Extracting Slopes from B-spline Models with Interaction Terms

Brief Answer: To extract slopes from a B-spline model with interaction terms, you need to calculate the derivatives of the prediction function with respect to the predictor variable for each category. For your specific model with linear splines (degree=1) and knots at 30 and 60, the slopes will be constant between knots but differ by health category due to the interaction. You can obtain these slopes by examining the model coefficients and their interactions, or by using numerical methods to calculate derivatives at specific points.

Contents

• Understanding B-spline Models with Interactions
• Approaches to Extracting Slopes
• Method 1: Direct Coefficient Interpretation
• Method 2: Numerical Differentiation
• Method 3: Marginal Effects Calculation
• Practical Implementation in R
• Interpreting the Results
• Recommendations

Understanding B-spline Models with Interactions

When you fit a B-spline model with interaction terms like in your example:

fit.spline <- glm(y ~ bs(age, knots = c(30, 60), degree = 1) * health, 
                  data = Wage, 
                  family = "binomial")

You’re creating a model where:

• The relationship between age and the response is piecewise linear between knots
• Each piece (segment between knots) can have a different slope
• The slopes vary by the levels of the health variable due to the interaction

With degree = 1, you’re using linear B-splines, which means the relationship is linear between knots but can change slope at each knot. The interaction with health allows these slopes to differ across health categories.

Approaches to Extracting Slopes

There are three primary approaches to extract slopes from B-spline models with interactions:

1. Direct coefficient interpretation: When using linear splines, the coefficients directly represent the slopes
2. Numerical differentiation: Calculating derivatives of the prediction function numerically
3. Marginal effects calculation: Using specialized R packages to compute marginal effects at different values

Each approach has its advantages and is suitable for different scenarios.

Method 1: Direct Coefficient Interpretation

For linear B-splines (degree=1), the model coefficients can be directly interpreted as slopes in each segment. Let’s examine how to extract these coefficients:

# Examine model structure
summary(fit.spline)

# Check the coefficient names
coef_names <- names(coef(fit.spline))
print(coef_names)

For linear B-splines with knots at 30 and 60, the age variable will be split into three segments:

1. Below 30
2. Between 30 and 60
3. Above 60

The coefficients related to the B-spline basis functions will represent the slopes in each segment. When you include an interaction with health, you’ll have separate slopes for each health category in each segment.

Key Insight: For linear B-splines, the slope coefficients are constant within each segment between knots, making interpretation straightforward compared to higher-degree splines.

Method 2: Numerical Differentiation

For more complex cases or when working with higher-degree splines, numerical differentiation is more robust:

library(numDeriv)

# Create a function to calculate the derivative at a specific age and health status
get_slope <- function(age, health_status, model, data) {
  # Create new data frame with specific values
  new_data <- data.frame(age = age, health = health_status)
  
  # Create model matrix with the same basis functions
  X <- model.matrix(formula(model)[-2], data = new_data)
  
  # Calculate derivative numerically
  grad <- grad(func = function(coef) {
    sum(X %*% coef)
  }, x = coef(model))
  
  return(grad)
}

# Example usage for health category "Good" at age 45
slope <- get_slope(age = 45, health_status = "Good", model = fit.spline, data = Wage)
print(slope)

This approach calculates the slope by taking the derivative of the linear predictor with respect to age at specific points.

Method 3: Marginal Effects Calculation

Several R packages can help calculate marginal effects for spline models:

# Using ggeffects package
library(ggeffects)

# Calculate marginal effects for different health categories
marginal_effects <- ggpredict(fit.spline, terms = c("age [30, 60]", "health"))
print(marginal_effects)

# Using margins package
library(margins)

# Calculate average marginal effects
marginal_effects <- margins(fit.spline, variables = "age")
summary(marginal_effects)

# By health category
marginal_effects_by_health <- margins(fit.spline, variables = "age", at = list(health = c("Good", "Bad", "Average")))
summary(marginal_effects_by_health)

These packages provide efficient ways to calculate slopes (marginal effects) at different points across the age range, stratified by health category.

Practical Implementation in R

Here’s a complete implementation to extract slopes for each health category between the knots:

library(splines)
library(ISLR)
library(dplyr)
library(tidyr)

# Prepare data
age.grid <- seq(range(Wage$age)[1], range(Wage$age)[2])
Wage$y <- ifelse(Wage$wage > 85, 1, 0)

# Fit the model
fit.spline <- glm(y ~ bs(age, knots = c(30, 60), degree = 1) * health, 
                  data = Wage, 
                  family = "binomial")

# Method 1: Direct coefficient interpretation
# Get health categories
health_categories <- unique(Wage$health)

# Create a data frame to store slopes
slopes_df <- data.frame()

# Extract coefficients
coefs <- coef(fit.spline)

# For linear splines with knots at 30, 60:
# Segment 1: age < 30 (reference)
# Segment 2: 30 <= age < 60 (coefficient for bsage1)
# Segment 3: age >= 60 (coefficient for bsage2)

# For each health category
for (health_cat in health_categories) {
  # Get slope in segment 2 (between knots 30-60)
  # Slope = coefficient for bsage1 + interaction coefficient
  slope_2 <- coefs[paste0("bs(age, knots = c(30, 60), degree = 1)1:", health_cat)]
  
  # Get slope in segment 3 (after knot 60)
  # Slope = coefficient for bsage2 + interaction coefficient
  slope_3 <- coefs[paste0("bs(age, knots = c(30, 60), degree = 1)2:", health_cat)]
  
  # Create a row for this health category
  row <- data.frame(
    Health = health_cat,
    Slope_20_30 = 0,  # Reference segment
    Slope_30_60 = slope_2,
    Slope_60_plus = slope_3
  )
  
  slopes_df <- rbind(slopes_df, row)
}

print(slopes_df)

# Method 2: Numerical differentiation at specific points
# Create a function to get slopes at any point
get_slope_at_point <- function(age, health_status, model) {
  # Create new data
  new_data <- data.frame(age = age, health = health_status)
  
  # Get prediction
  pred <- predict(model, newdata = new_data, type = "link")
  
  # Create slightly older age
  new_data$age <- new_data$age + 0.001
  
  # Get new prediction
  pred_new <- predict(model, newdata = new_data, type = "link")
  
  # Calculate slope (difference in predictions)
  slope <- (pred_new - pred) / 0.001
  
  return(slope)
}

# Test at different ages and health categories
test_ages <- c(25, 45, 65)  # Before, between, and after knots
slopes_numeric <- expand.grid(Health = health_categories, Age = test_ages)

slopes_numeric$Slope <- mapply(get_slope_at_point, 
                               slopes_numeric$Age, 
                               slopes_numeric$Health, 
                               MoreArgs = list(model = fit.spline))

print(slopes_numeric)

# Visualize the slopes
library(ggplot2)

# Create predictions across the age range
predictions <- data.frame()
for (health_cat in health_categories) {
  pred_data <- data.frame(age = age.grid, health = health_cat)
  pred_data$pred <- predict(fit.spline, newdata = pred_data, type = "response")
  pred_data$health <- health_cat
  predictions <- rbind(predictions, pred_data)
}

# Plot predictions
ggplot(predictions, aes(x = age, y = pred, color = health)) +
  geom_line() +
  geom_vline(xintercept = c(30, 60), linetype = "dashed") +
  labs(title = "B-spline Predictions with Health Interaction",
       x = "Age", y = "Predicted Probability", color = "Health Status") +
  theme_minimal()

Interpreting the Results

Once you’ve extracted the slopes, you can interpret them as follows:

1. Segment-specific slopes: Each row in your output represents the slopes for a specific health category in different age segments.

2. Statistical significance: Check the p-values associated with each coefficient to determine if the slope is significantly different from zero.

3. Comparative effects: Compare slopes across health categories to understand how the relationship between age and the outcome differs by health status.

4. Practical significance: Beyond statistical significance, consider whether the magnitude of the slope is practically meaningful in your context.

For example, you might find that:

• For individuals with poor health, the probability of high wage increases more rapidly with age between 30-60 compared to those with good health
• After age 60, the relationship might flatten or reverse for all health categories

Recommendations

1. For linear B-splines: Use the direct coefficient interpretation method as it’s most straightforward and efficient.

2. For higher-degree splines: Use numerical differentiation or marginal effects packages, as coefficients don’t directly represent slopes.

3. Visualization: Always visualize your B-spline fits with interaction terms to ensure the extracted slopes make sense in context.

4. Confidence intervals: When reporting slopes, include confidence intervals using bootstrapping or other methods to quantify uncertainty.

5. Model checking: Validate that your spline model adequately captures the relationship by examining residuals and considering alternative knot placements or degrees.

By following these methods, you can effectively extract and interpret slopes from B-spline models with interaction terms, providing valuable insights into how the relationship between your variables changes across different segments and categories.