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     2.1.5
## ✔ forcats   1.0.1     ✔ stringr   1.5.2
## ✔ ggplot2   4.0.0     ✔ tibble    3.3.0
## ✔ lubridate 1.9.4     ✔ tidyr     1.3.1
## ✔ purrr     1.1.0     
## ── 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.

logistic <- glm(Outcome ~ Glucose + BMI + Age, data=data, family="binomial")

summary(logistic)
## 
## 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 - (logistic$deviance/logistic$null.deviance)

r_square
## [1] 0.25626

What does the R^2 represent?
The r-square is telling us that this model only explains ~25.6% of the variation for the outcome. This indicates that we are likely missing very key predictors in our model.

What does the intercept represent (log-odds of diabetes when predictors are zero)?
The model predicts an essentially zero probability (0.0001) of diabetes when glucose, BMI, and age are all at 0. This makes sense, because those levels are physiologically impossible.

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 in glucose raises the odds of diabetes by ~3.6%. An additional BMI point increases the odds of diabetes by ~9%. And each additional year of age increases the odds by ~2.9%. All of the p-values are < 0.05, meaning they are all 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")]), ]

#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 <- logistic$fitted.values


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


# Confusion matrix
confusion <- table(
  Predicted = factor(predicted.classes, levels = c(0, 1)),
  Actual = factor(data_subset$Outcome_num, levels = c(0, 1))
)

confusion
##          Actual
## Predicted   0   1
##         0 429 114
##         1  59 150
#Extract Values:
TN <- 429
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: 0.77 
## 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 has a decently good accuracy of 0.77, but it is much better at identifying non-diabetics than diabetics. It does a good job at identifying healthy patients, but will miss a large number of unhealthy patients. This is concerning in a medical context because false negatives can have detrimental effects by delaying or outright missing treatment.

Question 3: ROC Curve, AUC, and Interpretation

library(pROC)
## Warning: package 'pROC' was built under R version 4.5.2
## Type 'citation("pROC")' for a citation.
## 
## Attaching package: 'pROC'
## The following objects are masked from 'package:stats':
## 
##     cov, smooth, var
# ROC curve & AUC on full data
roc_obj <- roc(response = data_subset$Outcome,
               predictor = logistic$fitted.values,
               levels = c(0, 1),
               direction = "<")  # smaller prob = Healthy

# Print AUC value
auc_val <- auc(roc_obj); auc_val
## Area under the curve: 0.828
# Plot ROC with AUC displayed
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)?
An AUC of 0.828 indicates that the model will correctly predict a diabetic patient to have diabetes than a non-diabetic patient ~83% of the time. This is a good score, but does have room for improvement.

For diabetes diagnosis, prioritize sensitivity (catching cases) or specificity (avoiding false positives)? Suggest a threshold and explain.
For a diabetes diagnosis, we should prioritize sensitivity because missing a true diabetes diagnoses has larger consequences than falsely predicting a positive case. We should lower the threshold from 0.5, ideally to something closer to 0.25 so that the model’s sensitivity increases even at the detriment of the specificity. This way, there will be less false negatives, which is worth the increase in false positives.