Quiz 2

Goodness of Fit

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# Goodness of Fit ## R-squared $$ R^2 = 1 - \frac{SS_{res}}{SS_{tot}} = \frac{SS_{reg}}{SS_{tot}} $$ - Proportion of variance explained by the model - Always increases when adding predictors ## Adjusted R-squared $$ R^2_{adj} = 1 - \frac{SS_{res}/(n-p)}{SS_{tot}/(n-1)} $$ - Penalizes model complexity - Can decrease...

Goodness of Fit

R-squared

R2=1−SSresSStot=SSregSStotR^2 = 1 - \frac{SS_{res}}{SS_{tot}} = \frac{SS_{reg}}{SS_{tot}}
  • Proportion of variance explained by the model
  • Always increases when adding predictors

Adjusted R-squared

Radj2=1−SSres/(n−p)SStot/(n−1)R^2_{adj} = 1 - \frac{SS_{res}/(n-p)}{SS_{tot}/(n-1)}
  • Penalizes model complexity
  • Can decrease when adding useless predictors

Partial F-Test

Test whether a subset of predictors collectively contribute:
F=(SSres,R−SSres,F)/(dfR−dfF)SSres,F/dfF∼FdfR−dfF,dfFF = \frac{(SS_{res,R} - SS_{res,F}) / (df_R - df_F)}{SS_{res,F} / df_F} \sim F_{df_R-df_F, df_F}
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