偏残差图无信息但需变换预测变量？分数多项式逻辑回归问询

Great question—this is a common scenario when working with fractional polynomials (FP) for logistic regression: you know transformations are improving model fit (via reduced deviance), but partial residual plots aren’t giving you clear visual cues to validate or refine those transformations. Here’s a practical, step-by-step approach to navigate this:

1. Use Alternative Residual Visualizations Tailored to Logistic Regression #

Partial residual plots can be misleading for binary outcomes because of the discrete nature of the response. Instead, try:

• Quantile residual plots: These are far more robust for generalized linear models (GLMs) with non-normal responses. Calculate and plot them with:

  quantile_resids <- qresid(your_mfp_model)
  # Check fit for x1's quadratic transformation
  plot(your_data$x1^2, quantile_resids, main = "Quantile Residuals vs. x1²")
  # Check fit for x2's combined transformation
  plot(6.3*your_data$x2^0.5 - 14.5*log(your_data$x2), quantile_resids, main = "Quantile Residuals vs. Combined x2 Transformations")
  

  Look for random, uniform scatter (no clear trends) to confirm your transformations are capturing the underlying relationship.
• Component-plus-residual plots: These add the fitted component of the predictor back to the residual, making it easier to spot missed non-linearities. Use the car package in R to generate these for your model.

2. Dig Into the MFP Model’s Internal Diagnostics #

Since you used mfp, leverage the tool’s built-in output to validate your chosen transformations:

• Run summary(your_mfp_model) to see all candidate FP powers considered for each variable, along with their AIC values. Check if the selected powers (e.g., 2 for x1, 0.5 and -1 for x2) are clearly the top performers, or if other powers are close—this might hint at alternative transformations to test.
• Use plot(your_mfp_model) to visualize the FP fits directly. This shows how the model’s predicted logit changes with the original variable, based on the selected transformation, which is often more intuitive than partial residuals.

3. Validate Transformations With Non-Parametric Grouped Analysis #

Take a raw-data approach to check if your transformations align with real patterns:

• Split x1 into 5-10 quantile groups. For each group, calculate the average adjusted logit (adjusted for the effect of x2’s transformations). Plot this average logit against the group’s average x1²—a strong linear relationship confirms the quadratic transformation is appropriate.
• For x2, repeat the process: group by x2 quantiles, calculate adjusted logit (adjusted for x1²), and plot against the group’s average 6.3*x2^0.5 -14.5*log(x2). A linear trend here reinforces that the combined transformation is capturing the true relationship.

4. Use Cross-Validation to Test Transformation Stability #

With 60,000 observations, you have plenty of data to validate transformations out-of-sample:

• Split your data into training (80%) and testing (20%) sets.
• Fit the linear model and your FP model on the training set.
• Compare performance metrics on the test set: AUC, Brier score, or deviance. If the FP model consistently outperforms the linear one, this confirms your transformations are meaningful, not just overfitting noise.
• Test alternative transformation combinations (e.g., x1^3 instead of x1², or x2^0.3 instead of x2^0.5) via cross-validation to see if you can refine the fit further.

5. Ground Transformations in Domain Knowledge #

When visual diagnostics fail, turn to subject-matter expertise:

• What do x1 and x2 represent? For example:
  • If x1 is age, a quadratic term makes intuitive sense (risk often rises then falls with age).
  • If x2 is income, a combination of square root (capturing diminishing returns at lower levels) and log (capturing saturation at higher levels) aligns with common economic patterns.
• If your transformations match domain expectations, that’s strong evidence they’re appropriate—even if partial residuals don’t show a clear pattern.

6. Check for Confounding From Outliers or Leverage Points #

Sometimes partial residual plots are uninformative because extreme points mask underlying patterns:

• Calculate Cook’s distance and hat values to identify high-leverage or influential observations.
• Temporarily remove these points and re-generate partial residual plots—you might uncover a clear pattern that was hidden before.

内容的提问来源于stack exchange，提问作者Danny