ANCOVA

What is ANCOVA?

ANCOVA is a one-way ANOVA augmented with a continuous covariate. It removes the covariate's predictable component from the response, then tests the group effect on the residuals — effectively a between-group test "at the same covariate level". Power gains over plain ANOVA can be large when the covariate is well-correlated with the outcome.

The critical assumption is homogeneity of regression slopes — the covariate's effect must be the same across groups. If slopes differ (test by including the interaction), the model is mis-specified and ANCOVA's group test isn't meaningful; move to a full interaction model and report simple slopes.

When should I use ANCOVA?

• Comparing groups while controlling for a baseline or pre-test measurement.
• Reducing residual variance to gain power when a covariate explains a lot of the outcome.
• Not a substitute for randomisation — in observational data ANCOVA controls for measured covariates only.

What data does it need?

Numeric response + categorical group + one or more numeric covariates.

What does it report?

Term table with F / p / partial η² / ω², adjusted group means (least-squares means) at the covariate mean.

What does it assume?

• Linear covariate–response relationship within each group.
• Equal slopes across groups (homogeneity of regression).
• Approximately normal residuals.
• Equal residual variance across groups.

How do I interpret the result?

Test the group × covariate interaction first. Significant ⇒ slopes differ ⇒ ANCOVA assumption violated ⇒ switch to the interaction model.

Adjusted (LS) means are what you should report alongside the group effect — they answer 'what would the group means be if all groups had the same covariate distribution'.

See also

• ANOVA (one-way)
• Linear regression
• Mixed-effects model (LMM / GLMM)