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British Journal of Cancer (2003) 89, 431 – 436& 2003 Cancer Research UK
All rights reserved 0007 – 0920/03 $25.00
Survival Analysis Part II: Multivariate data analysis – an introductionto concepts and methods
MJBradburn*,1,TGClark1,SBLove1 andDGAltman1
Cancer Research UK/NHS Centre for Statistics in Medicine, Institute of Health Sciences, Old Road, Oxford OX3 7LF, UK
British Journal of Cancer (2003) 89, 431 – 436. doi:10.1038/sj.bjc.6601119
& 2003 Cancer Research UK
Keywords: survival analysis; Cox model; AFT model; model selection
groups, it offers no estimate of the actual effect size; in otherwords, it offers a statistical, but not a clinical, assessment of the
Survival analysis involves the consideration of the time between a
factor's impact. The use of a statistical model improves on these
fixed starting point (e.g. diagnosis of cancer) and a terminating
methods by allowing survival to be assessed with respect to several
event (e.g. death). The key feature that distinguishes such data
factors simultaneously, and in addition, offers estimates of the
from other types is that the event will not necessarily have
strength of effect for each constituent factor. Therefore, statistical
occurred in all individuals by the time the study ends, and for
models are important and frequently used tools which, when
these patients, their full survival times are unknown. For instance,
constructed appropriately, offer valuable insight into the survival
in studies that measure the length of survival after diagnosis of
cancer, it is common for a proportion of individuals to remain
Several statistical methods have been proposed for modelling
alive and disease-free at the end of the follow-up period, and for
survival analysis data. We will describe the most important models
these patients, we know only a lower limit on their actual time to
and illustrate their application using example datasets described in
event. Thus, special methods are required for these type of data.

the previous paper (Clark et al, 2003). As before, we will assume
The explanation and demonstration of some of the methods
throughout that all survival times are independent of each other
proposed to analyse such data are the basis of this series.

and that censoring occurs solely as right-censoring and is
In the first paper of this series (Clark et al, 2003), we described
uninformative. The focus is on covariates that are measured at
initial methods for analysing and summarising survival data
the time of entry to the study, that may be continuous (e.g. the
including the definition of hazard and survival functions, and
patient age or tumour size), binary (e.g. gender), unordered
testing for a difference between two groups. We continue here by
categorical (e.g. histology) or ordered categorical or ordinal (e.g.

considering various statistical models and, in particular, how to
performance status or FIGO stage). In the next paper in this series,
estimate the effect of one or more factors that may predict survival.

we will discuss the statistical assumptions made when usingstatistical models, and provide advice on choosing the appropriatemodel and covariates therein. We will also consider how to model
THE NEED FOR MULTIVARIATE STATISTICAL
covariates that change values over time (called ‘time-dependent' or
The methods we present here may be divided into two broad
The previous paper demonstrated the construction of (Kaplan –
categories: proportional hazard approaches (including the semi-
Meier) survival curves for different patient groups, and introduced
parametric Cox model and fully parametric approaches) and
the logrank test to investigate differences between them. Both these
accelerated failure time models. These methods have different
methods are examples of univariate analysis; they describe the
properties and interpretations, but all may be used to summarise
survival with respect to the factor under investigation, but
survival data.

necessarily ignore the impact of any others. It is more common,at least in clinical investigations, to have a situation where several(known) quantities or covariates, potentially affect patient prog-nosis. For example, suppose two groups of patients are compared:
THE COX (‘SEMI-PARAMETRIC') PROPORTIONAL
those with and those without a specific genotype. If one of the
groups also contains older individuals, any difference in survival
The Cox (proportional hazards or PH) model (Cox, 1972) is the
may be attributable to genotype or age or indeed both. Hence,
most commonly used multivariate approach for analysing survival
when investigating survival in relation to any one factor, it is often
time data in medical research. It is a survival analysis regression
desirable to adjust for the impact of others. Moreover, while the
model, which describes the relation between the event incidence,
logrank test provides a P-value for the differences between the
as expressed by the hazard function and a set of covariates. A fullerexplanation of the hazard function was given in the previous
*Correspondence: Mr M Bradburn; E-mail: mike.bradburn@cancer.org.uk
article (Clark et al, 2003). Put briefly, the hazard is the
Received 6 December 2002; accepted 30 April 2003
instantaneous event probability at a given time, or the probability
Multivariate data analysis
MJ Bradburn et al
that an individual under observation experiences the event in a
stage (an ordinal covariate taking values of 1, 2 3 or 4), histology
period centred around that point in time.

(one of seven subtypes), grade (1, 2 or 3), ascites (yes/no) and
Mathematically, the Cox model is written as
patient age.

Table 1 shows the effect sizes (given as hazard ratios), 95%
hðtÞ ¼ h0ðtÞ expfb1x1 þ b2x2 þ þ bpxpg
confidence intervals (CI), regression coefficients and statisticalsignificance for each of these in relation to overall survival. Each
where the hazard function h(t) is dependent on (or determined by)
factor is assessed through separate univariate Cox regressions
a set of p covariates (x1, x2, y, xp), whose impact is measured by
(left-hand columns). However, the aim of the database is to
the size of the respective coefficients (b1, b2, y, bp). The term h0 is
describe how the factors jointly impact on survival, and so all five
called the baseline hazard, and is the value of the hazard if all the xi
factors were incorporated into the multivariate model (right-hand
are equal to zero (the quantity exp(0) equals 1). The ‘t' in h(t)
columns). It may be seen that higher FIGO stage, higher grade,
reminds us that the hazard may (and probably will) vary over time.

presence of ascites and increased age impaired survival to varying
An appealing feature of the Cox model is that the baseline hazard
(and statistically significant) degrees. The histology was also of
function is estimated nonparametrically, and so unlike most other
importance: the figures describe the survival of patients with each
statistical models, the survival times are not assumed to follow a
histology type in comparison with the serous type. In principle,
particular statistical distribution.

any type with a reasonable number of patients could be chosen as
The Cox model is essentially a multiple linear regression of the
the baseline of comparison. On multivariate analysis Mucinous
logarithm of the hazard on the variables xi, with the baseline
and serous were the tumour types with the best prognosis, whereas
hazard being an ‘intercept' term that varies with time. The
undifferentiated and mixed mesodermal were the worst. It is
covariates then act multiplicatively on the hazard at any point in
possible to present P-values for the comparisons between each
time, and this provides us with the key assumption of the PH
type and serous, but we have given an overall likelihood ratio test
model: the hazard of the event in any group is a constant multiple
for the differences between the categories as a whole. The FIGO
of the hazard in any other. This assumption implies that the hazard
stage could be modelled as a categorical variable in the same
curves for the groups should be proportional and cannot cross (see
manner as grade and histology, but assuming it is a continuous
Figure 1 for examples of each). Proportionality implies that the
variable with a linear trend across the four categories performed
quantities exp(bi) are called hazard ratios. A value of bi greater
sufficiently well.

than zero, or equivalently a hazard ratio greater than one, indicatesthat as the value of the ith covariate increases, the event hazardincreases and thus the length of survival decreases. Put another
PARAMETRIC PH MODELS
way, a hazard ratio above 1 indicates a covariate that is positivelyassociated with the event probability, and thus negatively
Parametric PH models are a class of models similar in concept and
associated with the length of survival. This proportionality
interpretation to the Cox (PH) model. The key difference between
assumption is often appropriate for survival time data but it is
the two is that the hazard is assumed to follow a specific statistical
important to verify that it holds. We discuss methods for assessing
distribution when a fully parametric PH model is fitted to the data,
proportionality in the next paper in this series.

whereas the Cox model enforces no such constraint. Other thanthis, the two model types are equivalent. Hazard ratios have thesame interpretation, whether derived from a Cox or a fully
The Cox PH model fitted to the ovarian cancer data
parametric regression model, and the proportionality of hazards is
This large database, as described in the previous paper of this
still assumed.

series (Clark et al, 2003), was used to derive a prognostic index for
A number of different parametric PH models may be derived by
overall survival among ovarian cancer patients in Clark et al
choosing different hazard functions. As shown previously, there is
(2001). Their analysis included 10 variables, but for simplicity we
a direct link between the survival and hazard, and the choice of
will consider five, all of which were measured at diagnosis: FIGO
hazard distribution determines that of the survival. In fact, themodels commonly applied, such as the Exponential, Weibull orGompertz models, take their names from the distribution that thesurvival times are assumed to follow, but the most distinguishing
features between them are in the hazard function. Examples of
survival and hazard functions derived from some of theseparametric models were presented in the previous paper (Clark
et al, 2003). Figure 1 shows increasing and decreasing Weibullhazard functions, as well as two groups with the latter that are
proportional to each other.

Parametric models fitted to the ovarian cancer data
The estimated hazard function of the ovarian cancer data asdisplayed in the previous paper (Clark et al, 2003) may be
consistent with that derived from a Weibull PH model with
decreasing hazard. Fitting this to the ovarian cancer database gives
similar results as the Cox model (see Table 2), and may beinterpreted in the same manner. Methods to check for the
appropriateness of the Weibull distribution will be discussed in
the next paper of this series.

Example of (non-) proportional hazards (groups (c) and (d)
only have proportional hazards) using the Weibull distribution. For theWeibull survival model, the hazard function h(t) ¼ ls(lt)s 1 for l, s40: (a)
COMPARISON OF THE TWO PH APPROACHES
increasing hazard (l ¼ 0.5, s ¼ 1.25); (b) decreasing hazard (l ¼ 0.25,s ¼ 0.75); (c) decreasing hazard (l ¼ 0.5, s ¼ 0.5); (d) decreasing hazard
The main drawback of parametric models is the need to specify the
(l ¼ 0.25, s ¼ 0.5).

distribution that most appropriately mirrors that of the actual
British Journal of Cancer (2003) 89(3), 431 – 436
& 2003 Cancer Research UK
Multivariate data analysisMJ Bradburn et al
Hazard ratios from the Cox PH model for the ovarian dataset
Univariate analysis
Multivariate analysis
Absence of ascites
Age (per 5-year increase)
HR ¼ hazard ratio, CI ¼ confidence interval.

Hazard ratios from the Weibull PH model for the ovarian dataset
Univariate analysis
Multivariate analysis
Absence of ascites
Age (per 5-year increase)
HR ¼ hazard ratio, CI ¼ confidence interval.

survival times. This is an important requirement that needs
INTERPRETING THE PH MODEL: BEYOND THE
to be verified and an appropriate distribution may be difficult
to identify. Where a suitable distribution can be found,however, the parametric model is more informative than the
In addition to the ratio of two hazards, it is possible to obtain other
Cox model. It is straightforward to derive the hazard function and
information from a PH regression model. One simple (and
to obtain predicted survival times when using a parametric model,
possibly underused) quantity that may be derived from a survival
which is not the case in the Cox framework (the use of such
model is the predicted survival proportion at any given point in
quantities is discussed in the next section). Additionally, the
time for a particular risk group. The survival proportion for a
appropriate use of these models offers the advantage of being
given risk group at any time, S(t), is equal to
slightly more efficient; they yield more precise estimates (i.e.

smaller standard errors).

SðtÞ ¼ S0ðtÞexpðgÞ
The results from the Cox or parametric PH models may be
compared directly, as the model types are merely different
where S0(t) is the baseline survival (the survival proportion when
approaches to assessing the same quantity. For either method to
be valid: (a) the covariate effect needs to be at least approximately
b1x1 þ b2x2 þ þbpxp. Once the value of the baseline survival
constant throughout the duration of the study, and (b) the
at a given time is derived, then the predicted survival probabilities
proportionality assumption must hold. These important issues will
for patients with any specified covariate values xi are easily
be addressed in the subsequent paper in this series.

obtained. This information could then be displayed via tabular or
& 2003 Cancer Research UK
British Journal of Cancer (2003) 89(3), 431 – 436
Multivariate data analysis
MJ Bradburn et al
Survival proportion
Survival probability
Patient age (years)
Illustration of the AFT model: (——), S0(t) the baseline survival
function; ( ), S(t1) ¼ S0(jt) for jo1; (- - - - - -), S(t2) ¼ S(jt) for
Predicted 5-year survival of ovarian cancer patients by age and
graphical displays. Figure 2 illustrates this by giving predicted 5-
The AFT model is commonly rewritten as being log-linear with
year survival according to patient age and FIGO stage. Further
respect to time, giving
examples are demonstrated by Christensen (1987) based on theCox model, but can also be used when fitting fully parametric
logðTÞ ¼ b0 þ b1x1 þ b2x2 þ þ bpxp þ e
models. In a previous analysis that involved some of the patients inthe present data, Clark et al (2001) produced a nomogram to
where e is a measure of (residual) variability in the survival times.

summarise the impact of these and other covariates, and thus
Thus, the survival times can be seen to be multiplied by a constant
allows the reader to predict the median survival and the 2- and 5-
effect under this model specification, and the exponentiated
year survival probabilities for patients with given prognostic
coefficients, exp(bi), are referred to as time ratios. A time ratio
above 1 for the covariate implies that this ‘slows down', or
The advantage of fitting a parametric survival model is that
prolongs the time to the event, while a time ratio below 1 indicates
predictions of the event survival, event hazard, mean and median
that an earlier event is more likely.

survival times are readily available. For FIGO stages I – IV, the
When the survival times follow a Weibull distribution, it can be
median survival times are estimated to be 7.8, 4.0, 2.0 and 1.0
shown that the AFT and PH models are the same. However, the
AFT family of models differs crucially from the PH model types interms of their interpretation of effect sizes as time ratios asopposed to hazard ratios.

ACCELERATED FAILURE TIME MODELS
The survival times are usually assumed to follow a specific
The accelerated failure time (AFT) model is a different type of
distributional form in the AFT framework. Distributions such as
model that may be used for the analysis of survival time data. For a
the Log-Normal, Log-Logistic, Generalised Gamma and Weibull
group of patients with covariates (x
may be used to represent such survival data. Alternative methods
1, x2, y xp), the model is
written mathematically as
include the method of Buckley and James (1979), which isdiscussed by Stare et al (2000), and semiparametric AFT models,
SðtÞ ¼ S0ðjtÞ
in which the baseline survivor function is estimated nonparame-trically (see Wei, 1992, for an overview), but have not yet been
where S0(t) is the baseline survivor function and j is an
widely implemented in statistical software.

‘acceleration factor' that depends on the covariates according to
As with the PH approach, other quantities such as projected
survival probabilities may be derived. Also in keeping with PH
models is the fact that AFT models make assumptions; the
1x1þb2x2 þ þ bpxpÞg:
appropriate choice of statistical distribution needs to be made, andalso the covariate effects are assumed to be constant and
The principle here is that the effect of a covariate is to stretch or
multiplicative on the timescale, that is, that the covariate impacts
shrink the survival curve along the time axis by a constant relative
on survival by a constant factor.

amount j. Figure 3 demonstrates this for the case of a singlecovariate (x1) with two levels, for example, x1 ¼ 0 for a placebo
Parametric AFT models fitted to the lung cancer trial data
group and x1 ¼ 1 for a new treatment group. The survivalprobabilities, S(t), for the placebo and new treatment groups are
We use the non-small cell lung cancer dataset to illustrate the AFT
S0(t) and S0(jt), respectively. The proportion of patients who are
model, focusing on the relapse-free survival (i.e. , the time from
event-free in the placebo group at any time point t1 is the same as
diagnosis to the reappearance of cancer, with patients censored at
the proportion of those who are event-free in the new treatment
time of death if no recurrence had appeared). Again, we present
group at a time t2 ¼ jt1. Figure 3 shows the cases where j41 and
both the univariate and multivariate effect sizes in Table 3. The
jo1, which represent situations where the length of survival is
specific comparison of interest was the effect of adjuvant
increased and decreased in the new treatment group compared
(platinum-based) chemotherapy and radiotherapy compared with
with the placebo, respectively.

radiotherapy alone. The unadjusted treatment effect may be
British Journal of Cancer (2003) 89(3), 431 – 436
& 2003 Cancer Research UK
Multivariate data analysisMJ Bradburn et al
Time ratios from the generalised gamma AFT model for the lung cancer trial
Univariate analysis
Multivariate analysis
Treatment (RT+CAP vs RT alone)
Cell type (Sq vs non-Sq)
Performance status (8 – 10 vs 5 – 7)
Nodal involvement
Age at diagnosis (/years)
Gender (male vs female)
Weight loss (X10 vs o10%)
Race (white vs non-white)
TR ¼ time ratio, CI ¼ confidence interval, RT ¼ radiotherapy, CAP ¼ cytoxan, doxorubicin and platinum-based chemotherapy, Sq ¼ squamous.

summarised by a time ratio of 1.91 (95% CI: 1.21 – 3.01; P ¼ 0.005),
The strength of this method is in its simplicity: as the logrank
which, having allowed for other covariates increased slightly to
test is nonparametric, few distributional assumptions are made,
2.05. Therefore, we can conclude that the time to recurrence was
and its interpretation is straightforward. Its main limitation is that
significantly prolonged (approximately doubled) among patents
it is only applicable when the covariate is categorical (or with
given adjuvant chemotherapy in comparison with those who were
continuous variables that have been arbitrarily categorised).

Further, this method does not perform well with several covariates,
Again, we can derive model-based predictions: overall, patients
as the number of individuals in each stratum quickly becomes too
allocated to receive adjuvant chemotherapy had a predicted
small to allow reasonable comparisons. In addition, it does not
median survival time of approximately 16 months, as opposed to
quantify the strength of effect of each variable, or even offer a P-
8 months among those treated with radiotherapy alone. Other
value for factors other than the one of primary interest. This
factors are also significant and would influence these times, but
method is not generally regarded as a formal statistical model, but
these are of less importance in the context of the comparative trial.

is of use where a very small number of covariates are to be
We will return to this example in the next paper of this series.

considered, if only as an exploratory method of analysis.

Aalen's additive model
WHICH MODEL SHOULD WE USE: PH OR AFT?
Another approach to modelling the relationship between survival
From a statistical viewpoint, an obvious way to choose between the
and covariates is to assume that the covariates act additively on the
two model types is to fit a type that is in keeping with the data. If
hazard. Aalen's additive hazard model (Aalen, 1989) is one method
the AFT model clearly fits the data better than the PH model, or
that has been suggested for this, but its properties are rather unlike
vice versa, this model may be adopted as being the more
any other model described in this paper. The covariates are
appropriate. However, in some cases, either type of model may
assumed to impact additively upon a (unknown) baseline hazard,
appear to fit the data adequately. In such instances, the choice of
but the effects are not constrained to be constant. The impact is
model may be influenced by other factors. For instance, if other
therefore allowed to vary freely over time according to the
studies of a similar nature had all used the Cox regression and
underlying equation
reported the results as hazard ratios, one may be tempted to followsuit to aid comparability. Against this, the parametric approach
hðtÞ ¼ h0ðtÞ þ b1ðtÞx1 þ b2ðtÞx2 þ þ bpðtÞxp
offers more in the way of predictions, and the AFT formulationallows the derivation of a time ratio, which is arguably more
where h(t) is the hazard, h0(t) is the baseline hazard and the bi(t)
interpretable than a ratio of two hazards. As yet, however, AFT
are coefficients, which may change in magnitude and even sign
models are relatively unfamiliar and seen rarely in medical
with time. Compare this with the Cox regression, where h0(t) is
research papers (see Kay and Kinnersley, 2002).

also estimated nonparametrically, but the bi quantify the multi-plicative effect of covariate i on the hazard and are assumedconstant at all times.

As it is not straightforward to estimate h0(t) nonparametrically,
the cumulative baseline hazard is used and the regression
Stratified survival analysis
coefficients that are actually estimated from the data are also the
A more straightforward way to incorporate covariates into a
cumulative (additional) hazard
survival analysis is to use a stratified survival analysis. For
example, suppose the covariate of primary interest is treatment,
but we wish to control for the clinical stage of the tumour whenmaking the comparison. Here, the survival in each treatment
group can be compared within each stage of disease (the ‘strata')
The usual method of representing these effects is to graph them
by the logrank or some other method, and the differences within
against time. The further Bi(t) is from zero at time t, the greater the
each stratum are then combined to give an overall comparison of
effect the covariate has had on the hazard over the course of the
treatments that has been adjusted for the stage.

study up to t. The values of bi(t), the absolute increase in hazard at
& 2003 Cancer Research UK
British Journal of Cancer (2003) 89(3), 431 – 436
Multivariate data analysis
MJ Bradburn et al
time t, are not actually observed, but their relative size may be
relevant one, because it needs to be kept in mind that all the
inferred from the slope of the line. These plots are sometimes
models introduced here make certain distributional assumptions
called Aalen plots, and they are also used to provide an informal
of the survival times that will not always be met.

assessment of the adequacy of the proportional hazards assump-
We have focused on the Cox model, the class of parametric PH
tion in the Cox model, although Aalen considered its primary role
models and AFT models as tools with which to analyse survival
as an alternative model in its own right (Aalen, 1993).

time data. Other models exist (see, e.g., Collett (1994) for a more
The flexibility of this approach is tempered by the lack of an
practical demonstration of some alternatives and Bagdonavicˇius
easy interpretation. The Bi(t) coefficients are not easy to under-
and Nikulin (2001) for the theoretical background), but many are
stand, and as they change repeatedly over time, can offer no single
similar to, if not extensions of, the approaches we have discussed.

quantifiable effect size. Formal tests of statistically significant
The use of the Cox model offers greater flexibility than parametric
covariate effects may be carried out, but Aalen plots are essentially
alternatives and, in particular, does not require the direct
the only manner with which to interpret the effect sizes. These
estimation of the baseline hazard function (i.e. it avoids the need
reasons, together with the relative lack of statistical software, are
to specify the distribution of the survival times). However, the
probably the deciding factors in the relatively minimal use of
assumption of proportional hazards is a crucial one that needs to
Aalen's model.

be fulfilled for the results to be meaningful, and will not always besatisfied. Further, while the Cox PH model may be valid, other
Classification trees and artificial neural networks
parametric models will produce more precise estimates where thedistribution is specified correctly.

Two relatively recent developments are classification trees and
A further concern is that the choice of covariates to include is
artificial neural networks. These methods differ substantially in
also far from simple. In the third paper of this series, we will
their complexity and interpretation to the methods presented here
consider ways to choose between the various model types, to
and to each other. Both approaches are described in more detail in
identify and assess the importance of covariates, and to verify that
a later paper of this series.

the final model is adequate.

The principal strength of statistical models is their ability to assessseveral covariates simultaneously. The strengths of the stratified
We wish to thank John Smyth for providing the ovarian cancer
logrank test and other such methods are their obvious simplicity
data, and Victoria Cornelius and Peter Sasieni for invaluable
and the fact that they make fewer parametric assumptions of the
comments on an earlier manuscript. Cancer Research UK
data. Although these reasons are usually insufficient to suggest that
supported all authors. Taane Clark holds a National Health Service
the stratified method be used more widely, this second feature is a
(UK) Research Training Fellowship.

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& 2003 Cancer Research UK

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