class: center, middle, inverse, title-slide # Goodness of Fit ### Dr. D’Agostino McGowan --- layout: true <div class="my-footer"> <span> Dr. Lucy D'Agostino McGowan </span> </div> --- ## Partitioned variablity  --- ## Why? * `\(y - \bar{y}=(\hat{y}-\bar{y})+(y-\hat{y})\)` -- * `\(\sum(y-\bar{y})^2=\sum(\hat{y}-\bar{y})^2+\sum(y-\hat{y})^2\)` -- * **Total Sum of Squares = Model Sum of Squares + Residual Sum of Squares** -- * **TSS = ModelSS + RSS** --- ## Goodness of Fit * `\(R^2\)`: coefficient of determination, percentage of the variance explained -- `$$R^2=1-\frac{\sum(y_i-\hat{y}_i)^2}{\sum(y_i-\bar{y})^2}= 1 - \frac{RSS}{TSS}$$` -- * Between 0 and 1, higher is better --- ## Goodness of Fit * `\(\hat\sigma\)` is also used as a goodness of fit measure -- * **Advantage**: This is measured in the units of the outcome, so it can be interpreted in the context of the data at hand -- * **Disadvantage**: Makes it harder to compare across contexts, for this same reason