Weighted Least Squares Regression

Weighted least squares regression estimates linear model coefficients by minimizing a weighted sum of squared residuals, giving more influence to observations with smaller error variance.

가중최소제곱회귀는 오차분산이 작은 관측에 더 큰 가중치를 주어 가중 잔차제곱합을 최소화하는 선형회귀입니다.

My observations have different precision, such as averages based on different group sizes; how should I weight them in a regression?

관측마다 정밀도가 다른데(예: 표본 수가 다른 평균값) 회귀에서 가중치를 어떻게 주어야 하나요?

Weighted least squares gives each observation a weight proportional to the inverse of its error variance, which restores efficient estimates and correct standard errors when the variances differ in a known way. If the weights are well chosen, the weighted residuals have roughly constant spread.

When to use Weighted Least Squares Regression

• Each observation's error variance is known up to a constant, for example a mean of n_i cases has variance proportional to 1/n_i
• Residual spread grows with a measured variable and you can model that pattern
• You analyse aggregated data with unequal group sizes

When to use something else

• The variance pattern is unknown or weights come from survey design; use regression with robust standard errors instead
• The outcome is a count or a proportion with variance tied to its mean; use a generalized linear model such as Poisson or fractional logit regression

Assumptions and requirements

• Weights are positive and proportional to the inverse error variance
• The mean model is linear in the coefficients
• Errors are independent
• Weighted residuals have constant variance

What the analysis reports

• Weighted coefficients with standard errors and confidence intervals
• Unweighted OLS coefficients for comparison
• Weighted R² and overall F test
• Diagnostics on weighted residuals

How to report it

A weighted least squares regression with weights [weight variable] showed that [predictor] was associated with [outcome], b = [value], SE = [value], t([df]) = [value], p = [value]; R² = [value].

Related methods

• Regression with Robust Standard Errors
• Multiple Linear Regression

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