621-FinalProject
Ahmed Elsaeyed
2022-11-30
Probability of Cardiovascular Disease
This research centers around building a model that predicts the probability of cardiovascular disease given some prior health information.
Abstract
For our final assignment, we will be looking at a dataset of physical characteristics, behavioral characteristics, and health results of patients aged 30-59 in the 1940’s in the United States. The data was gathered by the Framingham Heart Study, which is a long-term ongoing study. 5209 subjects were monitored to see if cardiac health can be influenced by lifestyle and environmental factors. It is mainly due to studies like this one, and indeed this one in particular due to its large scope, that we take that conclusion for granted. Before this study, the contributing factors of cardiac health were not clear at all.
A result of this study is the 10-year cardiovascular risk score was developed, which is an estimation of how likely it is that someone will have a coronary disease based on the findings of the study. The objective of this project is to build a model to predict this to take their research a step further.
Keywords - Coffy
Introduction - Coffy
Literature Review
Framingham Study is a study initiated by the United State Public Health Service, that started in 1948 and has been ongoing since. The origin of the study has been closely linked with the cardiovascular health of President Franklin D. Roosevelt and his death from hypertensive heart disease and stroke in 1945. It aims to investigate the epidemiology and factors that are involved in the development of cardiovascular disease. The plan of the study was to track a large cohort of patients over time. The study initially recruited the original cohort of 5209 participants from the town if Framingham, MA. The age group for this study ranges from 30-59 years old. The patients were given questionnaires and exams every 2 years and these questions and exams expanded with time. This study was continued over three generations of the original participants.
The data collected by the study was used to develop the Framingham Risk Score. The Framingham Risk Score is a sex-specific algorithm that is used to estimate the 10 year cardiovascular risk of an individual. This score gives an estimate probability of a person developing cardiovascular disease within a specified amount of time, the range of which can be from 10 to 30 years.
The study collect a range of 16 variables measurements from the patients in order to create the database. The variables include: Male, Age, Education, Current Smoker, Cigs Per Day, BP Medications, Prevalent Stroke, Prevalent Hypertension, Diabetes, Total Cholesterol, Systolic Blood Pressure, Diastolic Blood Pressure, Heart Rate, Glucose, CHD in 10 years. Most of the variable are categorized by their role is a person’s cardiovascular health.
Demographic Risk Factors: - Male - Age - Education
Behavioral Risk Factors: - Current Smoker - Cigs Per Day
Medical History Factors: - BP Medications - Prevalent Stroke - Prevalent Hypertension - Diabetes
Physical Exam Risk: - Total Cholesterol - Systolic Blood Pressure - Diastolic Blood Pressure - Heart Rate - Glucose
Methodology
Our methodology for creating a predictive model for future coronary
heart disease begins with cleaning and preprocessing our subject
dataset. The dataset itself contains 16 fields, one of which is the
target: TenYearCHD. While the dataset was relatively
complete to begin with, we needed to employ multiple operations in order
to transform it into a format which subsequent models could interpet. In
this vain, we addressed missing values, multicollinearity, outliers, and
mis-shaped data using a combination of self-developed functions and
transformations. After cleaning / preprocessing, we split the new
dataframe into training and testing sets in order to create and test
various machine learning models. While developing models, we decided to
try multiple approaches, including Binary Logistic Regression, standard
and with log transforms, and Step-AIC. After each model, we calculated
the relvant performance metrics and stored them in a tracker.
Ultimately, our selection criteria was heavily influenced by the models’
recall and f1 scores. With medical diagnosis, it’s imperative that
individuals with an illness be properly classified.
| Variable | Description |
|---|---|
| Sex | Participant Sex (Male or Female) |
| Age | Age at exam (years) |
| Education | Attained Education |
| Current Smoker | Whether or not the patient is a current smoker |
| Cigs Per Day | The number of cigarettes that the person smoked on average in one day |
| BP Meds | Whether or not the patient was on blood pressure medication |
| Prevelant Stroke | Whether or not the patient had previously had a stroke |
| Prevalant Hyp | Whether or not the patient was hypertensive |
| Diabetes | Whether or not the patient had diabetes |
| Tot Chol | Total cholesterol |
| Sys BP | Systolic blood pressure |
| Dia BP | Diastolic blood pressure |
| BMI | Body Mass Index |
| Heart Rate | Heart rate |
| Glucose | Glucose level |
| Ten Year CHD | 10 year risk of coronary heart disease, ‘TARGET: 1 = Yes | 2 = No’ |
Experimentation and Results
[1] 4240 16Data Exploration
Our data set had 4,240 observations and 16 variables.
Sex, education, currentSmoker,
BPMeds, prevalentStroke,
prevalentHyp, diabetes and
TenYearCHD in the analysis carried out in this report are
categorical variables, while the rest are numeric. The histograms for
the numeric variables revealed that BMI,
glucose, and totChol were skewed, and a
Box-Cox transformation was undertaken for these variables in order to
create new variables that contained the transformed versions of
BMI, glucose, and totChol. An
analysis into the correlation was done in order to remove variables that
were highly correlated with another. Also, the data had several
observations with missing data. The methodology that was undertaken to
account for these missing values is shown in the “Missing
Values” section. For comparison purposes, two different datasets
were created; one containing the original data with observations with
missing values omitted, and another containing manipulated data, where
the data went through several transformations as explained in this
report.
Sex age education currentSmoker cigsPerDay
female:2420 Min. :32.00 1 :1720 No :2145 Min. : 0.000
male :1820 1st Qu.:42.00 2 :1253 Yes:2095 1st Qu.: 0.000
Median :49.00 3 : 689 Median : 0.000
Mean :49.58 4 : 473 Mean : 9.006
3rd Qu.:56.00 NA's: 105 3rd Qu.:20.000
Max. :70.00 Max. :70.000
NA's :29
BPMeds prevalentStroke prevalentHyp diabetes totChol
0 :4063 0:4215 0:2923 No :4131 Min. :107.0
1 : 124 1: 25 1:1317 Yes: 109 1st Qu.:206.0
NA's: 53 Median :234.0
Mean :236.7
3rd Qu.:263.0
Max. :696.0
NA's :50
sysBP diaBP BMI heartRate
Min. : 83.5 Min. : 48.0 Min. :15.54 Min. : 44.00
1st Qu.:117.0 1st Qu.: 75.0 1st Qu.:23.07 1st Qu.: 68.00
Median :128.0 Median : 82.0 Median :25.40 Median : 75.00
Mean :132.4 Mean : 82.9 Mean :25.80 Mean : 75.88
3rd Qu.:144.0 3rd Qu.: 90.0 3rd Qu.:28.04 3rd Qu.: 83.00
Max. :295.0 Max. :142.5 Max. :56.80 Max. :143.00
NA's :19 NA's :1
glucose TenYearCHD
Min. : 40.00 0:3596
1st Qu.: 71.00 1: 644
Median : 78.00
Mean : 81.96
3rd Qu.: 87.00
Max. :394.00
NA's :388 
Multicollinearity Analysis
Correlation refers to the strength of a relationship between two or more variables. Problems arise when there significant correlation between two or more predictor variables within a dataset, as this can lead to solutions that are wildly varying and possibly numerically unstable and there is redundancy between predictor variables. One solution to this is to drop one variable that is highly correlated with the other. Calkins explains that we can describe the correlation values between variables in the follwing manner:
- Correlation coefficients between 0.9 and 1 = very highly correlated
- Correlation coefficients between 0.7 and 0.9 = highly correlated
- Correlation coefficients between 0.5 and 0.7 = moderately correlated
- Correlation coefficients under 0.5 = low correlation
The correlations between the variables in the dataset were
calculated. What was found was that the vast majority of the variables
had a correlation that was less than 0.25. sysBP and
diaBP had a correlation of 0.79, meaning that these two
variables were highly correlated. One of these variables was removed
from the dataset. When a binary logistic regression model with the
original data was constructed, it was found that sysBP had
a p-value of 0.00015 indicating a high degree of statistical
significance, while diaBP had a p-value of 0.55. Therefore,
the diaBP variable was removed from the dataset. The second
highest correlation value after this was 0.38, and that was for the
BMI and diaBP variables, but with the removal
of the diaBP variable, and the fact that 0.38 would be
considered a low correlation, BMI was kept in.
Missing Values
The dataset itself contains several observations containing missing values. The percentage of missing data for each of the predictor variables is shown in Table 1.
| Variable Name | Percentage of Missing Data |
|---|---|
| education | 2.48 |
| cigsPerDay | 0.68 |
| BPMeds | 1.25 |
| totChol | 1.18 |
| BMI | 0.45 |
| heartRate | 0.02 |
| glucose | 9.15 |
Table 2: Percentage of missing data for each of the variables that had missing data.
Ultimately, it was decided to remove the observations with the missing values instead of imputing. The reason for this is explained in the “Removing Outlier Values” section.
# Data Preparation
Removing Outlier Values
The boxplots, summaries, and histograms that were generated revealed that some of these heart rates, diastolic, and systolic values as well as total cholesterol do not seem realistic. Imputing the values would not be appropriate because this is medical data and would harm the authenticity of the dataset, and so we have decided to simply remove the bad data.
We conducted research on suspected variables to further examine the authenticity of the data points in the dataset. Using multiple credited medical sources we created plausible ranges for each variable and omitted extreme or impossible values. Brief discussions and analysis of the process for each variable are below.
Heart Rate
An article by Cardiologist Jim Liu of Ohio State University states a normal resting heart rate, measured when the person is not exercising, is between 60 and 100 bpm (beats per minute). The further you move from 100 bpm increases the level of risk on the high sided. On the decreasing side going lower than 60 deviates from the safe levels. The risk levels are assigned by taking into account the heart rate along with the heart rhythm. Taking this into account we removed records with heart rates over a 100.
Systolic Blood Pressure
The FDA states that a normal blood pressure reading is 120/80 with 120 being the systolic measurement. If the systolic reading is 130, the blood pressure is considered high at Stage 1, and a reading of 140 is considered Stage 2. A systolic reading of 180 or greater is considered ‘hypertensive crisis’ and needs immediate medical attention. Taking into account the different stages of risk we declared a range of 100 to 160 an acceptable Systolic Blood Pressure reading.
Total Cholesterol
The classification of the levels of total cholesterol are as follows. Less than 200 is optimal, between 200 and 240 is elevated, and greater than 240 is high. Upon further researched that higher levels of total cholesterol, such as 500 are possible. For this reason, we decided to cap our total cholesterol maximum at 500.
Box-Cox Transformation for Skewed Variables
For the variables that exhibited skewness as explained in the “Data Exploration” section of this paper, A Modern Approach to Regression with R explains the following:
…if the skewed predictor can be transformed to have a normal distribution conditional on Y, then just the transformed version of X should be included in the logistic regression model. (Sheather, 284)
In order to address this skewness and attempt to normalize these features for future modeling, we employed Box-Cox transformations. Because some of these values include 0, we replaced any zero values with infinitesimally small, non-zero values.
The \(\lambda\)’s that were used to transform the skewed variables are shown on Table 3.
| Column Name | \(\lambda\) |
|---|---|
| BMI | -0.309 |
| glucose | -1.279 |
| totChol | 0.069 |
Table 3: \(\lambda\)’s for transforming skewed variables to normal distribution.
Split Data Into Testing and Training
The data was into testing and training subsets such that 70% of it will be used to train, and 30% to test. The first row shows the split for the testing data while the second row shows the split for the training data. Tables 4 and 5 show the distributions of the testing and training data using the original and modified datasets.
Number of Observations where TenYearCHD = 1 |
Number of Observations where TenYearCHD =
0 |
|
|---|---|---|
| Test | 930 | 167 |
| Train | 2171 | 390 |
Table 4: Distribution of testing and training data using original dataset
Number of Observations where TenYearCHD = 1 |
Number of Observations where TenYearCHD =
0 |
|
|---|---|---|
| Test | 736 | 109 |
| Train | 1716 | 255 |
Table 5: Distribution of testing and training data using modified/transformed dataset
Building Models
Binary Logistic Regression Model with Original Data
A binary logistic regression model was generated using the original dataset. The observations with missing values were omitted. The final binary logistic regression takes on the following form:
\[ \begin{aligned} logit(p) = -9.155 + (0.504*Sex_{male}) + (0.066*age) + \\ (-0.290 * education_2) + (-0.223 * education_3) + (-0.120*education_4) + \\ (0.086 * currentSmoker_{Yes}) + (0.019 * cigsPerDay) + (0.123 * BPMeds_1) + \\ (0.982 * prevalentStroke_1) + (0.091 * prevalentHyp_1) + (-0.435 * diabetes_{Yes}) + \\ (0.002 * totChol) + (0.018 *sysBP) + (-0.005 * diaBP) + (0.02 * BMI) + \\ (-0.005 * heartRate) + (0.01 * glucose) \end{aligned} \] Equation 1: Binary logistic regression model using original data.
In this model, the variables that had a p-value greater than 0.05
were Sex age, cigsPerDay,
sysBP and glucose. The sex of the person
definitely seems to play a significant factor in determining whether or
not a person will develop heart disease. Figure XX revealed that men
that were surveyed on average smoked more cigarettes per day than women.
In fact, Figure XX reveals that within the sample, women on average do
not smoke. Since smoking increases the formation of plaque in the blood
vessels which leads to coronary heart disease according to the CDC, the
fact that men smoke more than women is why the sex variable in
particular is of great statistical significance. The National Institute
on Aging points out the following:
People age 65 and older are much more likely than younger people to suffer a heart attack, to have a stroke, or to develop coronary heart disease (commonly called heart disease) and heart failure. Heart disease is also a major cause of disability, limiting the activity and eroding the quality of life of millions of older people. (NIH)
This explains why age was statstically significant in
this model as well. The Brisighella Heart Study conducted in 2003 was
able to conclude that within their sample that systolic blood pressure
was a strong predictor of coronary heart disease, while diastolic blood
pressure was not statistically significant. This result in this study is
also seen in this model; sysBP has a much lower p-value
(0.0001) than diaBP (0.55). cigsPerDay is tied
to Sex as shown on Figure xx, which is why the
cigsPerDay variable is also statistically significant. Park
was able to conclude the following about glucose levels and coronary
heart disease based on a study which had a sample size of 1,197,384
adults:
Both low glucose level and impaired fasting glucose should be considered as predictors of risk for stroke and coronary heart disease. The fasting glucose level associated with the lowest cardiovascular risk may be in a narrow range. (Park)
This provides evidence as to probably why glucose is
also statistically significant within the model.
Call:
glm(formula = TenYearCHD ~ ., family = binomial(link = "logit"),
data = original_train)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.1920 -0.5860 -0.4192 -0.2732 2.7225
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -9.154885 0.877043 -10.438 < 2e-16 ***
Sexmale 0.503829 0.132129 3.813 0.000137 ***
age 0.066400 0.008136 8.161 3.32e-16 ***
education2 -0.290393 0.149348 -1.944 0.051846 .
education3 -0.223186 0.184932 -1.207 0.227486
education4 -0.119852 0.199401 -0.601 0.547800
currentSmokerYes 0.086271 0.189409 0.455 0.648768
cigsPerDay 0.018727 0.007488 2.501 0.012384 *
BPMeds1 0.123212 0.286340 0.430 0.666977
prevalentStroke1 0.982234 0.595212 1.650 0.098897 .
prevalentHyp1 0.090940 0.168521 0.540 0.589447
diabetesYes -0.434817 0.409077 -1.063 0.287818
totChol 0.001996 0.001396 1.431 0.152566
sysBP 0.017671 0.004672 3.783 0.000155 ***
diaBP -0.004660 0.007811 -0.597 0.550789
BMI 0.021327 0.015395 1.385 0.165957
heartRate -0.004900 0.005042 -0.972 0.331105
glucose 0.011013 0.002930 3.759 0.000171 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 2185.3 on 2560 degrees of freedom
Residual deviance: 1903.2 on 2543 degrees of freedom
AIC: 1939.2
Number of Fisher Scoring iterations: 5Binary Logistic Regression Model with Modified Data
A binary logistic regression model was generated using the modified dataset. The observations with missing values were omitted. The final binary logistic regression model that used modified data takes on the following form:
\[ \begin{aligned} logit(p) = -128.3 + (0.52*Sex_{male}) + (0.073*age) + \\ (-0.105 * education_2) + (-0.55 * education_3) + (-0.074*education_4) + \\ (0.277 * currentSmoker_{Yes}) + (0.013 * cigsPerDay) + (0.091 * BPMeds_1) + \\ (0.651 * prevalentStroke_1) + (0.145 * prevalentHyp_1) + (-0.026 * diabetes_{Yes}) + \\ (0.009 * totChol) + (0.013 * sysBP) + (0.18 * BMI) + (-0.009 * heartRate) + \\ (0.013 * glucose) + (-13 * tf\_BMI) + (-140.9 * tf\_glucose) + (-1.39 * tf\_totChol) \end{aligned} \] Equation 2: Binary logistic regression model using modified data.
The parameters in this model that had p-values less than 0.05 were
Sex, age, education_3,
sysBP and glucose. None of the Box-Cox
transformed variables had p-values less than 0.05. We see the same
variables (except cigsPerDay) that were statistically
significant when the original data was used.
Call:
glm(formula = TenYearCHD ~ ., family = binomial(link = "logit"),
data = modified_train)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.9142 -0.5564 -0.3987 -0.2737 2.8641
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.283e+02 1.349e+02 0.951 0.341801
Sexmale 5.164e-01 1.550e-01 3.331 0.000865 ***
age 7.270e-02 8.949e-03 8.123 4.54e-16 ***
education2 -1.051e-01 1.683e-01 -0.625 0.532142
education3 -5.491e-01 2.282e-01 -2.406 0.016110 *
education4 -7.443e-02 2.193e-01 -0.339 0.734331
currentSmokerYes 2.774e-01 2.160e-01 1.284 0.199058
cigsPerDay 1.302e-02 8.515e-03 1.529 0.126229
BPMeds1 9.097e-02 5.050e-01 0.180 0.857039
prevalentStroke1 6.507e-01 6.103e-01 1.066 0.286348
prevalentHyp1 1.450e-01 1.959e-01 0.740 0.459116
diabetesYes -2.551e-01 4.898e-01 -0.521 0.602544
totChol 8.562e-03 9.697e-03 0.883 0.377252
sysBP 1.310e-02 6.409e-03 2.044 0.040943 *
BMI 1.805e-01 1.129e-01 1.599 0.109869
heartRate -9.006e-03 7.019e-03 -1.283 0.199437
glucose 1.331e-02 5.245e-03 2.538 0.011138 *
tf_BMI -1.300e+01 9.534e+00 -1.364 0.172658
tf_glucose -1.409e+02 1.738e+02 -0.811 0.417489
tf_totChol -1.390e+00 1.854e+00 -0.750 0.453482
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 1653.4 on 2127 degrees of freedom
Residual deviance: 1467.5 on 2108 degrees of freedom
AIC: 1507.5
Number of Fisher Scoring iterations: 5Step-AIC Binary Linear Regression Model with Original Data
Step AIC works by deselecting features that negatively affect the
AIC, which is a criterion similar to the R-squared. It selects the model
with not only the best AIC score but also a model with less predictors
than the full model, since the full model may have predictors that do
not contribute or negatively contribute to the model’s performance. The
direction for the Step AIC algorithm was set to both,
because this implements both forward and backward elimination in order
to decide if a predictor negatively affects the model’s performance. The
final step-AIC binary logistic regression model that used the original
data takes on the following form:
\[ \begin{aligned} logit(p) = -9.589 + (0.503*Sex_{male}) + (0.072*age) + (0.02 * cigsPerDay) + \\ (1.004 * prevalentStroke_1) + (0.017 * sysBP) + (0.023 * BMI) + (0.009 * glucose) \end{aligned} \] Equation 3: Step-AIC binary logistic regression model using the original data.
The p-values for this model, which is far more parsimonious than the
original binary logistic regression model that used all of the
predictors, were all below 0.05 except for prevalentStroke1
and BMI, which had p-values of 0.0861 and 0.1179
respectively.
Call:
glm(formula = TenYearCHD ~ Sex + age + cigsPerDay + prevalentStroke +
sysBP + BMI + glucose, family = binomial(link = "logit"),
data = original_train)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.2231 -0.5892 -0.4224 -0.2792 2.6812
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -9.589258 0.603218 -15.897 < 2e-16 ***
Sexmale 0.503148 0.127044 3.960 7.48e-05 ***
age 0.072072 0.007718 9.339 < 2e-16 ***
cigsPerDay 0.020838 0.005012 4.158 3.21e-05 ***
prevalentStroke1 1.003640 0.584843 1.716 0.0861 .
sysBP 0.017106 0.002706 6.321 2.60e-10 ***
BMI 0.022856 0.014619 1.563 0.1179
glucose 0.008856 0.002112 4.193 2.75e-05 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 2185.3 on 2560 degrees of freedom
Residual deviance: 1912.5 on 2553 degrees of freedom
AIC: 1928.5
Number of Fisher Scoring iterations: 5Step-AIC Binary Logistic Regression Model with Modified Data
The final step-AIC binary logistic regression model that used the modified data takes on the following form:
\[ \begin{aligned} logit(p) = -14.41 + (0.52 * Sex_{male}) + (0.073 * age) + (-0.09 * education_2) + (-0.55 * education_3) + \\ (-0.056 * education_4) + (0.02 * cigsPerDay) + (0.015 * sysBP) + (0.2 * BMI) + \\ (0.009 * glucose) + (-14.76 * tf\_BMI) \end{aligned} \] Equation 4: Step-AIC binary logistic regression model using the modified data.
The p-values for this model, which is far more parsimonious than the
original binary logistic regression model that used all of the
predictors, were all below 0.05 except for Intercept,
education2, education4, BMI and
tf_BMI; 3 more than the amount of predictors that had
values above 0.05 for the step-AIC binary logistic regression model
using the original predictors.
Call:
glm(formula = TenYearCHD ~ Sex + age + education + cigsPerDay +
sysBP + BMI + glucose + tf_BMI, family = binomial(link = "logit"),
data = modified_train)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.7476 -0.5552 -0.4010 -0.2753 2.7893
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 14.410817 14.969749 0.963 0.335717
Sexmale 0.520105 0.150793 3.449 0.000562 ***
age 0.072950 0.008761 8.326 < 2e-16 ***
education2 -0.093929 0.167136 -0.562 0.574121
education3 -0.545300 0.226890 -2.403 0.016245 *
education4 -0.056454 0.217162 -0.260 0.794892
cigsPerDay 0.020256 0.005666 3.575 0.000350 ***
sysBP 0.015889 0.004996 3.180 0.001471 **
BMI 0.198747 0.109916 1.808 0.070579 .
glucose 0.009194 0.002376 3.870 0.000109 ***
tf_BMI -14.756198 9.247248 -1.596 0.110547
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 1653.4 on 2127 degrees of freedom
Residual deviance: 1474.6 on 2117 degrees of freedom
AIC: 1496.6
Number of Fisher Scoring iterations: 5Model Selection
The various metrics that were used to determine the best model are shown in Table 6. Figure XX is a bar chart which plots all of the metrics for all of the models. Table 6 and Figure xx reveal that the accuracy, precision, recall, AUC, and F-score are higher when the original data was used as opposed to the modified. Table 6 revealed that the step-AIC binary logistic regression model using the original data has the second highest AIC, but it also has the highest precision and F-score out of all of the models. Taking all of this into consideration, and the fact that the step-AIC binary logistic regression model using the original data is the most parsimonious out of all of the models, this model is the best performing model out of all of them. The AUC curve for the step-AIC binary logistic regression with original data is provided in Figure xx.
| Model | Precision | Recall | AIC | AUC | F-score | Accuracy | Error |
|---|---|---|---|---|---|---|---|
| Bin. Log. w/ Original Data | 0.68 | 0.11 | 1939.25 | 0.71 | 0.195 | 0.86 | 0.14 |
| Bin. Log. w/ Modified Data | 0.67 | 0.02 | 1507.5 | 0.7 | 0.033 | 0.87 | 0.13 |
| Step AIC Bin. Log. w/ Original Data | 0.7 | 0.11 | 1928.55 | 0.71 | 0.196 | 0.86 | 0.14 |
| Step AIC Bin. Log. w/ Modified Data | 0.67 | 0.02 | 1496.61 | 0.69 | 0.033 | 0.87 | 0.13 |
Table 6: Model metrics for binary logistic regression models
Discussion and Conclusions
In this paper, 4 different binary logistic regression models were generated in order to predict the 10-year risk of chronic heart disease. The results from the analysis carried out in this report indicate that the transformation of the skewed variables to a normal distribution and the removal of outliers resulted in worse model performance when comparing the metrics to these models and the models that used the original unaltered dataset. With that being said, the AUC’s for all of the models were relatively the same. The final model that was selected in order to predict the 10-year risk of chronic heart disease was the best performing and the most parsimonious. We were able to test the validity of this particular model from when the data was split into testing and training datasets.
References
Center for Drug Evaluation and Research. (2021a, January 21). High Blood Pressure– Understanding the Silent Killer. U.S. Food And Drug Administration. https://www.fda.gov/drugs/special-features/high-blood-pressure-understanding-silent- killer
Center for Drug Evaluation and Research. (2021b, January 21). High Blood Pressure– Understanding the Silent Killer. U.S. Food And Drug Administration. https://www.fda.gov/drugs/special-features/high-blood-pressure-understanding-silent- killer
Framingham Study | Boston Medical Center. (n.d.). https://www.bmc.org/stroke-and- cerebrovascular-center/research/framingham-study
High cholesterol - Symptoms and causes. (2021, July 20). Mayo Clinic. https://www.mayoclinic.org/diseases-conditions/high-blood-cholesterol/symptoms- causes/syc-20350800
Liu, J., MD. (2022a, July 19). What’s a dangerous heart rate? What’s a Dangerous Heart Rate? | Ohio State Health & Discovery. Retrieved December 5, 2022, from https://health.osu.edu/health/heart-and-vascular/what-is-dangerous-heart-rate
NCBI - WWW Error Blocked Diagnostic. (n.d.). https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4159698/
NHS website. (2022, July 4). Low blood pressure (hypotension). nhs.uk. https://www.nhs.uk/conditions/low-blood-pressure-hypotension/
Tachycardia: Symptoms, Causes & Treatment. (n.d.). Cleveland Clinic. https://my.clevelandclinic.org/health/diseases/22108-tachycardia
Appendix
FIGURES AND CODE GO HERE.