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title: “Predicting Maximum Heart Rate” subtitle: “Cross-Sectional Analysis of Heart Health Data” author: “Your Group Names” format: html ————

Introduction

Heart health can be influenced by many different factors, including age, blood pressure, cholesterol, and a person’s response to exercise. Understanding the relationships between these characteristics can help us better understand cardiovascular health.

For this project, we analyze a cross-sectional heart health dataset. Each observation represents a patient, and the dataset contains both quantitative and qualitative variables related to cardiovascular health.

Our main research question is:

What factors are associated with a patient’s maximum heart rate during exercise?

Our dependent variable, or Y variable, is thalach, which represents the maximum heart rate achieved by a patient.

We will investigate whether characteristics such as age, heart disease status, sex, chest pain type, exercise-induced angina, and ST depression can help explain differences in maximum heart rate.

Setup

First, we load the packages that will be used throughout our analysis.

The tidyverse package is used for data cleaning and visualization. The knitr package is used to create cleaner tables in our final report.

Importing the Data

We begin by importing the original heart.csv file.

We can view the first six observations to make sure the dataset imported correctly.

  age sex cp trestbps chol fbs restecg thalach exang oldpeak slope ca thal
1  52   1  0      125  212   0       1     168     0     1.0     2  2    3
2  53   1  0      140  203   1       0     155     1     3.1     0  0    3
3  70   1  0      145  174   0       1     125     1     2.6     0  0    3
4  61   1  0      148  203   0       1     161     0     0.0     2  1    3
5  62   0  0      138  294   1       1     106     0     1.9     1  3    2
6  58   0  0      100  248   0       0     122     0     1.0     1  0    2
  target
1      0
2      0
3      0
4      0
5      0
6      1

Next, we check the dimensions of the dataset.

[1] 1025   14

The first number represents the number of observations, while the second number represents the number of variables.

Understanding the Data

We can examine the structure of the original dataset using str().

'data.frame':   1025 obs. of  14 variables:
 $ age     : int  52 53 70 61 62 58 58 55 46 54 ...
 $ sex     : int  1 1 1 1 0 0 1 1 1 1 ...
 $ cp      : int  0 0 0 0 0 0 0 0 0 0 ...
 $ trestbps: int  125 140 145 148 138 100 114 160 120 122 ...
 $ chol    : int  212 203 174 203 294 248 318 289 249 286 ...
 $ fbs     : int  0 1 0 0 1 0 0 0 0 0 ...
 $ restecg : int  1 0 1 1 1 0 2 0 0 0 ...
 $ thalach : int  168 155 125 161 106 122 140 145 144 116 ...
 $ exang   : int  0 1 1 0 0 0 0 1 0 1 ...
 $ oldpeak : num  1 3.1 2.6 0 1.9 1 4.4 0.8 0.8 3.2 ...
 $ slope   : int  2 0 0 2 1 1 0 1 2 1 ...
 $ ca      : int  2 0 0 1 3 0 3 1 0 2 ...
 $ thal    : int  3 3 3 3 2 2 1 3 3 2 ...
 $ target  : int  0 0 0 0 0 1 0 0 0 0 ...

The dataset contains the following variables:

age: Age of the patient in years
sex: Sex
cp: Chest pain type
trestbps: Resting blood pressure
chol: Serum cholesterol
fbs: Fasting blood sugar category
restecg: Resting ECG result
thalach: Maximum heart rate achieved
exang: Exercise-induced angina
oldpeak: ST depression induced by exercise
slope: Slope of the peak exercise ST segment
ca: Number of major vessels
thal: Thalassemia category
target: Heart disease status

Quantitative Variables

Our dataset contains several quantitative variables, including:

age — age in years
trestbps — resting blood pressure
chol — cholesterol
thalach — maximum heart rate
oldpeak — ST depression

This gives us at least four quantitative variables as required by the project.

Qualitative Variables

Several variables are represented by numbers in the raw dataset but actually represent categories.

These include:

sex
cp
fbs
restecg
exang
slope
thal
target

This gives us at least four qualitative variables as required by the project.

Data Cleaning

Before analyzing the data, we check for missing observations.

     age      sex       cp trestbps     chol      fbs  restecg  thalach 
       0        0        0        0        0        0        0        0 
   exang  oldpeak    slope       ca     thal   target 
       0        0        0        0        0        0

If every variable returns zero, there are no missing values.

Checking for Duplicates

We also check whether there are any exact duplicate rows.

[1] 723

Duplicate rows can cause certain observations to have more influence on our results.

We therefore remove exact duplicates.

Now we check the dimensions again.

[1] 302  14

Converting Qualitative Variables

Some variables are stored as numbers even though they actually represent categories.

We convert these variables into factors so R treats them as qualitative variables.

We can check the structure again to make sure the conversion worked.

'data.frame':   302 obs. of  14 variables:
 $ age     : int  52 53 70 61 62 58 58 55 46 54 ...
 $ sex     : Factor w/ 2 levels "0","1": 2 2 2 2 1 1 2 2 2 2 ...
 $ cp      : Factor w/ 4 levels "0","1","2","3": 1 1 1 1 1 1 1 1 1 1 ...
 $ trestbps: int  125 140 145 148 138 100 114 160 120 122 ...
 $ chol    : int  212 203 174 203 294 248 318 289 249 286 ...
 $ fbs     : Factor w/ 2 levels "0","1": 1 2 1 1 2 1 1 1 1 1 ...
 $ restecg : Factor w/ 3 levels "0","1","2": 2 1 2 2 2 1 3 1 1 1 ...
 $ thalach : int  168 155 125 161 106 122 140 145 144 116 ...
 $ exang   : Factor w/ 2 levels "0","1": 1 2 2 1 1 1 1 2 1 2 ...
 $ oldpeak : num  1 3.1 2.6 0 1.9 1 4.4 0.8 0.8 3.2 ...
 $ slope   : Factor w/ 3 levels "0","1","2": 3 1 1 3 2 2 1 2 3 2 ...
 $ ca      : int  2 0 0 1 3 0 3 1 0 2 ...
 $ thal    : Factor w/ 4 levels "0","1","2","3": 4 4 4 4 3 3 2 4 4 3 ...
 $ target  : Factor w/ 2 levels "0","1": 1 1 1 1 1 2 1 1 1 1 ...

Summary Statistics

Before creating our regression model, we examine summary statistics for the quantitative variables.

Summary Statistics for Quantitative Variables
Variable	Mean	Median	SD	Minimum	Maximum
Age	54.42	55.5	9.05	29	77.0
Resting Blood Pressure	131.60	130.0	17.56	94	200.0
Cholesterol	246.50	240.5	51.75	126	564.0
Maximum Heart Rate	149.57	152.5	22.90	71	202.0
Oldpeak	1.04	0.8	1.16	0	6.2

We display the table using kable().

Summary Statistics for Quantitative Variables
Variable	Mean	Median	SD	Minimum	Maximum
Age	54.42	55.5	9.05	29	77.0
Resting Blood Pressure	131.60	130.0	17.56	94	200.0
Cholesterol	246.50	240.5	51.75	126	564.0
Maximum Heart Rate	149.57	152.5	22.90	71	202.0
Oldpeak	1.04	0.8	1.16	0	6.2

We can also examine a basic summary of every variable.

      age        sex     cp         trestbps          chol       fbs    
 Min.   :29.00   0: 96   0:143   Min.   : 94.0   Min.   :126.0   0:257  
 1st Qu.:48.00   1:206   1: 50   1st Qu.:120.0   1st Qu.:211.0   1: 45  
 Median :55.50           2: 86   Median :130.0   Median :240.5          
 Mean   :54.42           3: 23   Mean   :131.6   Mean   :246.5          
 3rd Qu.:61.00                   3rd Qu.:140.0   3rd Qu.:274.8          
 Max.   :77.00                   Max.   :200.0   Max.   :564.0          
 restecg    thalach      exang      oldpeak      slope         ca        
 0:147   Min.   : 71.0   0:203   Min.   :0.000   0: 21   Min.   :0.0000  
 1:151   1st Qu.:133.2   1: 99   1st Qu.:0.000   1:140   1st Qu.:0.0000  
 2:  4   Median :152.5           Median :0.800   2:141   Median :0.0000  
         Mean   :149.6           Mean   :1.043           Mean   :0.7185  
         3rd Qu.:166.0           3rd Qu.:1.600           3rd Qu.:1.0000  
         Max.   :202.0           Max.   :6.200           Max.   :4.0000  
 thal    target 
 0:  2   0:138  
 1: 18   1:164  
 2:165          
 3:117

Visualizing Quantitative Variables

We now visualize each of our main quantitative variables.

Age

This graph shows the age distribution of patients in our sample.

Resting Blood Pressure

Cholesterol

Maximum Heart Rate

Maximum heart rate is particularly important because it is the Y variable that we will attempt to predict with our regression model.

Oldpeak

Visualizing Qualitative Variables

For qualitative variables, bar charts allow us to compare the number of observations in each category.

Sex

Chest Pain Type

Fasting Blood Sugar

Resting ECG

Exercise-Induced Angina

ST Slope

Number of Major Vessels

Although ca is stored numerically, it only takes a small number of discrete values. A bar chart makes the distribution easy to interpret.

Thalassemia Category

Heart Disease Status

Exploring Maximum Heart Rate

Now that we have examined each variable individually, we look more closely at variables that may help explain our Y variable, maximum heart rate.

Age vs. Maximum Heart Rate

`geom_smooth()` using formula = 'y ~ x'

The fitted line allows us to see the general direction of the relationship between age and maximum heart rate.

Heart Disease Status vs. Maximum Heart Rate

The boxplots allow us to compare the distribution of maximum heart rate between patients in the two heart disease categories.

Exercise-Induced Angina vs. Maximum Heart Rate

Chest Pain Type vs. Maximum Heart Rate

Multiple Linear Regression

We now use multiple linear regression to investigate which characteristics are associated with maximum heart rate.

Our dependent variable is:

Y = thalach (Maximum Heart Rate)

Our explanatory variables are:

age
target
sex
cp
exang
oldpeak

The model can be represented generally as:

Maximum Heart Rate = Age + Heart Disease Status + Sex + Chest Pain Type + Exercise-Induced Angina + Oldpeak

We estimate the model using the lm() function.

We then use summary() to view the regression results.


Call:
lm(formula = thalach ~ age + target + sex + cp + exang + oldpeak, 
    data = heart)

Residuals:
    Min      1Q  Median      3Q     Max 
-65.241 -11.189   2.319  12.487  41.435 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 188.4373     8.1034  23.254  < 2e-16 ***
age          -0.7532     0.1257  -5.992 6.08e-09 ***
target1       7.1626     2.8895   2.479   0.0137 *  
sex1          0.7259     2.4751   0.293   0.7695    
cp1           7.8258     3.5514   2.204   0.0283 *  
cp2           4.4549     2.9868   1.492   0.1369    
cp3           9.0015     4.4793   2.010   0.0454 *  
exang1       -8.9980     2.7072  -3.324   0.0010 ** 
oldpeak      -2.4568     1.0687  -2.299   0.0222 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 18.73 on 293 degrees of freedom
Multiple R-squared:  0.3492,    Adjusted R-squared:  0.3314 
F-statistic: 19.65 on 8 and 293 DF,  p-value: < 2.2e-16

Understanding the Regression Output

There are several important parts of the regression output.

Estimate

The estimate shows the predicted relationship between an explanatory variable and maximum heart rate while holding the other variables constant.

For example, the coefficient on age tells us how much maximum heart rate is predicted to change when age increases by one year, holding the other explanatory variables constant.

P-value

The p-value helps us determine whether there is statistical evidence that an explanatory variable is associated with maximum heart rate.

For this project, we use a significance level of:

0.05

A p-value below 0.05 will therefore be considered statistically significant.

R-Squared

R-squared tells us what proportion of the variation in maximum heart rate is explained by the explanatory variables included in our model.

A higher R-squared means that the model explains more of the variation in our Y variable.

Regression Diagnostics

Linear regression relies on several assumptions.

We can examine diagnostic plots to determine whether our model appears reasonable.

Residuals vs. Fitted Values

Ideally, the residuals should appear randomly scattered around zero without a strong curved pattern.

Normal Q-Q Plot

If the residuals are approximately normally distributed, most of the points should be reasonably close to the diagonal line.

Actual vs. Predicted Maximum Heart Rate

Our regression model can generate a predicted maximum heart rate for each patient.

First, we add the predictions to our dataset.

We can then compare the model’s predictions with the actual maximum heart rates.

`geom_smooth()` using formula = 'y ~ x'

If our model predicts maximum heart rate well, we should see a positive relationship between the predicted and actual values.

Key Findings

Our analysis examined the relationship between maximum heart rate and several patient characteristics.

When presenting the regression results, we will focus on:

Which explanatory variables have p-values below 0.05.
Whether statistically significant relationships are positive or negative.
The magnitude of important regression coefficients.
The R-squared value.
Whether the diagnostic plots suggest that our linear regression model is reasonable.

The regression model allows us to examine each variable’s relationship with maximum heart rate while controlling for the other variables included in the model.

Limitations

Our analysis has several limitations.

First, the dataset is cross-sectional and observational. Therefore, the regression results show associations between variables, but they do not necessarily establish cause and effect.

Second, maximum heart rate can be affected by factors that are not included in this dataset.

Third, our results depend on the patients included in this particular dataset, so they may not perfectly represent every population.

Finally, linear regression makes assumptions about the relationships between the variables. Our diagnostic plots help us evaluate whether these assumptions are reasonable.

# Conclusion

The purpose of this project was to investigate the factors associated with maximum heart rate during exercise.

Our dependent variable was maximum heart rate (thalach), and we used age, heart disease status, sex, chest pain type, exercise-induced angina, and ST depression as explanatory variables.

Through data cleaning, summary statistics, visualization, and multiple linear regression, we can identify which characteristics have the strongest relationships with maximum heart rate.

The regression results provide a way to estimate these relationships while holding the other explanatory variables constant.

Overall, this project demonstrates how cross-sectional health data and multiple linear regression can be used to investigate relationships between patient characteristics and cardiovascular exercise response.