An evaluation of the Cox-Snell residuals

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Department of Statistics

Master Thesis

2015

An evaluation of the Cox-Snell residuals

Elin Ansin

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Abstract

It is common practice to use Cox-Snell residuals to check for overall goodness of fit in survival models. We evaluate the presumed relation of unit exponentially dis-tributed residuals for a good model fit and evaluate under some violations of the model. This is done graphically with the usual graphs of Cox-Snell residual and formally using Kolmogorov-Smirnov goodness of fit test. It is observed that residu-als from a correctly fitted model follow unit exponential distribution. However, the Cox-Snell residuals do not seem to be sensitive to the violations of the model.

KEYWORDS: Cox-Snell residuals, Martingale residuals, Proportional Hazards Model, Survival models

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1 Introduction

Residuals are a widely used tool to assess the adequacy of a model. When modelling survival data, it is not as easy to define a residual as for a general linear model. A set of different residuals has therefore been proposed. In this thesis aims at evaluating Cox-Snell residuals (Cox and Snell, 1968) and their comparison with Martingale and Deviance residuals for different survival models. Even though these residuals are widely used, we couldn’t find any evaluation of their performance in the literature. An important criterion for the Cox-Snell residuals is that for a good model fit the residuals are unit exponentially distributed. Using this criterion the Cox-Snell residuals are evaluated both graphically and formally using a test of goodness of fit. For these evaluations event times following exponential, Weibull and gamma distri-butions are used. Two different types of models are considered: a simple parametric model without any covariates and a Cox proportional hazards model (CPHM) which takes covariates in to account.

The outline of the thesis is as follows. In Chapter 2, preliminary setup for the two cases is discussed, followed by a description of the simulation setup and results in Chapter 3. A brief summary and discussion is given in Chapter 4.

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2 Preliminaries

In section 2.1 the general setup to evaluate residuals from a simple parametric model are given and in section 2.2 this setup is extended to cover residuals from CPHM.

2.1 Models Without Covariates

Let ti, i = 1, ..., n, denote the event times. To include censoring, let ti* be the right

censored time point where ti ≤ ti* if we observe an actual event and ti > ti* if ti was

censored and the event could not be observed. Let δi be the event indicator where

δi = 1 if ti ≤ ti* and δi = 0 if ti > ti*. Let f (ti), F (ti), S(ti), h(ti) and H(ti) be

the usual probability density, distribution, survival, hazard and cumulative hazards functions see (Klein and Moeschberger, 2003, pp.22-31; Collett, 1994, pp.10-13). The survival function is the probability of an individual to experience the event after time t. Then for T as a continuous random variable

S(t) = 1 − F (t), (2.1.1)

where S(T ) is also connected to the cumulative hazards function as

H(t) = − log(S(t)). (2.1.2)

Assuming a certain distribution of ti, the distribution parameters can be

esti-mated using maximum likelihood estimation (MLE) and the survival function can then be estimated using Equation (2.1.1). In MLE the parameter estimates are found by maximizing the likelihood function

L(θ; ti) =

n Y

i=1

f (ti; θ).

As an example consider ti ∼ Exp(λ)with pdf f (t) = λe−λ for 0 < t < ∞. To

estimate λ, the likelihood function

L(λ; ti) = n Y i=1 λe−λti _{= λ}n_exp _−λ n X i=1 ti !

with the log likelihood

l(λ; ti) = n log(λ) − λ

n X

i=1

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The log likelihood equation maximized by d dλn log(λ) − λ n X i=1 ti = 0 gives b λ = _P_nn i=1ti ,

which, using the usual theory of MLE (Casella and Berger, 2002, pp.315-16) can be verified as the MLE of λ. In the presence of censored observations in the data set, partial MLE can be used to estimate the parameter. Consider our dataset

containing right-censored time points with (xi, δi) where xi = min(ti, ti*). The

likelihood function will now be defined as

L(θ; xi, δ) =

n Y

i=1

[f (xi; θ)]δi[S(xi; θ)]1−δi.

Since h(xi; θ) = f (xi; θ)/S(xi; θ), the likelihood can be simplified as

L(θ; xi, δi) =

n Y

i=1

[h(xi; θ)]δiS(xi; θ).

As an example, let again ti ∼ Exp(λ), with some ti right-censored. With h(xi) = λ

and S(xi) = e−λx, we have L(θ; xi, δi) = n Y i=1 λδie−λxi _{and l(λ; x} i, δi) = X δilog(λ) − λ n X i=1 xi. So that d dλr log(λ) − λ n X i=1 xi = 0, gives b λ = Pr r i=1ti Pn i=r+1t ∗ i ,

where r denotes the number of uncensored observations, we get the partial maxi-mum likelihood estimate.

When the survival function has been estimated, the Cox-Snell residuals (Cox and Snell, 1968) defined as

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can be computed. The Cox-Snell residuals have the main property that if the model

fits the data, rCi follow the standard exponential distribution, exp(1). This means

that rCi are expected to have a mean of one. Since the rCi for a censored distribution

is

rCi = bH(t∗i) = − log( bS(t

∗

i)),

the residual for a censored observation will be smaller than if the actual event time had been observed. To make the residuals for censored observations compatible with the ones for the uncensored observations a modification of the Cox-Snell residuals

could be used. Since rCi has an unit exponential distribution a suitable modification

is to add the expected value, one, to the censored observation, that is

r0_Ci =

 



rCi for uncensored observations

rCi+ 1 for censored observations

(2.1.4)

From the Cox-Snell residuals the related Martingale and Deviance residuals can also be computed. The Martingale residuals (Collett, 1994, p.151),

rMi = δi− rCi (2.1.5)

are a reallocation of the Cox-Snell residuals to a mean of zero for uncensored obser-vations, so that the Martingale residuals follow similar properties as the residuals in linear regression analysis. Martingale residuals for censored observations take

negative values. rMi are however not symmetrically distributed around zero which

makes plots of these residuals difficult to interpret. More symmetrically distributed around zero are the deviance residuals, defined as (Collett, 1994, p.153)

rDi = sgn(rMi)[−2{rMi+ δilog(δi− rMi)}]

1

2 (2.1.6)

The sgn() function makes sure that the deviance residuals has the same sign as the martingale residuals. Since these residuals are more symmetrically distributed around zero, plots of these residuals against event times are easier to interpret, a good model fit is indicated with a random scatter around zero. We should notice that even though these residuals are more symmetrically distributed around zero they do not necessarily sum to zero.

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The fact that rCi ∼ exp(1) can be used to evaluate the good fit of the

sur-vival model. Obviously, given rCi ∼ exp(1) their pdf is f (rC) = e−rC and using

Equation (2.1.1)

S(rC) = e−rC,

which, using Equation (2.1.2) further implies that

H(rC) = − log S(rC) = rC.

Hence, a plot of rCi against the estimated cumulative hazard rate of the

Cox-Snell residuals ( bH(rCi)) is expected to show a straight line with zero intercept and

unit slope.

The two most commonly used estimators of H(r) is Kaplan-Meier (KM) and

Nelson-Aalen (NA). Let t1 < t2 < ... < tD be the distinct event times, and let Yi be

the number of individual at risk at time ti when di events occur. The quantity

di

Yi

is an estimate of the probability that an individual who were alive at a time point

just prior to time ti will experience the event at time ti. The KM estimator is then

defined as (Klein and Moeschberger, 2003, p.92)

b S(t) =    1 if t1 > t Q ti≤t h 1 − di_Yii if t1 ≤ t (2.1.7)

By Equation (2.1.2) the KM estimator can be used to estimate bH(t). The NA

estimator is defined as (Klein and Moeschberger, 2003, p.94):

e H(t) =    0 if t1 > t P ti≤t di Yi if t1 ≤ t (2.1.8)

Here the cumulative hazard is estimated as the sum of the probability of having an event at each point in time. This estimator is said to have a better small sample performance than the Kaplan-Meier estimator (Klein and Moeschberger, 2003, p.94).

Using these estimators gives us two estimates of H(rCi), bH(rCi) = − log( bS(rCi)) and

e

H(rCi).

In addition we use Kolmogorov-Smirnov goodness of fit test to formally evaluate if the residuals follow a unit exponential distribution. The K-S test statistic is

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defined (Hollander and Wolfe, 1999, p.529) as D = sup

rC

|Fn(rC) − F0(rC)| (2.1.9)

Where F0(rC) is the assumed specific cumulative distribution function and Fn(rC)

is the corresponding empirical distribution function.

2.2 Models With Covariates using CPHM

As before, let ti, i = 1, ..., n denote the event time, ti* denote censored times and δi

the event indicator, where δi = 1 if ti ≤ ti* and δi = 0 if ti > ti*. Let f (ti), F (ti),

S(ti), h(ti) and H(ti) be the usual probability density, distribution, survival, hazard

and cumulative hazards functions. The proportional hazards model proposed by Cox (1972) specifies that the hazard rate at time t for an individual with covariate X are

h(t | X) = h0(t)c(β

0

X), (2.2.1)

where h0(t) is an arbitrary baseline hazard rate, β is a vector of parameters and

c(β0X) is a specified function that in practice is usually taken to be eβ0X_{. The}

function eβ0X _{can be interpreted as the hazard at time t for a specific individual}

with covariates values Xirelative to the hazard for a individual with X = 0 (Collett,

1994). For the survival function it holds that

S(t|X) = [S0(t)]e

β0X

. (2.2.2)

The Cox-Snell residuals for CPHM are defined as

rCi = − log bS(ti|X) = bH(ti|X), (2.2.3)

such that β and S0(t) is needed to be estimated. The parameter vector β can be

estimated using partial MLE. In case of no ties i.e. only one event at each time t

the r event times can be denoted as t(1) < t(2) < ... < t(r), so that t(i) is the ith

ordered event time. The individuals still at risk at the instant before time t(i) are

denoted as R(t(i)). The probability that a specific individual have an event at time

ti given one of the individuals in R(t(i)) has an event at time ti is given by

h(ti | x(i)) P j∈R(ti)h(ti | xj) = h0(ti)e β0x(i) P j∈R(ti)h0(ti)eβ 0_xj = eβ0x(i) P j∈R(ti)eβ 0_xj.

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By taking the product over all death times we get the partial likelihood (Cox, 1972) L(β) = n Y i=1 eβ0x(i) P j∈R(ti)eβ 0_xj

with the corresponding log-likelihood function

l(β) = k X i=1 βx(i)− k X i=1 ln   X j∈R(ti eβ0xj  .

The partial MLE of β can be obtained by maximizing l(β) numerically. Note that

we do not need any assumption of h0(t) to estimate β. Once a model is fitted, the

baseline hazard function h0(t) and the corresponding survival function S0(t) can be

estimated. Most commonly the Breslow and the Kalbfleisch & Prentice estimators

are used to estimate h0(t).

The Breslow estimator is defined as (Klein and Moeschberger, 2003, p.283) e H0(t) = X ti≤t di P j∈R(ti)exp( bβ 0 xj) (2.2.4) If no covariates are present this reduces to the Nelson-Aalen estimator in Equa-tion (2.1.8). The estimator of the baseline survival funcEqua-tion is given by

e

S0(t) = exp[− eH0(t)]. (2.2.5)

The Kalbfleisch and Prentice estimator is defined as (Collett, 1994, p.96)

b S0(t) = k X j=1 b ξj, (2.2.6)

where bξj is the solution of the equation

X l∈D(t(j)) exp( bβ0xl) 1 − bξ_jexp( bβ0xl) = X l∈R(t(j)) exp( bβ0xl) (2.2.7)

When there are no tied event times, the left-hand side of this equation becomes a single term and the equation can then be solved to give

b ξj =  1 − exp( bβ 0_x (j)) P l∈R(t(j))exp( bβ 0_x l)   exp(− bβ0_x (j)) (2.2.8)

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If there are no covariates this estimator reduces to the KM estimator in Equa-tion (2.1.7).

For both of these estimators of the baseline survival, the estimated survival function for the ith individual can be found using Equation (2.2.2).

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3 Results and Discussion

3.1 Simulation Setup

Some of the distributions known to be appropriate for survival times include expo-nential, Weibull and gamma distributions. The exponential distribution serves as a basic model for survival time and is a special case of several other distributions as Weibull and gamma distribution. These three distributions are used to simulate the survival times. Table 1 gives the pdfs of these distribution.

Distribution, notation Density function

standard exponential, Exp(1) f (t) = e−t, t > 0

standard Weibull, W (γ, 1) f (t) = γt(γ−1)_e−tγ

, γ > 0, t > 0

standard gamma, Γ(γ, 1) f (t) = γt(γ−1)_Γ(γ)e−t, γ > 0, t > 0

Table 1: Probability density functions of event times.

For γ = 1 both the Weibull and the gamma distributions simplifies to standard exponential distribution. We use γ = 2 for Weibull and gamma distribution giving a linear increasing hazard h(t) = 2t for Weibull distribution and an increasing hazard

h(t) = _1−Γt(2)te−t , where Γt(2) =

Rt

0 ue

−u _{du, for gamma distribution. The standard}

exponential distribution have a constant hazard, h(t) = 1. Then the exponential and Weibull distribution share the crucial assumption of proportional hazards with CPHM (Lee and Go, 1997).

The datasets are generated in R using the rexp(), rweibull() and rgamma() functions. Sample sizes 10, 20, 50 and 100 are selected to cover the sample size spec-trum of interest. To include different degrees of censoring we use 10 and 20 percent censoring. For the CPHM a potential impact of event per independent variable were covered by including models with 1 to 5 covariates. Covariates were simulated, independent of event times, using rmvnorm() from the mvtnorm package. For comparison covariates with a 0.5 correlation with the event times were simulated as well. The entire simulation scheme is summarized in Table 2.

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Model without covariates

Distribution n censoring

Exp, W, Gamma 10, 20, 50, 100 0, 10 %, 20 %

Model with covariates Distribution

n

cens.

0 10 % 20 %

Exp, W, Gamma

10 1 cov 1 cov 1 cov

3 cov 5 cov

20

1 cov 1 cov 1 cov

3 cov 5 cov

50

1 cov 1 cov 1 cov

3 cov 5 cov

100

1 cov 1 cov 1 cov

3 cov 5 cov

Table 2: Simulation scheme

For each of the cases described in Table 2, 1000 samples were drawn. Models were fitted to each sample and the Cox-Snell, Martingale and Deviance residuals were averaged. In the samples where the Breslow and Kalbfleisch & Prentice estimates of the parameters did not converge, causing extreme values (0 and 1) of the baseline survival estimates, the sample was discarded and a new one was drawn.

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3.2 Graphical evaluation

In this section the results from the graphical evaluation is presented, starting with the comparison of Cox-Snell, Martingale and Deviance residuals. The distribution of the event times do not appear to have an impact of the residuals, only graphs with exponential event times are therefore presented here. When no censoring is added to the samples the models with no covariates have Cox-Snell residuals with a mean of approximately one as expected for a good model fit, see Fig. 1. The martin-gale residuals have an average of approximately zero as expected and the Deviance residuals has an average about 0.4. The Martingale and deviance residuals are not affected when censoring is added to the samples (Figs. 2 and 3), but the average of the Cox-Snell residuals tends to decrease slightly when censoring is present.

The residual from the CPHM with a time-independent covariate (Fig. 4) are equivalent to the ones from the model with no covariates, indicating a good model fit even though no relation to the event time is assumed. The averages do not change for any of the residuals discussed when adding more independent covariates, (Fig. 6 and 7). As expected, the residuals for the CPHM with time-covariate relationship (Fig. 5) are also equivalent to the no covariate case.

For the CPHM model, the residuals for the small samples are unstable when censoring is present, especially in the samples with only 10 observations (Figs. 8 and 9). For larger sample sizes, the Cox-Snell residuals are stable with mean one, both with 10 and 20 percent of censoring.

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* * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Exponential, n=10 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Exponential, n=20 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * ** ** * * ** * ** ** * ** ** ** * * ** ** * ** * ** ** * ** * ** ** * ** * * * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Exponential, n=50 Inde x residual * ** ** * ** * * * * ** * ** ** ** ** ** ** ** * ** * ** * ** ** * ** ** * * ** * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * ** * ** * * * * ** ** * ** ** * ** * ** * * * ** ** ** ** * ** ** * ** ** ** 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance ** ** **** ** ** ** * *** ** ** ** * *** * * **** * *** ** * * ** **** ** * ** * ** **** * * **** * *** **** ** ** * * * *** * * * *** * *** * * ** * * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Exponential, n=100 Inde x residual ** * *** * * **** * *** * * **** * *** **** * * * *** ** * *** * ** * ** * ** * * * *** ** ** * *** ** * *** * ** * ** * ** ** **** ** * *** ** ** ** ** 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * *** ** **** * ** * ** **** * *** * *** * * * *** ** *** * ** * * ** * *** * * *** * * *** ** ** ** * * * * ** * * * *** ** **** ** * *** ** **** * * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance Figure 1: Comparison of Co x-Snell, Martingale and D eviance residuals from mo del without co v ariates.

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* * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Exponential, n=10 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Exponential, n=20 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * ** ** ** ** ** * ** ** ** ** ** * ** ** * ** ** * ** ** ** * * ** * * ** ** 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Exponential, n=50 Inde x residual ** * * * ** ** * * ** * * * ** * * ** * ** ** ** * * * * * * * * ** * ** * * * * * ** * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * ** ** ** ** * * ** * * ** * * ** * * ** ** ** * * * * * * * * ** * ** * * * ** ** * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance ** * * **** * ** * **** * * * * * * ** ** **** * *** * * * *** ** * *** * *** ** *** *** ** * ** * * *** * * **** ** *** * **** * *** ** * * ** * * ** * * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Exponential, n=100 Inde x residual * *** ** * *** ** * * * *** ** ** ** **** ** ** * * *** **** ** ** * *** ** * *** * * *** ** * ** ** * * * *** ** **** **** ** **** ** * *** * *** * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * *** ** **** * * ** ** *** * ** * ** * *** ** * *** * * *** ** ** **** * * **** * * * *** ** * ** ** * * * *** ** **** **** ** **** * *** * ** **** 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance Figure 2: Co mparison of Co x-Snell, Martingale and Deviance residuals from mo del without co v ariates and 10 p ercen t censoring (mean corrected residuals)

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* * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Exponential, n=10 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Exponential, n=20 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance ** ** ** ** * * ** * ** ** ** * ** * * ** * ** * ** * * * ** * * * * ** * * * * ** * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Exponential, n=50 Inde x residual * * * ** * * ** * * * ** * * ** ** * * * ** * * * ** * * ** ** * ** ** * * * ** ** ** 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * ** ** * * ** * ** * * ** ** * ** * * ** * * * ** * * ** * * * * * ** * * ** * ** * * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * ** **** * * **** ** * * * * * ** * *** * *** ** * ** * ** * *** ** * *** **** * * * *** * * * *** ** * *** * ** * ** ** * *** * * * ** * **** ** **** * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Exponential, n=100 Inde x residual *** * ** * *** * * * * * *** ** ** * * *** * *** * * * * *** ** **** ** * *** ** ** * ** * ** * * ** * *** *** * * * * *** * * * *** * *** * ** ** * * * *** * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual **** ** * ** * ** ** * *** **** * * *** * *** * ** * *** ** **** ** * *** ** ** * ** * * * * * ** ***** * *** * * *** * * * *** * *** * ** ** * * * *** * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance Figure 3: Co mparison of Co x-Snell, Martingale and Deviance residuals from mo del without co v ariates and 20 p ercen t censoring (mean corrected residuals)

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* * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=10 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=20 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance ** ** ** ** ** ** * ** * ** ** ** ** ** * * ** * * * * * ** * ** ** ** ** * * * * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=50 Inde x residual * ** * * * * ** ** ** * * * * * * * * ** ** ** * * * ** ** ** ** * ** ** ** ** * * * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * ** * * * * ** ** ** * * * * * * * * ** ** ** * * * ** ** ** ** * ** ** ** ** ** * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * *** * *** * * * *** ** * ** * * ** * *** * ** * ** * ** * ** * *** ** * ** * * *** ** * *** *** * * * ** **** ** * *** ** ** ** ** ** **** * *** **** * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=100 Inde x residual ** * ** * *** * *** ** * * ** * *** * *** * ** * ** * ** *** * **** * *** * * **** * * *** * * ** *** * * * * ** * ** * * * * * ** * * ** * ** **** **** * ** 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual ** **** * * **** * * * *** **** * *** * * ** * ** * ** * * * *** * * * * * * * * **** * * *** * * ** **** * * * *** ** ** * * * ** * * * **** * *** ** ** * ** 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance Figure 4: Comparison of Co x-Snell, Martingale and Deviance residuals, CPHM with one co v ariate

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* * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=10 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=20 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance ** * ** * ** * ** * * ** * ** * ** * * ** ** * * ** * ** * * * ** ** ** * ** * * ** 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=50 Inde x residual * ** * ** * * * * ** * * ** * * ** ** * * ** * ** * ** * ** ** * ** ** ** * ** * * * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * ** * ** * * * ** * * * ** * * ** ** * * ** * ** * * ** ** ** * ** ** ** * ** * * * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance *** * * * * *** **** ** ** ** **** * *** ** **** * *** * * * * *** ** ** * ** * * ** * * * * *** ** * * ** * *** ** * *** **** ** ** ** * ** * * *** ** * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=100 Inde x residual ** **** ** * * * * * *** * *** ** * ** * *** * *** * * ** * * ** * ** ** ** * * * * * ** * * * * *** ** ** * *** ** * ** * * ** * ** ** **** * * * ** * * *** ** 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual ** **** ** * * * * * *** * *** ** **** ** * *** * *** * * * ** * ** * *** * ** * * * * ** * * *** ** * * **** ** * * * * * ** * ** ** **** ** * ** * * *** ** 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance Figure 5: Comp arison of Co x-Snell, Martingale and Deviance residuals, CPHM with one co v aria te correlated with ev en t time.

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* * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=10 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=20 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * ** * ** * ** ** ** ** ** * ** ** ** * * ** * * ** * ** ** * ** ** * ** ** * * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=50 Inde x residual ** * * ** * ** ** ** ** ** * ** ** ** ** * * * ** * * ** ** * ** ** ** * ** * ** 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual ** * * ** * ** ** ** ** * * * ** ** ** ** * * * ** * * * * * * ** * * * ** * ** * ** 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance **** ** * * * ** * * *** * * ** * * * *** ** *** * ** * ** * * * ** * * * *** * **** * ** ** ** ** * *** ** * *** **** * ** * ** **** ** ** * *** ** ** ** 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=100 Inde x residual ** ** * *** ** **** * * **** **** ** * ** * * *** ** * ** * * * * * * *** * *** * * **** * ** * ** * ** * ** * ** * *** * *** * *** ** * *** **** **** * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual ** ** * *** ** **** * * ** * *** ** * * * ** * * *** ** * * * *** * * * *** * ** * *** * * **** ** ** * *** **** ** * * * *** * *** ** *** * * * ** **** * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance Figure 6: Comparison of Co x-Snell, Martingale and Deviance residuals, CPHM with three co v ariates.

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* * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=10 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=20 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * * * ** * ** ** * ** * * ** ** * ** ** ** ** ** * * * ** * ** ** * * * * ** ** * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=50 Inde x residual * * ** * ** * ** ** * ** ** ** ** * ** ** ** * * * ** ** * ** ** ** * * ** ** ** 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * ** * ** * * * ** * * ** * * * ** * ** ** ** ** * ** ** ** * * * ** * * ** ** ** 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance ** * * * *** ** **** * * *** * *** ** **** * *** * * * *** ** **** * ** ** * * **** * *** * ** * *** * ** * ** * *** *** * *** * *** * * * *** * * ** * * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=100 Inde x residual **** ** * * * ** * * *** * * * * * *** **** ** **** * *** * * *** **** * ** * * * **** * ** * **** * *** ** * *** *** * * * * *** * *** ** * ** * * *** * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual **** ** ** * ** * ** ** * * * * * *** **** ** ** ** * *** * * * ** * *** * ** * * ** * ** ** * * ** ** * *** ** * * ** **** * *** * * * *** ** * *** * *** * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance Figure 7: Comparison of Co x-Snell, Martingale and Deviance residuals, CPHM with fiv e co v ariates.

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* * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=10 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=20 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance ** * ** ** ** ** ** * ** * ** ** * ** * ** ** ** ** * * ** * ** * ** ** ** * * * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=50 Inde x residual * ** ** * ** * * ** * ** ** * ** ** * * ** ** * * * * ** ** ** * * ** * * ** ** * * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * ** ** * ** ** ** * ** * ** ** ** * * ** * * * * * ** * ** ** * * * * ** ** ** * * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance **** * *** **** ** **** * * *** * *** **** * * * *** * * * *** * * * ** * * * * *** ** * * * ** * **** * * *** * ** * * * * * *** * *** **** **** * ** * * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=100 Inde x residual ** * *** * * ** * ** * * * * ** * **** * *** ** * * * *** ** * *** * ** ** * * ** * * *** * ** * ** * *** ** * *** * *** * *** * ** * ** ** * ** ** * *** ** * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual ** * *** * * ** * ** * ** * ** * * ** * * *** ** * * * *** * * * *** * ** ** * * ** * **** * ** * * **** * *** * *** * * **** ** **** * ** * ** ** * *** ** * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance Figure 8: Com parison of Co x-Snell, Martingale and Deviance residuals, Co x mo del with one co v ariate and 10 p ercen t censoring.

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* * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=10 Inde x residual * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * 2 4 6 8 10 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=20 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * * * * * * * * * * * * * * * * * * 5 10 15 20 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance ** ** * ** * ** ** * ** ** * ** ** ** * ** * * * * * * * ** * ** ** * * * ** * ** * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=50 Inde x residual * ** * ** ** * * * ** * * * * ** * * ** ** * * ** * ** ** * * * * ** ** ** * ** * * * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * ** * * * ** * * * ** * * * * ** * * * * * * ** ** * ** * ** * * * ** ** ** * ** * * * 0 10 20 30 40 50 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance * * * * * ** * *** **** ** **** ** *** * ** * *** * * * *** * *** * * * *** ** ** * * * * * *** * * ** * *** *** * **** ** * * * ** * * * *** ** * *** * *** 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 n=100 Inde x residual ** ** ** * *** ** * * ** * ** ** * ** * ** * *** **** ** * ** **** * *** ** **** * ** * **** * ** * ** ** ** * *** ** * * * *** *** ** ** * *** * * ** * 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual ** ** ** ** *** * * * ** * * * *** * * * ** * **** * * * ** * ** * * ** * ** * * *** * * * ** * **** ** * * ** ** ** * *** ** * * * *** ** ** ** **** **** ** 0 20 40 60 80 100 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Inde x residual * * * Co x−Snell Mar tingale De viance Figure 9: Com parison of Co x-Snell, Martingale and Deviance residuals, Co x mo del with one co v ariate and 20 p ercen t censoring.

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Cox-Snell residuals

Here graphs of rCi against bH(rCi) and eH(rCi) are presented along with their

log-transformed confidence intervals. The graphs based on exponential event times are presented here and the corresponding graphs for Weibull and gamma distributed event times are given in Figs 20 to 34 in Appendix A. For models without covariates the estimates in Fig. 10 follow the expected 45 degree line closely for all sample sizes, indicating unit exponential distribution of the residuals when no censoring is

present. For the CPHM Fig. 13, indicate a good fit to unit exponential rCi, for both

estimators used. Comparing this to the results of CPHM with a correctly fitted covariate in Fig. 14, we see that the model with incorrectly fitted covariate has an even closer fit than the model with a correctly fitted covariate. It is not clear but the reason might be that the survival times for the model with a correct fitted covariate are normally distributed and not exponential as for the other models. This indicates that the Cox-Snell residuals are not sensitive to the violation of the model. When a larger number of independent covariates are included in the model, the fit is not as close for a small sample size (Fig. 15 and 16).

When censoring is added to the samples and mean correction is used for censored

observations, the resulting rCi for models without covariates show a slight deviation

from the reference line, indicating rCi ∼ exp(1), increasing whit the degree of

cen-soring (Figs. 11 and 12). Figs. 17 and 18 give the corresponding graphs for rCi from

CPHM. With censoring present, the rCi for CPHM deviates from exp(1) for small

sample sizes, especially for sample size 10. But as n increases they behave similarly to the no covariate case.

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0 1 2 3 4 5 0 1 2 3 4 5 Exponential, n=10 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Exponential, n=20 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Exponential, n=50 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Exponential, n=100 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI Figure 10: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals for mo dels without co v ariates with exp onen tially distributed surviv al times.

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0 1 2 3 4 5 0 1 2 3 4 5 Exponential, n=10 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Exponential, n=20 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Exponential, n=50 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Exponential, n=100 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI Figure 11: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals for mo dels without co v ariates with exp onen tially distributed surviv al times with 10 p ercen t censoring.

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0 1 2 3 4 5 0 1 2 3 4 5 Exponential, n=10 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Exponential, n=20 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Exponential, n=50 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Exponential, n=100 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI Figure 12: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals for mo dels without co v ariates with exp onen tially distributed surviv al times with 20 p ercen t censoring.

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0 1 2 3 4 5 0 1 2 3 4 5 n=10 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates

KM Bres NA Bres KM Kalbf NA Kalbf KM CI NA CI

Figure 13: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals, CPHM with one unrelated co v ariate.

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Figure 14: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals, CPHM with one related co v ariate.

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Figure 15: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals, CPHM with three unrelated co v ariates.

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Figure 16: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals, CPHM with fiv e unrelated co v ariate s.

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Figure 17: Plot of Co x-Snell residua ls against cum ulativ e hazards of residuals, CPHM with one unrelate d co v ariate and 10% censoring

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Figure 18: Plot of Co x-Snell residua ls against cum ulativ e hazards of residuals, CPHM with one unrelate d co v ariate and 10% censoring

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3.3 Goodness of fit

All though a deviation from expected appearance of graphs for rCi ∼ exp(1) is found

for the censored cases, and for small samples with larger numbers of covariates in the CPHM, none of the cases did however lead to rejection of the hypothesis of unit exponentially distributed residuals when evaluated by Kolmogorov-Smirnov goodness of fit test. For most cases the resulting p-values are approximately one. In Tables 3 and 4, we can see that when 20 percent censoring is used the p-values are slightly smaller for sample size 100, with p-values of approximately 0.8 for both models, still far from rejection. More importantly both models show equivalent results. Remaining results can be found in Tables 5 to 10 in Appendix B. The results of the K-S test presented for the CPHM are the ones obtained by the Breslow estimator. The results are similar for the Kalbfleisch and Prentice estimator, even though the p-values are slightly lower.

n D p-value

10 0.1044 0.9992

20 0.0788 0.9987

50 0.065 0.9752

100 0.0594 0.8719

Table 3: K-S tests for models with no covariates; exponentially distributed; 20 percent censoring. n D p-value 10 0.1272 0.9901 20 0.0941 0.987 50 0.0733 0.9332 100 0.0659 0.778

Table 4: K-S tests for CPHM with one unrelated covariate; exponentially dis-tributed; 20 percent censoring.(Breslow estimator).

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4 Summary and Outlook

We observed that the main property of unit exponentially distributed rCi for a

good model fit is reasonable. However the residuals from models with violations do not deviate much from their usual behaviour, leading to the conclusion that the Cox-Snell residuals are not sensitive to model violations.

Even though no difference in residual behaviour could be detected in the hypoth-esis testing, the residuals graphically showed moderate signs of a different behaviour for violated models when the sample size is small and censoring is present. This in-dicates that for small samples with a higher degree of censoring the residuals could be sensitive for model violations. Search for a possible threshold of sample size and degree of censoring where residuals become sensitive to model violations could therefore be a scope for further research.

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References

Casella, G. and Berger, R. L. (2002). Statistical inference. Duxbury press, CA, 2nd edition.

Collett, D. (1994). Modelling survival data in medical research. Chapman & Hall, London, 1st edition.

Cox, D. R. (1972). Regression models and life-tables. Journal of the Royal Statistical Society, series B, 34(2):187–220.

Cox, D. R. and Snell, E. J. (1968). A general definition of residuals. Journal of the Royal Statistical Society, series B, 30(2):248–275.

Hollander, M. and Wolfe, D. A. (1999). Nonparametric statistical methods. Wiley, New York, 2nd edition.

Klein, J. P. and Moeschberger, M. L. (2003). Survival analysis: techniques for censored and truncated data. Springer, New York, 2nd edition.

Lee, E. T. and Go, O. T. (1997). Survival analysis in public health research. Annual review of public health, 18:105–134.

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Appendix A

0 1 2 3 4 5 0 1 2 3 4 5 W eib ull, n=10 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Gamma, n=10 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 W eib ull, n=20 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Gamma, n=20 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI

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0 1 2 3 4 5 0 1 2 3 4 5 W eib ull, n=50 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Gamma, n=50 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 W eib ull, n=100 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5

Gamma, n=100 _Cox−Snell Residuals

Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI Figure 19: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals for mo dels without co v ariates with W eibull and gamma distributed surviv al times with sample sizes 50 and 100.

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0 1 2 3 4 5 0 1 2 3 4 5 W eib ull, n=10 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Gamma, n=10 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 W eib ull, n=20 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Gamma, n=20 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI Figure 20: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals for mo dels without co v ariates with W eibull and gamma distributed surviv al times with sample sizes 10 and 20 and 10 p ercen t censoring.

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Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI Figure 21: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals for mo dels without co v ariates with W eibull and gamma distributed surviv al times with sample sizes 50 and 100 an d 10 p ercen t censoring.

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0 1 2 3 4 5 0 1 2 3 4 5 W eib ull, n=10 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Gamma, n=10 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 W eib ull, n=20 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI 0 1 2 3 4 5 0 1 2 3 4 5 Gamma, n=20 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI Figure 22: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals for mo dels without co v ariates with W eibull and gamma distributed surviv al times with sample sizes 10 and 20 and 20 p ercen t censoring.

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Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates KM NA KM CI NA CI Figure 23: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals for mo dels without co v ariates with W eibull and gamma distributed surviv al times with sample sizes 50 and 100 an d 20 p ercen t censoring.

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0 1 2 3 4 5 0 1 2 3 4 5 W eib ull, n=10 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates

0 1 2 3 4 5 0 1 2 3 4 5 Gamma, n=10 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates

Figure 24: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals for mo dels with one co v ariate with W eibull and gamma distributed surviv al times with sample sizes 10 and 20.

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0 1 2 3 4 5 0 1 2 3 4 5

Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates 0 1 2 3 4 5 0 1 2 3 4 5 Co x−Snell Residuals Estimated Cumulativ e Hazard Rates

Figure 25: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals for mo dels with one co v ariate with W eibull and gamma distributed surviv al times with sample sizes 50 and 100.

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Figure 26: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals for mo dels with one co v ariate with W eibull and gamma distributed surviv al times and 10 p ercen t censoring for sample sizes 50 and 100.

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0 1 2 3 4 5 0 1 2 3 4 5

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0 1 2 3 4 5 0 1 2 3 4 5

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Figure 30: Plot of Co x-Snell residuals against cum ulativ e hazards of residuals for mo dels with three co v ariate with W eibull and gamma distributed surviv al times with sample sizes 10 and 20.

An evaluation of the Cox-Snell residuals

Department of Statistics

Master Thesis

2015

An evaluation of the Cox-Snell residuals

Elin Ansin

Abstract

Contents

1

Introduction

2

Preliminaries

2.1

Models Without Covariates

2.2

Models With Covariates using CPHM

3

Results and Discussion

3.1

Simulation Setup

3.2

Graphical evaluation

3.3

Goodness of fit

4

Summary and Outlook

References

Appendix A