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     1.1.4     ✔ readr     2.1.6
## ✔ forcats   1.0.1     ✔ stringr   1.6.0
## ✔ ggplot2   4.0.2     ✔ tibble    3.3.1
## ✔ lubridate 1.9.4     ✔ 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.

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

summary(model)
## 
## Call:
## glm(formula = Outcome ~ Glucose + BMI + Age, 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 ***
## BMI          0.089753   0.014377   6.243  4.3e-10 ***
## Age          0.028699   0.007809   3.675 0.000238 ***
## ---
## 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 the odds of having diabetes when your glucose, BMI, and age are all zero. An impossible scenario, so it has no real-world interpretation on its own

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)?

All three predictors increase the odds of diabetes and are statistically significant:

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")]), ]

data_subset$Outcome_num <- ifelse(data_subset$Outcome == "1", 1, 0)


# Predicted probabilities
predicted_probs <- predict(model, newdata = data_subset, type = "response")

# Predicted classes
predicted_classes <- ifelse(predicted_probs > 0.5, 1, 0)

conf_matrix <- table(
  Predicted = factor(predicted_classes, levels = c(0, 1)),
  Actual    = factor(data_subset$Outcome_num, levels = c(0, 1))
)

conf_matrix
##          Actual
## Predicted   0   1
##         0 429 114
##         1  59 150
#Extract Values:
TN <- conf_matrix["0", "0"]
FP <- conf_matrix["1", "0"]
FN <- conf_matrix["0", "1"]
TP <- conf_matrix["1", "1"]

#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: 0.77 
## Sensitivity: 0.568 
## Specificity: 0.879 
## 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?

The model is better at detecting things that aren’t diabetes, which means it misses actual diabetes cases. In medical diagnosis, false negatives are dangerous and a missed diabetes diagnosis delays treatment, making high sensitivity more important in a screening context.

Question 3: ROC Curve, AUC, and Interpretation

library(pROC)
## Warning: package 'pROC' was built under R version 4.5.3
## Type 'citation("pROC")' for a citation.
## 
## Attaching package: 'pROC'
## The following objects are masked from 'package:stats':
## 
##     cov, smooth, var
# ROC curve & AUC
roc_obj <- roc(response  = data_subset$Outcome_num,
               predictor = predicted_probs,
               levels    = c(0, 1),
               direction = "<")   # lower probability = no diabetes

# Print AUC value
auc_val <- auc(roc_obj)
auc_val
## Area under the curve: 0.828
# Plot ROC curve with AUC displayed
plot.roc(roc_obj,
         print.auc   = TRUE,
         legacy.axes = TRUE,
         xlab = "False Positive Rate ",
         ylab = "True Positive Rate ",
         main = "ROC Curve – Logistic Regression: Glucose, BMI, Age")

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

The AUC measures how well the model separates diabetic patients from non diabetic patients. Our AUC of ~0.83 means the model did its job and ranked diabetic patient as higher risk most of the time.

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

I would prioritize sensitivity because a lowering the threshold to 0.3 creates less missed diabetes cases, and false positives simply lead to more testing.