1. What is Standard Error in Regression?

The standard error (SE) in regression analysis measures the accuracy with which the coefficients are estimated. Smaller SE values indicate more precise estimates of the true population parameters.

2. How to Calculate Standard Error

The standard error can be extracted from a linear model summary in R using:

Where:

• model — A linear model object created with lm()
• [,2] — Extracts the second column (standard errors) from the coefficients matrix

3. Interpretation of Standard Error

Details: The standard error is used to calculate confidence intervals and p-values for the coefficients. A rule of thumb is that the estimate ± 2×SE gives approximately a 95% confidence interval.

4. Practical Example

Example: Using the mtcars dataset to predict MPG (miles per gallon) from weight (wt) and horsepower (hp):

# Fit linear model
model <- lm(mpg ~ wt + hp, data = mtcars)

# Extract standard errors
SE <- summary(model)$coefficients[,2]

# Results:
# (Intercept)          wt          hp 
#   1.8778473   0.3026911   0.0076757

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between standard error and residual standard error?
A: The standard error refers to coefficient estimates, while residual standard error measures the quality of the overall model fit.

Q2: How does sample size affect standard error?
A: Standard error decreases as sample size increases, following a square root relationship (SE ∝ 1/√n).

Q3: Can standard error be zero?
A: In practice, almost never. A zero SE would indicate perfect estimation, which is unrealistic with real-world data.

Q4: How is standard error related to p-values?
A: p-values are calculated using the ratio of the coefficient estimate to its standard error (t-value).

Q5: What's a "good" standard error value?
A: There's no universal threshold - it depends on the scale of your variables and the context of your analysis.