Residual diagnostics

Residual plots and standardised residuals check the linearity, independence, normality and equal-variance conditions behind regression inference.

r_i = \frac{e_i}{\hat\sigma\sqrt{1-h_i}}

Internally studentised (standardised) residual

$e_i$: raw residual $y_i - \hat y_i$
$h_i$: leverage of observation i
$\hat\sigma$: residual standard error

t_i = r_i\sqrt{\frac{n-p-1}{n-p-r_i^2}}

Externally studentised residual (observation i left out of σ̂)

$t_i$: follows a t distribution with n − p − 1 df if the model holds

d = \frac{\sum_{i=2}^{n}(e_i - e_{i-1})^2}{\sum_{i=1}^{n} e_i^2}

Durbin–Watson statistic (rows in time order)

$d$: near 2: no lag-1 autocorrelation; near 0: positive autocorrelation

When to use

After every regression fit, before interpreting p-values or intervals.

Assumptions checked (LINE)

Linearity: residuals vs fitted show no curve.
Independence: no pattern against time or collection order (Durbin–Watson for time-ordered rows).
Normality: Q–Q plot of residuals roughly straight (matters most for small n and for prediction intervals).
Equal variance: no funnel shape in residuals vs fitted (Breusch–Pagan gives a formal test).

How to interpret

A random horizontal band around 0 is what you want.
A curve suggests a missing term or a transformation; a funnel suggests transforming y or using methods for unequal variance.
Studentised residuals beyond about ±3 flag unusual observations; check them, do not delete them automatically.

Common mistakes

Checking normality of y instead of the residuals.
Relying on a normality test alone in large samples, where trivial departures are "significant".
Treating diagnostics as pass/fail rather than as evidence about how much to trust the results.

Example

y = 10, 12, 15, 18, 22, 27 at x = 0…5 fitted with a straight line gives $R^2 = 0.978$ , but residuals 1.10, −0.28, −0.65, −1.02, −0.39, 1.24: positive at both ends, negative in the middle. The U shape reveals curvature that the high $R^2$ hides; a log transformation of y fixes it.