Conditions for Regression Inference

You’ll be able to

State the conditions (LINER) required for inference about a slope
Use residual plots to check the conditions
Explain the consequence of a violated condition

The LINER conditions

Regression inference requires five conditions, remembered as LINER: Linear (the true relationship is linear — check that the residual plot has no curved pattern), Independent observations (the 10% condition when sampling without replacement), Normal (the response varies Normally around the line at each x — check a graph of the residuals for strong skew/outliers), Equal variance (the scatter of residuals is roughly constant across x — no fanning), and Random (data from a random sample or randomized experiment).

Reading the residual plot

The residual plot is the workhorse for checking conditions. No curved pattern supports Linearity. Roughly constant vertical spread (no funnel or fan shape) supports Equal variance. A residual plot that curves signals a nonlinear relationship, and one that fans out signals non-constant variance — in either case the slope inference is not trustworthy.

LINER checklist

L linear · I independent · N Normal · E equal variance · R random

Linearity and Equal variance are judged from the residual plot; Normality from a graph of the residuals; Independence from the 10% condition; Random from the study design.

Worked example

A student fits a regression line and the residual plot shows a clear U-shaped (curved) pattern. Which LINER condition is violated, and what does it mean for the inference?

1.A residual plot should show random scatter if the relationship is truly linear.
2.A U-shaped (curved) residual plot means the relationship is not linear — the Linear condition fails.
3.Because a core condition is violated, the slope inference (interval and test) is not valid.

Answer: The curved residual plot violates the Linear condition, indicating the true relationship is nonlinear. When the Linear condition fails, the confidence interval and significance test for the slope are not trustworthy, and a transformation or nonlinear model should be considered.

Checkpoint

A residual plot for a regression shows the points fanning out — the vertical spread of residuals increases as x increases. Which condition is violated?

On the exam

On free response, name each condition and cite the specific evidence you used: "the residual plot shows no leftover curve (Linear) and roughly constant spread (Equal variance)." Vague statements like "the conditions are met" without evidence do not earn the point.

Checkpoint

Which graphical tool is most useful for checking the Linear and Equal-variance conditions for regression inference?

Answer the 2 checkpoints as you read.