Residuals

Residuals is the difference between actual values and predicted values

$i$th residual is:

  • $e_i = y_i - (b_0 + b_1 x_i) = y_i - b_0 - b_1 x_1 $

Mean of $e$:

  • $\bar{e} = \bar{y} - b_0 - b_1 \bar{x} = \bar{y} - (\bar{y} - b_1 \bar{x}) - b_1 \bar{x} = 0$


What about $\text{Var}(e)$?

  • $e_i = y_i - b_0 - b_1 x_i = y_i - (\bar{y} - b_1 \bar{x}) - b_1 x_i = (y_i - \bar{y}) - b_i (x_i - \bar{x}) = (y_i - \bar{y}) - R \cfrac{s_y}{s_x}(x_i - \bar{x})$
  • so,
$(e_i - \bar{e})^2 = (y_i - \bar{y})^2 + (R \cfrac{s_y}{s_x})^2 (x_i - \bar{x})^2 - 2 (\bar{y} - b_1 \bar{x}) \cdot R \cfrac{s_y}{s_x}(x_i - \bar{x})$
  • $\text{Var}(e) = \cfrac{1}{n - 1} \sum (e_i - \bar{e})^2 = s_y^2 + (R \cfrac{s_y}{s_x})^2 s_x^2 - 2R \cfrac{s_y}{s_x} \cfrac{1}{n - 1} \sum (x_0 - \bar{x})(y_i - \bar{y}) = $ (note the orrelation coefficient again!)
  • $= s_y^2 + R^2 s_y^2 - 2R\cfrac{s_y}{s_x} s_x s_y R = s_y^2 + R^2 s_y^2 - 2R^2 s_y^2 = s_y^2 (1 - R^2)$

So

$\text{Var}(e) = \text{Var}(y)(1 - R^2)$


Coefficient of Determination

The regression multiplies the variance of $y$ by $(1 - R^2)$

  • Or, the regression line removes (or reduces) a fraction of $R^2$ of the variance of $y$
  • Or we say it "explains a fraction of $R^2$ of the variation"

$R^2$ is called coefficient of determination - and says what fraction of $\text{Var}(Y)$ has been explained by the linear relationship


Examples:

  • $R^2 = 0$: the linear relationship explains nothing (so no linear relationship between $X$ and $Y$)
  • $R^2 = 1$: the linear relationship explains everything - no left-overs, no uncertainty
  • $R^2 = 0.0186$: only 1.86% of variation was explained by the linear model - so there hardly is a linear relation. The rest of the variance (98%) is due to something else


Let's take a look at the example again:

  • $R^2$ = 0.4033
  • so it means quite a bit of variance there is explained by linear model
  • but still it doesn't explain everything - indeed the real data doesn't seem to have linear relationship


Residual Analysis

Are there any other kinds of relationships between $X$ and $Y$, not captured by regression?

Ideal case

  • This is a good case because after taking out linear relationship there's no particular pattern in residuals: only independent errors are left
  • So overall there's no particular trend and that means that the regression really tells us something about the relationships between $X$ and $Y$


Another Example

And the same here

In both cases the linear relationship doesn't describe the whole story and we see there are apparent patterns in the residuals in both cases


Logarithmic Transformation

  • To improve the situation we could try to transform the variables before applying regression.
  • Most common transformation is logarithmic

So we have the following:

Recall that in this case $R^2 = 0.40$


If we calculate $\log_{10} x$ what we get is

Now we're able to fit a better regression line and in this case $R^2 = 0.6576$

Here we interpret a slope of 14.93 as

  • if $\log_b x$ increases by $1$, $y$ increases by 14.93
  • or if $x$ is multiplied by $b$, $y$ increases by 14.93


Sources

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