HW 10

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):

Pregnancies: Number of times pregnant
Glucose: Plasma glucose concentration (2-hour test)
BloodPressure: Diastolic blood pressure (mm Hg)
SkinThickness: Triceps skin fold thickness (mm)
Insulin: 2-hour serum insulin (mu U/ml)
BMI: Body mass index (weight in kg/(height in m)^2)
DiabetesPedigreeFunction: Diabetes pedigree function (a function scoring genetic risk)
Age: Age in years
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   1.5.1
## ✔ 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.

Provide the model summary.
Calculate and interpret R²: 1 - (model\(deviance / model\)null.deviance). What does it indicate about the model’s explanatory power?

## Enter your code here
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 intercept represent (log-odds of diabetes when predictors are zero)?

The intercept (–9.03) represents the log-odds of diabetes when glucose, BMI, and age are all zero.

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 glucose, BMI, and age have positive coefficients, indicating that if any of them increase so does the odds of having diabetes. Each of these predictors is statistically significant (p < 0.05), meaning they contribute meaningfully to the model. Glucose has the biggest impact, followed by BMI and then age.

Question 2: Confusion Matrix and Important Metric

Predict probabilities using the fitted model.
Create predicted classes with a 0.5 threshold (1 if probability > 0.5, else 0).
Build a confusion matrix (Predicted vs. Actual Outcome).

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: 1 if prob > 0.5, else 0
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 <- 114
FN <- 59
TP <- 150

#Metrics    
accuracy    <- (TP + TN) / (TP + TN + FP + FN)
sensitivity <- TP / (TP + FN)    # recall, true positive rate
specificity <- TN / (TN + FP)    # true negative rate
precision   <- TP / (TP + FP)    # positive predictive value
f1_score    <- 2 * (precision * sensitivity) / (precision + sensitivity)


cat("Accuracy:", round(accuracy, 3), "\nSensitivity:", round(sensitivity, 3), "\nSpecificity:", round(specificity, 3), "\nPrecision:", round(precision, 3))

## Accuracy: 0.77 
## Sensitivity: 0.718 
## Specificity: 0.79 
## Precision: 0.568

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 well, with an accuracy of about (77%). However, it’s behavior in comparison of the two classes differs. Specificity at (79%) is much higher than sensitivity at (71.8%), meaning the model is less accurate when identifying people with diabetes than those without diabetes.
In healthcare, this would pose an issue. Having a lower sensitivity means missing cases of diabetes which is more dangerous because misdiagnosis of diabetes can lead to serious health complications.

Question 3: ROC Curve, AUC, and Interpretation

Plot the ROC curve, use the “data_subset” from Q2.
Calculate AUC.

#Enter your code here
# install.packages("pROC") # if needed
library(pROC)

## 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_num,
               predictor = predicted.probs)

## Setting levels: control = 0, case = 1

## Setting direction: controls < cases

# 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 has excellent ability when differentiating between diabetics and non-diabetics. This means the model will assign a higher predicted probability to the diabetic case about (83%) of the time.

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

For diagnosing diabetes, sensitivity should be prioritized, because missing a diabetic patient can be harmful. As a suggestion for a threshold,lowering the probability cutoff could be (0.3) rather than (0.5). This will increase sensitivity and helps identify more people who actually have diabetes. However, it will also cause more false alarms.

HW 10

Marvellous Onajobi

Introduction: