曲线回归模型可靠性评估方法咨询（n=60，无法交叉验证）

Great question—when working with a small sample like n=60, it makes total sense to look beyond just the initial ANOVA F-test to validate your curve regression model's reliability. Bootstrapping is a solid choice, but there are several other practical methods you can use to build confidence in your results:

• Refine your bootstrapping approach
  Don’t stop at basic case resampling. For regression models, two tailored variants work well for small samples:

  • Residual bootstrapping: If your model’s residuals meet independence and homoscedasticity assumptions, resample from the residuals instead of raw data. This preserves the original predictor values while testing how well the model captures the error structure.
  • Bootstrap confidence intervals: Generate hundreds of bootstrap samples, refit your curve model each time, and calculate confidence intervals for your regression coefficients. Narrow intervals mean stable estimates, while wide ones signal more uncertainty to report.
    You can also bootstrap your model’s prediction intervals to see how consistent its predictive performance is across resampled data.

• Leave-One-Out Cross-Validation (LOOCV)
  You mentioned concerns about sample size for cross-validation, but LOOCV is perfect for small datasets. Instead of splitting into k folds, you leave out one observation at a time, fit the model on the remaining 59, and predict the excluded point. Repeat this 60 times, then calculate the average mean squared error (MSE) or mean absolute error (MAE) across all predictions. LOOCV uses nearly all your data for each fit, so it’s more stable than k-fold cross-validation for small n—and with n=60, the computational load is totally manageable.

• Jackknife Resampling
  A simpler alternative to bootstrapping, the jackknife works by systematically removing one observation at a time, recalculating key model metrics (coefficients, R², prediction error) each time. You can then use the distribution of these metrics to estimate variability and approximate confidence intervals. It’s less computationally intensive than bootstrapping and great for quick checks of how sensitive your model is to individual data points.

• Rigorous Model Diagnostics
  Reliability starts with valid assumptions. Spend time checking:

  • Residual plots: Plot residuals against fitted values to check for homoscedasticity (no funnel shape) and non-linear patterns (which would mean your curve still isn’t capturing the relationship).
  • Q-Q plots: Assess if residuals follow a normal distribution—critical for trusting your ANOVA results and prediction intervals.
  • Outlier/influence tests: Use metrics like Cook’s distance to identify points that disproportionately affect your model. Removing these and refitting can tell you how robust your curve is to extreme values.

• Sensitivity Analysis
  Test how your model holds up when you tweak inputs or assumptions:

  • Remove 1-2 of the most influential points (identified via diagnostics) and refit the model. If your coefficients and fit metrics (R², AIC) barely change, your model is robust. If they shift drastically, you’ll need to address those outliers or consider a more robust regression method.
  • Try alternative curve specifications (e.g., a cubic instead of quadratic, if you used quadratic). If the core findings hold across similar models, that’s a good sign your results aren’t dependent on one arbitrary curve choice.

• Information Criteria & Adjusted Fit Metrics
  Beyond the ANOVA F-test, use metrics that account for model complexity:

  • AIC/BIC: Lower values indicate a better balance of fit and parsimony. If your curve model has a lower AIC/BIC than the linear model, it confirms the better fit isn’t just due to adding extra terms.
  • Adjusted R²: Unlike raw R², this penalizes you for adding more parameters. A meaningful increase in adjusted R² from linear to curve model means the extra complexity is justified by improved explanatory power.

内容的提问来源于stack exchange，提问作者HIL