Analysis of survival data with cross-effects of survival functions

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Biostatistics (2004), 5, 3, pp. 415–425 doi: 10.1093/biostatistics/kxg045

Analysis of survival data with cross-effects of survival functions ˇ VILIJANDAS BAGDONAVICIUS Department of Mathematical Statistics, University of Vilnius, Naugarduko 24, Vilnius, Lithuania MOHAMED A. HAFDI Mohammed V University of Rabat, Maroc MIKHAIL NIKULIN† Statistique Math´ematique et ses applications, IFR 99 Sant´e Publique, Victor Segalen University, Bordeaux, France and Steklov Mathematical Institute, RAN, Saint Petersburg, Russia [email protected]

S UMMARY We present a model and semiparametric estimation procedures for analyzis of survival data with crosseffects (CE) of survival functions. Finite sample properties of the estimators are analyzed by simulation. A goodness-of-fit test for the proportional hazards model against the CE model is proposed. The well known data concerning effects of chemotherapy and radiotherapy on the survival times of gastric cancer patients is analyzed as an example. Keywords: CE model; Chemotherapy; Chi-squared type test; Cross-effects; Gastric cancer data; Gastrointestinal tumor; GPH model; Hougaard–Aalen model; Hsieh Model; PH model; Radiotherapy; Semiparametric likelihood; Survival.

1. I NTRODUCTION When analyzing survival data from clinical trials, cross-effects of survival functions are sometimes observed. A classical example is the well-known data set of the Gastrointestinal Tumor Study Group (1982), concerning effects of chemotherapy and radiotherapy on the survival times of gastric cancer patients (Stablein and Koutrouvelis, 1985; Klein and Moeschberger, 1997). Hsieh (2001) considered a model for analyzis of survival regression data with cross-effects of survival functions and used the over-identified estimating equation (OEE) approach and the method of sieves for parametric estimation of unknown parameters. We give a cross-effect (CE) model different from the Hsieh model, which has the advantage that the ratio of hazard rates under different covariates at time zero is finite. We use efficient semiparametric estimation based on the likelihood. In Hsieh (2001) rough parametric estimation procedures are used for estimation of the baseline cumulative hazard, approximating it by a piecewise-constant function with several jumps as unknown parameters. We propose a natural generalization of the Nelson–Aalen estimator of this baseline function to the case of regression data with cross-effects of survival functions. † To whom correspondence should be addressed.

c Oxford University Press 2004; all rights reserved. Biostatistics Vol. 5 No. 3


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Fig. 1. Comparison of Kaplan–Meier (KM) and OEE (Hsieh model) estimates of survival functions; · · · · · · KM chemotherapy and radiotherapy; —— KM, chemotherapy; — — (smoothed) OEE, chemotherapy and radiotherapy; — - — (smoothed) OEE, chemotherapy.

Finite sample properties of the regression parameter estimators under non-random and random covariates with and without censoring it are analyzed by simulation. Consistence and n −1 rate of convergence to zero of the variances of estimators is confirmed by simulation results. We also propose a goodness-of-fit test for the proportional hazards model of Cox (1972) against the CE model. We analyze the radio-chemo data of Stablein and Koutrouvelis (1985), studied also by Kleinbaum (1996) and Klein and Moeschberger (1997). These data concern the survival of 90 patients in two groups, one group receiving chemo-therapy and the other chemo- and radio-therapy. This example is also analyzed in Hsieh (2001), Kleinbaum (1996) and Klein and Moeschberger (1997). By plotting the two Kaplan–Meier (KM) estimators of survival functions pertaining to both treatment groups, a crossingeffect phenomenon is clearly manifest. Figure 1 shows KM estimators and OEE smoothed estimators of survival functions (Hsieh, 2001) of the two groups. The resulting inference is that radiotherapy would first be detrimental to a patient’s survival but becomes beneficial later on. We should stress that our purpose is not only analyzis of data with dichotomous covariates, as in the case of the radio-chemo data. Data with continuous or poly-tomous covariates can also be analyzed with our methods. To show that our estimation procedure works well for finite samples in the case of continuous covariates we give the results of a small simulation study. Our analyzis of two groups’ radio-chemo data shows that KM estimates of survival functions and estimates of the same functions obtained using the CE regression model and our method of estimation agree very closely. In particular, the fit to our model is much better than the fit to the Hsieh model.

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2. M ODELING Let Sx (t) and λx (t) be the survival and the hazard rate functions under an m-dimensional possibly time dependent explanatory variable x. Denote by x (t) = − log{Sx (t)} the cumulative hazard under x. The generalized proportional hazards (GPH, Bagdonaviˇcius and Nikulin, 1999) model holds on a set of explanatory variables E if for all x ∈ E λx (t) = ψ {x(t), Sx (t)} λ0 (t).

(2.1)

This model implies that for different explanatory variables x1 and x2 the ratio of hazard rates at any moment t is a function of the values x1 (t) and x2 (t) and the probabilities of survival up to t. The well known proportional hazards (PH) or Cox model (Cox, 1972) is a particular case of the GPH model when the function ψ does not depend on Sx (t). In terms of the cumulative hazard the model (2.1) can be written in the form λx (t) = u {x(t), x (t)} λ0 (t).

(2.2)

Various forms of the function u(·, ·) give submodels which can include effects when the ratio of hazards rates increases, decreases or shows cross-effects (see Bagdonaviˇcius and Nikulin, 2002). We consider here the so-called CE model λx(·) (t) = eβ

T x(t)

γ T x(t)

{1 + x(·) (t)}1−e

λ0 (t),

(2.3)

proposed by Bagdonaviˇcius and Nikulin (2002), where β and γ are m-dimensional unknown parameters and 0 is an unknown baseline cumulative hazard. If x(t) ≡ x is constant in time then resolving the differential equation (2.3) with respect to x and differentiating, the CE model can be written in the explicit form λx (t) = e where 0 (t) =

t 0

βT x


e−γ T x −1 (β+γ )T x 1+e 0 (t) λ0 (t),

(2.4)

λ0 (u) du is the baseline cumulative hazard.

P ROPOSITION 1 Suppose that λ0 is a continuous function, β T x < β T y and γ T x < γ T y, 0 (∞) = ∞, 0 (t) < ∞ for any t > 0. Then under the CE model and for any constant covariates x, y ∈ E 0 the survival functions Sx and S y intersect once in the interval (0, ∞). The proof is given in the supplementary material which can be found at www.biosta\-tistics. oupjournals.org. The CE model resembles (in form but not in content) the extension of the positive stable frailty model (Hougaard, 1986) given by Aalen (1992). Indeed, the Hougaard–Aalen (HA) model with cross-effects of hazard rates (Aalen, 1994) for constant covariates x has one of two following forms: λx (t) = ηr (x)(1 ± r (x)α −1 δη0 (t))−α λ0 (t),

α, δ, η, r (x) > 0.

If we take the minus sign then for any constant covariates x1 and x2 the hazard rates λx1 (t) and λx2 (t) cross once. An unpleasant property of this model is that the survival distributions have finite supports which are different for different values of covariates. Estimation procedures are always complicated in such cases. If we take the plus sign then the hazard rates cross for values α > 1. In such a case the supports are [0, ∞). The HA also resembles the CE model in the sense that if we replace the constant α by the function T T T 1 − e−γ x , the function ηr (x) by eβ x and ±r (x)α −1 δη by e(β+γ ) x in the HA model with cross-effects

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then we obtain the CE model with constant covariates (2.4). Note that in the CE model (2.4) the power T e−γ x − 1 can take either sign and the supports of the survival distributions are [0, ∞) for all values of the covariates. Another difference between the HA and the CE models is that the HA model includes cross-effects of the hazard rates but does not include cross-effects of the survival functions. In the case of the HA model with covariates, the power α is constant and the survival functions Sx1 (t) and Sx2 (t) do not cross. For α = 1 and r (x2 ) > r (x1 ),    α Sx2 (t) (1 ± r (x2 )α −1 δη0 (t))1−α − (1 ± r (x1 )α −1 δη0 (t))1−α < 1, = exp Sx1 (t) ±δ(α − 1) whilst if α = 1, then

Sx2 (t) = Sx1 (t)



1 − r (x2 )α −1 δη0 (t) 1 − r (x1 )α −1 δη0 (t)

α/δ

< 1.

Suppose that the model (2.2) holds on a set E 0 of constant explanatory variables. Solving (2.2) with respect to x (t) we obtain that for any x ∈ E 0 the cumulative hazard has the form x (t) = H {x, 0 (t)}.

(2.5)

Hsieh (2001) considered the following model with cross-effects: T

λx (t) = e(β+γ ) x {0 (t)}e

γ T x −1

λ0 (t),

(2.6)

which is a generalization of the PH model taking a power function H (x, s) = r (x)s ρ(x) in (2.5) with the T T natural parametrizations r (x) = eβ x and ρ(x) = eγ x , and the PH model emerges when γ = 0. Note that in the case of the Hsieh model (2.6) the ratios of the hazard rates and even the ratios of the cumulative hazards go to ∞ (or 0) as t → 0. In the case of the CE model these ratios are defined and finite at t = 0. This property of the CE model is more natural and helps avoid complications when seeking efficient estimators. 3. S EMIPARAMETRIC ESTIMATION We shall consider the GPH model with time dependent covariates in the general form λx (t) = g{x, 0 (t), θ} λ0 (t),

(3.1)

with a specified parametrization g(x, s, θ ) of the function g via parameters θ and an unknown baseline function λ0 (t). Suppose that n patients are observed. The ith of them is observed under the explanatory variable xi . Denote by Ti and Ci the failure and censoring times for the ith patient and set X i = min(Ti , Ci ), δi = 1{Ti
Ci } , Ni (t) = 1{Ti
t,δi =1} , Yi (t) = 1{X i t} ,

n

n where 1 A denotes the indicator of the event A. Set N (t) = i=1 Ni (t) and Y (t) = i=1 Yi (t). The partial likelihood function (Andersen et al., 1993), δi n ∞  g{xi , 0 (v), θ}

n dNi (v) , L(θ) = 0 j=1 Y j (v)g{x j , 0 (v), θ} i=1

(3.2)

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419

depends on the unknown cumulative hazard 0 . The score function for θ is U (θ) =

n 

j=1 0

∞

{w (i) (u, 0 , θ ) − E(u, 0 , θ)} d Ni (u),

where w(i) (t, θ, 0 ) = E(v, 0 , θ) =

S (1) (v, 0 , θ ) , S (0) (v, 0 , θ ) S (1) (v, 0 , θ ) =

(3.3)

∂ log{g(xi , 0 (v), θ)}, ∂θ n  S (0) (v, 0 , θ ) = Yi (v)g{xi , 0 (v), θ} i=1

n  i=1

Yi (v)

∂ g{xi , 0 (v), θ}. ∂θ

˜0 The score function depends on unknown function 0 , so it is replaced in (9) by its estimator  (depending on θ) which is defined recurrently from the equation

t dN (u) ˜ 0 (t, θ) = . (3.4) (0) ˜ 0, θ ) 0 S (u−,   We note that this estimator is obtained using the martingale property of the difference Ni − Yi dx (i) . The modified score function is then n ∞  ˜ ˜ 0 , θ ) − E(u,  ˜ 0 , θ)} dNi (u), U (θ) = {w(i) (u,  (3.5) j=1 0

ˆ 0 of θ and 0 , respectively, satisfy the system of equations and the estimators θˆ and  

 ∞ (i) n ˆ ˆ ˆ ˆ j=1 0 {w (u, 0 , θ) − E(u, 0 , θ )} dNi (u) = 0, ˆ ˜ ˆ 0 (t) = 0 (t, θ).

(3.6)

√ ˜ 0 , the asymptotic covariance matrix of n(θˆ − θ) is obtained by standard Given the consistency of  methods using the functional delta method and the central limit theorem for martingales. For consistency proofs of estimators, given by the equations of the type (3.5), see Ceci and Mazliak (2002). ˆ 0 of the baseline cumulative hazard 0 generalizes the Nelson–Aalen Under our model the estimator  estimator, just as in the case of the PH model the Breslow estimator (Andersen et al., 1993) generalizes the Nelson–Aalen estimator. T ˆ 0 ), Note that in the case of the PH model, i.e. when g(x, s, θ) = eθ x , the solution of (3.6) is (θˆ ,  ˆ ˆ where θ is the semiparametrically efficient estimator of the regression parameters θ and 0 is the Breslow estimator of 0 . This suggests that in the case of the CE model, i.e. when g(x, s, θ) = eβ

Tx


e−γ T x−1 T 1 + e(β+γ ) x s ,

the estimator θˆ will also be semiparametrically efficient. Investigation of semiparametric efficiency is not within the scope of this paper. We give only some suggestions. An estimator is semiparametrically efficient if there exists a sequence of parametric models such that the limit covariance matrix of semiparametric estimators coincides with the limit Fisher

V. BAGDONAVI Cˇ IUS ET AL.

420

information matrix of the sequence of parametric estimators, corresponding to this sequence of parametric models. This should hold in our case because the parametric score functions obtained by the method of the maximum likelihood and the semiparametric score function (3.5) are asymptotically equivalent: the parametric score function for the model (3.1) is ∗

U (θ) =

=

n  i=1

∞ 0

n 

∞

j=1 0

∂ log λxi (v, θ){dNi (v) − Yi (v)λxi (v, θ) dv} ∂θ

w (i) (u, 0 , θ )[dNi (v) − Yi (v)g{xi , 0 (v), θ} d0 (v)].

(3.7)

˜ 0 satisfying (3.4) then the modified score function (3.5) is If the function 0 is replaced in (3.7) by  obtained. The estimator of the survival function under any value x of the explanatory variable is ˆ Sˆ x (t) = e−x (t) ,

where

(3.8)


e−γˆ T x ˆ γˆ )T x ˜ ˆ x (t) = 1 + e(β+  0 (t, θ) − 1.

Hsieh (2001) approximated 0 by a piecewise-constant function with jumps as unknown parameters (so defining a parametric model). A unified approach of estimation for different models with continuously varying covariates is given in Bagdonaviˇcius and Nikulin (2004) 4. G OODNESS - OF - FIT FOR THE PH MODEL AGAINST THE CE MODEL We consider a test for checking the adequacy of the PH model versus model (2.3): H0 : λx (t) = eβ

Tx

H1 : λx (t) = eβ

λ0 (t) versus

Tx


e−γ T x −1 T 1 + e(β+γ ) x 0 (t) λ0 (t).

If the baseline cumulative hazard function 0 is completely known then the model (2.3) is parametric and the maximum likelihood estimator for θ satisfies the system of equations U j (τ ; θ, 0 ) = where U j (t; θ, 0 ) =

n  i=1

t 0

˜ ∂ log L(θ) = 0, ∂θ j

( j = 1, . . . , 2m),

∂ log{λi (t, θ)}{dNi (u) − Yi (u)λi (u, θ ) du}. ∂θ j

If 0 is unknown, then the partial derivative (i)

w j (t, θ, 0 ) =

∂ log{λi (t, θ)} ∂θ j

depends on a finite-dimensional parameter θ = (β T , γ )T and an infinite-dimensional parameter 0 . ˆ 0 ), and βˆ is the Under the PH model γ = 0. Let Uˆ j = Uˆ j (τ ), where Uˆ j (t) = U j (t, (βˆ T , 0)T , 

Analysis of survival data with cross-effects of survival functions

421

ˆ 0 is the Breslow (1975) estimator of 0 partial likelihood estimator of the regression parameter β and  under the PH model. Under the PH model Uˆ j = 0, ( j = 1, . . . , m) and Uˆ j (t) =

n  i=1

t 0

(i) wˆ j (u) d Mˆ i (u),

( j = m + 1, . . . , 2m),

(4.1)

where Mˆ i (t) are the martingale residuals corresponding to the PH model,

t

t dN (u) βˆ T x (i) βˆ T x (i) ˆ ˆ , Mi (t) = Ni (t) − e Yi (u) d0 (u) = Ni (t) − e Yi (u) (0) ˆ S (u, β) 0 0 where S (0) (t, β) = and

n 

Yi (t) eβ

T x (i) )

,

i=1

  (i) (i) ˆ 0 (t)) . ˆ 0 ) = −x (i) 1 + eβˆ T x (i) log( wˆ j (t) = w j (t, (βˆ T , 0)T ,  j

The test is based on the statistics

Uˆ = (Uˆ m+1 , . . . , Uˆ 2m )T .

(4.2)

To construct a chi-squared type test (Greenwood and Nikulin, 1996), we need the asymptotic distribution of Uˆ under the PH model, which is given by the following result. T HEOREM 1 Under Assumptions A given in the supplementary material D

Yn2 = n −1 Uˆ T Dˆ −1 Uˆ → χm2 ,

(4.3)

where Dˆ is a consistent estimator of the limit covariance matrix of the random vector n −1/2 Uˆ . The proof and the expression of Dˆ are given in the supplementary material which can be found at www.biostatistics.oupjournals.org. 5. I TERATIVE PROCEDURE FOR COMPUTING THE ESTIMATOR To calculate the estimate θˆ we do not need the score equation (3.5). We use Splus program and the general quasi-Newton optimization algorithm seeking the value of θ which maximizes the modified partial likelihood (MPL) function δi n ∞  ˜ 0 (v), θ } g{x ,  i ˜ L(θ) = dNi (v) (5.1)

n ˜ 0 j=1 Y j (v)g{x j , 0 (v), θ} i=1

with respect to θ. ˜ 0 can be found as follows. Let T ∗ < · · · < Tr∗ be observed and ordered For fixed θ the estimator  1 distinct failure times, r
n. Note by di the number of failures at the moment Ti . Then ˜0 (0; θ ) = 0,

˜0 (T1∗ ; θ ) =

d1 (0) ˜ S (0, 

0, θ )

,

V. BAGDONAVI Cˇ IUS ET AL.

422

Table 1. Simulated means and variances of estimators (dichotomous covariates)

n 100

200

400

β=1

γ =1

β=1

γ =2

p 0

βˆ 1.068 (0.261)

γˆ 0.992 (0.080)

βˆ 1.054 (0.554)

γˆ 2.008 (0.081)

0.2

1.065 (0.324)

0.956 (0.189)

1.128 (0.806)

1.947 (0.217)

0

1.052 (0.133)

1.005 (0.038)

1.051 (0.277)

2.003 (0.038)

0.2

1.030 (0.154)

0.973 (0.070)

1.063 (0.330)

2.010 (0.067)

0

1.017 (0.064)

1.001 (0.018)

1.040 (0.142)

2.001 (0.018)

0.2

1.009 (0.074)

0.985 (0.034)

1.062 (0.168)

2.003 (0.031)

∗ ˜0 (T j+1 ; θ ) = ˜0 (T j∗ ; θ) +

d j+1 (0) ˜ 0, θ ) S (T j∗ , 

( j = 1, . . . , r − 1).

(5.2)

In the case of the CE model S (0) (v, 0 , θ ) =

n 

Yi (v) eβ

T x (i)

i=1

hence ˜0 (T1∗ ; θ) = n

{1 + e(β+γ )

T x (i)

−γ T x (i) −1

0 (v)}e

d1

∗ β T x (i) i=1 Yi (T1 ) e

,

.

The iterative procedure is very simple. We use the initial value θ0 = (β0 , γ0 ), where β0 is an estimator of ˜ θ0 ) given by recurrence formula (5.2) and the β using the PH model, and γ0 = 1. Then the estimator (t, initial estimator θ0 is inserted into the MPL function (5.1), which we maximize to give θ1 . The value θ1 is ˜ θ1 ), and so on. then used to obtain (t, 6. P ROPERTIES OF ESTIMATORS : A SIMULATION STUDY We did a simulation study for scalar explanatory variables of the following two types: (a) dichotomous (two groups of equal size with x (i) = 0 and x (i) = 1); (b) uniformly distributed random variables (x (i) ∼ U (0, 1)). We specified a baseline cumulative hazard function 0 (t) = t and values (1, 1) and (1, 2) for the parameter θ = (β, γ ).

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Table 2. Simulated means and variances of estimators (U(0,1) covariables)

n 100

200

400

β=1

γ =1

β=1

γ =2

p 0

βˆ 1.069 (0.598)

γˆ 0.899 (0.588)

βˆ 1.097 (1.104)

γˆ 1.963 (0.574)

0.2

1.071 (0.750)

0.780 (1.060)

1.119 (1.130)

1.841 (0.948)

0

1.041 (0.274)

0.982 (0.092)

1.041 (0.456)

1.995 (0.072)

0.2

1.044 (0.425)

0.901 (0.368)

1.012 (0.602)

1.906 (0.242)

0

1.009 (0.143)

0.987 (0.047)

1.027 (0.229)

2.004 (0.035)

0.2

1.012 (0.173)

0.975 (0.150)

1.011 (0.248)

1.981 (0.078)

The simulation consisted of 1000 replications. The simulated means and variances (in parentheses) of βˆ and γˆ are given in the tables. We considered the cases of non-censored samples and independently censored samples with censoring proportion p = 0.2. The simulation results show that in all cases the estimators are consistent and their variances decrease with the rate n −1 when n is large. The asymptotic standard errors are smaller in the case of dichotomous covariates. 7. A NALYSIS OF RADIO - CHEMOTHERAPY DATA In this section we give an analyzis of the two-sample data of Stablein and Koutrouvelis (1985) concerning the effects of chemotherapy and chemotherapy plus radiotherapy on the survival times of gastric cancer patients. This example is also analyzed in Hsieh (2001), Kleinbaum (1996) and Klein and Moeschberger (1997). The number of patients is 90. KM estimators of survival functions pertaining to both treatment groups (Figure 1) clearly show a crossing-effect phenomenon. The two estimated curves indicate that radiotherapy would initially be detrimental to a patient’s survival but becomes beneficial later on. To confirm this we first test the adequacy of the PH model versus the CE model. The test statistic takes the value Yn2 = 13.131 whereas the critical value of χ12 is 5.0239 for a significance level α = 0.025. So the proportional hazards hypothesis is rejected. Note that in Klein and Moeschberger (1997) the null hypothesis of no difference in survival function between the two groups is not rejected, since they used a weak test. See also the discussion in Kleinbaum (1996). We then applied the CE model to estimate the influence of covariates on the survival. The modified partial likelihood estimator of θ = (β, γ ) is (1.894, 1.384). We used coding 0 for chemo-therapy and 1

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V. BAGDONAVI Cˇ IUS ET AL.

Fig. 2. Comparison of KM and MPL estimates of survival functions; − − −, KM, chemotherapy and radiotherapy; ——, KM, chemotherapy; — — MPL, chemotherapy and radiotherapy; — - — MPL, chemotherapy.

for chemo+radio-therapy. For both groups of patients the graphs of the KM estimators and the smoothed Hsieh estimators of the survival functions are presented in Figure 1. Smoothing was necessary because the Hsieh estimators are step-functions with only five steps (if the number of steps is larger, the estimators of step height may be bad). The graphs of the KM estimators and our estimators of the survival functions are presented in Figure 2. The estimators obtained from all of the data using the regression model (2.4) and our estimation method give exellent fits to the KM estimators obtained from the two subsamples. Our estimation procedures should be especially useful for the analyzis with continuously varying covariables, for which the KM estimators are not applicable. ACKNOWLEDGEMENTS The authors thank two anonymous referees and Peter J. Diggle for suggestions that have improved our paper. R EFERENCES A ALEN , O. O. (1992). Modelling heterogeneity in survival analyzis by the compound Poisson distribution. Annals of Applied Probability 2, 951–972. A ALEN , O. O. (1994). Effects of frailty in survival analyzis. Statistical Methods in Medical Research 3, 227–243. A NDERSEN , P. K., B ORGAN , O., G ILL , R. D. Processes. Springer: New York.

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