Chapter 4: Survival analysis 42
4.5 The Cox proportional hazards (PH) model
4.5.5 Residuals for the Cox regression model
Several residuals have been proposed for the assessment of the Cox PH model ad- equacy. For this project we will briefly discuss the Martingale residuals, Deviance residuals, Schoenfeld residuals and Score residuals. Others include Cox-Snell resid- uals and Modified Cox-Snell residuals but we will not discuss these here.
• Martingale residuals
rMi =δi−rCi (4.36)
Eqauation 4.36is known as a martingale residuals, since they can be derived using martingale methods (Collett, 2015). Following Collett (2015), Martingale residuals can take values between−∞and∞, with the residuals for censored observations, whereδi = 0being negative. It can be shown that these residu- als sum to zero and in large samples the Martingale residuals are uncorrelated with one another and have an expected value of zero (Collett, 2015). In this re- spect, they have properties similar to those possessed by residuals encountered in linear regression analysis, however they are not symmetrically distributed about zero, even when the fitted model is correct.
4.5. The Cox proportional hazards (PH) model
• Deviance residuals
The Deviance residuals are symmetrically distributed about zero and were first introduced by Therneau et al. (1990)
rDi =sgn(rMi)
−2 rMi+δilog(δi−rMi)12
, (4.37)
Where rMi is the martingale residual for the ith individual, and the function sgn(·) is the sign function. This is the function that takes the value +1 if its argument is positive and −1 if negative. Thussgn(rMi) ensures that the de- viance residuals have the same sign as the Martingale residuals (Collett, 2015).
The deviance is a statistic that is used to summarize the extent to which the fit of a model of current interest deviates from that of a model which is a perfect fit to the data. This latter model is called the saturated or full model in which β coefficients are allowed to be different for each individual. The statistic is given by
D=−2(logLˆc−logLˆf),
whereLˆcis the maximized partial likelihood under the current model andLˆf is the maximized partial likelihood under the full model. The smaller the value of the deviance, the better the model (Collett, 2015).
• Schoenfeld Residuals
These residuals are different to the ones mentioned above in a sense that, there is not a single value of the residual for each individual, but a set of values, one for each explanatory variable included in the fitted Cox regression model.
Schoenfeld Residuals were originally known as partial residuals. Theith par- tial or Schoenfeld Residual forXj, theXj explanatory variable in the model is given by
rpji =δi(xji−ˆaji), (4.38) where xji is the value of the jth explanatory variable, j = 1, . . . , pfor theith individual in the study,
ˆ aji=
P
l∈R(ti)xjiexp(βˆ0xl) P
l∈R(ti)exp(βˆ0xl) (4.39)
and R(ti) is the set of all individuals at risk at time ti. Note that non-zero
4.5. The Cox proportional hazards (PH) model
the largest observation in a sample of survival times is uncensored, the value of ˆ
ajifor that observation, from equation 4.39, will be equal toxjiand sorpji= 0 (Collett, 2015).
The ithSchoenfeld residual, for the explanatory variableXj is an estimate of theith component of the first derivative of the logarithm of the partial likeli- hood function with respect toβj which is given by
∂logL(β)
∂βj =
n
X
i=1
δi(xji−ˆaji), (4.40) where
aji= P
lxjlexp(β0xl) P
l∈R(ti)exp(β0xl) (4.41)
Theithterm in this summation, evaluated atβˆis then the Schoenfeld residual forXj given in equation 4.38. Since the estimates of theβ’s are such that
∂logL(β)
∂βj
|βˆ= 0,
the Schoenfeld residuals must sum to zero. These residuals also have the prop- erty that, in a large sample, the expected values of rpji is zero and they are uncorrelated with one other. The scaled version of the Schoenfeld residuals, proposed by Grambsch & Therneau (1994) is more effective in detecting de- partures from the assumed model and is given by
r∗pi =rV ar( ˆβ)rpi, where
rpi = (rp1i, . . . , rppi)0
and r is the number of deaths among the n individuals, and V ar( ˆβ) is the variance-covariance matrix of the parameter estimates in the fitted Cox regres- sion model. These scaled Schoenfeld residuals are therefore quite straight for- ward to compute (Collett, 2015).
• Score Residuals
Just like the Schoenfeld Residuals, the Score Residuals are also obtained from the first derivative of the logarithm of the partial likelihood function with re-
4.5. The Cox proportional hazards (PH) model
spect to the parameter βj, j = 1,· · ·, p. However, equation 4.40 is now ex- pressed as
∂logL(β)
∂βj
=
n
X
i=1
δi(xji−aji) +exp(β0xi) X
tr≤ti
(ajr−xji)δr P
l∈R(tr)exp(β0xl)
(4.42)
wherexjivalue of thejthexplanatory variable.δiis the event indicator which is zero for censored observations and unity otherwise,ajiis given in equation 4.41, and Rtr is the risk set at time tr. In this formulation, the contribution of the ith observation to the derivative only depends on information up to time ti. In other words, if the study was actually concluded at timeti theith component of the derivative would be unaffected. Residuals are then obtained as the estimated value of the ncomponent of the derivative. From equation 4.42the first derivative of the logarithm of the partial likelihood function, with respect toβj, is the efficient score forβj and so by Collett (2015) these residuals are known as score residuals. From equation 4.42, theith score residual,i = 1,· · ·, nfor thejthexplanatory variable in the model,Xj, is given by
rSji =δi(xji−aji) +exp(βˆ0xi) X
tr≤ti
(ˆajr−xji)δr
P
l∈R(tr)exp(β0xl) (4.43) Using equation 4.38, this may be written in the form
rSji =rpji+exp(βˆ0xi) X
tr≤ti
(ˆajr−xji)δr P
l∈R(tr)exp(β0xl) (4.44) Which according to Collett (2015) shows that the score Residuals are modifica- tions of the Schoenfeld Residuals.