11.6 Residual standard deviation $\sigma$ and explained variance $R^2$

• $\sigma$ = residual standard deviation.

• $R^2$ = fraction of variance explained by the model:

$$R^2 = 1 - \left(\sigma^2/s_y^2\right)$$

• For least squares, can compute $R^2$ directing from ‘explained variance’:

$$R^2 = V_{i=1}^n \hat{y}_i/s_y^2$$

Where the $V$ operator capture the sample variance. I.e. this is the ratio of the variance in fitted outcome values to the variance in the unfitted outcome values.

11.6.1 Bayesian $R^2$

• Bayesian inference is not least squares, so the two formulas given previously can disagree. (Influence of the prior.)

• Bayesian inference includes additional uncertainty beyond the point estimate that should be included

• This leads to an alternative definition of $R^2$ for each posterior draw $s$:

$$\text{Bayesian } R_s^2= \frac{V_{i=1}^n \hat{y}_i^s}{V_{i=1}^n \hat{y}_i^s + \sigma_s^2}$$

This can computed in R using bayes_R2(fit). This will give draws of $R^2$ which expresses also the uncertainty in $R^2$.

For example for the model predicting child’s test score from given mothers score on IQ and mother’s high school status:

fit_iq <- stan_glm(kid_score ~ mom_hs + mom_iq, data = kidiq, refresh=0)
r2 <- bayes_R2(fit_iq)
hist(r2)