AJR 2005; 185:19-22
© American Roentgen Ray Society
Fundamentals of Clinical Research for Radiologists |
Survival Analysis
Harald O. Stolberg1,2,
Geoffrey Norman3 and
Isabelle Trop4
1 Department of Radiology, McMaster University Medical Centre, 1200 Main St. W,
Hamilton, ON, L8N 3Z5 Canada.
2 Deceased.
3 Department of Educational Research, Clinical Epidemiology and Biostatistics,
McMaster University, Hamilton, ON, Canada.
4 Department of Radiology, Hospital St.-Luc, 1058 St-Denis St., Montreal, QC,
H2X 3J4 Canada.
Received November 17, 2004;
accepted after revision November 23, 2004.
Address correspondence to I. Trop.
Series editors: Nancy Obuchowski, C. Craig Blackmore, Steven Karlik, and
Caroline Reinhold.
This is the 19th in the series designed by the American College of
Radiology (ACR), the Canadian Association of Radiologists, and the
American Journal of Roentgenology. The series, which will ultimately
comprise 22 articles, is designed to progressively educate radiologists in the
methodologies of rigorous clinical research, from the most basic principles to
a level of considerable sophistication. The articles are intended to
complement interactive software that permits the user to work with what he or
she has learned, which is available on the ACR Web site
(www.acr.org).
Project coordinator: G. Scott Gazelle, Chair, ACR Commission on Research
and Technology Assessment.
Staff coordinator: Jonathan H. Sunshine, Senior Director for Research,
ACR.
Introduction
The breadth of radiology research is expanding. Previously, a large
proportion of radiology research projects were observational studies.
Increasingly, research now involves groups of patients to whom specific
interventions are administered in a randomized fashion. Analysis of data
obtained from these experimental studies varies, depending on the end point of
interest. Research protocols that are designed to evaluate the interval
between entry of a patient into the study and the time until the event of
interest are referred to as time-to-event studies, a form of follow-up study
[1]. The event may be death in
a diagnostic study of cancer or a progression of various chronic disease
entities to a defined stage. In interventional studies, such as vascular and
neuroradiologic procedures, the fate of grafts, stents, and other devices may
be followed through time. Survival analysis, also called "life
table" analysis, refers to the methodology of analysis of data gathered
in such protocols. Survival analysis, then, is the topic of this article
[2].
Overview
Under ideal circumstances, a study would enroll all its subjects
simultaneously and follow them either for a fixed period of time or until they
all reach some end point, such as recovery or death. However, more commonly,
studies require a large number of subjects or look at relatively rare
conditions, and so must enter subjects over a period of several months or even
years. When the study finally ends, the subjects will have been followed for
varying lengths of time, during which a number of different outcomes have to
be considered: the event has not yet occurred (outcome A), some patients are
lost to follow-up (outcome L), or the event has occurred (an example of the
event or end point is death) (outcome D).
Figure 1 shows how we can
illustrate these different outcomes, indicating what happened to the first 10
patients in a study. Subjects A, C, D, and F died during the trial; they are
labeled "D" for dead. Subjects B, G, and I were lost to follow-up,
hence the label "L," at various times after they started the drug.
The other subjects, E, H, and J (labeled "C"), were still alive at
the time the trial ended. These last three data points are called
"right-censored." Subjects are considered "censored"
when their data are incomplete. They are said to be right-censored because
they have been followed to the end of the study (the "right-hand
part" of the graph), but the outcome of interest has not occurred to
them. To be more quantitative about the data,
Table 1 shows how long each
person was in the study and what the outcome was.
The Kaplan-Meier Approach to Survival Analysis
To do a survival analysis, we must figure out how many people survive for
at least 1 year, for at least 2 years, and so on, in what is called a
"life table" technique. There are two ways to go about calculating
a life table: the actuarial approach and the Kaplan-Meier approach
[3]. The Kaplan-Meier approach
is far more common in medical literature, so we will describe it.
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Fig. 6 —Cumulative survival of patients with intramural hematoma
(IMH) with (experimental group) and without (control group) treatment with
ß-blockers. Upper curve (triangles) indicates treated patients;
small squares indicate censored cases. Difference between two subgroups was
statistically significant (p = 0.004).
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The first step involves redrawing the graph, so that all the people appear
to start at the same time. Figure
2 shows the same data as Figure
1; however, instead of the x-axis being Calendar Year, it
is now Number of Years in Study. The lines are all the same length as in
Figure 1; they have just been
shifted to the left so that they all begin at time 0.
The Kaplan-Meier approach uses the exact time of death in the calculation
of survival. It also computes the survival function only when an outcome
occurs. To show how this is done, let us use the data for the 10 subjects in
Table 1. The first step is to
rank-order the length of time in the trial and flag which entries reflect the
outcome of interest (death in this case) and which are due to withdrawal or
censoring. We have done this by putting an asterisk after the data for
subjects who were lost to follow-up or were censored by the termination of the
study:
This data set would generate a life table
(Table 2) with only four rows,
one for each of the four patients who died.
One person was lost to follow-up before the first person died, so the
number of remaining patients at risk at 22 months is only nine. Death rate,
survival rate, or any other statistical estimate is calculated on the basis of
the population at risk (Table
2). At 46 months, two people had died and three were lost to
follow-up, so the number of patients at risk is five, and so on. This little
data set would generate a survival curve like that shown in
Figure 3 except for fewer
steps.
Comparing Two (or More) Groups with the Log-Rank Test
Although the survival curve shown in
Figure 3 tells us what happened
to patients over time, we often want to compare two or more groups of
patients—for example, patients with different kinds of stents, or
patients who were screened (experimental group) versus patients who were not
screened (control group). So we will create an expanded survival table with
250 experimental subjects and 250 control patients. These data are presented
in Figure 4. This graph shows
that the survival curve for the treatment group dropped at a faster rate than
that for the control group. But is the difference statistically
significant?
The best approach for evaluating whether the difference is indeed
significant is to use the Mantel-Cox log-rank test, which is a modification of
the Mantel-Haenszel chi-square test
[4]. This test is a powerful
method for analyzing data when the time to the outcome is important; it deals
with censored data and differential length of follow-up of different subjects.
As with most chi-square tests, the log-rank test compares the observed number
of events with the number expected, under the assumption that the null
hypothesis of no group differences is true. That is, if there were no
differences between the groups, then at any interval, the total number of
events should be divided between the groups roughly in proportion to the
number of subjects at risk. The test determines how much the observed event
rate differs from the expected rate.
The Cox Proportional Hazards Model
A more sophisticated method of analysis commonly used, which examines the
difference in the survival curves while also accounting for other variables
(covariates), is the Cox proportional hazards model
[5]. Unlike the log-rank test,
the proportional hazards model allows adjustment for any number of covariates,
whether they are discrete (e.g., the technique used [CT or MRI]) or continuous
(e.g., age or serum electrolyte level), and then computes a test for each,
including, of course, a statistical test of the difference overall between the
treatment and control groups. Both survival and hazard functions can refer to
outcomes other than death. In the Cox model, this hazard is assumed to be
separable into a product of one function that depends on time and another
function that captures all the other variables including, specifically, the
relative difference between treatment and control groups.
No matter which form of survival analysis statistical test is used, four
assumptions must be met:
- Each person must have an identifiable starting point. All subjects should
enter the trial at the same time in the course of their illness. Using
diagnosis as an entry time can be problematic, because people may have had the
disorder for varying lengths of time.
- A clearly defined and uniform end point is required. This is not a problem
if the end point is death, but it can be a problem if the end point is
recurrence of disease.
- The reasons that people drop out of the study cannot be related to the
outcome. If persons have dropped out because they can no longer travel to
their scheduled appointments as a result of the worsening of symptoms of the
disease under study, the chances of survival could be seriously overestimated.
Otherwise, any changes we see may be due to these secular changes, rather than
the intervention.
- Diagnostic and treatment practices must not change over the life of the
study.
We have said that survival or life table analysis allows us to look at how
long people are in one state (e.g., life) followed by a discrete outcome
(e.g., death). This analysis can handle situations in which the people enter
the trial at different times and are followed up for varying periods; it also
allows us to compare two or more groups
[4]. The methods of life table
(survival) analysis are increasingly used in diagnostic imaging research in
recent years, and we therefore offer a recent review of a relevant research
study [6].
This multicenter study evaluated patients with intramural hematoma of the
aorta and hospital admission less than 48 hr after onset of initial symptoms.
Patients were enrolled between January 1994 and December 2000 after
confirmation of intramural hematoma on two imaging studies (transesophageal
echocardiography, CT, or MRI). Sixty-six patients were consecutively enrolled
over the course of 7 years. They were subjected to medical treatment in an ICU
setting and surgical treatment if indicated (criteria for surgical
intervention are available in the original article). Follow-up of these
patients ranged from 6 to 123 months and included outpatient visits and CT 6
months after the event and yearly thereafter.
From the raw data collected from 66 patients, a Kaplan-Meier curve was
built (Fig. 5). Dissecting
Figure 5, we obtain the
following information: survival is set at 100% at the beginning of the study,
when patients initially present to the emergency department. Each ladder step
indicates a drop in survival—that is, the death of a patient because
that was the event defined as the main outcome. A rapid decline ensues because
close to 20% of patients die in the acute phase. The first loss of information
occurs around 6 months, when the first follow-up is scheduled. The triangles
indicate censored data, and the figure shows that at 20 months, 12 patients
have already been censored. Figure
5 shows that the drop in survival is faster in the initial months
after intramural hematoma: the curve drops faster between 20 and 60 months
than later in the study.
Differential survival of subgroups of the study was assessed using the
log-rank test. The resulting Kaplan-Meier curves obtained from comparison of
patients who received oral ß-adrenergic receptor blockers (experimental
group) and those who did not (control group) are displayed in
Figure 6. Visual analysis
easily reveals that patients taking ß-blockers (upper curve) enjoyed much
greater survival than patients who did not receive the medication (lower
curve). In fact, the upper curve shows that only one patient died early in the
study, and that subsequently all patients from whom information is available
are still alive. However, many censored data points are seen, but there is no
reason to believe these patients have died without knowledge of the study's
investigators, which would falsely lead to the conclusion that ß-blockers
have a protective effect. The log-rank test performed on these two subgroups
of patients revealed important information that was embedded in the initial
Kaplan-Meier curve (Fig. 5) and
could not have been obtained had it not been for this separate analysis.
Conclusion
In this article, we address life table and survival analysis and describe
life table techniques such as the Kaplan-Meier approach. For the comparison of
two or more groups, we describe the Mantel-Cox log-rank test. Finally, we
discuss the Cox proportional hazards model, which examines the difference in
the survival curves and also accounts for other variables (covariates). These
statistical methods allow one to work with nontraditional units of analysis:
person-time rather than person only. These tools are seen increasingly in the
research literature and are gaining popularity in radiology research. These
methods of data analysis have potential applications in many fields of
radiology, most notably in the analysis of screening techniques and
interventional studies.
Acknowledgments
Sadly, Dr. Stolberg passed away last January. His determination and
enthusiasm were key in seeing this project to completion, and we are indebted
to him for all he accomplished.
We express our appreciation and gratitude to Monika Ferrier for her
patience and support, which is always a difficult task with several authors.
She has kept us on track and prepared the manuscript.
References
- Norman GR, Streiner DL. PDQ statistics, 2nd
ed. St. Louis, MO: Mosby, 1997
- Altman DG, Machin D, Bagant TN, Gardner MJ. Statistics
with confidence, 2nd ed. London, UK: BMJ Books,2000
- Kaplan EL, Meier P. Nonparametric estimation from incomplete
observations. J Am Statist Assoc1958; 53:457
-481[CrossRef]
- Norman GR, Streiner DL. Biostatistics: the bare
essentials, 2nd ed. Hamilton, ON: B. C. Decker,2000
- Cox DR. Regression models and life tables. J Roy Statist
Soc 1972;34:187
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- Von Kodolitsch Y, Csosz SK, Koschyk DH, et al. Intramural hematoma
of the aorta: predictors of progression to dissection and rupture.
Circulation2003; 107:1158
-1163[Abstract/Free Full Text]
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Introduction
Overview
