Variables with timevarying effects and the Cox model: some statistical concepts illustrated with a prognostic factor study in breast cancer.
ABSTRACT The Cox model relies on the proportional hazards (PH) assumption, implying that the factors investigated have a constant impact on the hazard  or risk  over time. We emphasize the importance of this assumption and the misleading conclusions that can be inferred if it is violated; this is particularly essential in the presence of long followups.
We illustrate our discussion by analyzing prognostic factors of metastases in 979 women treated for breast cancer with surgery. Age, tumour size and grade, lymph node involvement, peritumoral vascular invasion (PVI), status of hormone receptors (HRec), Her2, and Mib1 were considered.
Median followup was 14 years; 264 women developed metastases. The conventional Cox model suggested that all factors but HRec, Her2, and Mib1 status were strong prognostic factors of metastases. Additional tests indicated that the PH assumption was not satisfied for some variables of the model. Tumour grade had a significant timevarying effect, but although its effect diminished over time, it remained strong. Interestingly, while the conventional Cox model did not show any significant effect of the HRec status, tests provided strong evidence that this variable had a nonconstant effect over time. Negative HRec status increased the risk of metastases early but became protective thereafter. This reversal of effect may explain nonsignificant hazard ratios provided by previous conventional Cox analyses in studies with long followups.
Investigating timevarying effects should be an integral part of Cox survival analyses. Detecting and accounting for timevarying effects provide insights on some specific time patterns, and on valuable biological information that could be missed otherwise.

Article: ADAMTS13 Predicts Renal and Cardiovascular Events in Type 2 Diabetics and Response to Therapy.
Erica Rurali, Marina Noris, Antonietta Chianca, Roberta Donadelli, Federica Banterla, Miriam Galbusera, Giulia Gherardi, Sara Gastoldi, Aneliya Parvanova Ilieva, Ilian Petrov Iliev, Antonio Bossi, Carolina Haefliger, Roberto Trevisan, Giuseppe Remuzzi, Piero Ruggenenti[Show abstract] [Hide abstract]
ABSTRACT: In diabetics, impaired ADAMTS13 proteolysis of highlythrombogenic VWF multimers may accelerate renal and cardiovascular complications. Restoring physiological VWF handling might contribute to ACE inhibitors (ACEi) reno and cardioprotective effects.To assess how Pro618Ala ADAMTS13 variants and related proteolytic activity interact with ACEi therapy in predicting renal and cardiovascular complications, we genotyped 1163 normoalbuminuric type 2 diabetics from the BENEDICT trial. Interaction between Pro618Ala and ACEi was significant in predicting both renal and combined renal and cardiovascular events. The risk [HR(95%CI)] for renal or combined events vs reference Ala carriers on ACEi progressively increased from Pro/Pro homozygotes on ACEi [2.80(0.8499.216) and 1.58(0.7373.379), respectively], to Pro/Pro homozygotes on nonACEi [4.77(1.48415.357) and 1.99(0.9444.187)], to Ala carriers on nonACEi [8.50(2.41629.962) and 4.00(1.7399.207)]. In a substudy, serum ADAMTS13 activity was significantly lower in Ala carriers than in Pro/Pro homozygotes and in cases with renal, cardiovascular or combined events than in uneventful diabetic controls. ADAMTS13 activity significantly and negatively correlated with all outcomes.In diabetics, ADAMTS13 618Ala variant associated with less proteolytic activity, higher risk of chronic complications and better response to ACEi therapy. Screening for Pro618Ala polymorphism may help identifying diabetics at highest risk who may benefit the most from early reno and cardioprotective therapy.Diabetes 06/2013; · 7.90 Impact Factor  SourceAvailable from: Chris Hudson[Show abstract] [Hide abstract]
ABSTRACT: The evergrowing volume of data routinely collected and stored in everyday life presents researchers with a number of opportunities to gain insight and make predictions. This study aimed to demonstrate the usefulness in a specific clinical context of a simulationbased technique called probabilistic sensitivity analysis (PSA) in interpreting the results of a discrete time survival model based on a large dataset of routinely collected dairy herd management data. Data from 12,515 dairy cows (from 39 herds) were used to construct a multilevel discrete time survival model in which the outcome was the probability of a cow becoming pregnant during a given two day period of risk, and presence or absence of a recorded lameness event during various time frames relative to the risk period amongst the potential explanatory variables. A separate simulation model was then constructed to evaluate the wider clinical implications of the model results (i.e. the potential for a herd's incidence rate of lameness to influence its overall reproductive performance) using PSA. Although the discrete time survival analysis revealed some relatively large associations between lameness events and risk of pregnancy (for example, occurrence of a lameness case within 14 days of a risk period was associated with a 25% reduction in the risk of the cow becoming pregnant during that risk period), PSA revealed that, when viewed in the context of a realistic clinical situation, a herd's lameness incidence rate is highly unlikely to influence its overall reproductive performance to a meaningful extent in the vast majority of situations. Construction of a simulation model within a PSA framework proved to be a very useful additional step to aid contextualisation of the results from a discrete time survival model, especially where the research is designed to guide onfarm management decisions at population (i.e. herd) rather than individual level.PLoS ONE 08/2014; 9(8):e103426. · 3.53 Impact Factor  SourceAvailable from: Ross Andel[Show abstract] [Hide abstract]
ABSTRACT: Although alcoholstroke association is well known, the agevarying effect of alcohol drinking at midlife on subsequent stroke risk across older adulthood has not been examined. The effect of genetic/earlylife factors is also unknown. We used cohort and twin analyses of data with 43 years of followup for stroke incidence to help address these gaps. All 11 644 members of the populationbased Swedish Twin Registry born 1886 to 1925 with alcohol data aged ≤60 years were included. The interaction of midlife alcohol consumption by age at stroke was evaluated in Coxregression and analyses of monozygotic twins were used. Covariates were baseline age, sex, cardiovascular diseases, diabetes mellitus, stress reactivity, depression, body mass index, smoking, and exercise. Altogether 29% participants developed stroke. Compared with verylight drinkers (<0.5 drink/d), heavy drinkers (>2 drinks/d) had greater risk of stroke (hazard ratio, 1.34; P=0.02) and the effect for nondrinkers approached significance (hazard ratio, 1.11; P=0.08). Age increased stroke risk for nondrinkers (P=0.012) and decreased it for heavy drinkers (P=0.040). Midlife heavy drinkers were at high risk from baseline until the age of 75 years when hypertension and diabetes mellitus grew to being the more relevant risk factors. In analyses of monozygotic twinpairs, heavy drinking shortened time to stroke by 5 years (P=0.04). Strokerisk associated with heavy drinking (>2 drinks/d) in midlife seems to predominate over wellknown risk factors, hypertension and diabetes, until the age of ≈75 years and may shorten time to stroke by 5 years above and beyond covariates and genetic/earlylife factors. Alcohol consumption should be considered an agevarying risk factor for stroke. © 2015 American Heart Association, Inc.01/2015;
Page 1
RESEARCH ARTICLEOpen Access
Variables with timevarying effects and the Cox
model: Some statistical concepts illustrated with
a prognostic factor study in breast cancer
Carine A Bellera1,5*, Gaëtan MacGrogan2, Marc Debled3, Christine Tunon de Lara4, Véronique Brouste1,
Simone MathoulinPélissier1,5
Abstract
Background: The Cox model relies on the proportional hazards (PH) assumption, implying that the factors
investigated have a constant impact on the hazard  or risk  over time. We emphasize the importance of this
assumption and the misleading conclusions that can be inferred if it is violated; this is particularly essential in the
presence of long followups.
Methods: We illustrate our discussion by analyzing prognostic factors of metastases in 979 women treated for
breast cancer with surgery. Age, tumour size and grade, lymph node involvement, peritumoral vascular invasion
(PVI), status of hormone receptors (HRec), Her2, and Mib1 were considered.
Results: Median followup was 14 years; 264 women developed metastases. The conventional Cox model
suggested that all factors but HRec, Her2, and Mib1 status were strong prognostic factors of metastases. Additional
tests indicated that the PH assumption was not satisfied for some variables of the model. Tumour grade had a
significant timevarying effect, but although its effect diminished over time, it remained strong. Interestingly, while
the conventional Cox model did not show any significant effect of the HRec status, tests provided strong evidence
that this variable had a nonconstant effect over time. Negative HRec status increased the risk of metastases early
but became protective thereafter. This reversal of effect may explain nonsignificant hazard ratios provided by
previous conventional Cox analyses in studies with long followups.
Conclusions: Investigating timevarying effects should be an integral part of Cox survival analyses. Detecting and
accounting for timevarying effects provide insights on some specific time patterns, and on valuable biological
information that could be missed otherwise.
Background
Survival analysis, or timetoevent data analysis, is
widely used in oncology since we are often interested in
studying a delay, such as the time from cancer diagnosis
or treatment initiation to cancer recurrence or death.
Thanks to the improvement of cancer treatments, and
the induced longer life expectancy, we observe an
increasing number of studies with long followup peri
ods. Statistical models to analyze such data should thus
adequately account for the increasing duration of fol
lowups. The Cox proportional hazards (PH) model
allows one to describe the survival time as a function of
multiple prognostic factors [1]. This model relies on a
fundamental assumption, the proportionality of the
hazards, implying that the factors investigated have a
constant impact on the hazard  or risk  over time. If
timedependent variables are included without appropri
ate modeling, the PH assumption is violated. As a result,
misleading effect estimates can be derived, and signifi
cant effect in the early (or late) followup period may be
missed. Checking the proportionality of the hazards
should thus be an integral part of a survival analysis by
a Cox model. The assumption, however, is not systema
tically verified. In a 1995 review of cancer publications
using a Cox model, Altman et al. reported that most
* Correspondence: bellera@bergonie.org
1Department of Clinical Epidemiology and Clinical Research, Institut
Bergonié, Regional Comprehensive Cancer Centre, Bordeaux, France
Bellera et al. BMC Medical Research Methodology 2010, 10:20
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© 2010 Bellera et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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studies did not report verifying this assumption [2];
similar findings were reported recently by one of the co
authors of the present work [3].
Although the Cox model has been widely used (more
than 25 000 citations since the publication of the origi
nal paper by Cox [4]), recent publications suggest a
growing interest in the quality of its applications. Special
papers in statistics have been published in the oncology
literature providing general introductions to survival
analysis [58]; topics covered included summarizing sur
vival data, testing for a difference between groups, pre
senting existing statistical models, or assessing the
adequacy of a survival model. Others works focused on
providing definition of specific survival endpoints [9], or
on the quality of reporting of survival events [3].
Assessing whether the assumption of proportional
hazards is a central theme in survival analysis, and as
such is discussed in several statistical textbooks [1014]
as well as in the general statistical literature [1518]. To
our knowledge however, this topic has been discussed in
few medical journals. Importantly, this strong assump
tion does not seem to be systematically assessed. For
illustration, a recent review of clinical trials with primary
analyses based on survival end points showed that only
one of the 64 papers that used a Cox model mentioned
verifying the PH assumption [3].
Our objective is to inform clinicians, as well as those
who read and write manuscripts in medical journals,
about the importance of the underlying PH assumption,
the misleading conclusions that can be inferred if it is
violated, as well as the additional information provided
by verifying it. After a theoretical introduction, we
describe techniques to assess if this assumption is vio
lated, and model strategies to account for, and describe
timedependency. We illustrate our discussion with a
study on prognostic factors in breast cancer.
Methods and results
Survival analysis
In many studies, the primary variable of interest is a
delay, such as the time from cancer diagnosis to a parti
cular event of interest. This event may be death, and for
this reason the analysis of such data is often referred to
as survival analysis. The event of interest may not have
occurred at the time of the statistical analysis, and simi
larly, a subject may be lost to followup before the event
is observed. In such case, data are said to be censored at
the time of the analysis or at the time the patient was
lost to followup. Censored data still bring some infor
mation since although we do not know the exact date of
the event, we know that it occurred later than the cen
soring time.
Both the KaplanMeier method and the Cox propor
tional hazards (PH) model allow one to analyze
censored data [1,19], and to estimate the survival prob
ability, S(t), that is the probability that a subject survives
beyond some time t. Statistically, this probability is pro
vided by the survival function S(t) = P (T > t), where T
is the survival time. The Kaplan Meier method estimates
the survival probability nonparametrically, that is,
assuming no specific underlying function [19]. Several
tests are available to compare the survival distributions
across groups, including the logrank and the Mann
WhitneyWilcoxon tests [20,21]. The Cox PH model
accounts for multiple risk factors simultaneously. It does
not posit any distribution, or shape for the survival
function, however, the instantaneous incidence rate of
the event is modeled as a function of time and risk
factors.
The instantaneous hazard rate at time t, also called
instantaneous incidence, death, or failure rate, or risk, is
the instantaneous probability of experiencing an event
at time t, given that the event has not occurred yet. It is
a rate of event per unit of time, and is allowed to vary
over time. Just as the risk of events per unit time, one
can make an analogy by considering the speed given by
a car speedometer, which represents the distance tra
velled per unit of time. Suppose, that the event of inter
est is death, and we are interested in its association with
n covariates, X1, X2, ..., Xn, then the hazard is given by:
hthtxxx
xnn
( ) ( )exp()
0 1 1
22
(1)
The baseline hazard rate h0(t) is an unspecified non
negative function of time. It is the timedependent part
of the hazard and corresponds to the hazard rate when
all covariate values are equal to zero. b1, b2, ..., bnare
the coefficients of the regression function b1x1+ b2x2+...
bnxn. Suppose that we are interested in a single covariate
then the hazard is:
hthtx
x( )( )exp()
0
(2)
The hazards for two subjects with covariate values x1
and x2are thus given respectively by hx1(t) = h0(t) exp
(bx1) and hx2(t) = h0(t) exp(bx2), and the hazard ratio is
expressed as:
HRhthtxx
xx
2121
( ) / ( )exp[ ( )]
(3)
Taking x2= x1+ 1, the hazard ratio reduces to HR =
exp(b) and corresponds to the effect of one unit increase
in the explanatory variable X on the risk of event. Since
b = log(HR), b is referred as the log hazard ratio.
Although the hazard rate hx(t) is allowed to vary over
time, the hazard ratio HR is constant; this is the
assumption of proportional hazards. If the HR is greater
than 1 (b > 0), the event risk is increased for subjects
with covariate value x2 compared to subjects with
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covariate value x1, while a HR lower than 1 (b < 0) indi
cates a decreased risk. When the HR is not constant
over time, the variable is said to have a timevarying
effect; for example, the effect of a treatment can be
strong immediately after treatment but fades with time.
This should not be confused with a timevarying covari
ate, which is a variable whose value is not fixed over
time, such as smoking status. Indeed, a person can be a
nonsmoker, then a smoker, then a nonsmoker. Note
however, that a variable may be both timevarying and
have an effect that changes over time.
In a Cox PH model, the HR is estimated by consider
ing each time t at which an event occurs. When esti
mating the overall HR over the complete followup
period, the same weights are given to the very early HR
which affect almost all individuals and to very late HR
affecting only the very few individuals still at risk. The
HR is thus averaged over the event times. In the case of
proportional hazards, the overall HR is not affected by
this weighting procedure. If, on the other hand, the HR
changes over time, that is, the hazard rates are not pro
portional, then equal weighting may result in a non
representative HR, and may produce biased results [22].
It should be noted that the HR is averaged over the
event times rather than over the followup time. It is
unchanged if the time scale is changed without disturb
ing the ordering of events.
Example
We applied some of the presented methods to breast
cancer patients as timevarying effects have been
reported, such as for nodal or hormone receptor status,
[2326]. We studied women with nonmetastatic, oper
able breast cancer who underwent surgery between 1989
and 1993 at our institution, and who did not receive
previous neoadjuvant treatment. Exclusion criteria
included a previous history of breast carcinoma, concur
rent contralateral breast cancer, and pathologic data
missing. Followup was performed according to the Eur
opean Good Clinical Practice requirements and con
sisted of regular physical examinations, and annual X
ray mammogram, and additional assessments in case of
suspected metastases. Clinical and pathological charac
teristics were analyzed according to the hospital
recorded file at the time of treatment initiation. Patholo
gical tumour size (≤ or > 20 mm) was measured on
fresh surgical specimens. A modified version of the
ScarffBloomRichardson grading system was used (SBR
grade I, II, or III). PVI (Yes, No) was defined as the pre
sence of neoplastic emboli within unequivocal vascular
lymphatic or capillary lumina in areas adjacent to the
breast tumour. Exploratory immunohistochemical ana
lyses were performed on a tissue microarray (TMA) to
assess hormone receptor (HRec) status (positive if ER
positive and/or progesterone receptor [PgR]positive).
ER and PgR expression levels were evaluated semiquan
titatively according to a standard protocol with cutoff
values at 10% positive tumor cells. Her2 expression level
was evaluated according to the Herceptest scoring sys
tem [27]. Mib1 expression level was evaluated semi
quantitatively. Information on all factors was available
for 979 women (Table 1). The median followup time
was 14 years (95% confidence interval: 13.7  14.2) and
264 women developed metastases.
Working example
The prognostic factors were initially selected based on
current knowledge regarding risk of metastases. They
were next analyzed using a conventional Cox regression
model; all were statistically significant at the 5% level in
Table 1 Characteristics of the study population.
N(%)
Year of diagnosis
1989
1990
1991
1992
1993
231
207
182
189
170
23.6
21.1
18.6
19.3
17.4
Metastases following surgery
Yes
No
264
715
27.0
73.0
Age at diagnosis
≤ 40 years
> 40 years
76
903
7.8
92.2
SBR Grade
Grade I
Grade II
Grade III
275
444
260
28.1
45.3
26.6
Tumor size
≤ 20 mm
> 20 mm
753
226
76.9
23.1
Lymph node involvement
No
Yes
554
425
56.6
43.4
Peritumoral vascular invasion
No
Yes
700
279
71.5
28.5
Hormone Receptor status
Both ER and PR
At least ER+ or PR+
178
801
18.2
81.8
Her2 status
Positive
Negative
100
879
10.2
89.8
Mib1 status
Negative
Positive
691
288
70.6
29.4
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the univariate analyses, and were then entered onto a
multivariate Cox model. The risk of metastases was
increased for women with younger age compared to
older age; grade II and III tumours compared to grade I
tumours; large compared to small tumour sizes; lymph
node involvement compared to no involvement; and
PVI compared to no PVI (Additional file 1: Estimated
log hazard ratios (log(HR)), and hazard ratios (HR = exp
( ˆ )) with 95% confidence intervals (95% CI) and p
values for model covariates when fitting a multivariate
conventional Cox model and a Cox model with timeby
covariate interactions.). Based on this model, all vari
ables, but hormone receptor, Her2 and Mib1 status, sig
nificantly affected the risk of metastases.
Assessing nonproportionality: Graphical strategy
In the presence of a categorical variable, one can plot
the KaplanMeier survival distribution, S(t), as a func
tion of the survival time, for each level of the covariate.
If the PH assumption is satisfied, the curves should stea
dily drift apart. One can also apply a transformation of
the KaplanMeier survival curves and plot the function
log(log(S(t))) as a function of the log survival time,
where log represents the natural logarithm function. If
the hazards are proportional, the stratum specific log
minuslog plots should exhibit constant differences, that
is be approximately parallel. These visual methods are
simple to implement but have limitations. When the
covariate has more than two levels, KaplanMeier plots
are not useful for discerning nonproportionality
because the graphs become to cluttered [10]. Similarly,
although the PH assumption may not be violated, the
logminuslog curves are rarely perfectly parallel in prac
tice, and tend to become sparse at longer time points,
and thus less precise. It is not possible to quantify how
close to parallel is close enough, and thus how propor
tional the hazards are. The decision to accept the PH
hypothesis often depends on whether these curves cross
each other. As a result, the decision to accept the PH
hypothesis can be subjective and conservative [28], since
one must have strong evidence (crossing lines) to con
clude that the PH assumption is violated. In view of
these limitations, some suggest providing standard
errors to these plots [29]. This approach however can
be computationally intensive and is not directly available
in standard computer programs. KaplanMeier and log
minuslog plots are available from most standard statis
tical packages (Table 2).
Working example (cont’)
KaplanMeier survival curves and logminuslog plots
are shown for some variables (Figures 1 and 2). The
KaplanMeier survival curves appeared to steadily drift
apart for all but the hormone receptor status, Her2 sta
tus, and mib1 status. The logminus log plots looked
approximately parallel for Age, size of the tumour,
lymph node involvement, and PVI. Again, plots for the
hormone receptor status, Her2 status, and mib1 status
tended to indicate a violation of the PH assumption.
There was also some suspicion with respect to the SBR
grade.
Assessing nonproportionality: Modelling and testing
strategies
Graphical methods for checking the PH assumption do
not provide a formal diagnostic test, and confirmatory
approaches are required. Multiple options for testing
and accounting for nonproportionality are available.
Cox proposed assessing departure from nonpropor
tionality by introducing a constructed timedependent
variable, that is, adding an interaction term that involves
time to the Cox model, and test for its significance [1].
Suppose one is interested in evaluating if some variable
X has a timevarying effect. A timedependent variable
is created by forming an interaction (product) term
between the predictor, X (continuous or categorical),
and a function of time t (f(t) = t, t2, log(t), ...). Adding
this interaction to the model (equation 2), the hazard
then becomes:
hthtx x f t . . ( )]
x( )( )exp[
0
(5)
The hazard ratio is given by HR(t) = hx+1(t)/hx(t) =
exp[b + g.x.f(t)] for a unit increase in the variable X,
and is timedependent through the function f(t). If g > 0
(g < 0), then the HR increases (decreases) over time.
Testing for nonproportionality of the hazards is
Table 2 Statistical software
R/Splus©
SAS©
SPSS©
Stata©
Graphical checks survfit functionlifetest procedure Survt commandsts command
Timeby
covariate
interactions
programming required.phreg procedure (definition of
interactions)/test statement.
time program command (definition of
interactions)/cox reg command.
tvc option/stcox
command
Scaled Schonfeld
residuals
cox.zph function phreg procedure/ressch optionNot directly available/programming
required
stphtest command
Cumulative
residuals
Timereg/gof libraries/
cum.residuals function
phreg procedure/assess
statement/ph option
Not directly available/programming
required
Not directly available/
programming required
Bellera et al. BMC Medical Research Methodology 2010, 10:20
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Page 5
equivalent to testing if g is significantly different from
zero. One can use different time functions such as poly
nomial or exponential decay but often very simple fixed
functions of time such as linear or logarithmic functions
are preferred [28]. This modeling approach also provides
estimates of the hazard ratio at different time points
since values t of time can be fitted into the hazard ratio
function. Timedependent variables provide a flexible
method to evaluate departure from nonproportionality
and an approach to building a model for the depen
dence of relative risk over time. This approach however
should be used with caution. Indeed, if the function of
time selected is misspecified, the final model will not
be appropriate. This is a disadvantage of this method
over more flexible approach.
Working example (cont’)
We created timebycovariate interactions for each vari
able of the model, by introducing products between the
variables and a linear function of time. As shown in
Additional File 1 (Estimated log hazard ratios (log(HR)),
and hazard ratios (HR = exp( ˆ )) with 95% confidence
intervals (95% CI) and pvalues for model covariates
when fitting a multivariate conventional Cox model and
a Cox model with timebycovariate interactions.), sig
nificant timebycovariate interactions involved the SBR
grade, hormone receptor status, Her2 status, and PVI (p
< 0.05). Thus these results indicated that the hazard
ratios associated with these factors were not constant
over time. The parameters ( ˆ ) associated with most
interactions were negative, suggesting that the hazard
ratios were decreasing over time. The estimated hazard
ratio associated with an SBR grade II (versus grade I) as
a function of time t was given by: HR(t) = exp(1.71 
0.14t). Hazard ratios were 4.8, 3.6, and 2.7 at respec
tively 1, 3, and 5 years. Similarly, the estimated hazard
ratio associated with the hormone receptor status was:
HR(t) = exp(0.73  0.14t), that is hazard ratios of 1.8,
1.3, and 1.0 at respectively 1, 3, and 5 years. While the
conventional Cox model did not show any significant
effect for hormone receptors, Her2 and Mib1, these
variables had a significant effect once timebycovariate
interactions were included.
Departure from nonproportionality can also be inves
tigated using the residuals of the model. A residual
Figure 1 KaplanMeier survival curves for SBR grade, tumour size, PVI, hormone receptor status.
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measures the difference between the observed data, and
the expected data under the assumption of the model.
Schoenfeld residuals are calculated and reported at
every failure time under the PH assumption, and as
such are not defined for censored subjects [15,30]. They
are defined as the covariate value for the individual that
failed minus its expected value assuming the hypotheses
of the model hold. There is a separate residual for each
individual for each covariate. A smooth plot of the
Schoenfeld residuals can then be used to directly visua
lize the log hazard ratio [15]. Assuming proportionality
of the hazards, the Schoenfeld residuals are independent
of time. Thus, a plot suggesting a nonrandom pattern
against time is evidence of nonproportionality. Graphi
cally, this method is more reliable and easier to interpret
than plotting the log(log(S(t)) function presented ear
lier. The presence of a linear relationship with time can
be tested by performing a simple linear regression and a
test trend. A slope significantly different from zero
would be evidence against proportionality: an increasing
(decreasing) trend would indicate an increasing
(decreasing) hazard ratio over time. It is recommended
to carefully look at the residual plot in addition to
performing this test as some patterns may be apparent
on the plots (quadratic, logarithmic), but remain unde
tected by the statistical test. Moreover, undue influence
of outliers might become obvious [10]. Although, the
method based on the smoothed Schoenfeld residuals
provides timedependent estimates, it can have some
drawbacks [14,18]. The uncertainty estimates associated
with the resulting timedependent estimates can be diffi
cult to use in practice, and the estimator provided may
not have good statistical properties, such as consistency.
Importantly, pvalues resulting from trend tests based
on the Schoenfeld residuals are obtained independently
for each covariate of the model, assuming the Cox
model is justified for the other covariates of the model;
as such, results should be interpreted carefully. Tests
based on the Schoenfeld residuals can be easily imple
mented in most standard statistical packages (Table 2).
Working example (cont’)
For each covariate, scaled Schoenfeld residuals were
plotted over time, and tests for a zero slope were per
formed. The corresponding pvalues, as well as the p
value associated with a global test of nonproportionality
are reported in Table 3. The global test suggested strong
Figure 2 Log(log(survival)) curves as a function of time (log scale) for SBR grade, tumour size, PVI, hormone receptor status.
Bellera et al. BMC Medical Research Methodology 2010, 10:20
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evidence of nonproportionality (p < 0.01). Variables
that deemed most likely to contribute to nonpropor
tionality were the SBR grade (p < 0.01), PVI (p = 0.05)
and hormone receptor status (p = 0.05). These numeri
cal findings suggest a non constant hazard ratio for
these variables. Residuals help visualizing the log hazard
ratio
ˆ over time for each covariate (figure 3). We
added dashed and dotted lines representing respectively
the null effect (null log hazard ratio) and the averaged
log hazard ratio estimated by the conventional Cox
model. With respect to the SBR grade, the plots sug
gested strong effect over the first five years. This effect
tended to diminish afterwards. Similarly, the impact of
PVI changed over time, with again higher risks of
metastases in the early years, and then this effect tended
to vanish. Concerning hormone receptor status, plots
suggested that a negative status increased the risk of
metastases early on, and became protective afterwards.
The cumulative sum of Schoenfeld residuals, or
equivalently the observed score process can also be used
to assess proportional hazards [31]. Graphically, the
observed score process is plotted versus time for each
variable of the model, together with simulated processes
assuming the underlying Cox model is true, that is,
assuming proportional hazards. Any departure of the
observed score process from the simulated ones is evi
dence against proportionality. These plots can then be
used to assess when the lack of fit is present. In particu
lar, an observed score well above the simulated process
is an indication of an effect higher than the average one,
and conversely. This method is particularly well illu
strated in a recent publication by Cortese et al. [18].
Goodnessoffit tests can be implemented based on the
cumulative residuals. The cumulative residuals based
approach overcomes some drawbacks encountered with
the Schoenfeld residuals, since resulting estimators tend
to have better statistical properties, and justified p
values are derived [14]. The cumulative residuals
approach is implemented in some standard statistical
packages (Table 2).
Working example (cont’)
Tests based on cumulative residuals are presented in
Table 4. At the 5% significance level, test statistics sug
gest nonconstant effect over time for the grade of the
tumor, as well as the status of the hormone receptors,
her2, and Mib1. For illustration, we also plotted the
resulting score process for some variables (Figure 4). In
accordance with the test statistics based on the cumula
tive residuals, we observe strong departure of the
observed processes from the simulated curves under the
model for the grade and hormone receptor status. These
plots are particularly useful in identifying where the lack
of fit is present. For example, the initial positive score
process associated with hormone receptors, suggests
that the effect of this variable is initially higher than the
average effect, and thus lower than the average effect
afterwards. That is, the risk of metastases is increased
initially for women with both negative hormone recep
tors compared to the average risk, and decreased
afterwards.
Another simple approach for testing timevarying
effects of covariates involves fitting different Cox models
for different time periods. Indeed, although the PH
assumption may not hold over the complete followup
period, it may hold over a shorter time window. Unless
there is an interest in a particular cutoff time value,
two subsets of data can be created based on the median
event time [10]. That is, a first analysis is conducted by
censoring everyone still at risk beyond this time point,
and a second one by considering only those subjects
still at risk thereafter. In such case, the interpretation of
the models is conditional on the length of the survival
time, and results should thus be interpreted with cau
tion. Even if the period of analysis is shortened, one
should still ensure that the PH assumption is not vio
lated within these reduced time periods. Moreover,
since fewer event times are considered, analyses can suf
fer from a decreased power. Finally, although this
method is particularly simple to implement and might
provide sufficient information in some settings, that is if
one is interested in a short time window, it should be
noted that this method is not directly testing the PH
assumption, and a different parametrization would be
needed to perform such a test.
Working example (cont’)
The median event time was 4.3 years. A Cox model was
applied censoring everyone still at risk after 4.3 years,
while only those subjects still at risk beyond this time
point were included in another model (Additional file 2:
Estimated hazard ratios (exp( ˆ )) with 95% confidence
intervals (95% CI) and pvalues for model covariates in
two independent Cox models for two different time
Table 3 Test for nonproportionality based on the scaled
Schoenfeld residuals from the conventional Cox model
(see table 1).
Variablepvalue
Age
Grade II
Grade III
Size
Lymph node involvement
PVI
Hormone receptor
Her2
Mib1
0.10
<0.01
<0.01
0.32
0.22
0.05
0.05
0.08
0.07
GLOBAL
<0.01
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periods.). All variables but age were statistically signifi
cant in the first model as negative hormone receptor
status, positive Her2 status and Mib1 positive status
were associated with an increased risk of metastases. In
women still at risk past 4.3 years, younger age, greater
tumor size, and lymph node involvement were asso
ciated with an increased risk of metastases. The effects of
other variables have disappeared. Interestingly, hormone
receptor negative status had a significant protective effect
in this second model (HR = 0.5), while the first analysis
suggested a significant increased risk for (HR = 1.7).
Tests for nonproportionality based on the cumulative
residuals suggested a persistent timevarying effect of the
grade for the analysis restricted to the first 4.3 years.
It is also possible to account for nonproportionality by
partitioning the time axis as proposed by Moreau et al.
[32]. The time axis is partitioned and hazard ratios are
then estimated within each interval. Thus, testing for
nonproportionality is equivalent to testing if the time
specific HR are significantly different. Results can how
ever sometimes be driven by the number of time intervals
[33], and time intervals should thus be carefully selected.
Figure 3 Scaled Schoenfeld residuals for SBR grade, PVI, and hormone receptor status (with 95% confidence interval).
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Abandoning the assumption of proportional hazards,
and as such, the Cox model, is another option. Indeed,
other powerful statistical models are available to account
for timevarying effects, including additive models,
accelerated failure time models, regression splines mod
els or fractional polynomials [3336].
Finally, one can perform a statistical analysis stratified
by the variable suspected to have a timevarying effect;
this variable should be thus categorical or be categor
ized. Each stratum k has a distinct baseline hazard but
common values for the coefficient vector b, that is, the
hazard for an individual in stratum k is hk(t) = exp(bx)
Stratifying assumes that the other covariates are acting
in the same way in each stratum, that is, HRs are similar
across strata. Although stratification is effective in
removing the problem of nonproportionality and sim
ple to implement, it has some disadvantages. Most
importantly, stratification by a nonproportional variable
precludes estimation of its strength and its test within
the Cox model. Thus, this approach should be selected
if one is not directly interested in quantifying the effect
of the variable used for stratification. Moreover, a strati
fied Cox model can lead to a loss of power, because
more of the data are used to estimate separate hazard
functions; this impact will depend on the number of
subjects and strata [10]. If there are several variables
with timevarying risks, this would require the model to
Table 4 Test for nonproportionality based on the
Cumulative residuals from the conventional Cox model
(see table 1).
Variablepvalue
Age
Grade II
Grade III
Size
Lymph node involvement
PVI
Hormone receptor
Her2
Mib1
0.97
0.02
<0.01
0.16
0.75
0.11
<0.01
<0.01
<0.01
Figure 4 Observed score process for SBR grade, lymph node involvement, and hormone receptor status (with 95% confidence interval).
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be stratified on these multiple factors, which again is
likely to decrease the overall power.
Discussion
While ensuring that the PH assumption holds is part of
the modeling process, it is also useful in providing valu
able information on timevarying effects. In our illustra
tive example, the conventional Cox model suggested
that all factors but HRec, Her2, and Mib1 status were
strong prognostic factors of metastases. Additional tests
indicated that the PH assumption was not satisfied for
some variables of the model. Tumour grade had a sig
nificant timevarying effect, but although its effect
diminished over time, it remained strong. According to
the conventional model hormone receptor status did
not significantly impact relapses. Additional tests pro
vided strong evidence of a timevarying effect. Impor
tantly, both tests based on residuals suggested that
negative hormone receptor status increased the risk of
metastases early but became protective thereafter, in
accordance with the analysis partitioned on event time.
This reversal of effect may explain the nonsignificant
averaged hazard ratio provided by the conventional Cox
model and reported earlier [26].
Applying a Cox model without ensuring that its
underlying assumptions are validated can lead to nega
tive consequences on the resulting estimates [28,37]. For
variables not satisfying the nonproportionality assump
tion, the power of the corresponding tests is reduced,
that is, we are less likely to conclude for a significant
effect when there is actually one. If the hazard ratio is
increasing over time, the estimated coefficient assuming
PH is overestimating at first and underestimating later
on. For those variables of the model with a constant
hazard ratio, the power of tests is also reduced as a con
sequence of an inferior fit of the model.
Once nonproportionality is established, timedepen
dency can be accounted for in different ways. The strat
egy will depend on the study objectives. If there is no
interest in longer time periods, one can shorten the fol
lowup time as nonproportionality is less likely to be
an issue on short time intervals. If there is no particular
interest in the variable with the timevarying effect, one
could stratify on this variable in the statistical analysis,
however no association between the stratification vari
able and survival can be tested. If one wants to describe
the effect of the variable over time, it is possible to rely
on time by covariate interactions or on plots of residuals
to estimate of relative risks at different time points.
Methods to test and account for nonproportionality are
available in most standard statistical software (Table 2).
It is difficult to propose definite guidelines for the best
strategy for testing for nonproportionality. Each
method has its advantages and limitations, and
depending on the study objective some approaches
might be preferred. Before performing statistical model
ing, the study objectives should be clearly stated in
advance, as well as the statistical tests that will be
employed. Departure from nonproportionality can be
investigated using graphical and numerical approaches.
Plotting methods involve visualizing the KaplanMeier
survival curves for the variable tested for nonpropor
tionality. This graphical method requires categorical
variables, and is particularly appropriate for binary data;
however they do not provide formal diagnostic tests.
Numerical tests involve for example testing for covari
atebytime interactions or for the presence of a trend
in the residuals of the model. Including a covariateby
time interaction is particularly simple within the Cox
model; however, results are strongly dependent on the
choice of the functional form of the time function. Tests
based on cumulative residuals tend to have better statis
tical properties than those based on the Schoenfeld resi
duals. As a result, performing a test based on the
cumulative residuals seems to be a more powerful
approach in detecting covariates with timevarying
effects.
Note that the Cox model involves multiple types of
residuals including the martingale, deviance, score and
Schoenfeld residuals, which can be particularly useful as
additional regression diagnostics for the Cox model.
Martingale residuals are useful for determining the func
tional form of a covariate to be included in the model
and deviance residuals can be used to examine model
accuracy. Additional details can be found in [10,11].
Statistical testing raises the issue of power, that is, the
ability of tests to find true effects. We have seen for
example that some simple strategies, such as shortening
the observation period can suffer from reduced power
as fewer events are considered. This might be a limita
tion with small datasets. Simulations have shown that
stratified Cox modeling usually leads to wider confi
dence intervals, that is, reduced power compared to
unstratified analysis [38]. Statistical tests for timevary
ing effects have different power to detect nonpropor
tionality. It has been shown that tests requiring
partitioning of the failure time have less power than
other tests, while tests based on timedependent covari
ates or on the Schoenfeld residuals have equally good
power to detect nonproportionality in a variety of non
proportional hazards and are practically equivalent [17].
The issue of power naturally leads to the question of
sample size. Clinical trials are usually designed with just
enough power to detect the treatment effect. In this
context, one should not expect to have enough details
about the actual shape of the HR over time. Assuming a
trial designed with an 80% power to detect a treatment
effect, Therneau and Grambsch showed that the test
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based on the residuals was able to detect nonpropor
tionality, but could not distinguish between a linear and
a discrete increase of the hazard ratio over time [10].
Observational studies are usually designed for explora
tory analyses and do not rely on a formal estimation of
the sample size. There might not always be enough
power to detect a specific time trend. The question of
lack of power should not be interpreted as an argument
against testing for nonproportionality. Just as any other
statistical model, one should ensure that major assump
tions are not violated.
Since its original publication in 1972, the Cox propor
tionalhazards model has gained widespread use and has
become a popular tool for the analysis of survival data
in medicine. After performing an online search, we
found that the original paper by Cox had been cited
approximately 25, 000 times, with about 8, 000 citations
in oncology papers [4]. While time dependency has
been accounted for and reported in oncology publica
tions, such as in breast or colon cancer studies
[26,33,3942,42], the verification of the PH assumption
is unfortunately far from being systematic. In a 1995
review of five clinical oncology journals including about
130 papers, Altman et al. reported that only 2 out of the
43 papers which relied on a Cox model, mentioned that
the PH assumption was verified [2]. Similarly, about ten
years later Mathoulin et al. assessed the quality of
reporting of survival events in randomized clinical trials
in eight general or cancer medical journals [3]. The
authors reported that only one of the 64 papers that
used a Cox model mentioned verifying the PH
assumption.
Our objective was to familiarize the reader with the
PH assumption. We also highlighted that detecting and
accounting for timevarying effects provide insights on
some specific time patterns and valuable biological
information that could be missed otherwise. Given the
possible consequences on parameter estimates, checking
the proportionality of hazards should be an integral part
of a survival analysis based on a Cox model. In the pre
sence of variables with timevarying risks, plots should
be used to augment the results and indicate where non
proportionality is present. This seems particularly
appropriate in the context of oncology studies, as long
followups are common and nonconstant hazards have
already been reported.
Conclusions
Investigating timevarying effects should be an integral
part of Cox survival analyses. Detecting and accounting
for timevarying effects provide insights on some speci
fic time patterns, and on valuable biological information
that could be missed otherwise.
Additional File 1: Estimated log hazard ratios (log(HR)), and hazard
ratios (HR = exp( ˆ )) with 95% confidence intervals (95% CI) and p
values for model covariates when fitting a multivariate conventional Cox
model and a Cox model with timebycovariate interactions.
Click here for file
[http://www.biomedcentral.com/content/supplementary/1471228810
20S1.DOC]
Additional File 2: Estimated hazard ratios (exp( ˆ )) with 95%
confidence intervals (95% CI) and pvalues for model covariates in two
independent Cox models for two different time periods.
Click here for file
[http://www.biomedcentral.com/content/supplementary/1471228810
20S2.DOC]
Acknowledgements
The tissue microarray was financed by the Comités départementaux de la
Gironde, Dordogne, Charente, Charente Maritime, Landes, by la Ligue
Nationale contre le Cancer, and by Lyons Club de Bergerac, France.
Author details
1Department of Clinical Epidemiology and Clinical Research, Institut
Bergonié, Regional Comprehensive Cancer Centre, Bordeaux, France.
2Department of Pathology, Institut Bergonié, Regional Comprehensive
Cancer Centre, Bordeaux, France.3Department of Medical Oncology, Institut
Bergonié, Regional Comprehensive Cancer Centre, Bordeaux, France.
4Department of Surgery, Institut Bergonié, Regional Comprehensive Cancer
Centre, Bordeaux, France.5Unité INSERM 897, Université Victor Segalen
Bordeaux 2, Bordeaux, France.
Authors’ contributions
CB conceived the study, performed the statistical analysis and drafted the
manuscript. GMG carried out the immunoassays. MD provided clinical
expertise in oncology. CTL provided clinical expertise in surgery. VB was
responsible of the datamanagement. SMP participated in the design of
study. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Received: 2 November 2009 Accepted: 16 March 2010
Published: 16 March 2010
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doi:10.1186/147122881020
Cite this article as: Bellera et al.: Variables with timevarying effects and
the Cox model: Some statistical concepts illustrated with a prognostic
factor study in breast cancer. BMC Medical Research Methodology 2010
10:20.
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