Affilitaion: University of Chittagong

Loading the assigned dataset into R notebook:

Assigned dataset = diabetes.csv

This dataset originates from the National Institute of Diabetes and Digestive and Kidney Diseases. The aim is to forecast, based on diagnostic measurements, whether a patient is afflicted with diabetes. Stringent criteria were applied in selecting instances from a larger database, ensuring that all patients in this dataset are females, at least 21 years old, and of Pima Indian heritage. All columns in this dataset consist of numerical values.
Pregnancies: Number of occurrences of pregnancy
Glucose: Plasma glucose concentration two hours after an oral glucose tolerance test
BloodPressure: Diastolic blood pressure (mm Hg)
SkinThickness: Triceps skin fold thickness (mm)
Insulin: 2-Hour serum insulin (mu U/ml)
BMI: Body mass index (weight in kg/(height in m)^2)
DiabetesPedigreeFunction: Diabetes pedigree function
Age: Age (years)
Outcome: Class variable (0 or 1). This serves as the target column, where ‘0’ indicates no disease, and ‘1’ indicates the presence of disease.
1. The assigned dataset “diabetes” was downloaded and moved in the working directory of R studio.
2. Then, a notebook file is created in R studio and the dataset was loaded into a variable.
Diabetes =read.csv("diabetes.csv")
print(head(Diabetes))
##   Pregnancies Glucose BloodPressure SkinThickness Insulin  BMI
## 1           6     148            72            35       0 33.6
## 2           1      85            66            29       0 26.6
## 3           8     183            64             0       0 23.3
## 4           1      89            66            23      94 28.1
## 5           0     137            40            35     168 43.1
## 6           5     116            74             0       0 25.6
##   DiabetesPedigreeFunction Age Outcome
## 1                    0.627  50       1
## 2                    0.351  31       0
## 3                    0.672  32       1
## 4                    0.167  21       0
## 5                    2.288  33       1
## 6                    0.201  30       0

3. Some basic information about the “Diabates” dataset

Total number of columns and their names

column_names <- colnames(Diabetes)
num_columns <- length(column_names)

# column names and the number of columns
cat("Column names of Diabetes are", column_names, "\n","\n")
## Column names of Diabetes are Pregnancies Glucose BloodPressure SkinThickness Insulin BMI DiabetesPedigreeFunction Age Outcome 
## 
cat("Total number of columns is", num_columns)
## Total number of columns is 9

a) Finding out the numerical columns

numerical_columns <- sapply(Diabetes, is.numeric) #extraction of the numerical columns

numerical_column_names <- names(numerical_columns)[numerical_columns]

print(numerical_column_names)
## [1] "Pregnancies"              "Glucose"                 
## [3] "BloodPressure"            "SkinThickness"           
## [5] "Insulin"                  "BMI"                     
## [7] "DiabetesPedigreeFunction" "Age"                     
## [9] "Outcome"
All 9 columns of the datasets are numerical.

b) Finding out the character columns

character_columns <- sapply(Diabetes, is.character) # Extraction of the character columns

character_column_names <- names(character_columns)[character_columns]

print(character_column_names)
## character(0)

So, there is no character columns in the dataset.

c) Target variable of the dataset is “Outcome”.

4. There are no categorical columns in the dataset.

5.a) Scatter plot of Pregnancies and BloodPressure

Diabetes <- na.omit(Diabetes)   # data_cleaning
print(head(Diabetes))
##   Pregnancies Glucose BloodPressure SkinThickness Insulin  BMI
## 1           6     148            72            35       0 33.6
## 2           1      85            66            29       0 26.6
## 3           8     183            64             0       0 23.3
## 4           1      89            66            23      94 28.1
## 5           0     137            40            35     168 43.1
## 6           5     116            74             0       0 25.6
##   DiabetesPedigreeFunction Age Outcome
## 1                    0.627  50       1
## 2                    0.351  31       0
## 3                    0.672  32       1
## 4                    0.167  21       0
## 5                    2.288  33       1
## 6                    0.201  30       0
variety_color = as.numeric(factor(Diabetes$Outcome))

plot(
  x = Diabetes$Pregnancies,
  y = Diabetes$BloodPressure,
  col = variety_color,
  pch = 19,
  cex = 1.3,
  
  xlab = "Pregnancies",
  ylab = "Blood Pressure",
  main = "Scatter Plot of Pregnancies vs Blood Pressure",
)

# A legend with more descriptive labels
legend(
  "topright",
  legend = c("No Diabetes", "Diabetes"),  # Descriptive labels
  col= 1:2,
  pch = 19,
  title = "Outcome"
)

##### This scatter plot visualizes the relationship between Pregnancies and Blood . ##### Each point is colored based on the “Outcome” (0 or 1), representing different classes. The legend indicates which color corresponds to each class.

5.b) Histogram of Insulin column

hist(
  Diabetes$Insulin,
  breaks = 20,  # Increase the number of bins for better resolution
  main = "Histogram plot of Insulin Levels",
  
  col.main = "#008080",
  cex.main = 1.5,    # Adjust the size of the title
  cex.lab = 1.3,
  col.lab = c("Maroon"),
  font.main = 1,      # Use plain (non-bold) font for the title
  xlab = "Insulin",
  col = adjustcolor("purple", alpha = 0.7),  # Adjust alpha for transparency
  border = "white",  # White borders for better separation
  xlim = c(min(Diabetes$Insulin, na.rm = TRUE), max(Diabetes$Insulin, na.rm = TRUE))
)

# Add axis labels
axis(1, at = pretty(Diabetes$Insulin, n = 10), labels = TRUE, col.axis = "darkgray")
axis(2, labels = TRUE, col.axis = "darkgray")

# Addition of a grid for better readability
grid()

This histogram plot illustrates the distribution of the “Insulin” feature in the “Diabetes” dataset.
The x-axis represents different ranges or bins of insulin values, and the y-axis shows the frequency of observations within each bin.
The color “purple” is used to fill the bars for visual distinction.

5.c) Boxplot of SkinThickness column

boxplot(
  Diabetes$SkinThickness,
  ylab = "Skin Thickness",
  col = "#48b5c4",
  pch = 19,
  border = "black",
  main = "Boxplot of Skin Thickness"
)

##### This boxplot visualizes the distribution of “SkinThickness” values. ##### The box represents the interquartile range (IQR), the line inside the box is the median, and the whiskers extend to the minimum and maximum values providing insights into the central tendency and spread of the data. ##### There’s an outlier in the boxplot close to SKinThickness value, 100.

6.Using ggplot library functions to plot

a) Scatterplot of all combinations of two features- Pregnancies and Glucose

library(ggplot2)
## Warning: package 'ggplot2' was built under R version 4.3.2
# Scatter plot using ggplot2
ggplot(Diabetes, aes(x = Pregnancies, y = Glucose, color = factor(Outcome))) +
  geom_point(size=2.8, alpha = 1) +
  labs(
    x = "Pregnancies",
    y = "Glucose",
    title = "Scatter Plot of Pregnancies and Glucose",
    caption = "Source: Iskulghar",
    color = "Outcome"
  ) +
  theme_minimal() + 
  theme(
    text = element_text(color = "darkblue", size = 14),  # Adjust font color and size
    legend.position = "top",  # Place legends at the top
    axis.text.x = element_text(color = "#FF7F7F", size = 14),
    axis.text.y = element_text(color = "#C4A484", size = 14),
    plot.title = element_text(color= "maroon", hjust = 0.5),
  
    legend.title = element_text(color = "darkgreen")
  )

##### This scatter plot represents all combinations of the “Glucose” and “BMI” features, with points colored based on the “Outcome” class (0 or 1). ##### It includes proper axis labels, title, and a caption. Font colors and sizes are adjusted for readability, and the legend is positioned at the top.

b) Boxplot of all column following all instructions

library(reshape2)
## Warning: package 'reshape2' was built under R version 4.3.2
# Melting data into long format
melted_data <- melt(Diabetes)
## No id variables; using all as measure variables
# Boxplot of all columns
ggplot(melted_data, aes(x = variable, y = value, fill = variable)) +
  geom_boxplot() +
  labs(
    x = "Columns",
    y = "Values",
    title = "Boxplot of All Columns",
    caption = "Source: Iskulghar"
  ) +
  theme_bw() +
  theme(
    text = element_text(color = "darkblue", size = 10.5),
    legend.position = "top",
    legend.title = element_text(color = "#FF7F7F", size = 13),  # Legend title color and size
    plot.title = element_text(color= "maroon", hjust = 0.5)
  ) +
  facet_wrap(~variable, scale= "free")  # Creation of a facet grid for each column

##### This boxplot visualizes the distribution of values for all columns in the “Diabetes” dataset. ##### The data is melted into long format for easier plotting. Proper axis labels, title, and a caption are provided. ##### Font colors and sizes are adjusted, and the legend is positioned at the top for clarity.

Data Science Project - Part 2

1.a) Interactive violinplot of all features

library(plotly)
## Warning: package 'plotly' was built under R version 4.3.2
## 
## Attaching package: 'plotly'
## The following object is masked from 'package:ggplot2':
## 
##     last_plot
## The following object is masked from 'package:stats':
## 
##     filter
## The following object is masked from 'package:graphics':
## 
##     layout
Diabetes = read.csv("diabetes.csv")
fig = Diabetes %>%
  plot_ly(y = ~Pregnancies, type = 'violin')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~Glucose, type = 'violin')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~BloodPressure, type = 'violin')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~SkinThickness, type = 'violin')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~Insulin, type = 'violin')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~BMI, type = 'violin')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~DiabetesPedigreeFunction, type = 'violin')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~Age, type = 'violin')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~Outcome, type = 'violin')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~Pregnancies, type = 'box')
  
fig

1.b) Interactive boxplot of all features

library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~Glucose, type = 'box')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~BloodPressure, type = 'box')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~SkinThickness, type = 'box')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~Insulin, type = 'box')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~BMI, type = 'box')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~DiabetesPedigreeFunction, type = 'box')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~Age, type = 'box')
  
fig
library(plotly)

fig = Diabetes %>%
  plot_ly(y = ~Outcome, type = 'box')
  
fig

1.c) Calculation of correlation matrix

cor_matrix = cor(Diabetes[ ,1:9])
cor_matrix
##                          Pregnancies    Glucose BloodPressure SkinThickness
## Pregnancies               1.00000000 0.12945867    0.14128198   -0.08167177
## Glucose                   0.12945867 1.00000000    0.15258959    0.05732789
## BloodPressure             0.14128198 0.15258959    1.00000000    0.20737054
## SkinThickness            -0.08167177 0.05732789    0.20737054    1.00000000
## Insulin                  -0.07353461 0.33135711    0.08893338    0.43678257
## BMI                       0.01768309 0.22107107    0.28180529    0.39257320
## DiabetesPedigreeFunction -0.03352267 0.13733730    0.04126495    0.18392757
## Age                       0.54434123 0.26351432    0.23952795   -0.11397026
## Outcome                   0.22189815 0.46658140    0.06506836    0.07475223
##                              Insulin        BMI DiabetesPedigreeFunction
## Pregnancies              -0.07353461 0.01768309              -0.03352267
## Glucose                   0.33135711 0.22107107               0.13733730
## BloodPressure             0.08893338 0.28180529               0.04126495
## SkinThickness             0.43678257 0.39257320               0.18392757
## Insulin                   1.00000000 0.19785906               0.18507093
## BMI                       0.19785906 1.00000000               0.14064695
## DiabetesPedigreeFunction  0.18507093 0.14064695               1.00000000
## Age                      -0.04216295 0.03624187               0.03356131
## Outcome                   0.13054795 0.29269466               0.17384407
##                                  Age    Outcome
## Pregnancies               0.54434123 0.22189815
## Glucose                   0.26351432 0.46658140
## BloodPressure             0.23952795 0.06506836
## SkinThickness            -0.11397026 0.07475223
## Insulin                  -0.04216295 0.13054795
## BMI                       0.03624187 0.29269466
## DiabetesPedigreeFunction  0.03356131 0.17384407
## Age                       1.00000000 0.23835598
## Outcome                   0.23835598 1.00000000

A coefficient above 0.75 (or below -0.75) is generally considered a high degree of correlation, while one between -0.3 and 0.3 is a sign of weak or no correlation.

1.d) Plotting of correlation matrix (lower triangle) with values

library(ggcorrplot)
## Warning: package 'ggcorrplot' was built under R version 4.3.2
ggcorrplot(cor_matrix, 
           type = "lower",
           colors = c("purple", "white", "red"),
           lab = TRUE)

1.e) Pair plot of all feature

colnames(Diabetes) <- c("Prg", "Glu", "BP", "Skthic", "Ins", "BMI", "DPF", "Age", "OC")

library(ggplot2)
library(GGally)
## Warning: package 'GGally' was built under R version 4.3.2
## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2
Diabetes$OC <- factor(Diabetes$OC)

# Use the label_mapping in the ggpairs plot
ggpairs(
  Diabetes,
  aes(colour = OC),
  lower = list(
    combo = wrap("facethist", bins = 30),
    continuous = wrap("points", size = 2.5)
  ),
  upper = list(
    combo = wrap("facetdensity", bins = 30),
    continuous = wrap("cor", size = 2.5)
  )
) +
  theme(
    axis.text.x = element_blank(),  # Remove x-axis labels
    axis.text.y = element_blank(),  # Remove y-axis labels
    axis.title = element_blank(),   # Remove axis titles
    axis.text = element_text(size = 6)  # Adjust overall text size
  )
## Warning in stat_density(aes(y = after_stat(!!as.name("scaled")) * diff(range(x,
## : Ignoring unknown parameters: `bins`
## Warning in stat_density(aes(y = after_stat(!!as.name("scaled")) * diff(range(x, : Ignoring unknown parameters: `bins`
## Ignoring unknown parameters: `bins`
## Ignoring unknown parameters: `bins`
## Ignoring unknown parameters: `bins`
## Ignoring unknown parameters: `bins`
## Ignoring unknown parameters: `bins`
## Ignoring unknown parameters: `bins`

Bar plot of PCAs (Percentage of explained variance vs PCs)

Applying principal component analysis (PCA) and explaining the PCAs

library(stats)

Diabetes_pca = prcomp(Diabetes[ , -9], scale = TRUE, center = TRUE)
Diabetes_pca
## Standard deviations (1, .., p=8):
## [1] 1.4471973 1.3157546 1.0147068 0.9356971 0.8731234 0.8262133 0.6479322
## [8] 0.6359733
## 
## Rotation (n x k) = (8 x 8):
##               PC1        PC2         PC3         PC4        PC5          PC6
## Prg    -0.1284321  0.5937858 -0.01308692  0.08069115 -0.4756057  0.193598168
## Glu    -0.3930826  0.1740291  0.46792282 -0.40432871  0.4663280  0.094161756
## BP     -0.3600026  0.1838921 -0.53549442  0.05598649  0.3279531 -0.634115895
## Skthic -0.4398243 -0.3319653 -0.23767380  0.03797608 -0.4878621  0.009589438
## Ins    -0.4350262 -0.2507811  0.33670893 -0.34994376 -0.3469348 -0.270650609
## BMI    -0.4519413 -0.1009598 -0.36186463  0.05364595  0.2532038  0.685372179
## DPF    -0.2706114 -0.1220690  0.43318905  0.83368010  0.1198105 -0.085784088
## Age    -0.1980271  0.6205885  0.07524755  0.07120060 -0.1092900 -0.033357170
##                PC7          PC8
## Prg    -0.58879003 -0.117840984
## Glu    -0.06015291 -0.450355256
## BP     -0.19211793  0.011295538
## Skthic  0.28221253 -0.566283799
## Ins    -0.13200992  0.548621381
## BMI    -0.03536644  0.341517637
## DPF    -0.08609107  0.008258731
## Age     0.71208542  0.211661979
summary(Diabetes_pca)
## Importance of components:
##                           PC1    PC2    PC3    PC4     PC5     PC6     PC7
## Standard deviation     1.4472 1.3158 1.0147 0.9357 0.87312 0.82621 0.64793
## Proportion of Variance 0.2618 0.2164 0.1287 0.1094 0.09529 0.08533 0.05248
## Cumulative Proportion  0.2618 0.4782 0.6069 0.7163 0.81164 0.89697 0.94944
##                            PC8
## Standard deviation     0.63597
## Proportion of Variance 0.05056
## Cumulative Proportion  1.00000
pc_12 = data.frame(Diabetes_pca$x[ , 1:2])
head(pc_12)
pc_12_outcome = cbind(pc_12, Outcome = Diabetes$OC)
pc_12_outcome

f.ii) Bar plot of PCAs (Percentage of explained variance vs PCs)

library(factoextra)
## Warning: package 'factoextra' was built under R version 4.3.2
## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa
fviz_eig(Diabetes_pca, addlabels = TRUE)

#### f.ii) Contribution plot of PCs

library(factoextra)

# Plot PCA variable factor map
fviz_pca_var(
  Diabetes_pca,
  col.var = "contrib",
  repel = TRUE,  
  ggtheme = theme_minimal() 
)

f.iii) Contribution plot as Heatmap

library(factoextra)

contrib_matrix <- get_pca_var(Diabetes_pca)$contrib

heatmap(contrib_matrix,
        Colv = NA,  # Do not cluster columns
        Rowv = NA,  # Do not cluster rows
        main = "Contribution Plot - Heatmap",
        xlab = "Principal Components",
        ylab = "Variables",
        col = viridis::viridis(10),  # Use "viridis" color palette
        scale = "column",  # Scale by column 
        margins = c(5, 10))  # Adjust margins

f.iv) Cluster plot after PCA

fviz_pca_ind(Diabetes_pca,
             geom.ind = "point",
             col.ind = Diabetes$OC,
             addEllipses = TRUE)

Using SVM to train a model to classify the target variable

Result explanation

library(lattice)
library(e1071)
## Warning: package 'e1071' was built under R version 4.3.2
library(caret)
## Warning: package 'caret' was built under R version 4.3.2
train_ix = createDataPartition(Diabetes$OC, p = 0.8, list = FALSE)
train_data = Diabetes[train_ix, ]
test_data = Diabetes[-train_ix, ]

train_data
test_data
Diabetes$OC = as.factor(Diabetes$OC)
svm_model = svm(OC ~ Prg+Glu+BP+Skthic+Ins+BMI+DPF
+Age, data = train_data, kernel = "linear")

g.i)Explanation of the results

predictions = predict(svm_model, newdata = test_data)
confusion_mat = confusionMatrix(predictions, test_data$OC)
confusion_mat
## Confusion Matrix and Statistics
## 
##           Reference
## Prediction  0  1
##          0 92 24
##          1  8 29
##                                           
##                Accuracy : 0.7908          
##                  95% CI : (0.7178, 0.8523)
##     No Information Rate : 0.6536          
##     P-Value [Acc > NIR] : 0.0001499       
##                                           
##                   Kappa : 0.5028          
##                                           
##  Mcnemar's Test P-Value : 0.0080099       
##                                           
##             Sensitivity : 0.9200          
##             Specificity : 0.5472          
##          Pos Pred Value : 0.7931          
##          Neg Pred Value : 0.7838          
##              Prevalence : 0.6536          
##          Detection Rate : 0.6013          
##    Detection Prevalence : 0.7582          
##       Balanced Accuracy : 0.7336          
##                                           
##        'Positive' Class : 0               
##

g.ii) Plotting the confusion matrix

com = as.data.frame(confusion_mat$table)

ggplot(com, aes(Prediction, Reference, fill = Freq)) + 
  geom_tile() +
  geom_text(aes(label = Freq)) + 
  scale_fill_gradient(low="lightblue", high="maroon")

2.a) Removing the “Serial No” column from the dataset

US_Admission <- read.csv("US Admission.csv")

# Remove the "Serial No" column
US_Admission <- US_Admission[, -1]

2.b) Pair plot of all features

colnames(US_Admission) <- c("GRE", "TOEFL", "Rating", "SOP", "LOR", "CGPA", "Res", "Chance")
library(ggplot2)
library(GGally)

library(ggplot2)
library(GGally)

# Convert the "Res" variable to a factor
US_Admission$Res <- factor(US_Admission$Res)

# Check for missing values
if (any(is.na(US_Admission))) {
  # Remove rows with missing values
  US_Admission <- na.omit(US_Admission)
}

# Shorten column names
colnames(US_Admission) <- c("GRE", "TOEFL", "Rating", "SOP", "LOR", "CGPA", "Res", "Chance")

# Create a pair plot with ggplot2
ggpairs(
  US_Admission,
  aes(colour = Res),
  lower = list(
    combo = wrap("facethist", bins = 30),
    continuous = wrap("points", size = 2.5)
  ),
  upper = list(
    combo = wrap("facetdensity", bins = 30),
    continuous = wrap("cor", size = 2.5)
  )
) +
  theme(
    axis.text.x = element_blank(),  # Remove x-axis labels
    axis.text.y = element_blank(),  # Remove y-axis labels
    axis.title = element_blank(),   # Remove axis titles
    axis.text = element_text(size = 6)  # Adjust overall text size
  )
## Warning in stat_density(aes(y = after_stat(!!as.name("scaled")) * diff(range(x, : Ignoring unknown parameters: `bins`
## Ignoring unknown parameters: `bins`
## Ignoring unknown parameters: `bins`
## Ignoring unknown parameters: `bins`
## Ignoring unknown parameters: `bins`
## Ignoring unknown parameters: `bins`
## Ignoring unknown parameters: `bins`

2.c) Plotting linear regression with each feature

library(ggplot2)
US_Admission$Res = as.numeric(US_Admission$Res)

# Loop through each feature (excluding the target variable "Chance of Admit")
for (feature in setdiff(names(US_Admission), "Chance")) {
  # Create linear regression plot
  linear_plot <- ggplot(US_Admission, aes_string(x = "Chance", y = feature)) +
    geom_point() +
    geom_smooth(method = "lm", se = FALSE, color = "blue") +  # Add linear regression line
    labs(title = paste("Linear Regression -", feature),
         x = "Chance of Admit",
         y = feature) +
    theme_minimal()

  # Print the plot
  print(linear_plot)

  # Summarize linear regression model
  lm_model <- lm(paste(feature, "~ Chance"), data = US_Admission)
  lm_summary <- summary(lm_model)
  
  # Print regression summary
  cat("\nLinear Regression Summary for", feature, ":\n")
  print(lm_summary)

  # Interpretation
  cat("\nInterpretation for", feature, ":\n")
  if (lm_summary$coefficients[2, 4] < 0.05) {
    cat("The variable is a significant predictor of 'Chance of Admit'.\n")
    if (lm_summary$coefficients[2, 1] > 0) {
      cat("As", feature, "increases, 'Chance of Admit' tends to increase.\n")
    } else {
      cat("As", feature, "increases, 'Chance of Admit' tends to decrease.\n")
    }
  } else {
    cat("The variable is not a significant predictor of 'Chance of Admit'.\n")
  }

  cat("\n--------------------------------------\n")
}
## Warning: `aes_string()` was deprecated in ggplot2 3.0.0.
## ℹ Please use tidy evaluation idioms with `aes()`.
## ℹ See also `vignette("ggplot2-in-packages")` for more information.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
## `geom_smooth()` using formula = 'y ~ x'

## 
## Linear Regression Summary for GRE :
## 
## Call:
## lm(formula = paste(feature, "~ Chance"), data = US_Admission)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -19.5894  -4.7205   0.8904   4.4297  23.9084 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  270.033      1.776  152.06   <2e-16 ***
## Chance        64.574      2.406   26.84   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.853 on 398 degrees of freedom
## Multiple R-squared:  0.6442, Adjusted R-squared:  0.6433 
## F-statistic: 720.6 on 1 and 398 DF,  p-value: < 2.2e-16
## 
## 
## Interpretation for GRE :
## The variable is a significant predictor of 'Chance of Admit'.
## As GRE increases, 'Chance of Admit' tends to increase.
## 
## --------------------------------------
## `geom_smooth()` using formula = 'y ~ x'

## 
## Linear Regression Summary for TOEFL :
## 
## Call:
## lm(formula = paste(feature, "~ Chance"), data = US_Admission)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -10.6111  -2.5896   0.0145   2.4452  11.4961 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  83.0062     0.9623   86.25   <2e-16 ***
## Chance       33.6906     1.3036   25.84   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.713 on 398 degrees of freedom
## Multiple R-squared:  0.6266, Adjusted R-squared:  0.6257 
## F-statistic: 667.9 on 1 and 398 DF,  p-value: < 2.2e-16
## 
## 
## Interpretation for TOEFL :
## The variable is a significant predictor of 'Chance of Admit'.
## As TOEFL increases, 'Chance of Admit' tends to increase.
## 
## --------------------------------------
## `geom_smooth()` using formula = 'y ~ x'

## 
## Linear Regression Summary for Rating :
## 
## Call:
## lm(formula = paste(feature, "~ Chance"), data = US_Admission)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.40494 -0.57607 -0.06269  0.56478  2.64858 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -1.0444     0.2086  -5.006 8.35e-07 ***
## Chance        5.7042     0.2826  20.186  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.805 on 398 degrees of freedom
## Multiple R-squared:  0.5059, Adjusted R-squared:  0.5046 
## F-statistic: 407.5 on 1 and 398 DF,  p-value: < 2.2e-16
## 
## 
## Interpretation for Rating :
## The variable is a significant predictor of 'Chance of Admit'.
## As Rating increases, 'Chance of Admit' tends to increase.
## 
## --------------------------------------
## `geom_smooth()` using formula = 'y ~ x'

## 
## Linear Regression Summary for SOP :
## 
## Call:
## lm(formula = paste(feature, "~ Chance"), data = US_Admission)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.8315 -0.4976  0.0024  0.5029  3.2429 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.05579    0.19259   -0.29    0.772    
## Chance       4.77089    0.26088   18.29   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7431 on 398 degrees of freedom
## Multiple R-squared:  0.4566, Adjusted R-squared:  0.4552 
## F-statistic: 334.4 on 1 and 398 DF,  p-value: < 2.2e-16
## 
## 
## Interpretation for SOP :
## The variable is a significant predictor of 'Chance of Admit'.
## As SOP increases, 'Chance of Admit' tends to increase.
## 
## --------------------------------------
## `geom_smooth()` using formula = 'y ~ x'

## 
## Linear Regression Summary for LOR :
## 
## Call:
## lm(formula = paste(feature, "~ Chance"), data = US_Admission)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.31398 -0.44060  0.03334  0.49598  1.86129 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   0.3954     0.1731   2.284   0.0229 *  
## Chance        4.2205     0.2345  18.000   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6679 on 398 degrees of freedom
## Multiple R-squared:  0.4488, Adjusted R-squared:  0.4474 
## F-statistic:   324 on 1 and 398 DF,  p-value: < 2.2e-16
## 
## 
## Interpretation for LOR :
## The variable is a significant predictor of 'Chance of Admit'.
## As LOR increases, 'Chance of Admit' tends to increase.
## 
## --------------------------------------
## `geom_smooth()` using formula = 'y ~ x'

## 
## Linear Regression Summary for CGPA :
## 
## Call:
## lm(formula = paste(feature, "~ Chance"), data = US_Admission)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.89091 -0.16933  0.01528  0.16943  1.00290 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  5.95386    0.07538   78.98   <2e-16 ***
## Chance       3.65163    0.10212   35.76   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2909 on 398 degrees of freedom
## Multiple R-squared:  0.7626, Adjusted R-squared:  0.762 
## F-statistic:  1279 on 1 and 398 DF,  p-value: < 2.2e-16
## 
## 
## Interpretation for CGPA :
## The variable is a significant predictor of 'Chance of Admit'.
## As CGPA increases, 'Chance of Admit' tends to increase.
## 
## --------------------------------------
## `geom_smooth()` using formula = 'y ~ x'

## 
## Linear Regression Summary for Res :
## 
## Call:
## lm(formula = paste(feature, "~ Chance"), data = US_Admission)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.86774 -0.38443  0.04087  0.31109  1.15687 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   0.1472     0.1077   1.366    0.173    
## Chance        1.9332     0.1459  13.248   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4157 on 398 degrees of freedom
## Multiple R-squared:  0.306,  Adjusted R-squared:  0.3043 
## F-statistic: 175.5 on 1 and 398 DF,  p-value: < 2.2e-16
## 
## 
## Interpretation for Res :
## The variable is a significant predictor of 'Chance of Admit'.
## As Res increases, 'Chance of Admit' tends to increase.
## 
## --------------------------------------

Explanation of the results:

Significant Predictor: If the p-value for the coefficient of the feature is less than 0.05, it indicates that the variable is a significant predictor of “Chance of Admit.”

Positive/Negative Relationship: The sign of the coefficient indicates the direction of the relationship. A positive coefficient suggests that as the feature increases, “Chance of Admit” tends to increase, and vice versa.

2.d)Plotting polynomial regression of power 2 with each feature

library(ggplot2)


# Loop through each feature (excluding the target variable "Chance of Admit")
for (feature in setdiff(names(US_Admission), "Chance")) {
  # Create polynomial regression plot
  poly_plot <- ggplot(US_Admission, aes_string(x = "Chance", y = feature)) +
    geom_point() +
    geom_smooth(method = "lm", formula = y ~ poly(x, 2), se = FALSE, color = "blue") +  # Add polynomial regression line
    labs(title = paste("Polynomial Regression (Power 2) -", feature),
         x = "Chance of Admit",
         y = feature) +
    theme_minimal()

  # Print the plot
  print(poly_plot)
}

Explanation of the results:

Curvature: Polynomial regression of power 2 allows for a curve in the regression line. It can capture non-linear relationships between the feature and “Chance of Admit.”

Significant Deviation from Linearity: If the relationship between the feature and “Chance of Admit” is not well captured by a straight line, the polynomial regression of power 2 can provide a better fit.

Overfitting Warning: Polynomial regression of higher degrees may lead to overfitting. It’s essential to evaluate the model’s performance and consider a balance between complexity and goodness of fit.

Analyze each plot individually to understand the relationship between each feature and “Chance of Admit” when a quadratic (power 2) relationship is considered. Consider both the visual fit of the curve and the practical significance of the results.

# Create a regression model using all features
regression_model <- lm(Chance ~ ., data = US_Admission)

# Print the summary of the regression model
summary(regression_model)
## 
## Call:
## lm(formula = Chance ~ ., data = US_Admission)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.26259 -0.02103  0.01005  0.03628  0.15928 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.2839576  0.1217507 -10.546  < 2e-16 ***
## GRE          0.0017374  0.0005979   2.906  0.00387 ** 
## TOEFL        0.0029196  0.0010895   2.680  0.00768 ** 
## Rating       0.0057167  0.0047704   1.198  0.23150    
## SOP         -0.0033052  0.0055616  -0.594  0.55267    
## LOR          0.0223531  0.0055415   4.034  6.6e-05 ***
## CGPA         0.1189395  0.0122194   9.734  < 2e-16 ***
## Res          0.0245251  0.0079598   3.081  0.00221 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.06378 on 392 degrees of freedom
## Multiple R-squared:  0.8035, Adjusted R-squared:    0.8 
## F-statistic: 228.9 on 7 and 392 DF,  p-value: < 2.2e-16
# Print the coefficients and their significance
cat("\nCoefficients:\n")
## 
## Coefficients:
print(coef(regression_model))
##  (Intercept)          GRE        TOEFL       Rating          SOP          LOR 
## -1.283957585  0.001737412  0.002919577  0.005716658 -0.003305169  0.022353127 
##         CGPA          Res 
##  0.118939454  0.024525106
cat("\nSignificant Predictors:\n")
## 
## Significant Predictors:
significant_predictors <- names(summary(regression_model)$coefficients[,"Pr(>|t|)"] < 0.05)
print(significant_predictors)
## [1] "(Intercept)" "GRE"         "TOEFL"       "Rating"      "SOP"        
## [6] "LOR"         "CGPA"        "Res"

Explanation of the results:

Coefficients: The coefficients in the summary represent the estimated effect of each predictor variable on the target variable (“Chance of Admit”). A positive coefficient indicates a positive relationship, while a negative coefficient indicates a negative relationship.

P-values: The p-values associated with the coefficients test the null hypothesis that each coefficient is equal to zero. A low p-value (typically less than 0.05) suggests that the corresponding predictor variable is a significant predictor of the target variable.

Adjusted R-squared: The adjusted R-squared value takes into account the number of predictors in the model and provides an indication of the goodness of fit. A higher adjusted R-squared suggests a better fit.

Residuals: The residuals represent the differences between the observed and predicted values. Check for patterns in the residuals to assess the model’s assumptions.