模型残差对未纳入项作图的意义及二次项是否需纳入的技术问询

Hey there! Let's work through your two regression diagnostic questions—they're key to making sure your model is properly specified, so great call asking about them.

First, remember that residuals are the difference between your model’s predicted values and the actual observed values—they represent the variation in your dependent variable that your current model hasn’t explained. When you plot these residuals against a term you didn’t include in the model (like a quadratic term, interaction, or another predictor), here’s what you’re looking for and what it means:

• If the residuals are randomly scattered with no clear pattern (no upward/downward trends, curves, or clusters) across the range of the unincluded term, that tells you this term doesn’t add any meaningful explanatory power. Your current model is already capturing the key relationships with your dependent variable, so there’s no need to add it.
• If the residuals show a distinct systematic pattern (e.g., a U-shape, linear trend, or consistent rise/fall as the unincluded term increases), that’s a red flag: this term is related to the leftover variation in your dependent variable that your model missed. In other words, the unincluded term has a relationship with Y that your current model isn’t accounting for, so you should seriously consider adding it to improve model fit.
• Beyond that, this plot is a simple but powerful tool to diagnose model misspecification—like missing nonlinear relationships or important interactions—before you finalize your model.

Short answer: this is a strong signal that you should at least test adding the $X_1^2$ term, but you’ll want to back it up with a few checks:

• The non-random pattern (especially a curved one like U-shape) directly suggests that your linear model (only including $X_1$’s first-order term) isn’t capturing the true nonlinear relationship between $X_1$ and $Y$. The quadratic term is designed to fit exactly this kind of curved relationship, so it’s likely to reduce the unexplained variation in your model.
• That said, don’t just add it based on the plot alone:
  • Run a statistical significance test: After adding $X_1^2$ to the model, check if its regression coefficient is statistically significant (e.g., p-value < 0.05). Sometimes a visual pattern can be driven by a few extreme outliers, and the test will confirm if the term actually adds meaningful predictive power.
  • Consult domain knowledge: Does a quadratic relationship between $X_1$ and $Y$ make sense in your field? If your area of study has no theoretical reason for this kind of curve, you might need to dig deeper—maybe there’s another missing variable, or an issue with your data (like measurement error) causing the pattern.
  • Watch for multicollinearity: $X_1$ and $X_1^2$ are highly correlated, which can inflate standard errors for your coefficients. To fix this, center $X_1$ (subtract its mean from each value) before calculating $X_1^2$—this will reduce the correlation without changing the underlying relationship.

In most cases, though, that non-random residual plot is a clear indicator that adding the quadratic term will improve your model’s ability to capture the real relationship in your data.

内容的提问来源于stack exchange，提问作者Jax