线性回归残差标准误差过大求助：R语言建模及变量相关性问题

Hey there! Let's work through your linear regression questions one by one:

Is a Residual Standard Error (RSE) of 70830 "very bad"? #

Short answer: It depends entirely on the scale of your dependent variable. RSE is an absolute measure of the average error your model makes, so you need to compare it to the typical values of your outcome variable:

• If your dependent variable has a mean of, say, 100,000, an RSE of 70,830 means your model's predictions are off by ~70% of the average value—that's a sign of poor fit.
• If your dependent variable averages 500,000, the same RSE represents a ~14% error, which might be acceptable depending on your field.

A better way to contextualize this is to calculate the relative error: divide RSE by the mean of your dependent variable. For example:

# Assuming your model is stored as lm_model
dep_var_mean <- mean(lm_model$model[[1]])
relative_error <- summary(lm_model)$sigma / dep_var_mean
relative_error

What tests can you run to assess model quality? #

Here are key checks to evaluate your linear regression model in R:

1. Fit Metrics #

• R² and Adjusted R²: These measure the proportion of variance in the dependent variable explained by your model. Use adjusted R² for multi-variable models (it penalizes adding unnecessary variables):

  summary(lm_model)$r.squared       # Unadjusted R²
  summary(lm_model)$adj.r.squared   # Adjusted R²
  

  Values closer to 1 indicate better fit, but what's "good" varies by domain (e.g., social sciences often accept lower R² than hard sciences).

2. Significance Tests #

• Individual Variable Significance: Check the Pr(>|t|) column in summary(lm_model)—values < 0.05 mean the variable is statistically significant at the 95% confidence level. If many variables are insignificant, it could signal collinearity or poor variable selection.
• Overall Model Significance: The F-statistic and its p-value at the bottom of the summary tell you if the model as a whole explains more variance than a model with just an intercept. A p-value < 0.05 means your model is statistically meaningful.

3. Residual Diagnostics (Critical for Validating Assumptions) #

Linear regression relies on four key assumptions—residual plots help you verify them:

• Linearity: Plot residuals vs. fitted values to check for non-random patterns (e.g., U-shapes, increasing/decreasing spread):

  plot(lm_model, which = 1)
  

  If you see a clear trend, your model might be missing nonlinear terms or interactions.
• Normality of Residuals: Use a Q-Q plot to check if residuals follow a normal distribution:

  plot(lm_model, which = 2)
  

  You can also run a Shapiro-Wilk test for formal confirmation:

  shapiro.test(residuals(lm_model))
  

  A p-value > 0.05 suggests residuals are approximately normal.
• Homoscedasticity (Constant Variance): Look for a "funnel shape" in the residuals vs. fitted values plot. For a formal test, use the Breusch-Pagan test (requires the lmtest package):

  library(lmtest)
  bptest(lm_model)
  

  A p-value < 0.05 indicates heteroscedasticity (non-constant variance), which can bias your standard errors.
• No Autocorrelation: If your data is time-series, use the Durbin-Watson test to check for autocorrelated residuals:

  dwtest(lm_model)
  

4. Collinearity Check (Since You Know payment and Insure Are Correlated) #

High collinearity inflates standard errors and makes coefficient interpretations unreliable. Use Variance Inflation Factors (VIF) to quantify this (requires the car package):

library(car)
vif(lm_model)

• VIF values > 5 (or >10, depending on the source) indicate significant collinearity.

How to Handle High Collinearity Between payment and Insure? #

If VIF confirms severe collinearity, try these fixes:

• Drop one of the variables: Choose the one with less domain relevance or lower statistical significance.
• Combine the variables: Create a composite measure (e.g., sum, average, or ratio) if it makes sense for your data.
• Use regularized regression: Ridge regression or Lasso regression (via the glmnet package) can handle collinearity while retaining variables by shrinking coefficient estimates.

内容的提问来源于stack exchange，提问作者bdiucchio