WLS: WEIGHTED LEAST SQUARES One of the critical assumptions of ordinary least squares regression is homoscedasticity: that the variance of residual error should be constant for all values of the independent(s). If the independent(s) has/have different error variance at different ranges of their values, then the estimates of the regression coefficients will have unduly large standard errors for some ranges of the dependent and too small for other ranges. The power of significance tests will be reduced, which is to say regression estimates will be inefficient. Weighted least squares (WLS) regression compensates for violation of the homoscedasticity assumption by weighting cases differentially: cases whose value on the dependent variable corresponds to large variances on the independent variable(s) count less and those with small variances count more in estimating the regression coefficients. That is, cases with greater weights contribute more to the fit of the regression line. The result is that the estimated coefficients are usually very close to what they would be in OLS regression, but under WLS regression their standard errors are smaller. Apart from its main function in correcting for heteroscedasticity, WLS regression is sometimes also used to adjust fit to give less weight to distant points and outliers, or to give less weight to observations thought to be less reliable.

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Table of Contents

WLS Weighted Least Squares Regression	5
Overview	5
Key Terms and Concepts	6
The homoscedasticity assumption in regression	6
Violation of homoscedasticity	6
Weighted cases	7
WLS in SPSS	7
SPSS overview	7
Weighting cases in SPSS	8
Weighting with powers	8
The log-likelihood values table	9
Weighting with replicates	10
WLS regression output from SPSS Analyze, Regression, Weight Estimation	11
Running OLS regression on weighted cases	12
SPSS	12
WLS regression output from SPSS Analyze, Regression, Linear	13
Weighted predicted/residual plots	14
Obtaining weighted predicted/residual plots in SPSS	15
Assumptions	17
Proper specification	17
Data level	17
Multivariate normality	17
Linearity	17
Independence	17
Predictable variance	17
Frequently Asked Questions	18
Is WLS regression something that could be used with regression models other than OLS?	18
Can you do things in SPSS syntax not available in the Weight Estimation and WLS dialogs?	18
Bibliography	18