Identification of Individuals With
Covert Ischemic Thrombotic
Cerebrovascular Disease:
A Discriminant Function Analysis
BY MENARD M. GERTLER, M.D., JAMES L. ROSENBERGER, M.S.,
AND HILLAR E. LEETMA, M.D.
Abstract:
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Identification
of Individuals
With
Covert
Ischemic
Thrombotic
Cerebrovascular
Disease:
A Discriminant
Function
Analysis
• Sixty-one men with ischemic thrombotic cerebrovascular disease (ITCVD)
and 61 age-matched controls ( ± three years) were studied to select risk factors
which would identify ITCVD-prone individuals from a healthy population.
Numerous clinical and biochemical measurements were made on all subjects
including a three-hour oral glucose tolerance test concomitant with the
immunoreactive insulin response (IRI). The following parameters were selected
as risk factors for ITCVD from the many variables assessed: elevated systolic
blood pressure; triglycerides; uric acid; glucose one-hour, two-hour and threehour levels; and IRI two-hour and three-hour levels. There is also a higher
prevalence of abnormal lipoproteins (mostly type IV), abnormal EK.G, and
hypertension in ITCVD. A stepwise linear discriminant function analysis was
used to select the risk factors which best separate and classify the ITCVD and
control subjects. These primary risk factors are the IRI three-hour level, systolic
blood pressure, abnormal lipoproteins, and glucose two-hour level. Utilizing
only these four primary risk factors in a discriminant function, 84.4% of the
subjects were correctly classified. The contribution of the other variables is
minimal in the discriminant function, but they are applicable as individual risk
factors of ITCVD.
Additional Key Words
glucose tolerance test
immunoreactive insulin
stroke
lipoproteins
risk factors
Introduction
• The high incidence of ischemic thrombotic
cerebrovascular disease (ITCVD), particularly
in the elderly, and its debilitating effect in all
age groups has made early recognition and
prevention of this disease of paramount
importance. In the age decade of 60 to 70
years about three individuals per 1000, both
men and women, experience ITCVD each
From the Division of Cardiovascular Research,
Department of Rehabilitation Medicine, New York
University Medical Center, New York, New York.
Supported by grants from The John A. Hartford
Foundation, Inc., and the Social and Rehabilitation
Service of the Department of Health, Education and
Welfare, Grant No. 13-P-55064/2.
764
blood pressure
year.1 However, the long incubation period of
ITCVD makes it possible to recognize this
disease during its covert state. Thus primary
prevention can be instituted during the covert
stage long in advance of the earliest overt
symptomatology, e.g., transient ischemic attacks, in the attempt to prevent or delay the
acute episode. This study was designed on the
experience gained from a project which
resulted in the successful identification of
individuals prone to ischemic heart disease
(IHD). 2 Since IHD and ITCVD are both
atherothrombotic and possess some risk factors
in common (fig. 1), it was reasonable to
extend similar techniques to the identification
of individuals prone to ITCVD.
Strok; Vol. 3, Nor»mber-O»cember 7972
A DISCRIMINANT FUNCTION ANALYSIS
FIGURE 1
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Schematic representation of individual risk factors for
the classification of 1HD and ITCVD-prone individuals.
The purpose of this study was to develop
an effective tool for classifying individuals
prone to ITCVD. Risk factors were selected
from a large number of clinical and biochemical variables studied in a retrospective epidemiological study of ITCVD." 4 The technique
of multiple discriminant function analysis was
then employed to derive a function of the
selected risk factors which would best discriminate individuals with covert ITCVD from their
healthy counterparts. This communication will
report on these findings.
Methods
The study was based on 61 men with ischemic
thrombotic cerebrovascular disease (ITCVD)
selected from a retrospective stroke study of 364
patients admitted to the Institute of Rehabilitation
Medicine, New York University School of
Medicine, New York, New York. The criteria for
selection were absence of associated overt diseases
such as ischemic heart disease, hyperthyroidism,
hypothyroidism, diabetes mellitus, gout, and the
absence of medications known to influence the
carbohydrate metabolism (steroids or thiazides)
and lipid metabolism (hypolipemic agents). The
diagnosis of each case was confirmed by history,
clinical and neurological examinations, lumbar
puncture, electroencephalography and/or cerebral
angiography. All subjects were ambulatory and at
the time of study were participating in an active
rehabilitation program. The biochemical and
metabolic evaluations were made between one and
three months following the acute episode. The
healthy male controls were volunteer participants
from the New York metropolitan area and were
individually age-matched ( ± three years) with
Stroke, Vol. 3, Noramber-Otnmbar
1972
each ITCVD subject. In all, a data base of 61
ITCVD men and 61 age-matched male controls
was available for the ensuing analysis.
The following measurements were made on
all study subjects: systolic and diastolic blood
pressures, height, weight and ponderal index.
Biochemical determinations, accomplished on
fasting serum, were total cholesterol, lipid
phosphorus, triglycerides, and uric acid. A threehour oral glucose tolerance test (GTT) was
administered to each study subject. Blood glucose,
plasma immunoreactive insulin (IRI), free fatty
acid (FFA) and lactate levels were determined at
fasting, and following ingestion of the equivalent
of 75 gm of glucose in the form of Glucola (Ames
Co. Inc., Elkhart, Indiana) at one-half, one, two
and three hours. The methods for these determinations are described elsewhere.4' 8
The continuous variables, e.g., cholesterol,
uric acid, etc., were individually compared
between the ITCVD group and the age-matched
control group for a difference in the mean, using a
two-tailed /-test where applicable. The categorical
variables, e.g., presence or absence of hypertension, normal or abnormal electrocardiogram
(EKG) and lipoprotein patterns, were tested for
discriminating power with a chi-square analysis.
All study variables showing differences between
the two groups were considered risk factors of
ITCVD and were included in the multivariate
discriminant analysis. In cases where one or more
data items were missing, the grand mean of the
variable was inserted for the missing data.
The linear discriminant function analysis of
the risk factors was accomplished in a stepwise
manner, i.e., at each step of the analysis all the
variables not yet entered into the function were
tested to select the next most discriminating
variable. In addition, the F ratio was calculated
and tested to insure that each risk factor in the
function contributed significantly to the discrimination between the diseased and healthy groups.
The stepwise procedure was terminated when the
remaining variables provided no additional contribution to the overall discrimination.
Denoting the inverse of the pooled variancecovariance matrix by o-1J, the discriminant function coefficients (C | ) are calculated by
i
and the constant term
C(1 = -(1/2).2(X,( 2 )-1-X 1 ( 1 >) -C,,
_
I
where Xji-> is the mean of the jth variable in the
ITCVD group and Xj (1) in the control group.
The discriminant function is then written Y =
SC( • X, + Co where Y is the function (profile)
i
score and the X, are the values of the risk factors
765
GERTLER, ROSENBERGER,
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for an individual in the units of those variables
used to derive the function. The C, are the discriminant function coefficients described by
Fisher in 1936 which are derived to maximize the
ratio of variance between two samples relative to
the variance within the samples.6
The discriminant function is derived so that
the best criterion of classification is a function
(profile) score greater than zero for patients who
are "ITCVD prone" and less than zero for those
who are "non-ITCVD prone." However, if the a
priori probability of an individual belonging to
one of the two groups is known, i.e., the
probability of ITCVD in the general population,
then the criterion of classification is the loge
(PI/P-J)- where p2 is the probability of ITCVD
and Pi non-ITCVD, i.e., pj = 1 — p 2 . 7 An
individual is then classified as "ITCVD prone" if
his calculated function score is greater than loge
(Pi/P).
The transformation of a discriminant function into a multiple logistic function has been
described,8' ° resulting in the risk function
1
P = 1 +e(l0gc[Pl/P,.]-C0-2C1-X1)
1
where P denotes the conditional probability or
LEETMA
risk and Co and C, are as defined above. This
calculation of the multiple logistic function with
the term loge (Pi/p 2 ) is equivalent to the
classification criteria when the a priori probability
is that the individual will experience the overt
disease.
Results
SELECTION OF RISK FACTORS OF ITCVD
The 61 ITCVD men and 61 age-matched
controls were compared for significant differences in the mean in order to select the risk
factors of ITCVD from the clinical and
biochemical variables evaluated in this study
(table 1 ) . No significant differences were
found in height, weight, ponderal index,
cholesterol and lipid phosphorus levels between
the diseased and the control groups. The
variables showing a significant difference
between the ITCVD group and the agematched control group were considered individual risk factors and were entered consequently into the discriminant function analysis.
Significantly elevated systolic and diastolic
blood pressures, elevated glucose, triglyceride
and uric acid levels, abnormal IRI, FFA and
lactate response in ITCVD indicate that any
TABLE 1
Summary of Continuous Variables in ITCVD Males and Age-Matched Controls
ITCVD,
m*on
<N - 61)
Variablm
S.D.
Controls,
mtan
(N - «1)
S.D.
p<
Height
Weight
Systolic B. P.
Diastolic B. P.
Cholesterol, mg %
Lipid phosphorus, mg %
Triglycerides, mg %
Uric acid, mg %
GTT fasting, mg %
One-half hour
One hour
Two hours
Three hours
IRI fasting, p units/ml
One-half hour
One hour
Two hours
Three hours
Lactate* three hours, n mol/ml
FFA* three hours, fi Eq/L
63.8
67.6
159.0
147
86
228
10.0
149
6.1
95
137
162
146
116
18
55
83
96
74
1.13
162
9.1
2.6
19.9
23.2
13.6
40.9
1.5
41.1
1.1
15.2
24.7
41.5
47.9
49.2
10.0
36.6
60.5
56.7
45.4
0.36
146.1
62.7
68.2
166.0
133
81
228
9.7
119
5.4
92
141
143
105
75
15
55
76
60
26
0.92
246
8.9
2.4
21.8
16.1
9.3
41.3
1.4
64.8
1.1
9.5
22.2
32.9
31.9
19.2
7.1
34.9
49.8
46.0
19.0
0.32
151.0
NS
NS
NS
0.001
0.050
NS
NS
0.010
0.001
NS
NS
0.010
0.001
0.001
0.050
NS
NS
0.001
0.001
0.050
0.050
Age
'Measured on a subsample of 37 ITCVD subjects and 28 controls.
766
Slrokt, Vol. 3, Nov»mb«r-D«c«mber 1972
A DISCRIMINANT FUNCTION ANALYSIS
TABLE 2
Significant Chi-Square Comparisons in Categorical
Variables Between ITCVD Males and Age-Matched
Controls
Variable*
ITCVD
Lipoproteins
Normal
Abnormal
Hypertension
12 (33%)
24 (67%)
24 (77%)
7 (23%)
0.001
39 (64%)
22 (36%)
55 (90%)
6 (10%)
0.01
51 (84%)
10 (16%)
61 (100%)
0 (0%)
0.01
No
Yes
Control!
P<
EKG
Normal
Abnormal
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one of these variables can be considered as an
individual risk factor and may be predictive of
ITCVD. Chi-square analyses of lipoprotein
patterns, history of hypertension, and electrocardiograms revealed a significantly greater
prevalence of these abnormalities in ITCVD
than in control subjects (table 2 ) .
MULTIVARIATE ANALYSIS OF RISK FACTORS
The discriminant function analysis, calculated
in a stepwise manner as described in Methods,
selected six variables which in the function
analysis displayed the greatest discriminating
power (table 3). The order in which the
variables are entered into the discriminant
function is determined by their relative discriminating power dependent on the variables
previously entered.
The weight or contribution of each
variable in the discriminant function is expressed as a standardized coefficient, derived
by multiplying the function coefficient by the
standard deviation of that variable. This
reflects the positive or negative contribution of
each variable corresponding to a change of one
standard deviation.
In order to evaluate the discriminant
function of these six risk factors, the function
score was calculated for each ITCVD and
control subject. The scores for each group were
arranged in ascending order and the cumulative
percentage was calculated over the range of the
score values. Figure 2 shows the graph of
cumulative percentages for the ITCVD subjects and 100% minus the cumulative percentage for the controls. Using zero as the criterion
for separation of the two groups, subjects with
a positive score, i.e., greater than zero, were
classified as "ITCVD prone," whereas subjects
with a negative score were classified "nonITCVD prone." According to this discriminant
function seven of the controls and ten of the
ITCVD subjects were misclassified. This
resulted in correct classification of 86.1% of
the study population.
Since all the distributions of the continuous variables show a positive skew of varying
degree, a logarithmic transformation was
performed on each of these variables, which
eliminated the skewness from the distributions
and made them more normal or Gaussian. The
transformed data base then was used to
calculate a second discriminant function, which
included the same variables selected in the first
function with the addition of the three-hour
glucose level. However, the addition of this one
variable and the use of the logarithmic
transformation classified correctly only one
TABLE 3
Discriminant Function of Those Variables Selected by the Stepwite Procedure
Function
coetfWeirt<Ci)
VoriBkMXl)
0.033
2.350
0.040
3.300
)0.055
0.035
)9.200
IRI three hours (/* units/ml)
Hypertension (0, 1)*
GTT two hours (mg % )
Lipoprotein ( 1 , 2)f
GTT one-half hour (mg % )
Systolic B. P. (mm Hg)
Constant (Co)
S.D.
(42.2)
( 0.422)
(45.4)
( 0.502)
(23.5)
(21.1)
Standardized
coefficient
1.41
0.99
1.79
1.66
-1.29
073
*Code: 0 = none, 1 = hypertension.
fCode: 1 = normal, 2 = abnormal.
Discriminant function score: Y = 2C, • X, + Co.
l
Stroke, Vol. 3, November-December J972
767
GERTLER, ROSENBERGER,
CL'ML'LATIVE PERCENTAGES OF THE DISCRIMINANT
FUNCTION SCORES FOR ITCVD SUBJECTS AND CONTROL!
. 7 - t . - S - 4 - 3 - 2 . 1 0
1 2
3
4
5
b
7
8
DISCRIMINANT FUNCTION SCORE
FIGURE 2
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Cumulative percentages of the discriminant function
scores for ITCVD and control subjects.
additional subject. Thus, this function correctly
classified 86.9% of the subjects versus 86.1%
by the preceding function.
DEVELOPMENT OF A PROFILE
OF THE ITCVD-PRONE INDIVIDUAL
Since several variables reflecting glucose tolerance are included in the above functions, i.e.,
blood glucose one-half hour, two-hour and
three-hour levels, and two measures of hypertension, i.e., personal history of hypertension
and systolic blood pressure, it was thought that
little discriminating power would be lost by
employing only one measure each for the
LEETMA
functions. (2) Systolic blood pressure was
included rather than hypertension because
elevated levels are indicators of hypertension
and small changes are correspondingly reflected in the function score. (3) The lipoprotein
classification had a strong standardized coefficient in the preceding functions. (4) The
glucose two-hour level is the best indicator of
glucose intolerance since it has the strongest
standardized coefficient of all the glucose
values during the GTT. The discriminant
function analysis of the four variables, now
designated as primary risk factors of ITCVD,
is shown in table 4. Again, using zero as the
best criterion for discriminating between the
ITCVD and controls, only 11 ITCVD subjects
and eight control subjects were misclassified,
resulting in correct classification of 84.4% of
the subjects.
IDENTIFICATION OF THE ITCVD-PRONE INDIVIDUAL
The application of the discriminant function in
an actual screening program is demonstrated
by the following examples:
Subject: *1
IRI three-hour level
Systolic B. P.
Lipoproteins
Glucose two-hour level
Age: 66
32 fi units/ml
154 mm Hg
Normal
108 mg %
The discriminant function from table 4 is as
follows:
0.033 X (32) + 0.039 X (154) + 2.48 x (1) + 0.020 x (108) - 13.3 = - 1 . 5 .
glucose tolerance and the blood pressure.
Therefore, a third discriminant function was
calculated and the rationale for the inclusion of
only four variables is as follows: (1) IRI
three-hour level is the most powerful individual
discriminator shown by the previous two
Since the resulting function (profile) score
value is less than zero, the subject is classified
as "non-ITCVD prone." Employing this same
subject, the transformation of his discriminant
function score of —1.5 into an actual probability (P) of ITCVD is accomplished as follows:
TABLE 4
Discriminant Function of the Primary Ritk Factort
VarioM*<Xi)
IRI three hours In units/ml)
Systolic B. P. (mm Hg)
Lipoprotein ( 1 , 2)*
Glucose two hours (mg % )
Constant (Co)
Function
co*ffld»nt<Ci)
0.033
0.039
2.480
0.020
- 13.300
t.D.
(42.2)
(21.1)
(0.502)
(45.4)
Standardlxad
coefficient
1.40
0.83
1.24
0.91
•Code: 1 = normal, 2 = abnormal.
Discriminant function score: Y = 2C,X, + C o = O.O33x(IRI three hours) + 0.039x(syst.
B.P.) + 2.48x(lipoprotein) + 0.020x(glucose two hours) — 13.3.
768
Stroke, Vol. 3, November-December 1972
A DISCRIMINANT FUNCTION ANALYSIS
Assuming the unconditional probability per
year of ITCVD in a man in the age range of 60
to 70 years to be 0.003, then
P=1+ e
1
= 0.00067.
(log,, (0.997/0.003)+ 1.5)
Therefore the conditional probability of
ITCVD in this subject is 0.00067, which is
below the assumed risk for the entire population.
Subject: # 2
IRI three-hour level
Systolic B. P.
Lipoproteins
Glucose two-hour level
Age: 62
55 JJL units/ml
160 mm Hg
Abnormal
150 mg %
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The function (profile) score is again calculated
as follows:
such as diastolic blood pressure, uric acid,
triglycerides, FFA and lactate three-hour
levels, and EKG abnormalities demonstrate
univariate discriminating power, but when each
is included in the multivariate function as an
additional variable to the set of more powerful
discriminators, its additional contribution in
the classification of ITCVD-prone individuals
is not significant. However, it must be
remembered that each of these variables may
be considered clinically as an indicator of
covert ITCVD.
There are major differences as well as
some similarities in the risk factors which
enable one to classify ischemic heart disease
(IHD) and ITCVD from each other and from
other individuals in the population (fig. 1).
0.033 X (55) + 0 . 0 3 9 x (160) + 2.48 X (2) + 0.020 X (150) - 13.3 = 2.7.
The function score of 2.7 being greater than
zero classifies the subject as "ITCVD prone."
Transforming this score by the multiple logistic
function yields a probability of ITCVD of
0.04, i.e.,
1
p=1 +
(log, (0.997/0.003) - 2 . 7 )
= 0.04.
This value indicates that the subject is
(0.04/0.003) = 13 times more vulnerable to
ITCVD than the average individual in the
population.
Discussion
There have been various publications concerning the risk factors of ITCVD derived from
prospective studies. 1 0 ' u These publications
included as risk factors hypertension, diabetes
mellitus, hyperlipoproteinemia, and possibly
hyperuricemia, which are consistent with the
individual risk factors shown in tables 1 and 2.
Since a complete evaluation of carbohydrate
metabolism including a GTT was made in this
study, the role of impaired glucose tolerance
and elevated IRI response was found to be
most important in the delineation of the known
risk factors. Accordingly, only those variables
which contribute significantly to the overall
classification of ITCVD are included in the risk
function used for the selection of ITCVDprone individuals.
The entire set of variables in tables 1 and
2 which show significant differences can be
considered as risk factors of ITCVD. Variables
Stroke,
Vol. 3, November-December 1972
The differences are chiefly in the lipids, e.g.,
cholesterol and phospholipids, which are
elevated in IHD and in the two-hour postprandial glucose and the three-hour IRI levels
which are elevated to a greater extent in
ITCVD. Thus, it may be said that IHD is
primarily a disease of abnormal lipid metabolism, whereas ITCVD is more of a disease of
disordered carbohydrate metabolism. The elevation of systolic blood pressure is common to
both diseases, but the strength of its contribution varies, with the emphasis probably in
ITCVD. This may be due in part to age
differences inasmuch as the ITCVD patients
are on the average ten years older than patients
with IHD.
It appears from the standardized coefficients in table 3 that the history of hypertension is a more powerful discriminator than
systolic blood pressure. However, there are two
possible explanations for this: Since the
diseased and age-matched control subjects are
in the seventh decade and it is known that
some parameters approximate each other, e.g.,
cholesterol at older ages, thus a history of
hypertension which reflects the blood pressure
at a younger age is a stronger discriminator.
Furthermore, it is possible that blood pressure
is subjected to therapy either directly or
indirectly during the earliest part of hospitalization which tends to reflect in a lower mean
systolic blood pressure for the ITCVD subjects.
769
GERTLER, ROSENBERGER, LEETMA
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The impaired glucose tolerance and
abnormal IRI response in ITCVD are shown in
table 1, and both of these risk factors
contributed significantly to the discriminant
functions. Since carbohydrate metabolism had
not been evaluated in these subjects prior to
this study, there was little or no manipulation
which would affect the glucose tolerance of
these patients. Consequently the glucose and
IRI measurements are probably unbiased.
Furthermore, ITCVD subjects with a history of
diabetes mellitus were excluded from the
study.
Even though the triglyceride and uric acid
levels were significantly higher in ITCVD
subjects than in control subjects, these individual risk factors did not enter the discriminant
function. However, they are probably reflected
in the greater number of abnormal lipoprotein
patterns found in ITCVD of which they form a
strong component. Thus, the strong contribution of lipoproteins to the discriminant function is accounted for by the combined effect of
these risk factors.
Caution must be used, however, in
transferring these results to other populations.
The data utilized in constructing these discriminant functions were collected retrospectively
in the ITCVD subjects. Therefore, the sample
of ITCVD patients studied reflects the characteristics of those individuals who survived the
acute thrombotic cerebrovascular episode. Furthermore, they were within the age range and
demonstrated the physical and mental ability
required to undergo an active rehabilitation
program. However, care was taken to evaluate
the patients in this study after the acute
manifestations of the disease had subsided and
the patients were metabolically stable.
This presentation has demonstrated the
feasibility of classifying the individual with
covert ITCVD from the putatively healthy
population with the use of only four primary
risk factors. The importance of finding the
candidate for ITCVD should be evaluated in
three areas: (1) primary prevention, (2) secondary prevention, and (3) future research
studies.
Primary prevention can only be a reality
when the candidate for ITCVD is identified.
This communication offers such a possibility.
The method of intervention therapy will
depend upon the degree of abnormality in the
770
risk factors and in the risk factors themselves.
Thus, it may be necessary to reduce the blood
pressure and impose restrictions in order to
control the abnormal carbohydrate metabolism
by either diet, medication or exercise. It may
be necessary to initiate preventive anticoagulation therapy. In the cases of advanced cerebral
artery occlusion it also may be necessary to
employ surgical means to restore the blood
supply to its near-full capacity. This can be
achieved during the covert stage if symptoms
and signs suggest that this procedure is
indicated.
The problem of secondary prevention
revolves along similar principles as that
described under primary prevention. In this
instance, there is a completed incident of
ITCVD. The main purpose at hand is to
restore the most efficient functional level that
the individual can possibly achieve from the
resultant pathological changes. In addition to
initiating medical rehabilitative procedures, it
also is mandatory to institute secondary
prevention, the purpose of which is to delay the
onset of further acute episodes of ITCVD. It
should be emphasized that unless secondary
prevention is practiced concomitantly with an
ongoing active rehabilitation program, the
success of the latter will fail to reach its
objective in terms of long-term amelioration of
the illness and restoration of the optimal
physiological functions. Thus, every patient
undergoing rehabilitation should be evaluated
for any co-existing or underlying illness, e.g.,
diabetes mellitus or hypertension, as well as for
the presence of abnormal risk factors.
The identification of the ITCVD-prone
individual offers an additional discipline for
study. This would enable one to study and
assess additional factors which could be
associated with or perhaps even be causally
related to ITCVD in individuals who are
ITCVD prone and in whom one is reasonably
certain that the disease is incubating. This
offers a unique possibility to study still
unknown factors in a very efficient manner,
because it enables one to concentrate his efforts
in those cases with high probability of having
an overt incidence of ITCVD rather than
studying all healthy individuals as one would
attempt in a classical prospective epidemiological study.
Strok;
Vol. 3, Novembor-Decomber 1972
A DISCRIMINANT FUNCTION ANALYSIS
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Identification of Individuals With Covert Ischemic Thrombotic Cerebrovascular Disease: A
Discriminant Function Analysis
MENARD M. GERTLER, JAMES L. ROSENBERGER and HILLAR E. LEETMA
Stroke. 1972;3:764-771
doi: 10.1161/01.STR.3.6.764
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