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 ──
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## âś” ggplot2   4.0.0     âś” tibble    3.3.0
## âś” lubridate 1.9.4     âś” tidyr     1.3.1
## âś” purrr     1.1.0     
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## âś– dplyr::filter() masks stats::filter()
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## â„ą 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.

data_subset <- data[complete.cases(data[, c("Glucose", "BMI", "Age")]), ]
logistic <- glm(Outcome ~ Glucose + BMI + Age, data = data_subset, family = "binomial")

summary(logistic)
## 
## Call:
## glm(formula = Outcome ~ Glucose + BMI + Age, family = "binomial", 
##     data = data_subset)
## 
## 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
## AIC: 732.96
## 
## Number of Fisher Scoring iterations: 4

What does the intercept represent (log-odds of diabetes when predictors are zero)? The intercept is –9.03, and it represents the log odds of having diabetes when glucose, BMI, and age are all equal to zero. These values are not realistic in a medical place.

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)? For Glucose, the coefficient is positive (0.0355), which means that as glucose increases, there is a higher chance of diabetes. The p-value is very small (< 2e-16), so glucose is significant

For BMI, the coefficient is also positive (0.0898). This means that higher BMI is associated with a higher chance of diabetes. Its p-value (4.3e-10) shows that BMI is significant as well.

For Age, the coefficient is positive (0.0287), so as age increases the chance of diabetes also increases. The p-value (0.000238) is below 0.05, so age is significant too.

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 accuracy and shows a good balance between correct predictions and false alarms. It is especially strong at identifying individuals who do not have diabetes, as shown by its high specificity. The sensitivity is lower, meaning it catches many not all true diabetes cases. The model performs well for a simple logistic regression on medical data.

Question 3: ROC Curve, AUC, and Interpretation

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 = predicted.probs,
               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 of 0.828 shows the model does a good job finding people with and without diabetes. Since 0.5 is random and 1.0 is perfect, 0.828 means the model is good.

For diabetes diagnosis, prioritize sensitivity (catching cases) or specificity (avoiding false positives)? Suggest a threshold and explain. For diabetes, we should prioritize sensitivity so we don’t miss true cases. Lowering the threshold to around 0.4 helps find more positives, even if it adds some falses.