Revista Científica Interdisciplinaria Investigación y Saberes
2024, Vol. 14, No. 1 e-ISSN: 1390-8146
Published by: Universidad Técnica Luis Vargas Torres
How to cite this article (APA):
Ramos, D., Toapanta, M., Alcivar, C. (2024) Prediction of stroke using
logistic regression, Revista Científica Interdisciplinaria Investigación y Saberes, 14(1) 158-178
Predicción de accidente cerebrovascular utilizando regresión logística
Prediction of stroke using logistic regression
David Fernando Ramos Tomalá
Faculty of Industrial Engineering, University of Guayaquil, david.ramost@ug.edu.ec,
https://orcid.org/0009-0007-2702-8926
Mariuxi Del Carmen Toapanta Bernabé
Faculty of Industrial Engineering, University of Guayaquil, mariuxi.toapantab@ug.edu.ec,
https://orcid.org/0000-0002-4839-7452
Cesar Andrés Alcívar Aray
Faculty of Industrial Engineering, University of Guayaquil, cesar.alcivarar@ug.edu.ec,
https://orcid.org/0009-0004-0214-905X
Open clinical trial data provide a valuable opportunity for researchers
worldwide to evaluate new hypotheses, validate published results,
and collaborate for scientific advances in medical research. Here we
present a health dataset for noninvasive stroke prediction, containing
data from 219 subjects. The dataset covers an age range of 20-89
years and disease records including hypertension and diabetes. Data
acquisition was performed under the control of standard experimental
conditions and specifications. The response variable used is whether
or not stroke occurred, whether nonhemorrhagic or hemorrhagic. The
predictors used were age, body mass index, blood sugar level,
systolic and diastolic blood pressure, heart rate, and gender. Using
logistic regression, good modeling accuracy was obtained, where the
predictors with a significant effect (alpha <0.05) were age, body mass
index, diastolic blood pressure, and type of diabetes.
Abstract
Received 2023-11-18
Revised 2023-12-19
Published 2024-01-05
Corresponding Author
David Fernando Ramos Tomalá
david.ramost@ug.edu.ec
Pages: 158-177
https://creativecommons.org/licens
es/by-nc-sa/4.0/
Distributed under
Copyright: © The Author(s)
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Keywords:
stroke prediction, correlation matrix, ROC curve,
sensitivity and specificity, confounding matrix, logistic regression,
machine learning
Resumen
Los datos de ensayos clínicos abiertos proporcionan una valiosa
oportunidad para que los investigadores de todo el mundo evalúen
nuevas hipótesis, validen los resultados publicados y colaboren para
obtener avances científicos en la investigación médica. Aquí se
presenta un conjunto de datos de salud para la predicción no invasiva
de accidentes cerebrovasculares, que contiene datos de 219 sujetos.
El conjunto de datos cubre un rango de edad de 20-89 años y
registros de enfermedades incluyendo hipertensión y diabetes. La
adquisición de datos se llevó a cabo bajo el control de las condiciones
y especificaciones experimentales estándar. La variable de respuesta
utilizada es si se produjo o no el accidente cerebrovascular, sea este
no hemorrágico o hemorrágico. Los predictores utilizados fueron la
edad, el índice de masa corporal, el nivel de azúcar en la sangre, la
presión arterial sistólica y diastólica, la frecuencia cardíaca y el
género. Mediante el uso de la regresión logística, se obtuvo una
buena precisión de modelado, donde los predictores que tienen un
efecto significativo (alfa <0,05) son la edad, el índice de masa
corporal, la presión arterial diastólica y el tipo de diabetes.
Palabras clave:
predicción ictus, matriz de correlación, curva
ROC, sensibilidad y especificidad, matriz de confusión, regresión
logística, machine learning
Introduction
Stroke (CVA) is the leading cause of disability in adults and the third
leading cause of death in the world (Abubakar & Isezuo, 2012). It is a
disease that affects the arteries leading to and within the brain. The
impact of stroke on people's lives represents a major challenge to
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society. In addition to being a sudden event, stroke affects both the
individual and family members who are not prepared to deal with the
rehabilitation process or the disabilities that result from this condition.
As a result, a large number of people are unable to work and receive
financial assistance after suffering a stroke.
One way to prevent or reduce the serious impact of stroke is to first
know the risk factors that significantly affect it. To overcome these
problems, it is necessary to conduct an analysis that aims to
significantly determine the risk factors for stroke. Moreover, based on
the model formed, this analysis can predict the occurrence or non-
occurrence of stroke in a person. The analysis used is logistic
regression (LR), which is a statistical method to determine the
relationship between the dependent variable (response) that is
categorical and one or more independent variables (predictors) in the
form of categorical or continuous data (Agresti, 2002).
LR is one of the machine learning methods that are currently widely
used in many fields. The LR method allows performing classification
analysis and also providing information about variables that have a
significant effect (Fa'rifah & Poerwanto, 2019). This method is also
often used in health data, for example, research that modeled the
pathological diagnosis of metastatic breast cancer (Bustan &
Poerwanto, 2021). In another example, death from COVID 19 was
predicted using non-medical characteristics with logistic regression
(Josephus et al., 2021).
Rapidly developing clinical signs due to a local (or global) brain
disorder with symptoms that last 24 hours or more and may result in
death in the absence of obvious causes other than vascular disease
are signs of stroke (Bootkrajang & Kaban, 2014). Stroke, or ictus, is
divided into two types, namely ischemic or nonhemorrhagic stroke
(NHS) and hemorrhagic stroke (HS). Ischemic stroke is a category of
stroke that can occur due to blockage or clot in one or more large
arteries in the cerebral circulation. Patients with ischemic stroke often
relapse due to seizures, migraines, and other conversion disorders
that trigger relapse (Geyer et al., 2009). While hemorrhagic stroke is
a category of stroke that can occur if the intracerebral vascular lesion
ruptures, causing bleeding into the subarachnoid space or directly
into the brain tissue (Garg et al., 2019).
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In the last four decades, the incidence in low- and middle-income
countries has doubled. This disease, of course, in addition to having
an impact on socioeconomic, also causes permanent disability and
productivity of the sick (Kementerian Kesehatan Republik Indonesia,
2018).
Data from the World Stroke Organization show that each year there
are 13.7 million new cases of stroke and about 5.5 million deaths from
stroke. About 70 % of these strokes occur in low- and middle-income
countries (Bustan M. N., 2017).
Methodology
Many studies have been conducted using early diagnostic and
noninvasive screening techniques for cardiovascular diseases (CVD)
such as hypertension and coronary artery sclerosis in order to discover
more convenient and effective methods for early identification of
CVD. Of these methods, photoplethysmography (PPG) has been
widely recognized as a low-cost noninvasive screening technology for
CVD (Allen, 2007).
Data acquisition was conducted at the Guilin People's Hospital, China
(Liang et al., 2017).
The information from the working dataset is summarized in Table 1.
This dataset was collected from 219 subjects, aged 21 to 86 years,
with a median age of 58 years. Males accounted for 48%. The dataset
covers several diseases, including hypertension, diabetes, cerebral
infarction, and cerebral insufficient blood supply.
The data collection program involved acquiring information on the
basic physiology of the individuals, extracting cardiovascular disease
information from the hospital's electronic medical records, collecting
PPG waveform signals, and detecting instantaneous blood pressure
at the same time.
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The data set includes systolic and diastolic blood pressure information
from subjects who were diagnosed with normotension,
prehypertension, and stage I/II hypertension:
Table 1
Working dataset information
Article Site
US National Library of
Medicine National
Institutes of Health
Item name
A new short-log
photoplethysmogram data
set for blood pressure
monitoring in China.
Article website
https://www.ncbi.nlm.nih.
gov/pmc/articles/PMC582
7692/
Dataset
repository
https://figshare.com/articl
es/dataset/PPG-
BP_Database_zip/5459299
Remarks
It has a total of 219
records, The dataset also
covers several different
cardiovascular (e.v.)
diseases, including
hypertension, cerebral
infarction and insufficient
blood supply to the brain
and other related
diseases, such as diabetes.
Variables
Sex(M/F) = sex (Male /
Female)
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Age = Age
Height(cm) = Height
Weight(kg) = Weight
Systolic Blood
Pressure(mmHg) = Systolic
Blood Pressure
Diastolic Blood
Pressure(mmHg) =
Diastolic Blood Pressure
Heart Rate(b/m) = Heart
rate
BMI (kg/m^2) = Body
Mass Index
(v.e) Hypertension =
Hypertension
(v.e) Diabetes
(e.g.) cerebral infarction =
cerebral infarction
(e.g.) cerebrovascular
disease = cerebrovascular
accident
Note: This table explains where the information was obtained from,
the sample size, and the variables that were finally considered
relevant. Taken from PPG-BP Database, by Y. Liang et al., 2017,
Figshare.
Table 2 shows the data for all independent variables (𝑥
!"
) and those
of the dependent variable (𝑌
!
).
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Table 2
Sample data
Note. In total there are 219 rows and 8 columns: grouped in 7
independent variables (𝑥
"
) (Sex, Body Mass Index, Heart Rate, Systolic
Blood Pressure, Diastolic Blood Pressure, and Type of Diabetes), of
which 2 (Sex and Type of Diabetes) are binary (0 or 1); and all predict
Sexo Edad IMCo FrCa Sist Diast TD EnCe
1 45 27,27 97 161 89 0 0
1 50 20,28 76 160 93 0 0
1 47 20,89 79 101 71 0 0
0 45 21,97 87 136 93 0 0
. . . . . . . .
. . . . . . . .
. . . . . . . .
1 68 19,63 65 142 90 0 0
0 52 23,03 80 154 88 0 0
0 66 24,77 93 173 107 0 0
0 65 22,58 73 111 62 0 0
0 66 19,03 84 107 63 0 0
0 47 23,44 71 128 66 0 0
. . . . . . . .
. . . . . . . .
. . . . . . . .
1 24 20,7 74 108 65 0 0
1 25 20,81 64 84 56 0 0
0 25 23,46 63 104 70 0 0
1 24 19,49 87 109 68 0 0
0 24 21,6 77 111 70 0 0
1 25 19,31 79 93 57 0 0
0 25 17,76 72 120 69 0 0
0 25 21,05 67 106 69 0 0
0 24 18,94 65 108 68 0 0
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1 dependent variable (𝑌) (Brain Disease) which is also binary (0 or 1).
Taken from PPG-BP Database, by Y. Liang et al., 2017, Figshare.
Prediction of stroke occurrence in subjects:
The data set used for this study can be evaluated here with responses
from 219 patients of both sexes who had a stroke (1) and who did not
have a stroke (0).
The logistic regression model assumes that each response Yi is an
independent random variable with Bernoulli distribution (pi), where
the log-odds corresponding to pi are modeled as a linear
combination of the covariates plus a possible intercept term:
The intercept β0 represents the "baseline" log-odds of the subject
who will suffer a stroke, if all covariates take the value 0. Each
coefficient βj represents the amount of increase or decrease in log-
odds, if the value of covariate xij is increased by 1 unit.
LR is the most commonly used linear prediction method for binary
data. If the response used has two categories, the regression analysis
used is called binary LR [5]. The multiple binary logistic regression
model can be written as follows:
The β0, β1, ... βi are parameters of the model. These parameters are
estimated using the maximum likelihood estimation (MLE) method.
Basically, the MLE provides an estimated value of β to maximize the
likelihood function (Tampil et al., 2017). Systematically, the likelihood
function for the binary logistic regression model is as follows:
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Given that the responses Y1, ..., Yn are independent Bernoulli random
variables, the probability for the logistic regression model is given by:
To obtain the values of 𝛽, the equation is derived from$𝛽$and then
equals 0:
Logistic regression is still often used as a tool for binary classification
problems, even if the model does not give an extremely accurate fit
to the data, as long as the model has good classification accuracy. In
such fits, the βˆ MLE represents the "logistic regression model"
closest (on the given covariates) to the true distribution of Y1, . . ., Yn.
The standard error for βˆj can be robustly estimated using a sandwich
estimator or the nonparametric bootstrap. For the logistic regression
model, the sandwich estimation of the covariance matrix of βˆ is given
by:
where W = diag ((Y1 - pˆ1)2, . . . . (Yn - pˆn)2) and pˆi is the probability
of fit for the ith observation, defined by the right-hand side of formula
(1) with βˆ instead of β. The (j, j) element of this matrix gives an
estimate of the variance of βˆj. Alternatively, one can use the pairwise
bootstrap, which pairs the covariates and the response for each ith
observation into a single data vector (xi1, ..., xip, Yi), and then draws
bootstrap samples by randomly selecting, with replacement, n from
these vectors. The logistic regression model is fit to each bootstrap
sample to produce an MLE βˆˆ, and the standard error of βˆj is
estimated by the empirical standard deviation of βˆ across bootstrap
samples.
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ROC curves (receiver operating characteristic curve) are a very useful
tool in health sciences, basically under two circumstances. The first is
to be able to determine the best cut-off point for a diagnostic test;
that is, in this case, what will be the criterion for predicting whether or
not a subject will have a stroke. The second is to be able to determine,
from among two tests, with which one can obtain better results in
establishing a diagnosis; that is, in this case, with which one will be
able to better predict whether or not a subject will have a stroke.
A diagnostic test is basically a classifier that attempts to distinguish
individuals who have a pathology or condition from those who do not;
that is, in this case, individuals who will or will not suffer a stroke in the
future. However, like any detection system, it is susceptible to errors.
In this case, the possible errors would be: predicting that a person will
not have a stroke when he or she will; or predicting that a person will
have a stroke when he or she will not. In this sense we can say that a
diagnostic test can present four scenarios for each individual:
Individual without stroke classifies as without stroke (True negative:
TN). Individual with stroke classifies as having stroke (True positive:
TP). Individual with stroke classifies as without stroke (False negative:
FN). And, finally, an individual without stroke is classified as having a
stroke (False positive: FP).
Clearly the first pair is correct and the second pair is wrong. Thus,
diagnostic tests can be evaluated by quantifying how many times a
correct result is obtained with respect to the total number of tests.
The parameters that establish how accurate the diagnostic test is are
sensitivity and specificity, which are defined as follows:
Sensitivity: The probability of classifying an individual with stroke as
having a stroke. Specificity: The probability of classifying an individual
without stroke as not having a stroke.
For the calculation of these parameters it is very practical to organize
the information in a contingency matrix of size 2×2, as shown in Figure
1, where the columns are generated from the LCA variable, and the
rows from the Test variable:
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Figure 1
Confusion Matrix
Note: Where: TP: True Positive; TN: True Negative; FP: False Positive;
and FN: False Negative. Taken from Cómo interpretar la matriz de
confusión: ejemplo práctico, by P. Recuero de los Santos, 2021,
Telefónica Tech.
In terms of conditional probability, sensitivity is understood as the
probability that the test predicts stroke given that the individual will
actually have stroke (P (predict stroke | will have stroke)), and
specificity is the probability that the test predicts no stroke given that
the individual will not actually have stroke (P (predict no stroke | will
not have stroke)). Applying Bayes' Rule and the Law of Total
Probability we can arrive at the following equations for sensitivity and
specificity (Recuero de los Santos, 2021).
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𝑆𝑒𝑛𝑠𝑖𝑏𝑖𝑙𝑖𝑑𝑎𝑑 =
𝑇𝑃
𝑇𝑃 + 𝐹𝑁
$$$$$$$$$$$$$$$$$$$$𝐸𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑑𝑎𝑑 =
𝑇𝑁
𝑇𝑁 + 𝐹𝑃
It is important to clarify that in order to consider a person as having
or not having a stroke, it is necessary to have a reference test that
indicates in the first instance the state of the individual in question.
Suppose that the value of a biomarker X of a person is known. This
biomarker could have the property of defining the person as having
or not having stroke if it is above (or below) a certain cut-off point C
at first unknown.
An ROC curve is a graphical representation of the variation of
sensitivity versus (1-specificity) when the cut-off point changes. In
other words, for each possible cut-off point, a sensitivity and
specificity must be calculated and plotted.
One parameter to determine the capacity of a diagnostic test to
discriminate between possible or not strokes is the area under the
curve; that is, the area delimited by the (1-specificity) axis, the vertical
line passing through (1-specificity) =1, and the ROC curve. The
maximum value of this area is less than or equal to unity and a test is
said to be non-discriminative if it coincides with the straight line
joining the point (0,0) with (1,1) which defines an area equal to 0.5
(50%). The larger the area, the more discriminative it is considered to
be. Thus, the area under the curve can be a good parameter to
compare two diagnostic tests, considering that the best one will be
the one that covers the largest area.
Another aspect of interest is to determine the cut-off point that best
identifies those who will have a stroke from those who will not. In
terms of sensitivity and specificity, the best cut-off point would be the
one that classifies all those who will have a stroke as positive and all
those who will not have a stroke as negative. In practice, it is very
difficult to ensure that a diagnostic test is infallible, so obtaining
perfect sensitivity and specificity would be very complicated;
however, the cut-off point that best fits these conditions can be
sought.
One criterion for finding the best cut-off point is to identify the point
P for which the distance from point P to point (0,1) is minimum.
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Results
Table 3 shows that systolic blood pressure (Sist) is directly related to
diastolic blood pressure (Diast) (𝒓 = 𝟎. 𝟕𝟐); which was to be expected.
Table 3
Pearson correlation matrix
Note. The correlation, in absolute value, can be: null (0.00); very weak
(0.01-0.20); weak (0.21-0.40); moderate (0.41-0.60); strong (0.61-
0.80); very strong (0.81-0.99); or perfect (1.00).
4.2 Simultaneous significance test results
Simultaneous tests are conducted to see the effect of the global
predictor variables on the response variable. The hypothesis for this
test is as follows:
H0:$𝛽1$=$𝛽2$= = =$𝛽i$= 0
H1: there is at least one parameter 0
Sexo Edad IMCo FrCa Sist Di ast TD
Edad -0,07
IMCo 0,04 0,02
FrCa -0,01 -0,09 -0,11
Sist -0,09 0,41 0,23 0,14
Di ast -0,08 0,00 0,23 0,19 0,72
TD 0,00 0,06 0,13 -0,02 0,07 0,06
EnCe 0,01 0,38 0,08 -0,04 0,14 -0,11 -0,23
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The test statistics used for the likelihood ratio test are as follows
(David & Stanley, 2000):
As Table 4 shows, the result D is 69.85 while the p-Value is 1.59E-12.
Therefore, this step concludes that there is at least one parameter that
is not equal to zero.
Table 4
Simultaneous significance test result
Note. By using an alpha (𝛼) of 0.05, it means that the test criterion is
to reject H0 if D >$𝜒2.
Partial test results
This partial test was performed to identify predictors that had a
significant effect on the dependent variable (𝑌). Table 5 shows the
individual (partial) results for each of the 7 predictors, and for the
constant term.
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Table 5
Results of partial significance tests
Note. In the partial test, the test statistics used are the Wald test (𝑊).
The criterion for rejecting H0 is if W >$𝜒2$or p-value < 0.05.
The hypothesis used for each variable is as follows:
H0:$𝛽i$= 0 (the logit coefficient is not significant for the model).
H1:$𝛽i$0 (the logit coefficient is significant for the model).
To perform the Wald test, the following equation is used:
𝑊 =
𝛽
#
D
𝑠𝑒
$
!
%
The W value follows a Chi-square distribution with df = 1.
Using p-value, there are 4 predictors that have p-value < 𝛼; i.e., age
(𝑥
&
), body mass index (𝑥
'
), diastolic blood pressure (𝑥
(
) and type of
diabetes (𝑥
)
) (in addition to the constant term, which is also significant
b
0
b
1
b
2
b
3
b
4
b
5
b
6
b
7
-8,26444356 0,101862291 0,084110733 0,13140241 0,008189016 0,019807976 -0,06116992 -85,2100284
se
b
0
se
b
1
se
b
2
se
b
3
se
b
4
se
b
5
se
b
6
se
b
7
2,275298657 0,417872836 0,018041966 0,049060926 0,017092958 0,014433266 0,02983895 0,506265124
-81,9379589 -76,3393417 -87,0478724 -79,5971918 -76,3959504 -77,1174776 -78,4107112 -87,8653917
Z
b
0
Z
b
1
Z
b
2
Z
b
3
Z
b
4
Z
b
5
Z
b
6
Z
b
7
-3,63224561 0,243763848 4,661949427 2,678351627 0,479087092 1,372383495 -2,05000244 -168,311077
11,25917372 0,061939157 21,47900071 6,577639462 0,175156536 1,618211034 4,20467829 23,1140393
p -Value
b
0
p -Value
b
1
p -Value
b
2
p -Value
b
3
p -Value
b
4
p -Value
b
5
p -Value
b
6
p -Value
b
7
0,000280966 0,807413721 3,13228E-06 0,00739855 0,631876672 0,169944086 0,04036419 0
0,000792305 0,803456866 3,57725E-06 0,010326775 0,67556911 0,203341414 0,04031262 1,52672E-06
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at 5%). According to those significant predictors, the logistic
regression model can be seen in eq:
𝑃
[
𝑌
!
= 1
]
= 𝑝
!
=
𝑒
*+,&(-./,/+-0
"
./,1'10
#
*/,/(1&0
$
*+2,&10
%
1 + 𝑒
*+,&(-./,/+-0
"
./,1'10
#
*/,/(1&0
$
*+2,&10
%
The ROC curve (receiver operating characteristic curve) is shown in
Figure 2. Based on this curve, it is possible to determine the best cut-
off point for a diagnostic test for stroke, i.e., the criterion for
predicting whether or not a subject will have a stroke.
Figure 2
ROC curve
Note: The intersection of the dotted line (blue) with the ROC curve
(red) gives the cut-off point of the diagnostic test for stroke (the
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threshold). If the area under the ROC curve is close to 1.00, then this
test has a very good predictive ability for stroke.
Where the threshold is:
Therefore, under a probability of 0.224 it can be assumed that a stroke
will not occur, while above 0.224 it will be assumed that a stroke will
occur (Fawcett, 2006).
And the area under the curve (AUC):
Therefore, this test does have a very good discriminatory ability to
predict whether or not a person will suffer a stroke (Hanley & McNeil,
1982).
In this study, logistic regression was used to determine the factors that
significantly influence stroke so that people can prevent stroke as
early as possible. Four predictors have a significant effect in the
model, namely age, body mass index, diastolic blood pressure, and
type of diabetes.
It can be seen that an increase in age and body mass index will
increase the risk of stroke. Age and body mass index was positively
associated with stroke mortality (Yi et al., 2018), so it may be an early
warning to prevent stroke.
Thr e s hold
0,2241074
AUC
0,859
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Conclusions
Based on the analysis that has been done, it can be concluded that
age and body mass index are two predictors that significantly affect
the occurrence of stroke.
Diastolic blood pressure and the person's type of diabetes were also
shown to have an impact on the prognosis of a future stroke.
The logistic regression model was fairly accurate in classifying 219
medical record data.
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