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” forcats   1.0.1     âś” stringr   1.6.0
## âś” ggplot2   4.0.3     âś” tibble    3.3.1
## âś” lubridate 1.9.5     âś” tidyr     1.3.2
## âś” purrr     1.2.2     
## ── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
## âś– dplyr::filter() masks stats::filter()
## âś– dplyr::lag()    masks stats::lag()
## 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.

lg_model <- glm(Outcome ~ Glucose + BMI + Age,
               data = data,
               family = binomial)
summary(lg_model)
## 
## 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_squared <- 1 - (lg_model$deviance / lg_model$null.deviance)
r_squared
## [1] 0.25626
exp(-9.032377)
## [1] 0.0001194782

What does the intercept represent (log-odds of diabetes when predictors are zero)? The intercept represents the log-odds of the outcome when Glucose, BMI, and Age all equal 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)? Yes, all of the predictors in this model are significant, as they all increase the 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$prob <- predict(lg_model,
                            newdata = data_subset,
                            type = "response")


# Predicted classes
data_subset$predicted <- ifelse(data_subset$prob > 0.5, 1, 0)


# Confusion matrix
conf_matrix <- table(
  Predicted = data_subset$predicted,
  Actual = data_subset$Outcome_num
)

conf_matrix
##          Actual
## Predicted   0   1
##         0 429 114
##         1  59 150
#Extract Values:
TN <- conf_matrix["0", "0"]
FP <- conf_matrix["0", "1"]
FN <- conf_matrix["1", "0"]
TP <- conf_matrix["1", "1"]

#Metrics    
accuracy <- (TN + TP) / 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: 0.718 
## 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 does reasonably well for accuracy, sensitivity, and specificity, and is not very good at precision. The model is not very good for a medical diagnosis as for precision, around 57% of the predictions for diabetes are actually correct.

Question 3: ROC Curve, AUC, and Interpretation

#Enter your code here
library(pROC)
## Type 'citation("pROC")' for a citation.
## 
## Attaching package: 'pROC'
## The following objects are masked from 'package:stats':
## 
##     cov, smooth, var
roc_curve <- roc(data_subset$Outcome_num,
                 data_subset$prob)
## Setting levels: control = 0, case = 1
## Setting direction: controls < cases
plot(roc_curve,
     main = "ROC Curve")

auc(roc_curve)
## Area under the curve: 0.828

What does AUC indicate (0.5 = random, 1.0 = perfect)? The model has an AUC of 0.828, so it is generally good at distinguishing between the two outcomes.

For diabetes diagnosis, prioritize sensitivity (catching cases) or specificity (avoiding false positives)? Suggest a threshold and explain. For diabetes diagnosis, I would choose to prioritize sensitivity. Lowering the threshold to 0.4 would mean less people are diagnosed as not having diabetes when they do, and would mean theu don’t have treatment delayed for them. This would also increase the amount of false positives, but a false positive could be ruled out with more testing.