Regression Assumptions & Diagnostics

Info sheet · Statistics for Psychology & Neuroscience

Info sheet 0.2 (draft) · Prerequisites: multiple regression; residuals · Give feedback ↗

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What you’ll get from this sheet

Regression, like any test, rests on assumptions — some familiar from ANOVA, some new. By the end you should be able to:

1. List the core assumptions of regression.
2. Read the four diagnostic plots R gives you.
3. Name the formal tests (Shapiro–Wilk, NCV, VIF).

Regression assumes independence, normal residuals, constant variance (homoscedasticity), linearity, and no strong multicollinearity. Calling plot(model) gives four diagnostic plots to check them by eye; Shapiro–Wilk (normality), the NCV test (constant variance), and VIF (multicollinearity) back them with numbers.

The assumptions

Three are carried over from ANOVA: independence of observations, normality of the residuals, and homogeneity of variance (the residuals should have constant spread). Two are new to regression: linearity (the relationship really is a straight line / plane), and no strong multicollinearity (predictors shouldn’t be near‑duplicates of one another). Note it’s the residuals that must be normal — not the raw variables.

Read the diagnostic plots

The neat R trick: call plot(model) and you get four diagnostic plots. The most useful is Residuals vs Fitted — it should look like a formless, even band around zero. Departures have signatures. Pick a problem below and watch the fitted line (left) and its residual plot (right) change.

md`**What's wrong with the data?**`

viewof problem = html`<select style="font-size:15px;padding:4px 8px">
  <option value="ok" selected>Nothing — assumptions met</option>
  <option value="hetero">Heteroscedasticity (spread grows)</option>
  <option value="nonlin">Non-linearity (curved relationship)</option>
  <option value="outlier">An influential outlier</option>
</select>`

diagWidget = {
  const n=50; let s=5;
  const rand=()=>{s|=0;s=s+0x6D2B79F5|0;let t=Math.imul(s^s>>>15,1|s);t=t+Math.imul(t^t>>>7,61|t)^t;return((t^t>>>14)>>>0)/4294967296;};
  const nrm=()=>{const u=1-rand(),v=rand();return Math.sqrt(-2*Math.log(u))*Math.cos(2*Math.PI*v);};
  const xs=[],ys=[];
  for(let i=0;i<n;i++){const x=rand()*10; xs.push(x);
    let y;
    if(problem==="ok") y=2+1.5*x+2*nrm();
    else if(problem==="hetero") y=2+1.5*x+(0.3+0.55*x)*nrm();
    else if(problem==="nonlin") y=2+0.22*x*x+2*nrm();
    else y=2+1.5*x+2*nrm();
    ys.push(y);
  }
  if(problem==="outlier"){ xs[5]=9.3; ys[5]=6; }   // high-leverage, off-trend point
  const mx=xs.reduce((a,b)=>a+b,0)/n, my=ys.reduce((a,b)=>a+b,0)/n;
  let Sxy=0,Sxx=0; for(let i=0;i<n;i++){Sxy+=(xs[i]-mx)*(ys[i]-my);Sxx+=(xs[i]-mx)**2;}
  const b1=Sxy/Sxx, b0=my-b1*mx;
  const fit=xs.map(x=>b0+b1*x), res=ys.map((y,i)=>y-fit[i]);

  const W=470,H=250, midX=W/2;
  const pL=30, pR=12, pT=16, pB=26;
  // left: data + line
  const xmin=0,xmax=10, ymin=Math.min(...ys)-2, ymax=Math.max(...ys)+2;
  const LX=x=>pL+(x-xmin)/(xmax-xmin)*(midX-pL-pR), LY=y=>H-pB-(y-ymin)/(ymax-ymin)*(H-pT-pB);
  let dotsL=""; xs.forEach((x,i)=>dotsL+=`<circle cx="${LX(x)}" cy="${LY(ys[i])}" r="3" fill="rgba(31,78,121,0.7)"/>`);
  const lineL=`<line x1="${LX(xmin)}" y1="${LY(b0+b1*xmin)}" x2="${LX(xmax)}" y2="${LY(b0+b1*xmax)}" stroke="#1F4E79" stroke-width="2"/>`;
  // right: residuals vs fitted
  const fmin=Math.min(...fit), fmax=Math.max(...fit), rmax=Math.max(...res.map(Math.abs))*1.1;
  const RX=f=>midX+pL+(f-fmin)/(fmax-fmin)*(midX-pL-pR), RY=r=>pT+(H-pT-pB)/2 - r/rmax*((H-pT-pB)/2);
  let dotsR=""; fit.forEach((f,i)=>dotsR+=`<circle cx="${RX(f)}" cy="${RY(res[i])}" r="3" fill="rgba(192,80,30,0.7)"/>`);
  const zero=`<line x1="${midX+pL}" y1="${RY(0)}" x2="${W-pR}" y2="${RY(0)}" stroke="#94a3b8" stroke-dasharray="4 3"/>`;
  const labs=`<text x="${(pL+midX)/2}" y="12" text-anchor="middle" font-size="11" fill="#1F4E79">data + fitted line</text>`+
    `<text x="${(midX+W)/2}" y="12" text-anchor="middle" font-size="11" fill="#C0501E">residuals vs fitted</text>`;
  const caps={
    ok:"An even, patternless band around zero — assumptions look fine.",
    hetero:"The residuals fan out — spread grows with fitted value. That's heteroscedasticity (non-constant variance); the NCV test flags it. Fix: transform y (e.g. log) or use robust SEs.",
    nonlin:"The residuals bow into a U — a straight line can't capture a curved relationship. That's a linearity violation; add a quadratic term or transform.",
    outlier:"One point sits far from the rest with high leverage — it drags the line toward itself. Check Residuals vs Leverage / Cook's distance before trusting the fit."
  };
  const fig=document.createElement("figure"); fig.style.margin="0";
  fig.innerHTML=`<svg viewBox="0 0 ${W} ${H}" width="${Math.min(W,540)}" style="background:#f8fafc;border:1px solid #e2e8f0;border-radius:8px">`+
    `<line x1="${midX}" y1="${pT}" x2="${midX}" y2="${H-pB}" stroke="#e2e8f0"/>`+labs+lineL+dotsL+zero+dotsR+`</svg>`+
    `<figcaption style="margin-top:.5rem">${caps[problem]}</figcaption>`;
  return fig;
}

The trick is that each violation has a look. A good model shows a shapeless band. Heteroscedasticity fans out. Non‑linearity bends the residuals into a curve. An influential outlier sits alone, far from the pack, tugging the line. Learning to spot these by eye is most of the skill — the formal tests just confirm what the plot suggests.

See it in code

R

Python

import statsmodels.api as sm
import statsmodels.formula.api as smf
m = smf.ols("y ~ x", data=df).fit()
sm.qqplot(m.resid, line="s")                       # normality of residuals
sm.stats.diagnostic.het_breuschpagan(m.resid, m.model.exog)  # heteroscedasticity
# statsmodels.stats.outliers_influence.variance_inflation_factor(...) for VIF

MATLAB / Octave

m = fitlm(x, y);
plotDiagnostics(m, 'cooksdistance')   % influential points
plotResiduals(m, 'fitted')            % residuals vs fitted (spread / curvature)

The R tab runs live (including the four plot(m) diagnostics); Python and MATLAB are static references.

TipCheck your understanding

Your Residuals‑vs‑Fitted plot fans out — a narrow spread at low fitted values, a wide spread at high ones. Which assumption is broken, and what confirms it?

That fanning shape is heteroscedasticity — the constant‑variance (homoscedasticity) assumption is violated; the residuals’ spread depends on the fitted value. The NCV (non‑constant variance / Breusch–Pagan) test confirms it: p < .05 means non‑constant variance. Common fixes: transform the outcome (e.g. log price), or use heteroscedasticity‑robust standard errors.

The four plot(model) panels each target a different assumption — don’t read them all as “normality.” A curved Residuals vs Fitted signals non‑linearity (add a term or transform); a fanning one signals heteroscedasticity; the Normal Q‑Q plot checks residual normality; and a point far to the right on Residuals vs Leverage is influential (check Cook’s distance). Real datasets like London housing rarely pass everything cleanly — deciding what matters is judgment, not a pass/fail gate.

Where this shows up next

Next: choosing which predictors belong in the model, and evaluating overall fit. See Chapter (Correlation & Regression) for the full London‑housing diagnostics walkthrough.

Related topics

• Multiple Regression — the models these assumptions apply to.
• When Data Aren’t Normal — transformations for skew and non‑normal residuals.
• Why Reduce Dimensions? — multicollinearity and how PCA sidesteps it.
• Choosing Variables (over- vs under-fitting) — which predictors to keep.
• Evaluating Models — judging fit once the assumptions check out.