Week_12_Discussion
Saloua Daouki
2024-04-06
# Load the necessary packages
library(tidyverse)## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr 1.1.4 ✔ readr 2.1.5
## ✔ forcats 1.0.0 ✔ stringr 1.5.1
## ✔ ggplot2 3.4.4 ✔ tibble 3.2.1
## ✔ lubridate 1.9.3 ✔ tidyr 1.3.1
## ✔ purrr 1.0.2
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorslibrary(corrplot)## corrplot 0.92 loadedLoading the data
This data set provides different factors that may or may not influence the chances of having heart attack. Since there are many possible predictors that may explain the risk of having heart attack, I am going to use multiple regression model:
# Load Heart attack data into R
Heart_Attack <- read_csv("https://raw.githubusercontent.com/SalouaDaouki/Data605/main/heart_attack_prediction_dataset2.csv",
col_types = cols(
Age = col_double(),
Cholesterol = col_double(),
`Heart Rate` = col_double(),
Diabetes = col_double(),
`Family History` = col_double(),
Smoking = col_double(),
Obesity = col_double(),
`Alcohol Consumption` = col_double(),
`Exercise Hours Per Week` = col_double(),
BMI = col_double(),
Triglycerides = col_double(),
`Physical Activity Days Per Week` = col_double(),
`Sleep Hours Per Day` = col_double(),
`Heart Attack Risk` = col_double()
))head(Heart_Attack)## # A tibble: 6 × 26
## `Patient ID` Age Sex Cholesterol `Blood Pressure` `Heart Rate` Diabetes
## <chr> <dbl> <chr> <dbl> <chr> <dbl> <dbl>
## 1 BMW7812 67 Male 208 158/88 72 0
## 2 CZE1114 21 Male 389 165/93 98 1
## 3 BNI9906 21 Female 324 174/99 72 1
## 4 JLN3497 84 Male 383 163/100 73 1
## 5 GFO8847 66 Male 318 91/88 93 1
## 6 ZOO7941 54 Female 297 172/86 48 1
## # ℹ 19 more variables: `Family History` <dbl>, Smoking <dbl>, Obesity <dbl>,
## # `Alcohol Consumption` <dbl>, `Exercise Hours Per Week` <dbl>, Diet <chr>,
## # `Previous Heart Problems` <dbl>, `Medication Use` <dbl>,
## # `Stress Level` <dbl>, `Sedentary Hours Per Day` <dbl>, Income <dbl>,
## # BMI <dbl>, Triglycerides <dbl>, `Physical Activity Days Per Week` <dbl>,
## # `Sleep Hours Per Day` <dbl>, Country <chr>, Continent <chr>,
## # Hemisphere <chr>, `Heart Attack Risk` <dbl>tail(Heart_Attack)## # A tibble: 6 × 26
## `Patient ID` Age Sex Cholesterol `Blood Pressure` `Heart Rate` Diabetes
## <chr> <dbl> <chr> <dbl> <chr> <dbl> <dbl>
## 1 YDX2478 59 Female 378 93/78 99 0
## 2 MSV9918 60 Male 121 94/76 61 1
## 3 QSV6764 28 Female 120 157/102 73 1
## 4 XKA5925 47 Male 250 161/75 105 0
## 5 EPE6801 36 Male 178 119/67 60 1
## 6 ZWN9666 25 Female 356 138/67 75 1
## # ℹ 19 more variables: `Family History` <dbl>, Smoking <dbl>, Obesity <dbl>,
## # `Alcohol Consumption` <dbl>, `Exercise Hours Per Week` <dbl>, Diet <chr>,
## # `Previous Heart Problems` <dbl>, `Medication Use` <dbl>,
## # `Stress Level` <dbl>, `Sedentary Hours Per Day` <dbl>, Income <dbl>,
## # BMI <dbl>, Triglycerides <dbl>, `Physical Activity Days Per Week` <dbl>,
## # `Sleep Hours Per Day` <dbl>, Country <chr>, Continent <chr>,
## # Hemisphere <chr>, `Heart Attack Risk` <dbl>The data has 26 variables with 8763 observations.
Visualizing the relationships in the Heart Attack data
First, let’s take a look at the description of the data:
summary(Heart_Attack)## Patient ID Age Sex Cholesterol
## Length:8763 Min. :18.00 Length:8763 Min. :120.0
## Class :character 1st Qu.:35.00 Class :character 1st Qu.:192.0
## Mode :character Median :54.00 Mode :character Median :259.0
## Mean :53.71 Mean :259.9
## 3rd Qu.:72.00 3rd Qu.:330.0
## Max. :90.00 Max. :400.0
## Blood Pressure Heart Rate Diabetes Family History
## Length:8763 Min. : 40.00 Min. :0.0000 Min. :0.000
## Class :character 1st Qu.: 57.00 1st Qu.:0.0000 1st Qu.:0.000
## Mode :character Median : 75.00 Median :1.0000 Median :0.000
## Mean : 75.02 Mean :0.6523 Mean :0.493
## 3rd Qu.: 93.00 3rd Qu.:1.0000 3rd Qu.:1.000
## Max. :110.00 Max. :1.0000 Max. :1.000
## Smoking Obesity Alcohol Consumption Exercise Hours Per Week
## Min. :0.0000 Min. :0.0000 Min. :0.0000 Min. : 0.002442
## 1st Qu.:1.0000 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.: 4.981579
## Median :1.0000 Median :1.0000 Median :1.0000 Median :10.069559
## Mean :0.8968 Mean :0.5014 Mean :0.5981 Mean :10.014284
## 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:15.050018
## Max. :1.0000 Max. :1.0000 Max. :1.0000 Max. :19.998709
## Diet Previous Heart Problems Medication Use Stress Level
## Length:8763 Min. :0.0000 Min. :0.0000 Min. : 1.00
## Class :character 1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.: 3.00
## Mode :character Median :0.0000 Median :0.0000 Median : 5.00
## Mean :0.4958 Mean :0.4983 Mean : 5.47
## 3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.: 8.00
## Max. :1.0000 Max. :1.0000 Max. :10.00
## Sedentary Hours Per Day Income BMI Triglycerides
## Min. : 0.001263 Min. : 20062 Min. :18.00 Min. : 30.0
## 1st Qu.: 2.998794 1st Qu.: 88310 1st Qu.:23.42 1st Qu.:225.5
## Median : 5.933622 Median :157866 Median :28.77 Median :417.0
## Mean : 5.993690 Mean :158263 Mean :28.89 Mean :417.7
## 3rd Qu.: 9.019124 3rd Qu.:227749 3rd Qu.:34.32 3rd Qu.:612.0
## Max. :11.999313 Max. :299954 Max. :40.00 Max. :800.0
## Physical Activity Days Per Week Sleep Hours Per Day Country
## Min. :0.00 Min. : 4.000 Length:8763
## 1st Qu.:2.00 1st Qu.: 5.000 Class :character
## Median :3.00 Median : 7.000 Mode :character
## Mean :3.49 Mean : 7.024
## 3rd Qu.:5.00 3rd Qu.: 9.000
## Max. :7.00 Max. :10.000
## Continent Hemisphere Heart Attack Risk
## Length:8763 Length:8763 Min. :0.0000
## Class :character Class :character 1st Qu.:0.0000
## Mode :character Mode :character Median :0.0000
## Mean :0.3582
## 3rd Qu.:1.0000
## Max. :1.0000Then, we need to check for any missing data values to see if we need to do any necessary cleaning:
# Check for missing values
sum(is.na(Heart_Attack))## [1] 0Perfect! No missing values. Although, the “Blood Pressure” column needs some manipulations to make it included in the analysis; the original data was (160/70) in the form of (Systolic Pressure/Diastolic Pressure), let’s separate the column to two columns:
# Separate systolic and diastolic pressure
Heart_Attack <- separate(Heart_Attack, `Blood Pressure`, into = c("Systolic_Pressure", "Diastolic_Pressure"), sep = "/")
# Convert separated columns to numeric
Heart_Attack$Systolic_Pressure <- as.numeric(Heart_Attack$Systolic_Pressure)
Heart_Attack$Diastolic_Pressure <- as.numeric(Heart_Attack$Diastolic_Pressure)Let’s compare all the data that we have in the data set and get a glimpse of different relationships between different variables. Since the data includes qualitative data as well, we need to select only the numerical data to compare their relationships, otherwise we’ll encounter an error;
# Select only numerical columns from Heart_Attack data
numerical_data <- select_if(Heart_Attack, is.numeric)
# Use pairs function to create scatter plot matrix for numerical variables
pairs(numerical_data, gap = 0.5)
Since the plots are very tiny and hard to see, let’s select only the numerical variables from the “Heart_Attack” data, and assign it to a subset “numerical_var”;
# Splitting the data into subdata sets for better visualization
# Selecting only numerical variables
numerical_var <- Heart_Attack[, sapply(Heart_Attack, is.numeric)]
# View the subset
View(numerical_var)Then, split the “numercial_var” data into subdata sets where we include only 3 to 4 variables in each and apply pairs() function so we can visualize the plots better;
# Selecting columns from 1 to 3 and the last column (21)
subdata <- numerical_var[, c(1:3, 21)]
pairs(subdata)
# Selecting columns from 4 to 6 and the last column
subdata1 <- numerical_var[, c(4:6, 21)]
pairs(subdata1)
# Selecting columns from 7 to 9 and the last column
subdata2 <- numerical_var[, c(7:9, 21)]
pairs(subdata2)
# Selecting columns from 10 to 13 and the last column
subdata3 <- numerical_var[, c(10:13, 21)]
pairs(subdata3)
# Selecting columns from 14 to 17 and the last column
subdata4 <- numerical_var[, c(14:17, 21)]
pairs(subdata4)
# Selecting columns from 18 to 21
subdata5 <- numerical_var[, 18:21]
pairs(subdata5)
# Compute the correlation matrix
correlation_matrix <- cor(numerical_var)
# Print the correlation matrix
print(correlation_matrix)## Age Cholesterol Systolic_Pressure
## Age 1.0000000000 -9.107011e-03 0.0030702087
## Cholesterol -0.0091070106 1.000000e+00 0.0001326976
## Systolic_Pressure 0.0030702087 1.326976e-04 1.0000000000
## Diastolic_Pressure -0.0098256171 2.083335e-03 0.0133370093
## Heart Rate -0.0038440129 3.149083e-04 0.0084818882
## Diabetes -0.0141052134 -1.342760e-02 -0.0053061373
## Family History 0.0083525467 -2.160793e-02 -0.0097617900
## Smoking 0.3948909143 1.634161e-02 -0.0095344334
## Obesity -0.0081402534 -1.484265e-02 -0.0019181865
## Alcohol Consumption -0.0066655960 -7.261001e-03 0.0107636116
## Exercise Hours Per Week 0.0012056390 2.151714e-02 -0.0095055610
## Previous Heart Problems 0.0008683616 -6.070179e-03 -0.0119258199
## Medication Use 0.0009803087 -9.054425e-04 -0.0011815384
## Stress Level 0.0183066457 -2.448711e-02 0.0178478307
## Sedentary Hours Per Day 0.0172801340 1.891449e-02 0.0033926071
## Income -0.0017327899 6.750208e-06 0.0104143552
## BMI -0.0026118462 1.729187e-02 0.0042790432
## Triglycerides 0.0034149567 -5.453721e-03 0.0051206953
## Physical Activity Days Per Week 0.0013836679 1.605594e-02 -0.0075739574
## Sleep Hours Per Day -0.0021847037 4.456229e-03 -0.0046277633
## Heart Attack Risk 0.0064031871 1.933968e-02 0.0185852077
## Diastolic_Pressure Heart Rate Diabetes
## Age -0.009825617 -0.0038440129 -0.0141052134
## Cholesterol 0.002083335 0.0003149083 -0.0134276004
## Systolic_Pressure 0.013337009 0.0084818882 -0.0053061373
## Diastolic_Pressure 1.000000000 -0.0181130570 -0.0005115390
## Heart Rate -0.018113057 1.0000000000 0.0067637485
## Diabetes -0.000511539 0.0067637485 1.0000000000
## Family History 0.017817782 -0.0134695874 -0.0138436970
## Smoking -0.012293212 -0.0123306434 0.0005265323
## Obesity -0.020574211 0.0127248822 0.0128657609
## Alcohol Consumption -0.007282476 0.0034586719 0.0055512063
## Exercise Hours Per Week -0.003468859 0.0082763293 -0.0070138312
## Previous Heart Problems 0.008813417 -0.0049559066 0.0008667311
## Medication Use 0.004607425 0.0092439990 -0.0026558450
## Stress Level -0.008445057 -0.0045467686 0.0067191335
## Sedentary Hours Per Day -0.006606069 -0.0102320484 0.0047052959
## Income 0.008816276 0.0048734774 -0.0007589881
## BMI 0.000805537 0.0052985748 -0.0028518365
## Triglycerides 0.000544911 0.0122436948 0.0104313583
## Physical Activity Days Per Week 0.016294383 0.0008343817 -0.0024111151
## Sleep Hours Per Day 0.010679456 0.0018112469 -0.0124572712
## Heart Attack Risk -0.007509192 -0.0042512016 0.0172252957
## Family History Smoking Obesity
## Age 0.0083525467 0.3948909143 -0.008140253
## Cholesterol -0.0216079303 0.0163416079 -0.014842648
## Systolic_Pressure -0.0097617900 -0.0095344334 -0.001918186
## Diastolic_Pressure 0.0178177823 -0.0122932117 -0.020574211
## Heart Rate -0.0134695874 -0.0123306434 0.012724882
## Diabetes -0.0138436970 0.0005265323 0.012865761
## Family History 1.0000000000 0.0117482875 -0.001443614
## Smoking 0.0117482875 1.0000000000 0.003968986
## Obesity -0.0014436141 0.0039689860 1.000000000
## Alcohol Consumption 0.0127013167 0.0127537498 -0.024195420
## Exercise Hours Per Week -0.0063777562 -0.0001498847 0.002098949
## Previous Heart Problems -0.0045680682 -0.0005743184 0.005159194
## Medication Use 0.0009806963 -0.0108769653 -0.006267008
## Stress Level 0.0156365541 -0.0017567282 0.010625928
## Sedentary Hours Per Day 0.0025611570 0.0153110447 -0.001333256
## Income -0.0004012910 0.0030964480 -0.003870406
## BMI -0.0114921470 0.0076700789 -0.006058378
## Triglycerides -0.0019035518 0.0046500312 0.001466947
## Physical Activity Days Per Week 0.0095614489 -0.0064654906 0.005337385
## Sleep Hours Per Day -0.0111987966 -0.0054244718 -0.005313827
## Heart Attack Risk -0.0016519219 -0.0040512792 -0.013317553
## Alcohol Consumption Exercise Hours Per Week
## Age -0.0066655960 0.0012056390
## Cholesterol -0.0072610006 0.0215171356
## Systolic_Pressure 0.0107636116 -0.0095055610
## Diastolic_Pressure -0.0072824758 -0.0034688589
## Heart Rate 0.0034586719 0.0082763293
## Diabetes 0.0055512063 -0.0070138312
## Family History 0.0127013167 -0.0063777562
## Smoking 0.0127537498 -0.0001498847
## Obesity -0.0241954196 0.0020989489
## Alcohol Consumption 1.0000000000 -0.0085143204
## Exercise Hours Per Week -0.0085143204 1.0000000000
## Previous Heart Problems 0.0103951916 0.0052525457
## Medication Use 0.0033387438 -0.0071193499
## Stress Level -0.0050229973 -0.0091024194
## Sedentary Hours Per Day -0.0128281458 0.0087556011
## Income -0.0223957093 -0.0234138473
## BMI 0.0105624546 0.0037769215
## Triglycerides 0.0061686073 0.0017169491
## Physical Activity Days Per Week 0.0015934493 0.0077251861
## Sleep Hours Per Day -0.0008434151 -0.0012453363
## Heart Attack Risk -0.0137776983 0.0111328240
## Previous Heart Problems Medication Use
## Age 0.0008683616 0.0009803087
## Cholesterol -0.0060701787 -0.0009054425
## Systolic_Pressure -0.0119258199 -0.0011815384
## Diastolic_Pressure 0.0088134173 0.0046074246
## Heart Rate -0.0049559066 0.0092439990
## Diabetes 0.0008667311 -0.0026558450
## Family History -0.0045680682 0.0009806963
## Smoking -0.0005743184 -0.0108769653
## Obesity 0.0051591937 -0.0062670079
## Alcohol Consumption 0.0103951916 0.0033387438
## Exercise Hours Per Week 0.0052525457 -0.0071193499
## Previous Heart Problems 1.0000000000 0.0053361057
## Medication Use 0.0053361057 1.0000000000
## Stress Level -0.0176285817 0.0008628768
## Sedentary Hours Per Day -0.0026944572 0.0225130353
## Income -0.0032805602 -0.0034639224
## BMI 0.0157179207 0.0095140704
## Triglycerides -0.0190293799 -0.0110953595
## Physical Activity Days Per Week 0.0085367268 -0.0111388955
## Sleep Hours Per Day 0.0044604459 -0.0203925832
## Heart Attack Risk 0.0002735645 0.0022344065
## Stress Level Sedentary Hours Per Day
## Age 0.0183066457 1.728013e-02
## Cholesterol -0.0244871111 1.891449e-02
## Systolic_Pressure 0.0178478307 3.392607e-03
## Diastolic_Pressure -0.0084450573 -6.606069e-03
## Heart Rate -0.0045467686 -1.023205e-02
## Diabetes 0.0067191335 4.705296e-03
## Family History 0.0156365541 2.561157e-03
## Smoking -0.0017567282 1.531104e-02
## Obesity 0.0106259284 -1.333256e-03
## Alcohol Consumption -0.0050229973 -1.282815e-02
## Exercise Hours Per Week -0.0091024194 8.755601e-03
## Previous Heart Problems -0.0176285817 -2.694457e-03
## Medication Use 0.0008628768 2.251304e-02
## Stress Level 1.0000000000 -5.397241e-03
## Sedentary Hours Per Day -0.0053972409 1.000000e+00
## Income -0.0027604514 3.510621e-03
## BMI -0.0032504472 -2.356074e-05
## Triglycerides -0.0039213025 -5.784609e-03
## Physical Activity Days Per Week 0.0074046302 -6.178012e-03
## Sleep Hours Per Day -0.0142054068 4.792013e-03
## Heart Attack Risk -0.0041113217 -5.612975e-03
## Income BMI Triglycerides
## Age -1.732790e-03 -2.611846e-03 0.003414957
## Cholesterol 6.750208e-06 1.729187e-02 -0.005453721
## Systolic_Pressure 1.041436e-02 4.279043e-03 0.005120695
## Diastolic_Pressure 8.816276e-03 8.055370e-04 0.000544911
## Heart Rate 4.873477e-03 5.298575e-03 0.012243695
## Diabetes -7.589881e-04 -2.851837e-03 0.010431358
## Family History -4.012910e-04 -1.149215e-02 -0.001903552
## Smoking 3.096448e-03 7.670079e-03 0.004650031
## Obesity -3.870406e-03 -6.058378e-03 0.001466947
## Alcohol Consumption -2.239571e-02 1.056245e-02 0.006168607
## Exercise Hours Per Week -2.341385e-02 3.776921e-03 0.001716949
## Previous Heart Problems -3.280560e-03 1.571792e-02 -0.019029380
## Medication Use -3.463922e-03 9.514070e-03 -0.011095359
## Stress Level -2.760451e-03 -3.250447e-03 -0.003921303
## Sedentary Hours Per Day 3.510621e-03 -2.356074e-05 -0.005784609
## Income 1.000000e+00 8.835838e-03 0.010738559
## BMI 8.835838e-03 1.000000e+00 -0.005963607
## Triglycerides 1.073856e-02 -5.963607e-03 1.000000000
## Physical Activity Days Per Week 1.302733e-04 8.110375e-03 -0.007556419
## Sleep Hours Per Day -6.598343e-03 -1.003041e-02 -0.029215971
## Heart Attack Risk 9.627602e-03 2.027903e-05 0.010471454
## Physical Activity Days Per Week
## Age 0.0013836679
## Cholesterol 0.0160559355
## Systolic_Pressure -0.0075739574
## Diastolic_Pressure 0.0162943827
## Heart Rate 0.0008343817
## Diabetes -0.0024111151
## Family History 0.0095614489
## Smoking -0.0064654906
## Obesity 0.0053373847
## Alcohol Consumption 0.0015934493
## Exercise Hours Per Week 0.0077251861
## Previous Heart Problems 0.0085367268
## Medication Use -0.0111388955
## Stress Level 0.0074046302
## Sedentary Hours Per Day -0.0061780115
## Income 0.0001302733
## BMI 0.0081103748
## Triglycerides -0.0075564192
## Physical Activity Days Per Week 1.0000000000
## Sleep Hours Per Day 0.0140334379
## Heart Attack Risk -0.0050135111
## Sleep Hours Per Day Heart Attack Risk
## Age -0.0021847037 6.403187e-03
## Cholesterol 0.0044562286 1.933968e-02
## Systolic_Pressure -0.0046277633 1.858521e-02
## Diastolic_Pressure 0.0106794559 -7.509192e-03
## Heart Rate 0.0018112469 -4.251202e-03
## Diabetes -0.0124572712 1.722530e-02
## Family History -0.0111987966 -1.651922e-03
## Smoking -0.0054244718 -4.051279e-03
## Obesity -0.0053138266 -1.331755e-02
## Alcohol Consumption -0.0008434151 -1.377770e-02
## Exercise Hours Per Week -0.0012453363 1.113282e-02
## Previous Heart Problems 0.0044604459 2.735645e-04
## Medication Use -0.0203925832 2.234407e-03
## Stress Level -0.0142054068 -4.111322e-03
## Sedentary Hours Per Day 0.0047920126 -5.612975e-03
## Income -0.0065983433 9.627602e-03
## BMI -0.0100304097 2.027903e-05
## Triglycerides -0.0292159711 1.047145e-02
## Physical Activity Days Per Week 0.0140334379 -5.013511e-03
## Sleep Hours Per Day 1.0000000000 -1.852822e-02
## Heart Attack Risk -0.0185282174 1.000000e+00That’s lots of data to process (with my eyes), let’s visualize this in a better way:
# Compute the correlation matrix 1
correlation_matrix1 <- cor(subdata)
# Create a heatmap of the correlation matrix with annotations
corrplot(correlation_matrix1, method = "color", type = "upper", order = "hclust", tl.col = "black", tl.srt = 45,
mar = c(0, 0, 0, 5), # Increase spacing on the right side
tl.cex = 0.8, # Adjust label size
cl.ratio = 0.2) # Adjust the width of color legend
# Highlight variables with strong correlations (absolute correlation coefficient > 0.7)
high_correlation_indices1 <- which(abs(correlation_matrix1) > 0.7 & correlation_matrix1 != 1, arr.ind = TRUE)
points(high_correlation_indices1, pch = 16, col = "red")
Cholesterol is the predictor that has little stronger correlation with the hrat attack risk.
# Compute the correlation matrix 2
correlation_matrix2 <- cor(subdata1)
# Create a heatmap of the correlation matrix with annotations
corrplot(correlation_matrix2, method = "color", type = "upper", order = "hclust", tl.col = "black", tl.srt = 45,
mar = c(0, 0, 0, 5), # Increase spacing on the right side
tl.cex = 0.8, # Adjust label size
cl.ratio = 0.2) # Adjust the width of color legend
# Highlight variables with strong correlations (absolute correlation coefficient > 0.7)
high_correlation_indices2 <- which(abs(correlation_matrix2) > 0.7 & correlation_matrix2 != 1, arr.ind = TRUE)
points(high_correlation_indices2, pch = 16, col = "red")
Based on this heatmap, diabetes is the predictor that has stronger correlation with the heart attack risk.
# Compute the correlation matrix 3
correlation_matrix3 <- cor(subdata2)
# Create a heatmap of the correlation matrix with annotations
corrplot(correlation_matrix3, method = "color", type = "upper", order = "hclust", tl.col = "black", tl.srt = 45,
mar = c(0, 0, 0, 5), # Increase spacing on the right side
tl.cex = 0.8, # Adjust label size
cl.ratio = 0.2) # Adjust the width of color legend
# Highlight variables with strong correlations (absolute correlation coefficient > 0.7)
high_correlation_indices3 <- which(abs(correlation_matrix3) > 0.7 & correlation_matrix3 != 1, arr.ind = TRUE)
points(high_correlation_indices3, pch = 16, col = "red")
Heart attack risk can be influenced with obesity as well.
# Compute the correlation matrix 4
correlation_matrix4 <- cor(subdata3)
# Create a heatmap of the correlation matrix with annotations
corrplot(correlation_matrix4, method = "color", type = "upper", order = "hclust", tl.col = "black", tl.srt = 45,
mar = c(0, 0, 0, 5), # Increase spacing on the right side
tl.cex = 0.8, # Adjust label size
cl.ratio = 0.2) # Adjust the width of color legend
# Highlight variables with strong correlations (absolute correlation coefficient > 0.7)
high_correlation_indices4 <- which(abs(correlation_matrix4) > 0.7 & correlation_matrix4 != 1, arr.ind = TRUE)
points(high_correlation_indices4, pch = 16, col = "red")
The number hours one exercises can lower their heat attack risk.
# Compute the correlation matrix 5
correlation_matrix5 <- cor(subdata4)
# Create a heatmap of the correlation matrix with annotations
corrplot(correlation_matrix5, method = "color", type = "upper", order = "hclust", tl.col = "black", tl.srt = 45,
mar = c(0, 0, 0, 5), # Increase spacing on the right side
tl.cex = 0.8, # Adjust label size
cl.ratio = 0.2) # Adjust the width of color legend
# Highlight variables with strong correlations (absolute correlation coefficient > 0.7)
high_correlation_indices5 <- which(abs(correlation_matrix5) > 0.7 & correlation_matrix5 != 1, arr.ind = TRUE)
points(high_correlation_indices5, pch = 16, col = "red")
There is very weak association between the heart attack risk and the income.
# Compute the correlation matrix 6
correlation_matrix6 <- cor(subdata5)
# Create a heatmap of the correlation matrix with annotations
corrplot(correlation_matrix6, method = "color", type = "upper", order = "hclust", tl.col = "black", tl.srt = 45,
mar = c(0, 0, 0, 5), # Increase spacing on the right side
tl.cex = 0.8, # Adjust label size
cl.ratio = 0.2) # Adjust the width of color legend
# Highlight variables with strong correlations (absolute correlation coefficient > 0.7)
high_correlation_indices6 <- which(abs(correlation_matrix6) > 0.7 & correlation_matrix6 != 1, arr.ind = TRUE)
points(high_correlation_indices6, pch = 16, col = "red")
Let’s look at the numbers using the full regression model across all numerical variables of the data set;
Identifying Potential Predictors:
Let’s define the regression model to all numerical variables in the data:
Heart_Attack_lm.Full <- lm(`Heart Attack Risk` ~ Age + Cholesterol + Systolic_Pressure + Diastolic_Pressure + `Heart Rate` + Diabetes + `Family History` + Smoking + Obesity + `Alcohol Consumption` + `Exercise Hours Per Week` + `Previous Heart Problems` + `Medication Use` + `Stress Level` + `Sedentary Hours Per Day` + Income + BMI + Triglycerides + `Physical Activity Days Per Week` + `Sleep Hours Per Day`, data = Heart_Attack)Now, let’s look at the summary of the model:
summary(Heart_Attack_lm.Full)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Age + Cholesterol + Systolic_Pressure +
## Diastolic_Pressure + `Heart Rate` + Diabetes + `Family History` +
## Smoking + Obesity + `Alcohol Consumption` + `Exercise Hours Per Week` +
## `Previous Heart Problems` + `Medication Use` + `Stress Level` +
## `Sedentary Hours Per Day` + Income + BMI + Triglycerides +
## `Physical Activity Days Per Week` + `Sleep Hours Per Day`,
## data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4375 -0.3641 -0.3393 0.6279 0.7080
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.317e-01 6.378e-02 5.201 2.03e-07 ***
## Age 2.208e-04 2.626e-04 0.841 0.4004
## Cholesterol 1.156e-04 6.346e-05 1.822 0.0684 .
## Systolic_Pressure 3.421e-04 1.945e-04 1.758 0.0787 .
## Diastolic_Pressure -2.657e-04 3.494e-04 -0.761 0.4469
## `Heart Rate` -1.132e-04 2.494e-04 -0.454 0.6499
## Diabetes 1.794e-02 1.076e-02 1.667 0.0956 .
## `Family History` -5.067e-04 1.026e-02 -0.049 0.9606
## Smoking -1.271e-02 1.835e-02 -0.693 0.4884
## Obesity -1.303e-02 1.025e-02 -1.270 0.2040
## `Alcohol Consumption` -1.371e-02 1.046e-02 -1.311 0.1900
## `Exercise Hours Per Week` 9.257e-04 8.864e-04 1.044 0.2963
## `Previous Heart Problems` 9.846e-04 1.025e-02 0.096 0.9235
## `Medication Use` 2.130e-03 1.025e-02 0.208 0.8354
## `Stress Level` -7.319e-04 1.794e-03 -0.408 0.6832
## `Sedentary Hours Per Day` -9.079e-04 1.479e-03 -0.614 0.5394
## Income 5.523e-08 6.362e-08 0.868 0.3854
## BMI -3.231e-05 8.112e-04 -0.040 0.9682
## Triglycerides 2.096e-05 2.292e-05 0.914 0.3605
## `Physical Activity Days Per Week` -9.919e-04 2.246e-03 -0.442 0.6587
## `Sleep Hours Per Day` -4.314e-03 2.579e-03 -1.673 0.0944 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4795 on 8742 degrees of freedom
## Multiple R-squared: 0.002317, Adjusted R-squared: 3.408e-05
## F-statistic: 1.015 on 20 and 8742 DF, p-value: 0.4396Based on the summary of the full regression model, the only predictors that explain little bit (because the p value are greater than 0.05) the heart attack risk are: Cholesterol, Systolic_Pressure , Diabetes, and Sleep Hours Per Day. Now, let’s perform the backward process elimination:
Heart_Attack.lm.2 <- update(Heart_Attack_lm.Full, .~. - Age, data =
Heart_Attack)
summary(Heart_Attack.lm.2)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Cholesterol + Systolic_Pressure +
## Diastolic_Pressure + `Heart Rate` + Diabetes + `Family History` +
## Smoking + Obesity + `Alcohol Consumption` + `Exercise Hours Per Week` +
## `Previous Heart Problems` + `Medication Use` + `Stress Level` +
## `Sedentary Hours Per Day` + Income + BMI + Triglycerides +
## `Physical Activity Days Per Week` + `Sleep Hours Per Day`,
## data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4369 -0.3639 -0.3396 0.6280 0.7132
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.383e-01 6.330e-02 5.345 9.29e-08 ***
## Cholesterol 1.147e-04 6.345e-05 1.808 0.0706 .
## Systolic_Pressure 3.433e-04 1.945e-04 1.765 0.0777 .
## Diastolic_Pressure -2.674e-04 3.494e-04 -0.765 0.4441
## `Heart Rate` -1.129e-04 2.494e-04 -0.453 0.6507
## Diabetes 1.779e-02 1.076e-02 1.654 0.0982 .
## `Family History` -4.778e-04 1.026e-02 -0.047 0.9628
## Smoking -6.615e-03 1.685e-02 -0.393 0.6947
## Obesity -1.312e-02 1.025e-02 -1.280 0.2006
## `Alcohol Consumption` -1.382e-02 1.046e-02 -1.322 0.1863
## `Exercise Hours Per Week` 9.269e-04 8.864e-04 1.046 0.2957
## `Previous Heart Problems` 1.001e-03 1.025e-02 0.098 0.9222
## `Medication Use` 2.178e-03 1.025e-02 0.212 0.8318
## `Stress Level` -7.014e-04 1.793e-03 -0.391 0.6957
## `Sedentary Hours Per Day` -8.924e-04 1.479e-03 -0.603 0.5463
## Income 5.504e-08 6.362e-08 0.865 0.3870
## BMI -3.629e-05 8.111e-04 -0.045 0.9643
## Triglycerides 2.100e-05 2.292e-05 0.916 0.3596
## `Physical Activity Days Per Week` -9.829e-04 2.246e-03 -0.438 0.6616
## `Sleep Hours Per Day` -4.314e-03 2.579e-03 -1.673 0.0944 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4795 on 8743 degrees of freedom
## Multiple R-squared: 0.002236, Adjusted R-squared: 6.756e-05
## F-statistic: 1.031 on 19 and 8743 DF, p-value: 0.4197There is a very tiny improvement on the p-value; it goes from 0.4396 to 0.4197. Let’s continue with backward process elimination:
Heart_Attack.lm.3 <- update(Heart_Attack.lm.2, .~. - Diastolic_Pressure, data =
Heart_Attack)
summary(Heart_Attack.lm.3)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Cholesterol + Systolic_Pressure +
## `Heart Rate` + Diabetes + `Family History` + Smoking + Obesity +
## `Alcohol Consumption` + `Exercise Hours Per Week` + `Previous Heart Problems` +
## `Medication Use` + `Stress Level` + `Sedentary Hours Per Day` +
## Income + BMI + Triglycerides + `Physical Activity Days Per Week` +
## `Sleep Hours Per Day`, data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4328 -0.3640 -0.3400 0.6286 0.7095
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.156e-01 5.595e-02 5.642 1.74e-08 ***
## Cholesterol 1.146e-04 6.345e-05 1.807 0.0708 .
## Systolic_Pressure 3.412e-04 1.945e-04 1.754 0.0795 .
## `Heart Rate` -1.095e-04 2.494e-04 -0.439 0.6607
## Diabetes 1.779e-02 1.076e-02 1.653 0.0983 .
## `Family History` -6.201e-04 1.025e-02 -0.060 0.9518
## Smoking -6.458e-03 1.685e-02 -0.383 0.7016
## Obesity -1.296e-02 1.025e-02 -1.265 0.2061
## `Alcohol Consumption` -1.376e-02 1.046e-02 -1.316 0.1883
## `Exercise Hours Per Week` 9.290e-04 8.864e-04 1.048 0.2946
## `Previous Heart Problems` 9.315e-04 1.025e-02 0.091 0.9276
## `Medication Use` 2.138e-03 1.025e-02 0.209 0.8348
## `Stress Level` -6.894e-04 1.793e-03 -0.385 0.7006
## `Sedentary Hours Per Day` -8.846e-04 1.479e-03 -0.598 0.5498
## Income 5.462e-08 6.362e-08 0.859 0.3906
## BMI -3.677e-05 8.111e-04 -0.045 0.9638
## Triglycerides 2.097e-05 2.292e-05 0.915 0.3601
## `Physical Activity Days Per Week` -1.011e-03 2.245e-03 -0.450 0.6526
## `Sleep Hours Per Day` -4.335e-03 2.579e-03 -1.681 0.0928 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4795 on 8744 degrees of freedom
## Multiple R-squared: 0.002169, Adjusted R-squared: 0.0001149
## F-statistic: 1.056 on 18 and 8744 DF, p-value: 0.3916Heart_Attack.lm.4 <- update(Heart_Attack.lm.3, .~. - `Heart Rate`, data =
Heart_Attack)
summary(Heart_Attack.lm.4)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Cholesterol + Systolic_Pressure +
## Diabetes + `Family History` + Smoking + Obesity + `Alcohol Consumption` +
## `Exercise Hours Per Week` + `Previous Heart Problems` + `Medication Use` +
## `Stress Level` + `Sedentary Hours Per Day` + Income + BMI +
## Triglycerides + `Physical Activity Days Per Week` + `Sleep Hours Per Day`,
## data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4353 -0.3639 -0.3402 0.6285 0.7100
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.076e-01 5.287e-02 5.818 6.16e-09 ***
## Cholesterol 1.146e-04 6.344e-05 1.807 0.0708 .
## Systolic_Pressure 3.405e-04 1.945e-04 1.751 0.0801 .
## Diabetes 1.776e-02 1.076e-02 1.651 0.0989 .
## `Family History` -5.618e-04 1.025e-02 -0.055 0.9563
## Smoking -6.368e-03 1.685e-02 -0.378 0.7055
## Obesity -1.302e-02 1.025e-02 -1.270 0.2040
## `Alcohol Consumption` -1.378e-02 1.046e-02 -1.317 0.1877
## `Exercise Hours Per Week` 9.257e-04 8.863e-04 1.044 0.2963
## `Previous Heart Problems` 9.545e-04 1.025e-02 0.093 0.9258
## `Medication Use` 2.094e-03 1.025e-02 0.204 0.8381
## `Stress Level` -6.858e-04 1.793e-03 -0.382 0.7021
## `Sedentary Hours Per Day` -8.779e-04 1.479e-03 -0.594 0.5528
## Income 5.448e-08 6.362e-08 0.856 0.3918
## BMI -3.864e-05 8.111e-04 -0.048 0.9620
## Triglycerides 2.085e-05 2.291e-05 0.910 0.3628
## `Physical Activity Days Per Week` -1.012e-03 2.245e-03 -0.451 0.6523
## `Sleep Hours Per Day` -4.337e-03 2.579e-03 -1.682 0.0926 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4795 on 8745 degrees of freedom
## Multiple R-squared: 0.002147, Adjusted R-squared: 0.0002072
## F-statistic: 1.107 on 17 and 8745 DF, p-value: 0.3394Heart_Attack.lm.5 <- update(Heart_Attack.lm.4, .~. - `Family History`, data =
Heart_Attack)
summary(Heart_Attack.lm.5)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Cholesterol + Systolic_Pressure +
## Diabetes + Smoking + Obesity + `Alcohol Consumption` + `Exercise Hours Per Week` +
## `Previous Heart Problems` + `Medication Use` + `Stress Level` +
## `Sedentary Hours Per Day` + Income + BMI + Triglycerides +
## `Physical Activity Days Per Week` + `Sleep Hours Per Day`,
## data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4355 -0.3640 -0.3401 0.6286 0.7103
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.073e-01 5.254e-02 5.848 5.15e-09 ***
## Cholesterol 1.147e-04 6.343e-05 1.808 0.0706 .
## Systolic_Pressure 3.406e-04 1.945e-04 1.751 0.0799 .
## Diabetes 1.777e-02 1.076e-02 1.652 0.0986 .
## Smoking -6.379e-03 1.685e-02 -0.379 0.7049
## Obesity -1.302e-02 1.025e-02 -1.270 0.2040
## `Alcohol Consumption` -1.378e-02 1.046e-02 -1.318 0.1874
## `Exercise Hours Per Week` 9.260e-04 8.862e-04 1.045 0.2961
## `Previous Heart Problems` 9.570e-04 1.025e-02 0.093 0.9256
## `Medication Use` 2.094e-03 1.025e-02 0.204 0.8381
## `Stress Level` -6.873e-04 1.793e-03 -0.383 0.7014
## `Sedentary Hours Per Day` -8.781e-04 1.479e-03 -0.594 0.5526
## Income 5.448e-08 6.361e-08 0.857 0.3917
## BMI -3.813e-05 8.110e-04 -0.047 0.9625
## Triglycerides 2.085e-05 2.291e-05 0.910 0.3627
## `Physical Activity Days Per Week` -1.013e-03 2.245e-03 -0.451 0.6519
## `Sleep Hours Per Day` -4.336e-03 2.578e-03 -1.682 0.0927 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4794 on 8746 degrees of freedom
## Multiple R-squared: 0.002147, Adjusted R-squared: 0.0003212
## F-statistic: 1.176 on 16 and 8746 DF, p-value: 0.2786Heart_Attack.lm.6 <- update(Heart_Attack.lm.5, .~. - Smoking, data =
Heart_Attack)
summary(Heart_Attack.lm.6)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Cholesterol + Systolic_Pressure +
## Diabetes + Obesity + `Alcohol Consumption` + `Exercise Hours Per Week` +
## `Previous Heart Problems` + `Medication Use` + `Stress Level` +
## `Sedentary Hours Per Day` + Income + BMI + Triglycerides +
## `Physical Activity Days Per Week` + `Sleep Hours Per Day`,
## data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4363 -0.3641 -0.3402 0.6285 0.7097
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.017e-01 5.040e-02 5.985 2.25e-09 ***
## Cholesterol 1.143e-04 6.341e-05 1.803 0.0715 .
## Systolic_Pressure 3.413e-04 1.945e-04 1.755 0.0793 .
## Diabetes 1.777e-02 1.076e-02 1.652 0.0987 .
## Obesity -1.304e-02 1.025e-02 -1.272 0.2033
## `Alcohol Consumption` -1.384e-02 1.046e-02 -1.323 0.1857
## `Exercise Hours Per Week` 9.262e-04 8.862e-04 1.045 0.2960
## `Previous Heart Problems` 9.594e-04 1.025e-02 0.094 0.9254
## `Medication Use` 2.138e-03 1.025e-02 0.209 0.8347
## `Stress Level` -6.866e-04 1.793e-03 -0.383 0.7017
## `Sedentary Hours Per Day` -8.868e-04 1.478e-03 -0.600 0.5486
## Income 5.441e-08 6.361e-08 0.855 0.3924
## BMI -4.041e-05 8.109e-04 -0.050 0.9603
## Triglycerides 2.082e-05 2.291e-05 0.909 0.3636
## `Physical Activity Days Per Week` -1.007e-03 2.245e-03 -0.449 0.6537
## `Sleep Hours Per Day` -4.330e-03 2.578e-03 -1.680 0.0931 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4794 on 8747 degrees of freedom
## Multiple R-squared: 0.00213, Adjusted R-squared: 0.0004191
## F-statistic: 1.245 on 15 and 8747 DF, p-value: 0.2292Heart_Attack.lm.7 <- update(Heart_Attack.lm.6, .~. - Obesity, data =
Heart_Attack)
summary(Heart_Attack.lm.7)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Cholesterol + Systolic_Pressure +
## Diabetes + `Alcohol Consumption` + `Exercise Hours Per Week` +
## `Previous Heart Problems` + `Medication Use` + `Stress Level` +
## `Sedentary Hours Per Day` + Income + BMI + Triglycerides +
## `Physical Activity Days Per Week` + `Sleep Hours Per Day`,
## data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4295 -0.3635 -0.3411 0.6290 0.7161
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.946e-01 5.009e-02 5.880 4.25e-09 ***
## Cholesterol 1.155e-04 6.341e-05 1.821 0.0686 .
## Systolic_Pressure 3.417e-04 1.945e-04 1.757 0.0789 .
## Diabetes 1.759e-02 1.076e-02 1.636 0.1020
## `Alcohol Consumption` -1.351e-02 1.045e-02 -1.293 0.1961
## `Exercise Hours Per Week` 9.237e-04 8.862e-04 1.042 0.2973
## `Previous Heart Problems` 8.868e-04 1.025e-02 0.087 0.9311
## `Medication Use` 2.219e-03 1.025e-02 0.216 0.8286
## `Stress Level` -7.095e-04 1.793e-03 -0.396 0.6923
## `Sedentary Hours Per Day` -8.846e-04 1.478e-03 -0.598 0.5496
## Income 5.475e-08 6.361e-08 0.861 0.3894
## BMI -3.468e-05 8.109e-04 -0.043 0.9659
## Triglycerides 2.077e-05 2.291e-05 0.907 0.3646
## `Physical Activity Days Per Week` -1.023e-03 2.245e-03 -0.456 0.6486
## `Sleep Hours Per Day` -4.313e-03 2.578e-03 -1.673 0.0944 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4794 on 8748 degrees of freedom
## Multiple R-squared: 0.001946, Adjusted R-squared: 0.0003484
## F-statistic: 1.218 on 14 and 8748 DF, p-value: 0.2536Heart_Attack.lm.8 <- update(Heart_Attack.lm.7, .~. - `Alcohol Consumption`, data =
Heart_Attack)
summary(Heart_Attack.lm.8)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Cholesterol + Systolic_Pressure +
## Diabetes + `Exercise Hours Per Week` + `Previous Heart Problems` +
## `Medication Use` + `Stress Level` + `Sedentary Hours Per Day` +
## Income + BMI + Triglycerides + `Physical Activity Days Per Week` +
## `Sleep Hours Per Day`, data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4244 -0.3633 -0.3417 0.6302 0.7109
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.866e-01 4.972e-02 5.765 8.44e-09 ***
## Cholesterol 1.161e-04 6.341e-05 1.830 0.0672 .
## Systolic_Pressure 3.389e-04 1.945e-04 1.743 0.0814 .
## Diabetes 1.752e-02 1.076e-02 1.628 0.1035
## `Exercise Hours Per Week` 9.338e-04 8.862e-04 1.054 0.2920
## `Previous Heart Problems` 7.510e-04 1.025e-02 0.073 0.9416
## `Medication Use` 2.173e-03 1.025e-02 0.212 0.8321
## `Stress Level` -6.969e-04 1.793e-03 -0.389 0.6974
## `Sedentary Hours Per Day` -8.605e-04 1.478e-03 -0.582 0.5606
## Income 5.663e-08 6.360e-08 0.890 0.3732
## BMI -4.589e-05 8.109e-04 -0.057 0.9549
## Triglycerides 2.058e-05 2.291e-05 0.898 0.3690
## `Physical Activity Days Per Week` -1.028e-03 2.245e-03 -0.458 0.6470
## `Sleep Hours Per Day` -4.311e-03 2.578e-03 -1.672 0.0945 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4794 on 8749 degrees of freedom
## Multiple R-squared: 0.001755, Adjusted R-squared: 0.0002717
## F-statistic: 1.183 on 13 and 8749 DF, p-value: 0.2844Heart_Attack.lm.9 <- update(Heart_Attack.lm.8, .~. - `Exercise Hours Per Week`, data =
Heart_Attack)
summary(Heart_Attack.lm.9)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Cholesterol + Systolic_Pressure +
## Diabetes + `Previous Heart Problems` + `Medication Use` +
## `Stress Level` + `Sedentary Hours Per Day` + Income + BMI +
## Triglycerides + `Physical Activity Days Per Week` + `Sleep Hours Per Day`,
## data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4198 -0.3634 -0.3426 0.6304 0.7020
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.961e-01 4.891e-02 6.053 1.48e-09 ***
## Cholesterol 1.175e-04 6.340e-05 1.853 0.0640 .
## Systolic_Pressure 3.370e-04 1.945e-04 1.733 0.0831 .
## Diabetes 1.744e-02 1.076e-02 1.621 0.1050
## `Previous Heart Problems` 8.055e-04 1.025e-02 0.079 0.9374
## `Medication Use` 2.093e-03 1.025e-02 0.204 0.8382
## `Stress Level` -7.127e-04 1.793e-03 -0.398 0.6909
## `Sedentary Hours Per Day` -8.469e-04 1.478e-03 -0.573 0.5667
## Income 5.506e-08 6.358e-08 0.866 0.3865
## BMI -4.286e-05 8.109e-04 -0.053 0.9578
## Triglycerides 2.064e-05 2.291e-05 0.901 0.3677
## `Physical Activity Days Per Week` -1.011e-03 2.245e-03 -0.450 0.6526
## `Sleep Hours Per Day` -4.317e-03 2.578e-03 -1.674 0.0941 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4794 on 8750 degrees of freedom
## Multiple R-squared: 0.001628, Adjusted R-squared: 0.0002591
## F-statistic: 1.189 on 12 and 8750 DF, p-value: 0.2839Heart_Attack.lm.10 <- update(Heart_Attack.lm.9, .~. - `Previous Heart Problems`, data =
Heart_Attack)
summary(Heart_Attack.lm.10)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Cholesterol + Systolic_Pressure +
## Diabetes + `Medication Use` + `Stress Level` + `Sedentary Hours Per Day` +
## Income + BMI + Triglycerides + `Physical Activity Days Per Week` +
## `Sleep Hours Per Day`, data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4194 -0.3634 -0.3426 0.6305 0.7024
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.965e-01 4.861e-02 6.099 1.11e-09 ***
## Cholesterol 1.174e-04 6.339e-05 1.852 0.0640 .
## Systolic_Pressure 3.369e-04 1.944e-04 1.733 0.0832 .
## Diabetes 1.744e-02 1.076e-02 1.621 0.1050
## `Medication Use` 2.097e-03 1.025e-02 0.205 0.8379
## `Stress Level` -7.152e-04 1.792e-03 -0.399 0.6898
## `Sedentary Hours Per Day` -8.473e-04 1.478e-03 -0.573 0.5666
## Income 5.504e-08 6.358e-08 0.866 0.3866
## BMI -4.186e-05 8.108e-04 -0.052 0.9588
## Triglycerides 2.060e-05 2.291e-05 0.899 0.3684
## `Physical Activity Days Per Week` -1.009e-03 2.245e-03 -0.450 0.6530
## `Sleep Hours Per Day` -4.316e-03 2.578e-03 -1.674 0.0942 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4794 on 8751 degrees of freedom
## Multiple R-squared: 0.001628, Adjusted R-squared: 0.0003727
## F-statistic: 1.297 on 11 and 8751 DF, p-value: 0.2188Heart_Attack.lm.11 <- update(Heart_Attack.lm.10, .~. - `Medication Use`, data =
Heart_Attack)
summary(Heart_Attack.lm.11)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Cholesterol + Systolic_Pressure +
## Diabetes + `Stress Level` + `Sedentary Hours Per Day` + Income +
## BMI + Triglycerides + `Physical Activity Days Per Week` +
## `Sleep Hours Per Day`, data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4183 -0.3635 -0.3426 0.6303 0.7014
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.976e-01 4.831e-02 6.160 7.61e-10 ***
## Cholesterol 1.174e-04 6.339e-05 1.852 0.0640 .
## Systolic_Pressure 3.368e-04 1.944e-04 1.732 0.0832 .
## Diabetes 1.743e-02 1.076e-02 1.621 0.1051
## `Stress Level` -7.149e-04 1.792e-03 -0.399 0.6899
## `Sedentary Hours Per Day` -8.404e-04 1.478e-03 -0.569 0.5696
## Income 5.500e-08 6.357e-08 0.865 0.3870
## BMI -4.031e-05 8.107e-04 -0.050 0.9603
## Triglycerides 2.055e-05 2.290e-05 0.897 0.3696
## `Physical Activity Days Per Week` -1.014e-03 2.244e-03 -0.452 0.6514
## `Sleep Hours Per Day` -4.327e-03 2.578e-03 -1.679 0.0933 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4794 on 8752 degrees of freedom
## Multiple R-squared: 0.001623, Adjusted R-squared: 0.0004821
## F-statistic: 1.423 on 10 and 8752 DF, p-value: 0.1631Heart_Attack.lm.12 <- update(Heart_Attack.lm.11, .~. - `Stress Level`, data =
Heart_Attack)
summary(Heart_Attack.lm.12)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Cholesterol + Systolic_Pressure +
## Diabetes + `Sedentary Hours Per Day` + Income + BMI + Triglycerides +
## `Physical Activity Days Per Week` + `Sleep Hours Per Day`,
## data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4156 -0.3632 -0.3427 0.6305 0.7010
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.936e-01 4.725e-02 6.213 5.43e-10 ***
## Cholesterol 1.180e-04 6.337e-05 1.863 0.0626 .
## Systolic_Pressure 3.354e-04 1.944e-04 1.726 0.0845 .
## Diabetes 1.741e-02 1.076e-02 1.618 0.1056
## `Sedentary Hours Per Day` -8.375e-04 1.478e-03 -0.567 0.5709
## Income 5.507e-08 6.357e-08 0.866 0.3863
## BMI -3.930e-05 8.107e-04 -0.048 0.9613
## Triglycerides 2.059e-05 2.290e-05 0.899 0.3686
## `Physical Activity Days Per Week` -1.021e-03 2.244e-03 -0.455 0.6490
## `Sleep Hours Per Day` -4.312e-03 2.577e-03 -1.673 0.0943 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4794 on 8753 degrees of freedom
## Multiple R-squared: 0.001605, Adjusted R-squared: 0.0005781
## F-statistic: 1.563 on 9 and 8753 DF, p-value: 0.1201Heart_Attack.lm.13 <- update(Heart_Attack.lm.12, .~. - Income, data =
Heart_Attack)
summary(Heart_Attack.lm.13)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Cholesterol + Systolic_Pressure +
## Diabetes + `Sedentary Hours Per Day` + BMI + Triglycerides +
## `Physical Activity Days Per Week` + `Sleep Hours Per Day`,
## data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4154 -0.3632 -0.3426 0.6308 0.6966
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.019e-01 4.627e-02 6.524 7.22e-11 ***
## Cholesterol 1.180e-04 6.337e-05 1.862 0.0626 .
## Systolic_Pressure 3.371e-04 1.944e-04 1.735 0.0828 .
## Diabetes 1.740e-02 1.076e-02 1.618 0.1058
## `Sedentary Hours Per Day` -8.330e-04 1.478e-03 -0.564 0.5730
## BMI -3.313e-05 8.106e-04 -0.041 0.9674
## Triglycerides 2.080e-05 2.290e-05 0.908 0.3637
## `Physical Activity Days Per Week` -1.021e-03 2.244e-03 -0.455 0.6492
## `Sleep Hours Per Day` -4.326e-03 2.577e-03 -1.679 0.0933 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4794 on 8754 degrees of freedom
## Multiple R-squared: 0.001519, Adjusted R-squared: 0.0006066
## F-statistic: 1.665 on 8 and 8754 DF, p-value: 0.1015Heart_Attack.lm.14 <- update(Heart_Attack.lm.13, .~. - BMI, data =
Heart_Attack)
summary(Heart_Attack.lm.14)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Cholesterol + Systolic_Pressure +
## Diabetes + `Sedentary Hours Per Day` + Triglycerides + `Physical Activity Days Per Week` +
## `Sleep Hours Per Day`, data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4156 -0.3632 -0.3427 0.6307 0.6966
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.009e-01 4.001e-02 7.520 6e-14 ***
## Cholesterol 1.180e-04 6.335e-05 1.862 0.0626 .
## Systolic_Pressure 3.371e-04 1.944e-04 1.735 0.0829 .
## Diabetes 1.740e-02 1.075e-02 1.618 0.1057
## `Sedentary Hours Per Day` -8.329e-04 1.478e-03 -0.564 0.5730
## Triglycerides 2.081e-05 2.290e-05 0.909 0.3635
## `Physical Activity Days Per Week` -1.022e-03 2.244e-03 -0.455 0.6489
## `Sleep Hours Per Day` -4.325e-03 2.577e-03 -1.678 0.0933 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4793 on 8755 degrees of freedom
## Multiple R-squared: 0.001519, Adjusted R-squared: 0.0007206
## F-statistic: 1.903 on 7 and 8755 DF, p-value: 0.06487Heart_Attack.lm.15 <- update(Heart_Attack.lm.14, .~. - `Physical Activity Days Per Week`, data =
Heart_Attack)
summary(Heart_Attack.lm.15)##
## Call:
## lm(formula = `Heart Attack Risk` ~ Cholesterol + Systolic_Pressure +
## Diabetes + `Sedentary Hours Per Day` + Triglycerides + `Sleep Hours Per Day`,
## data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4148 -0.3633 -0.3428 0.6311 0.6976
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.974e-01 3.927e-02 7.573 4e-14 ***
## Cholesterol 1.175e-04 6.334e-05 1.855 0.0636 .
## Systolic_Pressure 3.378e-04 1.943e-04 1.738 0.0822 .
## Diabetes 1.741e-02 1.075e-02 1.619 0.1055
## `Sedentary Hours Per Day` -8.285e-04 1.478e-03 -0.561 0.5750
## Triglycerides 2.088e-05 2.290e-05 0.912 0.3618
## `Sleep Hours Per Day` -4.341e-03 2.576e-03 -1.685 0.0921 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4793 on 8756 degrees of freedom
## Multiple R-squared: 0.001495, Adjusted R-squared: 0.000811
## F-statistic: 2.185 on 6 and 8756 DF, p-value: 0.04139The p-value is less than 0.05, which is better. So, we can stop the backward process elimination and use this model even though all predictors that we have left have p-value that is greater than 0.05. The coefficients of each of the predictors are very small (the slope or the rate of change). That indicates that as any of those predictors change, there is a small change in the heart attack risk. Also, the Multiple R-squared value of 0.001495 indicates that only about 0.15% of the variance in Heart Attack Risk is explained by the predictor variables included in the model. In addition, the Adjusted R-squared value of 0.000811 means that approximately 0.0811% of the variance in the dependent variable (Heart Attack Risk) is explained by the predictors.
Residual Analysis:
par(mfrow=c(2,2))
plot(Heart_Attack.lm.15)
# Define the formula with additional terms
formula <- formula(`Heart Attack Risk` ~ Cholesterol + Systolic_Pressure + Diabetes +
`Sedentary Hours Per Day` + Triglycerides + `Sleep Hours Per Day` +
I(Cholesterol^2) + as.factor(Diabetes) + Cholesterol:as.factor(Diabetes))
# Fit the linear regression model
model <- lm(formula, data = Heart_Attack)
# Print the summary of the model
summary(model)##
## Call:
## lm(formula = formula, data = Heart_Attack)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.4116 -0.3638 -0.3433 0.6308 0.7029
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.848e-01 6.973e-02 4.084 4.47e-05 ***
## Cholesterol 1.797e-04 4.642e-04 0.387 0.6987
## Systolic_Pressure 3.380e-04 1.944e-04 1.739 0.0821 .
## Diabetes 3.321e-02 3.633e-02 0.914 0.3607
## `Sedentary Hours Per Day` -8.418e-04 1.478e-03 -0.570 0.5690
## Triglycerides 2.070e-05 2.290e-05 0.904 0.3662
## `Sleep Hours Per Day` -4.370e-03 2.578e-03 -1.695 0.0901 .
## I(Cholesterol^2) -4.336e-08 8.653e-07 -0.050 0.9600
## as.factor(Diabetes)1 NA NA NA NA
## Cholesterol:as.factor(Diabetes)1 -6.064e-05 1.332e-04 -0.455 0.6489
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4794 on 8754 degrees of freedom
## Multiple R-squared: 0.001519, Adjusted R-squared: 0.0006066
## F-statistic: 1.665 on 8 and 8754 DF, p-value: 0.1015# Conduct residual analysis
residuals <- residuals(model)
# Plot residuals vs. fitted values
plot(fitted(model), residuals)
Based on this residual plot, I think the predictors that are on the data already are not significantly not enough or the linear model is not good enough.
Different Models:
# Fit polynomial regression model
poly_model <- lm(`Heart Attack Risk` ~ poly(Cholesterol, 2), data = Heart_Attack)
# Create a data frame with predictor variable(s)
new_data <- data.frame(Cholesterol = Heart_Attack$Cholesterol)
# Predictions
predictions <- predict(poly_model, newdata = new_data)# Create a scatter plot of actual vs. predicted values
plot(Heart_Attack$`Heart Attack Risk`, predictions,
xlab = "Actual Heart Attack Risk", ylab = "Predicted Heart Attack Risk",
main = "Actual vs. Predicted Heart Attack Risk")
abline(a = 0, b = 1, col = "red") # Add a 45-degree line for reference
Conclusion:
In conclusion, based on the analysis, it seems like the predictors are not explaining the variability in heart attack risk. since the R-squared value is low, it means that the predictors exaplin only small percentage of the response variable.