MZB127 Chap.11 Linear Regression and Model Inference
Linear Regression and Model Inference
Simple linear regression fits an intercept and slope to paired numerical data by minimising squared vertical residuals. Predictions lie on the fitted line, while residuals measure observed minus fitted response.
This chapter interprets coefficients with units, distinguishes interpolation from extrapolation, tests population slopes and uses scatterplots plus residual patterns to decide whether a straight-line summary is adequate. The predictor and response have different roles, and reversing them defines a different fitting problem.
A slope describes estimated change in mean response for one predictor unit, while an intercept describes the fitted mean response at predictor zero only when that value is meaningful. Each residual is observed response minus fitted response, so points above the line have positive residuals and fitted plus residual reconstructs observed.
Predictions inside the observed predictor range are interpolations; predictions beyond it depend on an unobserved continuation of the linear pattern. Inference about the population slope uses its own standard error and n minus two degrees of freedom. A small p-value can support linear association, but neither statistical significance nor a fitted line establishes causation.
Scatterplots, residual plots, uncertainty and study design must agree before a regression claim is trusted.
What this chapter covers
- 01
Scatterplots and linear form
- 02
Least-squares slope and intercept
- 03
Fitted values and residuals
- 04
Slope and intercept inference
- 05
Residual diagnostics and extrapolation
Worked example · free
Compute a fitted value and residual
- 1Insert x into the fitted equation.
- 1Calculate y hat=7.5.
- 1Compute observed minus fitted to obtain 0.9.
- 1Interpret the positive vertical discrepancy.
Key terms
- Fitted value
- The response predicted by the estimated regression line at a stated predictor value.
- Residual
- The observed response minus its fitted response.
- Least squares
- The fitting rule that minimises the sum of squared vertical residuals.
- Interpolation
- Prediction within the observed predictor range.
- Extrapolation
- Prediction beyond the observed predictor range, where the linear pattern is unverified.
Linear Regression and Model Inference FAQ
How should a regression slope be stated?
Name both variables and units: a one-unit increase in the predictor is associated with the estimated slope change in the mean response under the linear model. Avoid automatic causal language.
What does a positive residual mean?
The observed response is above the fitted line because observed minus fitted is positive. The residual uses the response variable's units and belongs to one data point.
Why inspect residuals after fitting?
Systematic curvature, changing spread or influential points can show that a straight line or constant-variance assumption is inadequate. Coefficients alone cannot reveal all of those failures.
How can a residual sign be checked?
Locate the observed point relative to the fitted line. A point above the line must have a positive observed-minus-fitted residual, while a point below must have a negative one. Adding the residual back to the fitted value should reconstruct the observed response exactly and preserve response units.
Exam move
Begin with the scatterplot and observed predictor range. Write the fitted equation with coefficient units, then calculate predictions and residuals separately. For slope inference, use the slope estimate, its standard error and n minus two degrees of freedom. Link the p-value and confidence interval to the same population slope claim.
Finish by checking residual patterns and stating whether a prediction is interpolation or extrapolation. Start every exercise by naming x as predictor and y as response, recording both units and marking the observed x-range. Interpret the slope as response units per predictor unit and treat the intercept cautiously if zero is outside that range.
For numerical practice, substitute x into the fitted equation, place the observed y beneath the prediction and calculate observed minus fitted. Check the sign against whether the point lies above or below the line, then add residual back to fitted value to reconstruct the observation. Label each requested prediction as interpolation or extrapolation before calculating.
For inference, write the population slope hypothesis, coefficient estimate, coefficient standard error and n minus two degrees of freedom together. Interpret a confidence interval and p-value as statements about linear association under the model, not about causal effect or guaranteed predictive accuracy.
Train diagnostic reading with sketches showing curvature, fan-shaped spread, clusters and influential points; describe what each pattern challenges. Compare the scatterplot and residual plot rather than relying on a single summary such as correlation. Finish a timed response with four scope checks: variables and units named, predictor range respected, association separated from causation and any visible model concern reported.
These qualifications are part of the mathematical conclusion.
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