply a proportion (binomial) from the total n. Confidence intervals can be constructed at each time of death t using the variance estimate:
V arh ˆS(t)i
=
S(t)[1 − ˆˆ S(t)]
n (5.23)
When there are censored observations, the survival function is estimated by the Kaplan–Meier estimator, which extended the simple binomial estimate in Equation (5.22).
The estimator is presented in detail in Kaplan and Meier (1958). We briefly describe its computation here. First the observed survival times are ordered from smallest to the largest as t(1) ≤ t(2) ≤ · · · ≤ t(n), where t(j) is the jth ordered survival time. The Kaplan–Meier estimator of the survival function can be constructed as number of patients with events at time tj. The estimated variance of ˆS(t) can be obtained as
5.3.2 Cox Proportional Hazards Regression
The aforementioned methods do not account for covariate information that might be correlated with survival. In many clinical trials, there exists con-comitant, covariate or regressor information on patients in addition to their observed survival times. In such settings, there is interest in assessing the statistical significance of the concomitant information as it relates to the dis-tribution of times-to-death.
Generally this is accomplished using the proportional hazards model pro-posed by Sir David Cox (1972). The model is specified in terms of the hazard function instead of the survival function, and assumes that additive changes in the concomitant variables correspond to multiplicative changes in the hazard function or, equivalently, to additive changes in the log of the hazard.
In other words, the hazard function reflecting the proportional hazards model is defined as
h(t|X) = exp(Xβ)h0(t) (5.26)
where X is a vector of concomitant, covariate or regressor information X =
Analysis of Clinical Trials with Time-to-Event Endpoints 95 (x1, x2, · · · , xp), β is the column vector of parameters (β1, β2, · · · , βp) corre-sponding to X, and h0(t) is the time-specific baseline hazard function defined in Section 5.2.1, sometimes referred to as a homogeneous or baseline hazard function.
Based on this structure in Equation (5.26), the hazard ratio for any two patients with different covariants X1 and X2 is constant over time since
h(t|X1)
h(t|X2)= exp(X1β)
exp(X2β) (5.27)
h0(t) cancels from numerator and denominator of the ratio, and therefore the hazard for one subject is proportional to the hazard of another subject.
From the formulation in Equation (5.26), it is noted that the concomitant information acts in a multiplicative fashion on the time-dependent-only haz-ard function. Further, the term proportional hazhaz-ards, also arises by observing that if a xi is an indicator of treatment group membership (xi = 1 if treat-ment group 1; xi = 0 if treatment group 0), then the ratio of the hazard for treatment group 1 to the hazard of treatment group 0 is exp(βi); or the haz-ard for treatment group 1 is proportional to the hazhaz-ard of treatment group 0.
Therefore, exp(β) sometimes is referred to as relative risk.
As noted by Cox (1972), coefficients of the covariates in the Cox Propor-tional Hazards model may be estimated and used to provide inferences by maximizing the conditional likelihood arising from the Cox model. In doing so, the actual survival times are not explicitly used in the estimation process, rather they serve as bookkeepers of the covariate information; thus knowledge of the survival time ranks would equally suffice (Peace and Flora (1978)).
In the discussion of Cox’s paper, several authors note that the likelihood is not strictly a conditional likelihood. Kalbfleisch and Prentice (2002) use the method of marginal likelihood. Cox (1975) later presents the notion of partial likelihood. Regardless of nomenclature, estimating the covariate coefficients in the Cox model enables one to test the significance of the covariates as well as provides estimates of the relative hazard of one group to another or of one patient to another (using the covariate information of those patients in addition to the estimates of the coefficients).
The survival function arising from the Cox specification of the hazard func-tion in Equafunc-tion (5.26) may be estimated. If we define S(t|X) as the survival function (i.e., survival probability) at time t for patients with concomitant variable X, then
96 Clinical Trial Data Analysis Using R
where H0(t) is the baseline cumulative hazard. Therefore, when the estimates of ˆβ are obtained, survival probabilities can be estimated for any covariates from clinical trials if we can also estimate the baseline cumulative hazard.
The estimation is based on the principal of maximum partial likelihood.
We refer the reader to Cox (1975) or the excellent texts of Lawless (1982), Kalbfleisch and Prentice (2002) and Collett (2003). We demonstrate the im-plementation in the R system.