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     1.2.1     ✔ readr     2.2.0
## ✔ forcats   1.0.1     ✔ stringr   1.6.0
## ✔ ggplot2   4.0.3     ✔ tibble    3.3.1
## ✔ lubridate 1.9.5     ✔ tidyr     1.3.2
## ✔ purrr     1.2.2     
## ── 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

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

summary(linearreg)
## 
## Call:
## glm(formula = Outcome ~ Glucose + BMI + Age, family = binomial(link = "logit"), 
##     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
pseudo_r2 <- 1 - (linearreg$deviance / linearreg$null.deviance)
cat("Pseudo R-squared:", round(pseudo_r2, 3))
## Pseudo R-squared: 0.256

What does the intercept represent (log-odds of diabetes when predictors are zero)?

The intercept represents the log-odds of having diabetes when all predictors of the factors that we’re concerned about, which in this case is Glucose, BMI, and age, are equalled to zero. However, this is an impossible situation since glucose and BMI can’t ever be “zero” for a human person.

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)?

Glucose, a positive coefficient shows that higher glucose raises your odds of diabetes. BMI, positive coefficient shows higher BMI raises odds of diabetes. Age, the older you are, higher odds 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
data_subset$pred_prob <- predict(linearreg, newdata = data_subset, type = "response")

# Predicted classes
data_subset$pred_class <- ifelse(data_subset$pred_prob > 0.5, 1, 0)

# Confusion matrix
conf_matrix <- table(Predicted = data_subset$pred_class, Actual = data_subset$Outcome_num)
print(conf_matrix)
##          Actual
## Predicted   0   1
##         0 429 114
##         1  59 150
#Extract Values:
TN <- conf_matrix[1, 1]  # Predicted 0, Actual 0
FP <- conf_matrix[2, 1]  # Predicted 1, Actual 0
FN <- conf_matrix[1, 2]  # Predicted 0, Actual 1
TP <- conf_matrix[2, 2]  # Predicted 1, Actual 1

#Metrics    
accuracy <- (TP + TN) / sum(conf_matrix)
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?

Compare accuracy, which we expect to be anywhere from 75% to 77%, to baseline diabetes which is 35%. Logistic regression analyses often shows higher specificity, which here is non-diabetes, than sensitivity, which is diabetes.

So unforuntately, this may mean we’re really good at finding healthy cases, but we’ll misee from the actual diabetes cases with a false negative. This could be bad for a patient especially because untreated diabetes can lead to amputations, or even death. We want to have an analysis with better sensitivity.

Question 3: ROC Curve, AUC, and Interpretation

install.packages("pROC")
## Installing package into '/cloud/lib/x86_64-pc-linux-gnu-library/4.6'
## (as 'lib' is unspecified)
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(data_subset$Outcome_num, data_subset$pred_prob)
## Setting levels: control = 0, case = 1
## Setting direction: controls < cases
plot(roc_obj, main = "ROC Curve for Diabetes Prediction", 
     col = "blue", lwd = 2, legacy.axes = TRUE)
abline(a = 0, b = 1, lty = 2, col = "gray")  # Random classifier line

auc_value <- auc(roc_obj)
cat("AUC:", round(auc_value, 3))
## AUC: 0.828

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

AUC ranges from 0.5,random guessing, to 1.0, perfect prediction. The AUC here was 0.828. Typical values for this model are around 0.75-0.85, so we’re within range.

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

Because we want to make this more sensitive, we should make sure to lower the threshold a little bit so we can catch more of the false-negatives and diagnose people with diabetes so they can intervene with treatment.