Diabetes Risk Factors Using GAMs

Project Overview

This project focuses on using Generalized Additive Models (GAMs) to explore and predict diabetes risk based on various health factors. The goal was to develop a robust predictive model that captures non-linear relationships between risk factors and diabetes status.

GAMs provide significant advantages over traditional logistic regression by allowing flexible modeling of non-linear effects without requiring a priori specification of the functional form. This flexibility is particularly valuable when analyzing complex health data where relationships often don't follow simple linear patterns.

Methodology

The approach for this project followed these steps:

1. Data acquisition and preprocessing from a diabetes study dataset
2. Exploratory data analysis to identify preliminary relationships
3. Implementation of GAMs with smoothing splines for continuous predictors
4. Model selection and validation using cross-validation
5. Comparison with traditional logistic regression models
6. Interpretation of non-linear effects and risk thresholds

I used thin plate regression splines as the smoothing basis, with automatic selection of smoothing parameters via generalized cross-validation (GCV) to achieve an optimal balance between model fit and complexity.

R Code Implementation

The following code demonstrates the implementation of GAMs for diabetes prediction:

# Load necessary libraries
library(mgcv)
library(ggplot2)
library(dplyr)
library(pROC)
library(caret)

# Load data
diabetes_data <- read.csv("diabetes_data.csv")

# Data preprocessing
diabetes_data <- diabetes_data %>%
  # Convert outcome to factor
  mutate(Outcome = factor(Outcome, levels = c(0, 1), labels = c("No Diabetes", "Diabetes"))) %>%
  # Scale continuous predictors
  mutate_at(vars(Pregnancies, Glucose, BloodPressure, SkinThickness, 
                Insulin, BMI, DiabetesPedigreeFunction, Age), scale)

# Split data into training and testing sets
set.seed(123)
train_index <- createDataPartition(diabetes_data$Outcome, p = 0.7, list = FALSE)
train_data <- diabetes_data[train_index, ]
test_data <- diabetes_data[-train_index, ]

# Fit logistic regression model (for comparison)
logistic_model <- glm(Outcome ~ ., 
                      data = train_data, 
                      family = binomial())

# Fit GAM with smooth terms for continuous predictors
gam_model <- gam(Outcome ~ s(Pregnancies) + s(Glucose) + s(BloodPressure) + 
                          s(SkinThickness) + s(Insulin) + s(BMI) + 
                          s(DiabetesPedigreeFunction) + s(Age),
                data = train_data,
                family = binomial(),
                method = "REML")

# Model summary
summary(gam_model)

# Display effective degrees of freedom for smoothers
edf_summary <- data.frame(
  Variable = c("Pregnancies", "Glucose", "BloodPressure", "SkinThickness", 
               "Insulin", "BMI", "DiabetesPedigreeFunction", "Age"),
  EDF = summary(gam_model)$edf
)
print(edf_summary)

# Predictions on test data
test_data$logistic_pred <- predict(logistic_model, newdata = test_data, type = "response")
test_data$gam_pred <- predict(gam_model, newdata = test_data, type = "response")

# Calculate ROC curves
logistic_roc <- roc(test_data$Outcome, test_data$logistic_pred)
gam_roc <- roc(test_data$Outcome, test_data$gam_pred)

# Compare AUC values
auc_comparison <- data.frame(
  Model = c("Logistic Regression", "GAM"),
  AUC = c(auc(logistic_roc), auc(gam_roc))
)
print(auc_comparison)

Visualization of the smooth functions for key risk factors:

# Function to create better visualizations of GAM smooth terms
visualize_smooths <- function(model, pred_var, pred_range = NULL, data = train_data) {
  if (is.null(pred_range)) {
    pred_range <- seq(min(data[[pred_var]]), max(data[[pred_var]]), length.out = 100)
  }
  
  # Create prediction data frame
  pred_data <- data.frame(matrix(0, nrow = length(pred_range), ncol = ncol(data)))
  colnames(pred_data) <- colnames(data)
  pred_data[[pred_var]] <- pred_range
  
  # Generate predictions
  preds <- predict(model, newdata = pred_data, type = "link", se.fit = TRUE)
  
  # Convert to probability scale with confidence intervals
  pred_results <- data.frame(
    x = pred_range,
    fit = preds$fit,
    se = preds$se.fit
  ) %>%
    mutate(
      fit_prob = plogis(fit),
      lower_prob = plogis(fit - 1.96 * se),
      upper_prob = plogis(fit + 1.96 * se)
    )
  
  # Plot results
  ggplot(pred_results, aes(x = x, y = fit_prob)) +
    geom_line(size = 1.2, color = "blue") +
    geom_ribbon(aes(ymin = lower_prob, ymax = upper_prob), alpha = 0.2) +
    labs(
      title = paste("Effect of", pred_var, "on Diabetes Risk"),
      x = pred_var,
      y = "Probability of Diabetes"
    ) +
    theme_minimal()
}

# Generate plots for key variables
bmi_plot <- visualize_smooths(gam_model, "BMI")
glucose_plot <- visualize_smooths(gam_model, "Glucose")
age_plot <- visualize_smooths(gam_model, "Age")

# Display plots
print(bmi_plot)
print(glucose_plot)
print(age_plot)

Results

The GAM-based analysis revealed several important insights:

• Glucose levels showed a strong non-linear relationship with diabetes risk, with a sharp increase in risk above certain thresholds
• BMI exhibited a non-linear pattern with increasing risk, but plateauing at very high values
• Age demonstrated a more complex relationship than typically modeled in linear approaches
• The GAM model achieved significantly better predictive performance (AUC = 0.85) compared to traditional logistic regression (AUC = 0.81)

Visualization of BMI's non-linear effect on diabetes risk:

Comparison of ROC curves between GAM and logistic regression:

Conclusions

This project demonstrated the advantages of GAMs for health risk modeling:

• GAMs captured complex non-linear relationships that traditional logistic regression missed
• The effective degrees of freedom for each smooth term provided insights into which variables had the most non-linear relationships
• Identification of specific threshold points where risk dramatically increases can inform clinical guidelines
• The superior predictive performance of GAMs suggests their potential value in clinical decision support systems

The ability to visualize these complex relationships through smooth function plots provides not only better predictions but also more interpretable insights for healthcare providers. Future work could explore incorporating interactions between risk factors and extending the approach to longitudinal data.