How do I analyze multiple failure-time data using Stata?
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Title
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Analysis of multiple failure-time survival data
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Authors
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Mario Cleves, StataCorp
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Date
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November 1999; updated July 2009
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This FAQ first appeared as an article in STB-49, ssa13, under the
heading Analysis of multiple failure-time data with Stata. Minor
revisions have subsequently been made. In this article, when a subject
experiences one of the events, it still remains at risk for events of
different types. In section 3.1.2 of this FAQ, we show a method to fit a
Cox model to a dataset with unordered failure events of different types.
The examples described in this section are not exactly the same as what we
call “competing risk” in the [ST] Survival Analysis and
Epidemiological Tables Reference Manual. In the Stata documentation, we
follow the definition for competing risk used in Fine and Gray (1999), where
the competing event impedes the occurrence of the event of interest. If you
have competing-risk data in the sense of Fine and Gray, see the entry for
stcrreg
in the [ST] Survival Analysis and Epidemiological Tables Reference Manual.
1. Introduction
Multiple failure-time data or multivariate survival data are frequently
encountered in biomedical and other investigations. These data arise from
time-to-occurrence studies when either of two or more events (failures)
occur for the same subject, or from identical events occurring to related
subjects such as family members or classmates. In these studies, failure
times are correlated within cluster (subject or group), violating the
independence of failure times assumption required in traditional survival
analysis.
In this paper, we follow Therneau’s (1997) suggestion that for
analyses purposes, failure events should be classified according to (1)
whether they have a natural order, and (2) whether they are recurrences of
the same types of events. Failures of the same type include, for example,
repeated lung infections with pseudomonas in children with cystic fibrosis,
or the development of breast cancer in genetically predisposed families.
Failures of different types include adverse reactions to therapy in cancer
patients on a particular treatment protocol, or the development of
connective tissue disease symptoms in a group of third graders exposed to
hazardous waste.
Ordered events may result from a study that records the time to first
myocardial infarction (MI), second MI, and so on. These are ordered events
in the sense that the second event cannot occur before the first event.
Unordered events, on the other hand, can occur in any sequence. For example,
in a study of liver disease patients, a panel of seven liver function
laboratory tests can become abnormal in a specific order for one patient and
in a different order for another patient. The order in which the tests
become abnormal (fail) is random.
The simplest way of analyzing multiple failure data is to examine time to
first event, ignoring additional failures. This approach, however, is
usually not adequate because it wastes possibly relevant information.
Alternative methods have been developed that make use of all available data
while accounting for the lack of independence of the failure times. Two
approaches to modeling these data have gained popularity over the last few
years. In the first approach, the frailty model method, the association
between failure times is explicitly modeled as a random-effect term, called
the frailty. Frailties are unobserved effects shared by all members of the
cluster. These unmeasured effects are assumed to follow a known statistical
distribution, often the gamma distribution, with mean equal to one and
unknown variance. This paper will not consider frailty models further.
In the second approach, the dependencies between failure times are not
included in the models. Instead, the covariance matrix of the estimators is
adjusted to account for the additional correlation. These models, which we
will call “variance-corrected” models, are easily estimated in
Stata. In this paper we illustrate the principal ideas and procedures for
estimating these models using the Cox proportional hazard model. There is no
theoretical reason, however, why other hazard functions could not be used.
2. Methods
Let Xkiand Cki be the failure and
censoring time of the kth failure type (k = 1, ...,
K) in the ith cluster (i = 1, ..., m), and let
Zki be a p-vector of possibly time-dependent
covariates, for ith cluster with respect to the
kth failure type. "Failure type" is used here to mean both
failures of different types and failures of the same type. Assume that
Xkiand Cki are independent, conditional
on the covariate vector (Zki). Define
Tki = min(Xki; Cki)
and
where I(.) is the indicator function, and let
be a p-vector of unknown regression coefficients. Under the
proportional hazard assumption, the hazard function of the
ith cluster for the kth failure type is
if the baseline hazard function is assumed to be equal for every failure
type, or
if the baseline hazard function is allowed to differ by failure type (Lin
1994).
Maximum likelihood estimates of
for the above models are obtained from the Cox’s partial
likelihood function,
L(),
assuming independence of failure times. The estimator
has been shown to be a consistent estimator for
and is asymptotically normal as long as the marginal models
are correctly specified (Lin 1994). The resulting estimated covariance matrix
obtained as the inverse of the information matrix, however,
does not take into account the additional correlation in the data, and
therefore, it is not appropriate for testing or constructing confidence
intervals for multiple failure time data.
Lin and Wei (1989) proposed a modification to this naive estimate,
appropriate when the Cox model is misspecified. The resulting robust
variance–covariance matrix is estimated as
where U is a n x p matrix of efficient score residuals.
The above formula assumes that the n observations are independent. When
observations are not independent, but can be divided into m
independent groups (G1, G2, ...,
Gm), then the robust covariance matrix takes the form
where G is a m x p matrix of the group efficient score
residuals. In terms of Stata, V is calculated according to the first
formula when the robust option is specified and according to the
second formula when cluster() is also specified. (cluster()
implies robust in Stata, so specifying cluster() by itself is
adequate.)
3. Implementation and examples
All variance-adjusted models suggested to date can be estimated in Stata.
All that is required is some preliminary thought about the analytic model
required, the correct way to set up the data, and the command options to be
specified.
The examples in this section are presented under the following headings:
All the examples we will describe use the survival time (st) system, which is
to say, for instance, in terms of
stcox rather than
cox. Although it is
not necessary that the
st system be used, it is
recommended.
The steps for analyzing multiple failure data in Stata are (1) decide
whether the failure events are ordered or unordered, (2) select the proper
statistical model for the data, (3) organize the data according to the model
selected, and (4) use the proper commands and command options to
stset the data and fit
the model. Much of this paper deals with the appropriate method for setting
the data and the correct way of specifying the estimation command. The
examples are used solely to illustrate these processes. Consult the
references for more detailed discussions on these methods and the datasets
used.
3.1 Unordered failure events
The data setup for the analysis of unordered events is relatively simple.
One first decides if the failure events are of the same type or of different
type, or equivalently, whether the baseline hazard should be equal for all
event types or should be all owed to vary by event type. Failure events of
the same type are described in section 3.1.1. In section 3.1.2, the baseline
hazard is allowed to vary by failure type and is used to examine a
dataset with unordered failure events of different types.
3.1.1 Unordered failure events of the same type
A possible source of correlated failure times of the same event type are
familial studies, in which each family member is at risk of developing a
disease of interest. Failure times of family members are correlated because
they share genetic and perhaps environmental factors.
Another source of correlated failure times of the same type are studies
where the same event can occur on the same individual multiple times. This
is rare because we are also restricting the events to have no order. Lee,
Wei, and Amato (1992) analyzed data from the National Eye Institute study on
the efficacy of photocoagulation as a treatment for diabetic retinopathy. In
that study, each subject was treated with photocoagulation on one randomly
selected eye while the other eye served as an untreated matched control. The
outcome of interest was the onset of severe visual loss, and the study hoped
to show that laser photocoagulation significantly reduced the time to onset
of blindness. In this study, the sampling units, the eyes, are pairwise
correlated , the failure types are the same and unordered because the right
eye can fail before the left eye or vice versa.
These types of data are straightforward to setup and analyze in Stata. Each
sampling unit is entered once into the dataset. In the family data, each
family member appears as an observation in the dataset and an id variable
identifies his or her family. In the laser photocoagulation example, because
each eye is a sampling unit, each eye appears as an observation in the
dataset. Therefore, if there are n patients in the diabetic retinopathy
study then the resulting dataset would contain 2n observations. A variable
is used to identify the matched eyes.
We will illustrate using a subset of the diabetic retinopathy data. The data
from 197 high-risk patients was entered into a Stata dataset. The first four
observations are
. list in 1/4, noobs
+------------------------------------+
| id time cens agegrp treat |
|------------------------------------|
| 5 46.23 0 1 1 |
| 5 46.23 0 1 0 |
| 14 42.5 0 0 1 |
| 14 31.3 1 0 0 |
+------------------------------------+
Each patient has two observations in the dataset, one for the treated eye
(treat=1) and another for the "control" eye, treat=0. The
data, therefore, contain 394 observations. Each eye is assumed to enter the
study at time 0 and it is followed until blindness develops or censoring
occurs. The follow-up time is given by the variable time. The four
observations listed above correspond to patients with id=5 and
id=14.
After creating the dataset, it is then
stset as usual. The
id() option, however, is not specified. Specifying id() would
cause stset to interpret subjects with the same id() as the
same sampling unit and would drop them because of overlapping study times.
Thus, we type
. stset time, failure(cens)
failure event: cens != 0 & cens != .
obs. time interval: (0, time]
exit on or before: failure
-------------------------------------------------------------------------
394 total obs.
0 exclusions
-------------------------------------------------------------------------
394 obs. remaining, representing
155 failures in single record/single failure data
14018.24 total analysis time at risk, at risk from t = 0
earliest observed entry t = 0
last observed exit t = 74.97
Note that stset correctly reports that there are 394 observations.
The command for fitting the corresponding Cox model is
. stcox agegrp treat, cluster(id) efron nohr
Iteration 0: log likelihood = -867.98581
Iteration 1: log likelihood = -856.74901
Iteration 2: log likelihood = -856.74456
Refining estimates:
Iteration 0: log likelihood = -856.74456
Cox regression -- Efron method for ties
No. of subjects = 394 Number of obs = 394
No. of failures = 155
Time at risk = 14018.24001
Wald chi2(2) = 27.71
Log likelihood = -856.74456 Prob > chi2 = 0.0000
(standard errors adjusted for clustering on id)
------------------------------------------------------------------------------
_t | Robust
_d | Coef. Std. Err. z P>|z| [95% Conf. Interval]
---------+--------------------------------------------------------------------
agegrp | .0538829 .1790951 0.301 0.764 -.2971371 .4049028
treat | -.7789297 .1488857 -5.232 0.000 -1.07074 -.487119
------------------------------------------------------------------------------
The cluster(id) option specifies to
stcox which
observations are related. Stata knows to produce robust standard errors
whenever the cluster() option is used. The efron option
requests that Efron’s method for handling ties be used, and the
nohr option is used to request that coefficients, instead of hazard
ratios, be reported.
3.1.2 Unordered failure events of different types
A common data source of unordered failure events of different types are
studies where a patient can suffer several
outcomes of interest in random order. In the analysis of these data, the
baseline hazard function is allowed to vary by failure type. This is
accomplished by stratifying the data on failure type, allowing each stratum
to have its own baseline hazard function, but restricting the coefficients
to be the same across strata.
We illustrate the use of Stata in the analysis of this kind of model,
with a subset of the Mayo Clinic’s Ursodeoxycholic acid (UDCA) data
(Lindor et al. 1994). The dataset consists of 170 patients with primary
biliary cirrhosis randomly allocated to either the UDCA treatment group or a
group receiving a placebo. The times up to nine possible events were
recorded: death, liver transplant, voluntary withdraw, histologic
progression, development of varices, development of ascites, development of
encephalophathy, doubling of bilirubin, and worsening of symptoms. All times
were measured from the date of treatment allocation.
An important characteristic of these failure events is that each can occur
only once per subject. Note that all subjects are at risk for all events.
Also, when a subject experiences one of the events, he remains at risk for
all other events. Therefore, if there are k possible events, each
subject will appear k times in the dataset, once for each possible
failure. Here is the resulting data for two of the subjects.
. list id rx bili time status rec if id==5 | id==18, nod noobs
+-----------------------------------------------+
| id rx bili time status rec |
|-----------------------------------------------|
| 5 placebo .0953102 1875 0 1 |
| 5 placebo .0953102 1875 0 2 |
| 5 placebo .0953102 1875 0 3 |
| 5 placebo .0953102 1875 0 4 |
| 5 placebo .0953102 1875 0 5 |
|-----------------------------------------------|
| 5 placebo .0953102 1875 0 6 |
| 5 placebo .0953102 1875 0 7 |
| 5 placebo .0953102 1875 0 8 |
| 5 placebo .0953102 1875 0 9 |
| 18 placebo .1823216 391 1 9 |
|-----------------------------------------------|
| 18 placebo .1823216 391 1 8 |
| 18 placebo .1823216 763 1 5 |
| 18 placebo .1823216 765 0 2 |
| 18 placebo .1823216 765 0 1 |
| 18 placebo .1823216 765 0 6 |
|-----------------------------------------------|
| 18 placebo .1823216 765 0 7 |
| 18 placebo .1823216 765 1 3 |
| 18 placebo .1823216 765 0 4 |
+-----------------------------------------------+
Each patient appears nine times, once for each possible event. The event
type, rec, is coded as 1 through 9. Patient number 5 did not
experience any events during the 1,875 days of follow-up. Thus, he appears
censored nine times in the data, each observation recording the complete
follow-up period. Patient 18 experienced 4 events: rec=8 (doubling
of bilirubin), rec=9 (worsening of symptoms), rec=5
(development of varices) and rec=3 (voluntary withdraw).
The command to
stset the data is
used without specifying the id() option.
. stset time, failure(status)
failure event: status != 0 & status != .
obs. time interval: (0, time]
exit on or before: failure
----------------------------------------------------------------------------
1530 total obs.
0 exclusions
----------------------------------------------------------------------------
1530 obs. remaining, representing
145 failures in single record/single failure data
1808720 total analysis time at risk, at risk from t = 0
earliest observed entry t = 0
last observed exit t = 1896
It correctly reported 1,530 observations (170x9). The id variable
will be used to cluster the related observations when estimating the Cox
model. Additionally, it does not seem reasonable to assume that each
failure type should have the same baseline hazard, thus the Cox model will
be stratified by failure type.
. stcox rx bili hi_stage, nohr efron robust strata(rec) cluster(id) nolog
Stratified Cox regr. -- Efron method for ties
No. of subjects = 1530 Number of obs = 1530
No. of failures = 145
Time at risk = 1808720
Wald chi2(3) = 31.99
Log likelihood = -662.44704 Prob > chi2 = 0.0000
(standard errors adjusted for clustering on id)
------------------------------------------------------------------------------
_t | Robust
_d | Coef. Std. Err. z P>|z| [95% Conf. Interval]
---------+--------------------------------------------------------------------
rx | -.9371209 .240996 -3.889 0.000 -1.409464 -.4647774
bili | .5859002 .1491832 3.927 0.000 .2935065 .8782939
hi_stage | -.0754988 .2777845 -0.272 0.786 -.6199464 .4689488
------------------------------------------------------------------------------
Stratified by rec
The covariates are treatment group (rx), log(bilirubin)
(bili), and high histologic stage indicator (hi_stage).
3.2 Ordered failure events
There are several approaches to the analysis of ordered events. The
principal difference between these methods is in the way that the risk sets
are defined at each failure time. The simplest method to implement in Stata
follows the counting process approach of Andersen and Gill (1982). The basic
assumption is that all failure types are equal or indistinguishable. The
problem then reduces to the analysis of time to first event, time to second
event, and so on. Thus, the risk set at time t for event k is
all subjects under observation at time t. A major limitation of this
approach is that it does not allow more than one event to occur at a given
time. For example, in a study examining time to side effects of a new
medication, if a patient exhibits two side effects at the same time, the
corresponding observations are dropped because the time span between
failures is zero. This approach is illustrated in section 3.2.1.
A second model, proposed by Wei, Lin, and Weissfeld (1989), is based on the
idea of marginal risk sets. For this analysis, the data is treated
as if the failure events were unordered, so each
event has its own stratum and each patient appears in all strata. The
marginal risk set at time t for event k is made up of all
subjects under observation at time t that have not had event
k. This approach is illustrated in section 3.2.2.
A third method proposed by Prentice, Williams, and Peterson (1981) is known
as the conditional risk set model. The data are set up as for Andersen and
Gill’s counting processes method, except that the analysis is
stratified by failure order. The assumption made is that a subject is not at
risk of a second event until the first event has occurred and so on. Thus,
the conditional risk set at time t for event k is made up of
all subjects under observation at time t that have had event k
− 1. There are two variations to this approach. In the first
variation, time to each event is measured from entry time, and in the second
variation, time to each event is measured from the previous event. This
approach is illustrated in sections 3.2.3 and 3.2.4.
The above three approaches will be illustrated using the bladder cancer data
presented by Wei, Lin, and Weissfeld (1989). These data were collected from
a study of 85 subjects randomly assigned to either a treatment group
receiving the drug thiotepa or to a group receiving a placebo control. For
each patient, time for up to four tumor recurrences was recorded in months
(r1–r4). These are the first nine observations in the
data.
. list in 1/9, noobs
+-----------------------------------------------------------+
| id group futime number size r1 r2 r3 r4 |
|-----------------------------------------------------------|
| 1 placebo 1 1 3 0 0 0 0 |
| 2 placebo 4 2 1 0 0 0 0 |
| 3 placebo 7 1 1 0 0 0 0 |
| 4 placebo 10 5 1 0 0 0 0 |
| 5 placebo 10 4 1 6 0 0 0 |
|-----------------------------------------------------------|
| 6 placebo 14 1 1 0 0 0 0 |
| 7 placebo 18 1 1 0 0 0 0 |
| 8 placebo 18 1 3 5 0 0 0 |
| 9 placebo 18 1 1 12 16 0 0 |
+-----------------------------------------------------------+
The id variable identifies the patients, group is the treatment
group, futime is the total follow-up time for the patient,
number is the number of initial tumors, size is the initial
tumor size, and r1 to r4 are the times to first, second,
third, and fourth recurrence of tumors. A recurrence time of zero indicates
no tumor.
3.2.1 The Andersen–Gill model
To implement the Andersen and Gill model using the results from the bladder
cancer study, the data are set up as follows: for each patient there must be
one observation per event or time interval. For example, if a subject has
one event, then there will be two observations for that subject. The first
observation will cover the time span from entry into the study until the
time of the event, and the second observation spans the time from the event
to the end of follow-up. The data for the nine subjects listed above is
. list if id!=10, noobs
+------------------------------------------------------+
| id group time0 time status number size |
|------------------------------------------------------|
| 1 placebo 0 1 0 1 3 |
| 2 placebo 0 4 0 2 0 |
| 3 placebo 0 7 0 1 0 |
| 4 placebo 0 10 0 5 0 |
| 5 placebo 0 6 1 4 0 |
|------------------------------------------------------|
| 5 placebo 6 10 0 4 0 |
| 6 placebo 0 14 0 1 0 |
| 7 placebo 0 18 0 1 0 |
| 8 placebo 0 5 1 1 3 |
| 8 placebo 5 18 0 1 3 |
|------------------------------------------------------|
| 9 placebo 0 12 1 1 1 |
| 9 placebo 12 16 1 1 1 |
| 9 placebo 16 18 0 1 1 |
+------------------------------------------------------+
In the original data, subjects 1 through 4 had no tumors recur, thus, each
of these 4 patients has only one censored (status=0) observation
spanning from time0=0 to end of follow-up (time=futime}).
Patient 5 ( id=5) had one tumor recur at 6 months and was followed
until month 10. This patient has two observations in the final dataset; one
from time0=0 to tumor recurrence (time=6), ending in an event
(status=1), and another from time0=6 to end of follow-up
(time=10), ending as censored (status =0).
We stset the data
with the command
. stset time, fail(status) exit(time .) id(id) enter(time0)
id: id
failure event: status != 0 & status != .
obs. time interval: (time[_n-1], time]
enter on or after: time time0
exit on or before: time time
----------------------------------------------------------------------------
178 total obs.
0 exclusions
----------------------------------------------------------------------------
178 obs. remaining, representing
85 subjects
112 failures in multiple failure-per-subject data
2480 total analysis time at risk, at risk from t = 0
earliest observed entry t = 0
last observed exit t = 59
and we fit the Andersen–Gill Cox model as
. stcox group size number, nohr efron robust nolog
Cox regression -- Efron method for ties
No. of subjects = 85 Number of obs = 178
No. of failures = 112
Time at risk = 2480
Wald chi2(3) = 11.41
Log likelihood = -449.98064 Prob > chi2 = 0.0097
(standard errors adjusted for clustering on id)
------------------------------------------------------------------------------
_t | Robust
_d | Coef. Std. Err. z P>|z| [95% Conf. Interval]
---------+--------------------------------------------------------------------
group | -.464687 .2671369 -1.740 0.082 -.9882656 .0588917
size | -.0436603 .0780767 -0.559 0.576 -.1966879 .1093673
number | .1749604 .0634147 2.759 0.006 .0506699 .2992509
------------------------------------------------------------------------------
This time it was not necessary to specify the cluster() option. Because
stset’s id() option was used, Stata knows to cluster on the
id() variable when producing robust standard errors.
3.2.2 The marginal risk set model (Wei, Lin, and Weissfeld)
The setup for the marginal risk model is identical to the
model described in section 3.1.2. In essence the model ignores the ordering
of events and treats each failure occurrence as belonging in an independent
stratum.
The resulting data for the first six of the nine subjects listed above are
. list id group time status number size rec if id<7, noobs
+----------------------------------------------------+
| id group time status number size rec |
|----------------------------------------------------|
| 1 placebo 1 0 1 3 1 |
| 1 placebo 1 0 1 3 2 |
| 1 placebo 1 0 1 3 3 |
| 1 placebo 1 0 1 3 4 |
| 2 placebo 4 0 2 1 1 |
|----------------------------------------------------|
| 2 placebo 4 0 2 1 2 |
| 2 placebo 4 0 2 1 3 |
| 2 placebo 4 0 2 1 4 |
| 3 placebo 7 0 1 1 1 |
| 3 placebo 7 0 1 1 2 |
|----------------------------------------------------|
| 3 placebo 7 0 1 1 3 |
| 3 placebo 7 0 1 1 4 |
| 4 placebo 10 0 5 1 1 |
| 4 placebo 10 0 5 1 2 |
| 4 placebo 10 0 5 1 3 |
|----------------------------------------------------|
| 4 placebo 10 0 5 1 4 |
| 5 placebo 6 1 4 1 1 |
| 5 placebo 10 0 4 1 2 |
| 5 placebo 10 0 4 1 3 |
| 5 placebo 10 0 4 1 4 |
|----------------------------------------------------|
| 6 placebo 14 0 1 1 1 |
| 6 placebo 14 0 1 1 2 |
| 6 placebo 14 0 1 1 3 |
| 6 placebo 14 0 1 1 4 |
+----------------------------------------------------+
The data are then stset without specifying the id() option:
. stset time, failure(status)
failure event: status != 0 & status != .
obs. time interval: (0, time]
exit on or before: failure
----------------------------------------------------------------------------
340 total obs.
0 exclusions
----------------------------------------------------------------------------
340 obs. remaining, representing
112 failures in single record/single failure data
8522 total analysis time at risk, at risk from t = 0
earliest observed entry t = 0
last observed exit t = 59
and the Cox model is fitted by clustering on id and stratifying on
the failure occurrence variable (rec).
. stcox group size number, nohr efron strata(rec) cluster(id) nolog
Stratified Cox regr. -- Efron method for ties
No. of subjects = 340 Number of obs = 340
No. of failures = 112
Time at risk = 8522
Wald chi2(3) = 15.35
Log likelihood = -426.14683 Prob > chi2 = 0.0015
(standard errors adjusted for clustering on id)
------------------------------------------------------------------------------
_t | Robust
_d | Coef. Std. Err. z P>|z| [95% Conf. Interval]
---------+--------------------------------------------------------------------
group | -.5847935 .3097738 -1.888 0.059 -1.191939 .0223521
size | -.051617 .095148 -0.542 0.587 -.2381036 .1348697
number | .2102937 .0670372 3.137 0.002 .0789032 .3416842
------------------------------------------------------------------------------
Stratified by rec
3.2.3 The conditional risk set model (time from entry)
As previously mentioned, there are two variations of the conditional risk
set model. The first variation in which time to each event is measured from
entry is illustrated in this section.
The data are set up as for Andersen and Gill’s method, however, a
variable indicating the failure order is included. The resulting
observations for the first nine subjects are
. list id if id<10, noobs
+------------------------------------------------------------+
| id group time0 time status number size str |
|------------------------------------------------------------|
| 1 placebo 0 1 0 1 3 1 |
| 2 placebo 0 4 0 2 1 1 |
| 3 placebo 0 7 0 1 1 1 |
| 4 placebo 0 10 0 5 1 1 |
| 5 placebo 0 6 1 4 1 1 |
|------------------------------------------------------------|
| 5 placebo 6 10 0 4 1 2 |
| 6 placebo 0 14 0 1 1 1 |
| 7 placebo 0 18 0 1 1 1 |
| 8 placebo 0 5 1 1 3 1 |
| 8 placebo 5 18 0 1 3 2 |
|------------------------------------------------------------|
| 9 placebo 0 12 1 1 1 1 |
| 9 placebo 12 16 1 1 1 2 |
| 9 placebo 16 18 0 1 1 3 |
+------------------------------------------------------------+
The resulting dataset is identical to that used to fit Andersen and
Gill’s model except that the str variable identifies the
failure risk group for each time span. For the first 4 individuals, who have
not had a tumor recur, the str value is one, meaning that during
their total observed time they are at risk of first failure. The last
individual listed, id=9, was at risk of a first recurrence for 12
months (str=1), at risk of a second recurrence from 12 through 16
months (str=2), and at risk of a third recurrence from 16 months to
the end of follow-up (str=3).
The stset command is identical to that used for the Andersen and Gill
model.
. stset time, fail(status) exit(time .) id(id) enter(time0)
id: id
failure event: status != 0 & status != .
obs. time interval: (time[_n-1], time]
enter on or after: time time0
exit on or before: time time
----------------------------------------------------------------------------
178 total obs.
0 exclusions
----------------------------------------------------------------------------
178 obs. remaining, representing
85 subjects
112 failures in multiple failure-per-subject data
2480 total analysis time at risk, at risk from t = 0
earliest observed entry t = 0
last observed exit t = 59
The corresponding conditional risk model is
. stcox group size number, nohr efron robust nolog strata(str)
Stratified Cox regr. -- Efron method for ties
No. of subjects = 85 Number of obs = 178
No. of failures = 112
Time at risk = 2480
Wald chi2(3) = 7.17
Log likelihood = -315.99082 Prob > chi2 = 0.0665
(standard errors adjusted for clustering on id)
------------------------------------------------------------------------------
_t | Robust
_d | Coef. Std. Err. z P>|z| [95% Conf. Interval]
---------+--------------------------------------------------------------------
group | -.3334887 .2060021 -1.619 0.105 -.7372455 .070268
size | -.0084947 .062001 -0.137 0.891 -.1300144 .1130251
number | .1196172 .0516917 2.314 0.021 .0183033 .2209311
------------------------------------------------------------------------------
Stratified by strata
3.2.4 The conditional risk set model (time from the previous event)
The second variation of the conditional risk set model measures time to each
event from the time of the previous event. The data is set up as in 3.2.3,
except that time is not measured continuously from study entry, but the
clock is set to zero after each failure.
. list id if id!=10, noobs nod
+------------------------------------------------------------+
| id group time0 time status number size str |
|------------------------------------------------------------|
1. | 1 placebo 0 1 0 1 3 1 |
2. | 2 placebo 0 4 0 2 1 1 |
3. | 3 placebo 0 7 0 1 1 1 |
4. | 4 placebo 0 10 0 5 1 1 |
5. | 5 placebo 0 4 0 4 1 2 |
|------------------------------------------------------------|
6. | 5 placebo 0 6 1 4 1 1 |
7. | 6 placebo 0 14 0 1 1 1 |
8. | 7 placebo 0 18 0 1 1 1 |
9. | 8 placebo 0 5 1 1 3 1 |
10. | 8 placebo 0 13 0 1 3 2 |
|------------------------------------------------------------|
11. | 9 placebo 0 2 0 1 1 3 |
12. | 9 placebo 0 4 1 1 1 2 |
13. | 9 placebo 0 12 1 1 1 1 |
+------------------------------------------------------------+
Note that the initial times for all time spans are set to zero and that the
time variable now reflects the length of the time span. After creating the
new time variable, the data need to be stset again.
. stset time, fail(status) exit(time .) enter(time0)
failure event: status != 0 & status != .
obs. time interval: (0, time]
enter on or after: time time0
exit on or before: time time
----------------------------------------------------------------------------
178 total obs.
0 exclusions
----------------------------------------------------------------------------
178 obs. remaining, representing
112 failures in single record/single failure data
2480 total analysis time at risk, at risk from t = 0
earliest observed entry t = 0
last observed exit t = 59
The corresponding conditional risk model is
. stcox group size number, nohr efron robust nolog strata(str) cluster(id)
Stratified Cox regr. -- Efron method for ties
No. of subjects = 178 Number of obs = 178
No. of failures = 112
Time at risk = 2480
Wald chi2(3) = 11.70
Log likelihood = -358.96849 Prob > chi2 = 0.0085
(standard errors adjusted for clustering on id)
------------------------------------------------------------------------------
_t | Robust
_d | Coef. Std. Err. z P>|z| [95% Conf. Interval]
---------+--------------------------------------------------------------------
group | -.2790045 .2169035 -1.286 0.198 -.7041277 .1461186
size | .0074151 .0647143 0.115 0.909 -.1194226 .1342528
number | .1580459 .0512421 3.084 0.002 .0576133 .2584785
------------------------------------------------------------------------------
Stratified by strata
4. Conclusion
This paper details how Stata can be used to fit variance-corrected models
for the analysis of multiple failure-time data. The examples used to
illustrate the various approaches, although real, were simple. More
complicated datasets, however, containing time-dependent covariates, varying
time scales, delayed entry and other complications, can be set up and
analyzed following the guidelines illustrated in this paper.
The most important aspect in the implementation of the methods described is
the accurate construction of the dataset for analysis. Care must be taken to
correctly code entry and exit times, strata variables and failure/censoring
indicators. It is strongly recommended that, after creating the final
dataset and before analyzing and reporting results, the data be examined
thoroughly. Lists of all representative, and especially complex cases,
should be carefully verified. This step, although time consuming and
tedious, is indispensable, especially when working with complicated survival
data structures.
A second important aspect of the analysis is the proper use of the
stset command.
Become familiar and have a clear understanding of the id(),
origin(), enter() and time0() options. Review the
output from stset and confirm that the final data contain the
expected number of observations and failures. Check any records dropped and
verify the data, especially the stset created variables, by listing
and examining observations.
Lastly fit the model using the correct
stcox options to
produce robust standard errors and, if needed, the strata specific baseline
hazard.
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