MATH5806 Chap.5 Residual Diagnostics and Assumption Checks

Residual Diagnostics and Assumption Checks

Residual Diagnostics and Assumption Checks is a quantitative decision problem built from residual patterns, variance structure and influence and model revision. The aim is to use a diagnostic pattern to identify which assumption or observation needs investigation; a numerical result earns meaning only when the variables, units, assumptions and comparison are all explicit.

Begin with residual patterns.

State what quantity it represents, the scale on which it is measured and the condition under which it changes. Writing those details before substituting numbers prevents a familiar-looking formula from being used on the wrong object.

Next connect variance structure to the calculation. Show the transformation line by line, preserve units and signs, and make any denominator or baseline visible.

A calculator output is not a method; the reader must be able to reconstruct why that operation answers the question.

Use influence and model revision to interpret or stress-test the result. Ask whether the magnitude is plausible, whether a boundary case behaves as expected and which conclusion would reverse if an assumption changed.

This is where computation becomes analysis rather than arithmetic.

When the task is to use a diagnostic pattern to identify which assumption or observation needs investigation, separate inputs supplied by the problem from quantities you derive.

Then report the result in the language of the course and attach the relevant uncertainty, limitation or decision consequence.

Build a representation check before solving Residual Diagnostics and Assumption Checks.

Put residual patterns, variance structure and influence and model revision into a small symbol-and-units table, mark which values are observed and which are calculated, and predict the direction of the result before doing arithmetic. A sign, scale or unit mismatch then becomes visible at the setup stage instead of being hidden inside a polished final number.

Run one sensitivity test after the baseline answer.

Change the input most closely connected to variance structure, hold the remaining assumptions fixed and recompute only the affected steps. Explain whether the movement in influence and model revision matches the mechanism.

This shows which assumption controls the conclusion and prevents a single scenario from being presented as a universal result.

Use a three-column error log for MATH5806: translation error, calculation error and interpretation error. Record the exact line where the Residual Diagnostics and Assumption Checks solution first diverged, rewrite that line, and check it with a limiting case or an independent calculation.

Correcting the first failed move is more useful than copying the complete solution again.

A complete Residual Diagnostics and Assumption Checks response should make the task visible before the detail: identify what must be decided, define the relevant terms, connect the evidence to variance structure, and use influence and model revision to test the result.

The final sentence should answer the question actually asked rather than merely repeat the topic.

The controlling limit is specific: A clean diagnostic plot cannot prove the model is true, and one unusual point should not be deleted automatically.

Keep that limit beside the worked example, because it separates a careful MATH5806 answer from one that sounds confident but claims more than the task or evidence supports.

For revision, retrieve residual patterns, variance structure and influence and model revision without notes, explain their relationship aloud, then complete a changed version of the application: use a diagnostic pattern to identify which assumption or observation needs investigation.

Record the first point at which your reasoning fails and repair that move before attempting another case.

What this chapter covers

• 01

  residual patterns

• 02

  variance structure

• 03

  influence and model revision

• 04

  Applying residual patterns

• 05

  Limits of variance structure and influence and model revision

Worked example: Residual Diagnostics and Assumption Checks

Key terms

standardized vs studentized residuals

A standardised residual divides by a common estimated error scale adjusted for leverage, while a studentised deleted residual uses an error variance estimated with that observation omitted. In this chapter, use the concept when you use a diagnostic pattern to identify which assumption or observation needs investigation.

Gauss-Markov assumptions

The Gauss–Markov conditions require a linear-in-parameters model with full-rank regressors and errors having zero conditional mean, constant variance and no cross-observation covariance; then OLS is BLUE. In this chapter, use the concept when you use a diagnostic pattern to identify which assumption or observation needs investigation.

hat matrix, leverage h_ii and Cook's distance

The hat matrix H = X(X'X)⁻¹X' maps observed responses to fitted values, h_ii measures observation leverage, and Cook's distance measures how strongly deleting an observation changes the fitted model. In this chapter, use the concept when you use a diagnostic pattern to identify which assumption or observation needs investigation.

Residual Diagnostics and Assumption Checks FAQ

Exam move

Reconstruct the relationship among residual patterns, variance structure and influence and model revision; complete the chapter application without notes; then test the result against this limit: A clean diagnostic plot cannot prove the model is true, and one unusual point should not be deleted automatically.

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