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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## ✔ lubridate 1.9.4     ✔ tidyr     1.3.2
## ✔ purrr     1.2.1     
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library(pROC)
## Type 'citation("pROC")' for a citation.
## 
## Attaching package: 'pROC'
## 
## The following objects are masked from 'package:stats':
## 
##     cov, smooth, var
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.

Outcome = -9.032377 + 0.028699(Age) + 0.089753(BMI) + 0.035548(Glucose)

R² = 1 - (Deviance/ Null Deviance)

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

summary(logistic)
## 
## Call:
## glm(formula = Outcome ~ Age + BMI + Glucose, family = "binomial", 
##     data = data)
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -9.032377   0.711037 -12.703  < 2e-16 ***
## Age          0.028699   0.007809   3.675 0.000238 ***
## BMI          0.089753   0.014377   6.243  4.3e-10 ***
## Glucose      0.035548   0.003481  10.212  < 2e-16 ***
## ---
## 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

What does the intercept represent (log-odds of diabetes when predictors are zero)? The intercept means that when BMI, Glucose, and Age are zero, the log-odds of having diabetes is -9.032377

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 predictors raise the odds of diabetes when there is a one-unit increase because their coefficients are all positive. All of these have significant pvalues, and therefore are strong predictors of diabetes.

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

Outcomes:

429 people had no diabetes, and the model said they had no diabetes. (true negative) 114 people had diabetes, but the model said they had no diabetes. (false negative) 59 people had no diabetes, but the model said that they had diabetes. (false positive) 150 people had diabetes, and the model said that they had diabetes. (true positive)

#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 performs semi-accurately (77%), but needs to be improved. More specifically, the model can predict that someone does not have diabetes 87.9% of the time, but can only predict if someone has diabetes 56.8% of the time. This is harmful, as 43.8% of the time the model is unable to predict if someone has diabetes, leading to frequent false positives.

Question 3: ROC Curve, AUC, and Interpretation

# ROC curve & AUC on full data
roc_obj <- roc(response = data_subset$Outcome_num,
               predictor = logistic$fitted.values,
               levels = c("0", "1"),
               direction = "<")

# Print AUC value
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)?

AUC of 0.828 means that the model is 82.8% likely to rank a person of diabetes higher (when someone is randomly selected).

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

Sensitivity should be prioritized over specificity, as it allows people who actually have the disease to get treatment. Although False positives cause confusion, there is less danger compared to the risk of people who have diabetes not getting treatment.