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\begin{center}
ADVANTAGES OF MODELING OVER STRATIFYING PROGNOSIS:
EXTRACTING MAXIMUM INFORMATION FROM 
CONTINUOUS VARIABLES

\end{center}
Frank E. Harrell, Jr., Ph.D., Kerry L. Lee, Ph.D., Daniel B. Mark, M.D.,
M.P.H., Ray A. McKinnis, Ph.D., Robert M. Califf, M.D., Mark A. Hlatky,
M.D., Robert A. Rosati, M.D., David B. Pryor, M.D.

From the Division of Biometry, Department of Community and Family Medicine,
and the Division of Cardiology, Department of Medicine, Duke University
Medical Center.


This work was supported by: Research Grants HL-17670, HL-29436, HL-36587,
and HL-09315 from the National Heart, Lung and Blood Institute, Bethesda,
Maryland; Research Grants HS-03834, HS-04873, and HS-05635 from the
National Center for Health Services Research, Rockville, Maryland; Research
Grant HS-06503 from the Agency for Health Policy and Research; Training
Grant LM-07003 and Grants LM-03373 and LM-00042 from the National Library
of Medicine, Bethesda, Maryland; grants from the Robert Wood Johnson
Foundation, Princeton, NJ; and a grant from the Andrew W. Mellon
Foundation, New York, New York.

Address for correspondence and reprint requests:


Frank E. Harrell, Jr., Ph.D.
Box 3363
Duke University Medical Center
Durham, North Carolina 27710
E-mail: feh@biostat.mc.duke.edu

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\begin{center}
{\bf ABSTRACT

}\end{center}

\section{}Accurate estimates of the prognosis of individual patients with a given
disease are required for therapeutic decisions. Prognostic estimation is
also an implicit requirement in much of clinical and epidemiological
research. Prognosis can be estimated by one of several available methods. 
Some epidemiologists and clinicians prefer to stratify patients into risk
groups or prognostic strata to describe patterns of prognosis and to
control for confounding. To estimate the prognosis of a new patient, they
match that patient with "like" patients whose outcomes are already known.

In this paper, we describe the loss of information that results from using
subgrouping and matching methods. We demonstrate using two examples that
statistical modeling results in a more accurate estimate of the likely
outcome of an individual patient. We discuss how modeling can extract more
of the available information about continuous outcome measurements and
multiple explanatory variables, especially continuous ones such as age and
blood pressure.

Some background discussion relating to these main points is also provided,
including discussion of the coding of the outcome variable.


\underline{Key words}: prognosis, prediction, multivariable regression models, Cox
model, continuous variables, stratification, predictive accuracy.

\newpage
\begin{center}
{\bf INTRODUCTION


}\end{center}

\section{}Clinicians have traditionally used two methods to estimate a patient's
prognosis from accumulated clinical experience: \underline{prognostic stratification}
{}[1], and a related method, \underline{matching} [2]. In prognostic stratification, a
researcher forms strata by separating patients into groups having
dissimilar prognoses, often using clinical judgment with an ad hoc
statistical approach. The strata are fixed once they are formed, and the
prognosis of a new patient is estimated by computing the average outcome in
the strata that would contain that patient. The second method, matching,
finds all patients in a dataset who are "similar" to the new patient and
uses the average outcome of this subset to estimate what will happen to the
new patient. Whereas in prognostic stratification the subsets are fixed
before their use in predicting patients' prognoses, in matching each new
patient may generate a unique matched subset.

Clinicians are more comfortable with those approaches because "real"
patients are used, little mathematical manipulation is required, and the
resulting prediction rule is more "natural" [3]. As the number of
potential prognostic factors becomes large, however, the accuracy of both
methods can suffer. If one attempts to match or subset on many factors,
there will be too few patients from whom to derive accurate estimates of
outcome [4]. This problem is more pronounced when one or more of the
factors is a continuous measurement, as subsetting methods require the
factor to be divided into discrete intervals, effectively increasing the
number of variables to analyze. Clinical researchers may be tempted to
ignore some prognostic factors or to use broad intervals of continuous

variables to be able to obtain adequate subset sizes. A statistical
analysis may be required to decide which factors to ignore. Of course in
complex clinical problems involving primarily qualitative variables,
clinical knowledge can result in optimal predictions. See for example
Wells, Feinstein, and Walter [5].

Multivariable statistical analysis of accumulated clinical experience has
become a practical competitor of subsetting methods, largely because
efficient computer techniques for implementing these methods are now
available. Two analytical approaches are used increasingly in medicine for
this purpose. The first involves dividing the dataset into subgroups that
optimally stratify prognosis in some statistical sense and then deriving a
summary measure of prognosis for each subgroup. Several methods use this
approach, for example, recursive partitioning [6-8] and entropy minimax
{}[9]. A disadvantage of these methods, however, is that since they use
stratification, all predictor variables must ultimately be expressed in
polytomous form. Arbitrary cutpoints must be chosen so that continuous
predictor variables can be categorized into intervals.

The second analytical approach for deriving prognostic predictions,
multivariable regression modeling, has many advantages over other methods. 
Regression analysis makes efficient use of mixtures of dichotomous and
continuous predictor variables. Especially important for prognostic
analyses, regression analysis can also utilize continuous response
variables, such as the time until a clinical event. More information is
extracted from the data as compared with using the life/death status at a
fixed follow-up point. Multivariable regression models are being used with
increasing frequency in medical applications: the logistic regression
model [10] for binary or ordinal dependent variables is appropriate for

modeling diagnosis or the outcome of an acute illness, while the Cox
proportional hazards regression model [11] is the most widely applicable
method for modeling the time until a clinical endpoint in a chronic
disease. 

In the following sections, we will elaborate on the advantages of
multivariable modeling over stratification techniques for estimating the
prognosis of individual patients. A companion article [12] discusses how
to measure the worth of two competing predictive methods as well as the
pitfalls in regression modeling that must be overcome for its advantages to
be realized.

\begin{center}
{\bf SOME PROBLEMS WITH STRATIFICATION

}\end{center}
\section{\underline{Stratifying the Outcome}}

The choice of how the outcome is quantified has a significant impact on
the efficiency of the analysis. Many investigators choose to predict
whether or not a patient suffers an event within a specific time, for
example to predict death within one year after hospitalization. This
approach adequately describes the outcome of an acute illness, in which the
patient either dies quickly or recovers. In chronic illness, however,
prognostic information is wasted by not differentiating between a death at
1 day and a death at 11 months. In one general setting, Franco [13] showed
that the best possible dichotomization of event times (i.e., classification
as above or below the true median survival time) results in an efficiency
of 75\% relative to an analysis making full use of continuous failure times. 
That is, dichotomization of a continuous outcome variable can reduce the

effective sample size by at least 25\%.

Many analyses use hospital death as the endpoint. When comparing multiple
hospitals with different discharge patterns, the use of hospital death can
cause artificial hospital effects on outcome. 

Besides bias and the loss of statistical information and efficiency, there
are several practical reasons for not dichotomizing a continuous outcome
variable. First, if death within one year was used as the dependent
variable in a discriminant or logistic regression analysis, patients who
have not died and have not been followed a full year must be omitted from
the sample. Otherwise, their outcomes are not well defined-the one-year
status of a patient followed only two months is unknown. Yet, there is
information contained in knowing that the patient is alive after two
months. Second, if a model is developed to predict the probability that a
patient will live one year, it will not help the clinician who wants to
know the chances of a patient living for six months or for two years. 
Finally, dichotomization of a variable introduces arbitrariness into the
analysis. An analysis of death within one year may yield different answers
from an analysis of death within two years, often merely due to the fact
that the standard errors of regression coefficients vary with the
frequencies of the different outcomes.

A multiple regression model used in analyzing prognosis as a continuous
outcome can solve these problems, but it must be able to take censored
responses into account. For a patient who has been followed 5 months
without suffering the event of interest, the analysis should use that
information and treat the response as 5+, or censored at 5 months.

\underline{Stratifying Predictor Variables}


In many clinical or epidemiological studies, it is customary for
investigators to describe or predict some outcome variable by stratifying
patients with respect to other variables and computing descriptive
statistics in each stratum. However, as most biological phenomena act on a
continuous or dose-response relationship, such stratification results in
loss of information and precision.

Modeling biological outcomes always involves some degree of approximation,
so models can be considered only first-order approximations of reality in
most cases. Many investigators distrust modeling for this reason and
prefer to make estimates without assuming any functional form for the
relationships observed. However, a purely empirical approach often makes
poor use of the available information. Furthermore, deriving estimates by
stratifying patients into groups introduces new problems: the groups are
not homogeneous, and the connection between groups which allows one to gain
statistical information through interpolation is lost.
\underline{A Simple Example}

The loss of information by subsetting data rather than modeling can be
appreciated by a simple example. Suppose that a response variable $Y$ is
related to an independent variable $X$ non-linearly by the equation

Y = X^{2}/300-X/20+2+random error, (1) 
where the random error has a normal distribution with mean zero and
variance two. If we generate a random sample of size n=60 from this model
for X=1,22,\ldots,60 we might obtain data points as shown in Figure 1. The
true regression line (omitting the random error) given by (1) is shown as a
solid curve. The sample variance of $Y$ is 9.44.

A typical set of stratified estimates might be obtained by dividing the
original set of patients into three groups: X=1,2,\ldots,20, X=21,\ldots,40, and

X=41,\ldots,60. The mean $Y$ in each group will be taken as the estimate of the
expected outcome for that group. These means are displayed as dashed lines
in Figure 1. From the figure, it is apparent that as long as a new patient
has an $X$ value near the mean of one of the three $X$ groups, the estimate of
that patient's $Y$ will be accurate. The farther the new $X$ is from one of the
group means, the more biased is the estimate of Y. This is reflected by
the MSE of the 3-group mean method (sum of squared differences between
predicted and observed Ys divided by 60 minus number of estimated
parameters) being 3.48 (R^{2} = 0.65).

Before fitting an ordinary regression equation to the data, a
conscientious analyst would estimate the true form of the relationship
between $X$ and Y. Restricted cubic spline regression is a
minimal-assumption method of estimating shapes that is applicable to all
regression models [14]. The essence of the approach is to use a standard
estimation method to estimate the coefficients of a piecewise cubic
polynomial that is constrained to be smooth everywhere and for stability is
also constrained to be linear in the trials. The join points for the
polynomials are called "knots". A 4-parameter (omitting intercept) cubic
spline estimate of the X-Y relationship, using knots at X=4,17,30.5,44,57,
is shown as a dotted line (MSE=2.35, R^{2} = 0.77). The test of linearity
using the spline fit yields p<0.0001. The analyst who did not check for
nonlinearity, however, would fit a least squares linear regression line as
shown with the other dashed line in Figure 1. Let us suppose that we want
to use this improper linear regression model to predict $Y$ for a new patient
having a specific $X$ value, likely to be any integer in the range
1,2,\ldots,60. The MSE of the linear fit is 2.88 and R^{2} = 0.70. Had the
correct quadratic regression been fitted, the mean squared error would be

2.29 (R^{2} = 0.77). Thus, a simple but inappropriate linear model achieved
more accurate predictions (MSE = 2.88) than did stratification (MSE =
3.48). In this example, stratification resulted in biased estimates; since
only three groups were formed, the variance of estimates was not the major
problem.

\begin{center}
\section{RELATIVE EFFICIENCY OF SUBGROUPING VS. MODELING}

\end{center}

In the simple example above, stratification caused inaccurate estimation
primarily because of bias. In more complicated situations involving
multiple predictor variables, stratification causes two other problems. 
First, stratification by multiple variables causes instability of estimates
by effectively fitting too many parameters. For example, describing
patient outcomes according to sex and race of the patient would require the
formation of four patient groups if race were dichotomized. For a
continuous uncensored outcome variable, this is synonymous with fitting a
model containing four intercepts (representing the four group means). If
there is no significant sex-by-race interaction, a more stable model could
be derived that used only three parameters instead of four: an overall
intercept, an additive effect due to being male, and an additive effect due
to being non-white. Such a model would be expected to generate estimates
having lower variance than would the four parameter model.

The second problem caused by subgrouping is the small group size necessary
to obtain homogeneity, resulting in estimates with large variance. 
Modeling uses information from all patients to interpolate estimates for a
specific patient; it is not necessary to form small patient groups.
\section{\underline{Example}}


We performed an experiment to ascertain the relative predictive
discrimination of modeling and subgrouping for deriving prognostic
estimates in a chronic disease. The experiment used patient data stored in
a computerized information system described previously [15]. From 1969
through mid 1980, 1504 patients undergoing cardiac catheterization for
suspected coronary artery disease at Duke University Medical Center were
found to have significant coronary artery disease and were treated with
medical (nonsurgical) therapy for at least six months after
catheterization. Of the 1504 patients, 1485 had complete data for the ten
predictor variables used in the experiment, which are shown in Table 1. 
These variables were selected from a larger list using a Cox model and
stepwise variable selection. The variables are listed in order of
selection, according to statistical significance.

Follow-up for this cohort had been obtained at six months after
catheterization, at one year, and at yearly intervals until 1981; follow-up
is 99\% complete. The endpoint we used is time until cardiovascular death. 
For patients undergoing coronary artery bypass surgery or suffering
noncardiovascular death, the survival time was censored at the point of
surgery or death, respectively. There were 298 cardiovascular deaths in
the sample.

To derive prognostic predictions by a subsetting method, we chose matching
over stratification since it was expected to be more efficient. We took
each patient in turn and identified a matching subgroup from the remaining
1484 patients using as many variables in Table 1 as possible. For discrete
variables, an exact match was required. For ejection fraction, patients
were considered to match when the difference in ejection fractions was 0.10
or less. For arteriovenous oxygen difference, patients were considered

matched if the difference was 0.75 vol\% or less. Matching on a symmetric
interval about a given patient's values of these two variables solves much
of the bias problem mentioned in the earlier example. Matching is not used
frequently in practice since the composition of the subgroups varies with
the patient for whom a prediction is requested, requiring access to the
entire dataset for making each new prediction. Subsetting strategies
received two other advantages in this comparative study. First, patients
were matched on the variables selected to be important in a Cox model
analysis of all 1485 patients, and the matches were made on as many
variables as possible until a given minimum number of patients were
selected, with the variables used in order of significance as judged by the
Cox model. Second, matches were made from a potential number of 1484
patients, whereas the regression modeling strategy used only 725 patients
for model development (see below).

When forming patient subgroups, either a given number of variables on
which to match or given number of patients can be fixed. These quantities
can be varied to study their effect on the accuracy of predictions. We
chose to vary the minimum number of matched patients. First, an attempt
was made to match the patient with other patients using all 10 variables. 
If fewer than a chosen number of patients were found to match, patients
were then matched on only the first 9 variables. This process continued
until at least the specified number of patients matched. The number of
variables matched on decreases as the minimum subset size increases.

Once a matched patient sample had been found, the Kaplan-Meier [16]
estimate of the probability of living two years was calculated using the
observed survival and follow-up times for that group of patients. The
two-year point was selected to allow quantification of the subsetting

estimates into a single number and because most of the patients had been
followed two or more years. To assess the predictive accuracy of these
estimates, we compared the Kaplan-Meier two-year estimate with the observed
survival for the single patient being matched. We recommend the use of the
most general form of the rank index of predictive discrimination [12], c,
for this assessment, a measure of correlation between the predicted
survival and the observed survival time in days, which may be censored. 
However in this experiment, this requires over one million paired
comparisons for n=1485. To reduce computer time in the multiple analyses
involved in the experiment, we chose to compare the predicted value with
the binary outcome of whether or not the patient survived two years,
equivalent to calculating a Somers' rank correlation or area under a
receiver operating characteristic curve. Because the life/death status was
undefined for patients not dying within two years who had not been followed
at least two years, the outcomes of these patients were not predicted and
were not used in calculating the c index.

The results of the subsetting experiment are displayed in Table 2. A
minimum subset size of 20 patients gave the best predictive discrimination
with c=0.778. If one allowed matching on more variables by allowing
smaller subsets to be used, c decreased because of instability of the
estimates derived from small sample sizes. If one matched on fewer
variables by requiring the subsets to be larger, c also decreased due to
the greater heterogeneity of the matched set.

We next compared the prediction ability of the subsetting method with that
of a Cox proportional hazards model [11]. This model can be stated as
follows. Let the patient descriptors be denoted by X_{1}, X_{2}, \ldots, X_{p}. The
model assumes that the probability the patient will survive until time t,

S(t), is given by

S(t) = S_{0}(t)^{exp(B_{1}}^{X_{1+B2X2}}^{+ \ldots+B_{pXp) }}, (2)
where the Bs are regression coefficients or weights for the patient
variables, and S_{0}(t) is an unspecified survival curve for a "standard"
patient (typically chosen as a hypothetical patient having $X$ values equal
to the mean values over all patients). This model is a major advance over
older models because the S_{O}(t) part of the model is arbitrary and
unspecified; S_{O}(t) is determined empirically for each dataset. Thus one
does not need to assume a specific shape for the survival curves, only a
relative shape with respect to the "standard" curve S_{O}(t).

Since it is computationally expensive to leave out each patient in turn,
develop a Cox model, and predict the outcome of the omitted patient, we
chose to obtain a fair estimate of the accuracy of the Cox model by
dividing the patients at random into two groups of roughly equal size. The
first half or training set was used to derive the model. The second half
or test set was used for testing the model. Had we developed and assessed
the model using all patients, an over-optimistic result would have been
obtained [17]. The training sample contained 725 patients of whom 142 had
suffered cardiovascular death. The test sample consisted of 760 patients
and 156 cardiovascular deaths. The model we fitted contained only the
first three variables in Table 1. For calculating c, we estimated the
probability of surviving two years for each patient in the test sample
using the method of Kalbfleisch and Prentice [18] with the Cox model. Upon
comparing predicted and observed survival in the test sample, the result
was c=0.792, which is better than any of the discriminations achieved by
the subsetting method (best c=0.778). Using a data reduction method called
"incomplete principal components regression" discussed in Harrell et al.

{}[17], one can obtain a Cox model having predictive discrimination of
c=0.824 in the test sample using all 67 variables from which the 10
variables in Table 1 were selected.

Though the subsetting strategy was given every advantage, even a simple
Cox model achieved greater accuracy. This was due in large part to three
factors: (1) the efficient way the model used left ventricular ejection
fraction (a powerful continuous variable), (2) the use of more patients to
derive any one prediction, and (3) the imposition of structure (additivity)
on the relationship between independent variables and the response
variable.

\begin{center}
{\bf DISCUSSION
}
\end{center}

Multivariable statistical models, when used with care, can extract more
prognostic information from data than can stratification. This is
demonstrated by comparing the accuracy of predictions derived from modeling
with that obtained with stratification.

The advantages of modeling are pronounced when one of the most important
prognostic factors is a continuous variable. Failure to use information
from the entire range of such a variable can result in inaccurate
predictions and make a prognostic model more complicated than it need be to
make up for the loss of information. For example, the Multicenter
Postinfarction Research Group [19] derived a model for cardiac mortality
using the variables left ventricular ejection fraction (LVEF) below .40,
premature ventricular contractions at least 10/hr., presence of lung rales,
and New York Heart Association functional class II-IV. These four
variables were able to stratify one-year mortality from 2\% to 47\% (see

their Figure 3). However LVEF grouped into only four intervals was also
able to stratify mortality from 2\% to 47\% (see their Figure 1). Had all
the information been extracted from the LVEF measurement, that single
measurement alone would have provided roughly the same amount of prognostic
information as a more complicated four-variable model. In addition, the
independent effect of premature ventricular contractions might have been
found less important had full adjustment for LVEF been made.

As stated in the Introduction, clinicians and others often prefer a
stratification-based rule because each node on the stratification tree
represents an easily described patient group. But a price is often paid in
terms of accuracy of estimates. The choice between stratifying and
modeling to a large extent rests on the choice between bias and variance. 
Stratification is model-free and hence can reduce bias when compared to a
model whose assumptions are violated. Cutting continuous variables into
intervals increases the variance of estimates. If the intervals are wide,
stratification also suffers from bias when a patient's values are not near
the center of an interval. The mean squared error of an estimate, and the
probability that an estimate is close to the true value, are derived from
bias^{2} and variance. Individual patient matching can reduce bias but will
not reduce variance caused by subgrouping. However, unless the sample size
is extremely large, variance will usually dominate bias. Since models
impose structure (e.g., no interaction) to reduce variance, models
frequently yield more accurate estimates even when the structure imposed
(e.g., linearity in example 1) is not satisfied by the data. 

Stratified estimates are easier to use when there are continuous
predictors. However, with the use of a nomogram [20], even complex
nonlinear relationships with multiple continuous variables can be evaluated

quickly and easily to obtain a clinical prediction.

\newpage
\begin{center}
{\bf ACKNOWLEDGEMENTS
}
\end{center}

The authors wish to thank Ms. Penny Hodgson, Ms. Denise Pearsall and Ms.
Lisa Davis for their expert assistance in the preparation and editing of
this paper.

\begin{center}
\section{REFERENCES}
\end{center}

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\newpage
\begin{center}
{\bf TABLE 1


}\section{\underline{Prognostic Factors used for Predicting Time Until Cardiovascular Death}}

\end{center}
\section{\underline{}}
Left ventricular ejection fraction
Presence of significant (\underline{>}75\% diameter narrowing) left main
coronary artery occlusion
Number of major coronary arteries significantly occluded (0-3)
Presence of nocturnal chest pain
Arteriovenous oxygen difference (vol\%)
Apical left ventricular asynergy
History of peripheral vascular disease
History of Prinzmental's variant angina
Presence of cardiomegaly on chest X-ray
Presence of progressive angina


\newpage
\section{TABLE 2}

\section{\underline{Predictive Discrimination of Subsetting Method}}


 MinimumMedian Number ofMedian Number of
\underline{Subset SizeVariables MatchedPatients Matchedc-index}

\underline{}510150.724
109210.765
158270.769
208330.778
257400.773
306470.771
405620.768


\newpage

\section{\underline{Figure Legends}}


\section{\underline{}}Figure 1: Estimates obtained by stratifying and by fitting an
 oversimplified model.
 True model is given by (1). Quadratic fit is
optimal. Spline function does not assume form of true
regression curve other than its smoothness.
\end{document}
