Partitioning variability

Why?

• $y-\overline{y}=(\hat{y}-\overline{y})+(y-\hat{y})$

• $\sum (y-\overline{y}{)}^2=\sum (\hat{y}-\overline{y}{)}^2+\sum (y-\hat{y}{)}^2$

• SSTotal = SSModel + SSE

Coefficient of determination

• $r^2=\frac{\text{Variability explained by the model}}{\text{Total variability in}y}$
• $r^2=\frac{\text{SSModel}}{\text{SSTotal}}$
• $r^2=\frac{\sum (\hat{y}-\overline{y}{)}^2}{\sum (y-\overline{y}{)}^2}$
• $r^2=\frac{\text{SSTotal − SSE}}{\text{SSTotal}}$
• $r^2=1-\frac{\text{SSE}}{\text{SSTotal}}$

Let's do it in R!

lm(Weight ~ WingLength, data = Sparrows) %>%
  glance()

## # A tibble: 1 x 11
##   r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
##       <dbl>         <dbl> <dbl>     <dbl>    <dbl> <int>  <dbl> <dbl> <dbl>
## 1     0.614         0.611  1.40      181. 2.62e-25     2  -203.  411.  419.
## # … with 2 more variables: deviance <dbl>, df.residual <int>

61% of the variation in "Weight" is explained by "Wing Length".