class: center, middle, inverse, title-slide # Weighted Least Squares ### Dr. D’Agostino McGowan --- layout: true <div class="my-footer"> <span> Dr. Lucy D'Agostino McGowan </span> </div> --- ## Weighted Least Squares * This is a **special case** of GLS where the errors are _uncorrelated_ -- * Now `\(\Sigma\)` is a diagonal matrix with weights `\(w_i, \dots,w_n\)` on the diagonal -- * `\(\mathbf{S} = \textrm{diag}(\sqrt{1/w_1}, \dots,\sqrt{1/w_n})\)` -- * So we would regress `\(\sqrt{w_i}y_i\)` on `\(\sqrt{w_i}x_i\)` -- * **Note** the column of 1s in the design matrix should now be replaced with `\(\sqrt{w_i}\)` -- * cases with low variability get a high weight and those with high variability a low weight --- ## WLS Examples * You see a positive relationship when you plot the absolute value of the residuals, `\(|e_i|\)`, against a predictor, `\(x_i\)`. **Errors are proportional to a predictor** This suggests you should use `\(w_i = x_i^{-1}\)` -- * **Your outcome, `\(y_i\)` is an _average_ of `\(n_i\)` observations**. Here you may want to weight by `\(w_i = n_i\)` -- * **The observed responses are known to be of varying quality.** Here you may want to use `\(w_i = 1/\textrm{var}(y_i)\)` --- ## WLS in R * You can fit weighted least squares by fitting the _normal_ linear model using `lm` and adding the `weights` parameter ```r lm(y ~ x, data = data, weights = wts) ```