Análisis e interpretación de datos

Grambsch and Therneau (1994) proposed an approach to express nonproportional hazards by extending the Cox proportional hazards model to include time-varying coefficients. They described a regression approach to the test of nonproportionality for which a score test can be carried out on

the time-varying coefficients. This approach is reviewed as follows.

For the hazard function in (2.39), an alternative to the proportional hazards is the consideration of time-varying coefficients whereqj(t) is a predictable process. Thus the hazard function of the ith subject is

λi(t) =λ0(t) exp{β(t)TZi}, (2.41) in which β(t) = β+Q(t)θ where Q(t) isp×p diagonal matrix withQjj(t) = qj(t). Grambsch and Therneau (1994) showed that the score test for H0 :β(t) = β (i.e., θ = 0, confirming the attainment of proportional hazards assumption) is equivalent to a generalized least squares test on the Schoenfeld residuals.

All the definitions and notations provided in Section (2.5.1) stay the same withβ(t) substituted forβ. Letβb be the maximum likelihood estimate underH₀; andrˆ_k and ˆV_k are as defined earlier

evaluated atβb. Grambsch and Therneau (1994) defined weighted residuals,r∗_k=V−1

k rk, termed as scaled Schoenfeld residuals, and showed that E Vˆ_k−1ˆrk ' Qkθ where Qk = Q(tk). Since

P

s∈Drˆs= 0,the residuals are correlated with cov(ˆrk,rˆl) consistently estimated, underH0,by the variance-covarance matrix provided earlier. Thus,var Vˆ_k−1rˆk

'Vˆ_k−1− P

s∈DVˆs

−1

.Generalized least squares gives

b θ=B−1 X s∈D Qsˆrs, (2.42) with B= X s∈D QsVˆsQTs − X s∈D QsVˆs ˆ V−1 X s∈D QsVˆs T

. Under H0, the asymptotic variance of

n−12 P

s∈DQsrˆs can be consistently estimated by n−1B, leading to an asymptotic χ2 test on p degrees of freedom: T(Q) = X s∈D Qsrˆs T B−1 X s∈D Qsrˆs . (2.43)

The estimator (2.42) and the statistic (2.43) are, respectively, a one-step Newton-Raphson estimator of θ and the Rao score test ofH0 : (β,θ) = (β,0),based on the partial likelihood.

Zhou et al. (2013) extended Grambsch and Therneau’s (1994) approach by modifying the scaled Schoenfeld residuals to carry out the proportionality assumption test and diagnostics for proportional subdistribution hazards model in the presence of competing risks, i.e. the model introduced by Fine and Gray (1999). They developed a score test for the time-varying coefficients based on the

modified Schoenfeld residuals derived by considering a certain form of nonproportionality in the Fine-Gray model.

In order to test for non-proportionality due to a time-varying covariate effect in a proportional subdistribution hazards (PSH) model with competing risks, Zhou et al. (2013) considered a model given by

λ1(t;Z) =λ1·0(t) expβ(t)TZ , (2.44) where β(t) =β+D(t)θ,β= (β1, . . . , βp)T and θ= (θ1, . . . , θb)T are vectors of unknown time- constant and time-varying covariate effects, respectively. FurtherD(t) is ap×bmatrix of predictable processes, say djk(t), j = 1, . . . , p, k= 1, . . . , b, and each row corresponds to one component of

β. Note that djk(t) is a pre-specified function of time corresponding to the jth component of

Z. That is, when Zj is a time-varying covariate, then at least one component of the jth row in

D(t) (i.e., at least one component of (dj1, . . . , djb)) is a non-zero function of time. Otherwise, if

Zj is time-independent, then all components of thejth row in D(t) are zero. Multiple non-zero components in (dj1, . . . , djb) mean several time-varying aspects of covariateZj are to be tested simultaneously. In conclusion, those components ofZ being tested for time-varying effects will have at least some non-zero elements in D(t).

Non-proportionality assessment in the models aims at testing

H0 :θ=θ0=0vs. HA:θ6=0.

Estimation of the model parameters follows the approach in Fine and Gray (1999). Differentiating the log partial likelihood L(β,θ) with respect to L(θT,βT)T provides the estimating equation U1 = (U₁Tθ,U1Tβ)T.For a given θ,the restricted maximum partial likelihood estimator, βb_θ, can

be obtained by solving U1β(θ,βb_θ) =0.Under the null hypothesis that the covariate effects are

correctly specified as time-independent coefficients (i.e. the proportionality assumption of the PSH model holds true), the true values ofθ, denoted by θ0, equal to 0(and hence β(t) =β). Denote βb

as the maximum likelihood estimator of βunder H0, i.e., a solution toU1β(θ0,βb) =0.

Zhou et al. (2013) introduced a non-proportionality assessment method in PSH models using a score test for H0 : θ = 0 following the ideas in Grambsch and Therneau (1994). The score

test for H0 : θ = 0 is given by T(D) = n−1U1Tθ(θ0,βb)Ab−1U_1θ(θ₀,βb) whereAb is an estimator

of A, the covariance matrix of n−12U_1θ(θ,βb). Zhou et al. (2013) asserted that T(D) has an

asymptotic χ2-distribution with bdegrees of freedom which follows from the asymptotic normality of n−12UT

1 (θ0,β0) andn−

1 2UT

1θ(θ0,βb).

Following Schoenfeld’s (1982) introduction to partial residuals, Xue et al. (2013) defined ‘case- cohort Schoenfeld residuals’ that can be used to test the proportional hazards assumption in the analysis of case-cohort study data. They used the pseudolikelihood function defined by Prentice (1986) for proportional hazards model in case-cohort study designs and extended Schoenfeld’s (1982) approach to define partial residuals for such study designs. The proportional hazards assumption is then done by calculating a Pearson correlation coefficient and its significance for each variable in the model between its case-cohort Schoenfeld residual and a function of the corresponding event times, with detection of a significant correlation considered evidence of violation of the assumption.

CHAPTER 3: PROPORTIONAL SUBDISTRIBUTION HAZARDS MODEL