Residual plots and what they catch

The plot that checks the model

L13.1’s assumptions are about the errors, and the residuals are the errors’ stand-ins — so plot them. The standard picture is residuals against fitted values ($e$ up, $\hat{y}$ across). If the model holds, the plot is boring: a random horizontal band centred on zero, with no shape, no trend in spread and no point far from the rest. Anything with a pattern is a specific assumption failing, and the paper asks you to name which.

Four pictures, described

A random band. Points scattered evenly above and below zero across the whole range, about the same height everywhere. Nothing to fix: linearity and constant variance look fine.

A curve. Residuals negative at both ends and positive in the middle (or the reverse) — a U or an arch. The straight line is the wrong shape; linearity fails. The remedy the course names is a transformation of $x$ or $y$ (a log or a square root) or a curved model, and the fitted line’s $R^2$ can look respectable while this is happening, which is why the plot, not $R^2$, is the check.

A funnel. The band is narrow at one end and widens towards the other, typically wider as $\hat{y}$ grows — larger accounts, larger errors. Constant variance fails (heteroscedasticity). The single $s$ then overstates precision where the band is narrow and understates it where it is wide, so every SE and interval is misleading. The remedy named is a transformation of $y$, usually a log.

One far point. A single residual far outside the band. Either an outlier in $y$ (a client whose score is nothing like others of that income) or, if it also sits at an extreme $x$, an influential point that has pulled the line towards itself — Module 4’s L4.6 case. Check the record first; if it is a data error, correct it; if it is real, report the fit with and without it.

Normality: the QQ plot

Residual-vs-fitted cannot judge normality. A normal quantile–quantile plot puts the sorted residuals against the quantiles a normal sample of the same size would have: a straight line means normal errors, a systematic bend means skew or heavy tails. With $n=8$ the picture is rough at best; with $n$ in the dozens the assumption matters less anyway, because the CLT covers the slope.

The 2s rule

A quick numerical screen: any residual with $|e|>2s$ is flagged as a potential outlier — the residuals should be roughly $N(0,s^2)$, and about 95 % of a normal sample lies within two SDs.

Northfield: $2s=2\times 9.642=19.28$. The residuals are −6.5, 1.7, −11.6, 9.8, 11.2, 2.3, 4.6, −11.4; the largest in size is 11.6, well inside 19.28. No outliers. Client 3 (−11.6) and client 8 (−11.4) are the two furthest below the line, and neither is remarkable.

What the paper asks

The question describes a plot in words — "the residuals fan out as the fitted values increase" — and wants the assumption, the consequence and the remedy in one sentence each. It does not ask you to draw the plot, and it may print one; the four descriptions above are the vocabulary either way.