Introduction:

In this homework, you will apply logistic regression to a real-world dataset: the Pima Indians Diabetes Database. This dataset contains medical records from 768 women of Pima Indian heritage, aged 21 or older, and is used to predict the onset of diabetes (binary outcome: 0 = no diabetes, 1 = diabetes) based on physiological measurements.

The data is publicly available from the UCI Machine Learning Repository and can be imported directly.

Dataset URL: https://raw.githubusercontent.com/jbrownlee/Datasets/master/pima-indians-diabetes.data.csv

Columns (no header in the CSV, so we need to assign them manually):

  1. Pregnancies: Number of times pregnant
  2. Glucose: Plasma glucose concentration (2-hour test)
  3. BloodPressure: Diastolic blood pressure (mm Hg)
  4. SkinThickness: Triceps skin fold thickness (mm)
  5. Insulin: 2-hour serum insulin (mu U/ml)
  6. BMI: Body mass index (weight in kg/(height in m)^2)
  7. DiabetesPedigreeFunction: Diabetes pedigree function (a function scoring genetic risk)
  8. Age: Age in years
  9. Outcome: Class variable (0 = no diabetes, 1 = diabetes)

Task Overview: You will load the data, build a logistic regression model to predict diabetes onset using a subset of predictors (Glucose, BMI, Age), interpret the model, evaluate it with a confusion matrix and metrics, and analyze the ROC curve and AUC.

Cleaning the dataset Don’t change the following code

library(tidyverse)
## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
## ✔ dplyr     1.2.0     ✔ readr     2.2.0
## ✔ forcats   1.0.1     ✔ stringr   1.6.0
## ✔ ggplot2   4.0.2     ✔ tibble    3.3.1
## ✔ lubridate 1.9.5     ✔ tidyr     1.3.2
## ✔ purrr     1.2.1     
## ── 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 errors
url <- "https://raw.githubusercontent.com/jbrownlee/Datasets/master/pima-indians-diabetes.data.csv"

data <- read.csv(url, header = FALSE)

colnames(data) <- c("Pregnancies", "Glucose", "BloodPressure", "SkinThickness", "Insulin", "BMI", "DiabetesPedigreeFunction", "Age", "Outcome")

data$Outcome <- as.factor(data$Outcome)

# Handle missing values (replace 0s with NA because 0 makes no sense here)
data$Glucose[data$Glucose == 0] <- NA
data$BloodPressure[data$BloodPressure == 0] <- NA
data$BMI[data$BMI == 0] <- NA


colSums(is.na(data))
##              Pregnancies                  Glucose            BloodPressure 
##                        0                        5                       35 
##            SkinThickness                  Insulin                      BMI 
##                        0                        0                       11 
## DiabetesPedigreeFunction                      Age                  Outcome 
##                        0                        0                        0

Question 1: Create and Interpret a Logistic Regression Model - Fit a logistic regression model to predict Outcome using Glucose, BMI, and Age.

The model explains 25.6% of the variation in the diabetes diagnoses based on Glucose, BMI, and Age.

## Enter your code here
Model <- glm(Outcome ~ Glucose + Age + BMI , data=data, family="binomial")

summary(Model)
## 
## Call:
## glm(formula = Outcome ~ Glucose + Age + BMI, family = "binomial", 
##     data = data)
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -9.032377   0.711037 -12.703  < 2e-16 ***
## Glucose      0.035548   0.003481  10.212  < 2e-16 ***
## Age          0.028699   0.007809   3.675 0.000238 ***
## BMI          0.089753   0.014377   6.243  4.3e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 974.75  on 751  degrees of freedom
## Residual deviance: 724.96  on 748  degrees of freedom
##   (16 observations deleted due to missingness)
## AIC: 732.96
## 
## Number of Fisher Scoring iterations: 4
r_square <- 1 - (Model$deviance/Model$null.deviance)

r_square
## [1] 0.25626

What does the intercept represent (log-odds of diabetes when predictors are zero)?

The intercept is at -9.032377 which is where the log odds of the patient will have diabetes when all predictors are 0.

For each predictor (Glucose, BMI, Age), does a one-unit increase raise or lower the odds of diabetes? Are they significant (p-value < 0.05)? . A one unit increase will raise the odds of diabetes. BMI will have a 0.089753 increase, Age will have a 0.028699 increase and Glucose will have a 0.035548 increase. And they are all significant because BMI is 4.3e-10, Age is 0.000238, and Glucose is 2e-16.

Question 2: Confusion Matrix and Important Metric

Calculate and report the metrics:

Accuracy: (TP + TN) / Total Sensitivity (Recall): TP / (TP + FN) Specificity: TN / (TN + FP) Precision: TP / (TP + FP)

Use the following starter code

# Keep only rows with no missing values in Glucose, BMI, or Age
data_subset <- data[complete.cases(data[, c("Glucose", "BMI", "Age")]), ]

#Create a numeric version of the outcome (0 = no diabetes, 1 = diabetes).This is required for calculating confusion matrices.
data_subset$Outcome_num <- ifelse(data_subset$Outcome == "1", 1, 0)


# Predicted probabilities

predicted.probs <- Model$fitted.values

# Predicted classes

predicted.classes <- ifelse(predicted.probs > 0.5, 1, 0)

# Confusion matrix

confusion_matrix <- table(
  predicted = factor(predicted.classes, levels = c(0, 1)),
  actual = factor(data_subset$Outcome_num, levels = c(0, 1))
)

confusion_matrix
##          actual
## predicted   0   1
##         0 429 114
##         1  59 150
#Extract Values:
TN <-426
FP <- 59
FN <- 114
TP <- 150

#Metrics    
accuracy <-  (TP + TN) / (TP + TN + FP + FN)
sensitivity <- TP / (TP + FN)
specificity <-  TN / (TN + FP)
precision <- TP / (TP + FP)

cat("Accuracy:", round(accuracy, 3), "\nSensitivity:", round(sensitivity, 3), "\nSpecificity:", round(specificity, 3), "\nPrecision:", round(precision, 3))
## Accuracy: 0.769 
## Sensitivity: 0.568 
## Specificity: 0.878 
## Precision: 0.718

Interpret: How well does the model perform? Is it better at detecting diabetes (sensitivity) or non-diabetes (specificity)? Why might this matter for medical diagnosis?

Its accuracy is 0.769 is like a C+ which is decent but I wouldn’t recommend using it. The specificity is good though. And it is better at detecting non-diabetes(87.8%) than sensitivity(56.8%).

Question 3: ROC Curve, AUC, and Interpretation

#Enter your code here
library(pROC)
## Type 'citation("pROC")' for a citation.
## 
## Attaching package: 'pROC'
## The following objects are masked from 'package:stats':
## 
##     cov, smooth, var
roc_obj <- roc(response = data_subset$Outcome_num,
               predictor = Model$fitted.values,
               levels = c("0", "1"),
               direction = "<")

auc_val <- auc(roc_obj); auc_val
## Area under the curve: 0.828
plot.roc(roc_obj, print.auc = TRUE, legacy.axes = TRUE,
         xlab = "False Positive Rate (1 - Specificity)",
         ylab = "True Positive Rate (Sensitivity)")

What does AUC indicate (0.5 = random, 1.0 = perfect)?

The AUC shows that the model is good and has a 82% chance of finding a diabetic patient.

For diabetes diagnosis, prioritize sensitivity (catching cases) or specificity (avoiding false positives)? Suggest a threshold and explain.

It would be better to prioritize sensitivity than specificity. We are trying to help as many people as possible so focusing on sensitivity is more important. I think a .4 or .5 would be good because we are trying to catch them early. And it would be better to be safe than sorry.