ANCOVA (Analysis of Covariance)

What is an ANCOVA?

ANCOVA (Analysis of Covariance) extends ANOVA (Analysis of Variance) by statistically adjusting for one or more additional variables, called covariates. This allows group means to be compared at the same covariate value.

ANCOVA adjusts group comparisons by modeling the relationship between each covariate and the dependent variable (outcome).

In practice, ANCOVA combines an ANOVA with a regression.

ANCOVA Example

Imagine you want to compare three study methods (Groups A, B, and C) on exam performance. You also know that prior knowledge (e.g., a pre-test score) affects the exam results. Without controlling for that, Group B might simply have higher pre-test scores and therefore better results, making Method B appear more effective than it is.

In general, ANCOVA is useful when you want to compare groups while accounting for another variable that is related to the outcome.

One potential benefit of ANCOVA is greater statistical power when the covariate is strongly related to the outcome. By accounting for variation associated with the covariate, ANCOVA can reduce unexplained error variance.

ANOVA vs. ANCOVA

So what are the differences between ANOVA and ANCOVA?

• ANOVA compares group means of a dependent variable (e.g., exam score) without adjusting those comparisons for a continuous covariate.

• ANCOVA additionally adjusts for one or more covariates (e.g., pre-test score). By statistically accounting for these covariates, ANCOVA estimates adjusted group means. The adjusted means are the predicted group means at a common covariate value.

Assumptions of an ANCOVA

• The dependent variable is metric, the grouping variable is categorical, and each covariate is usually metric.
• The relationship between each covariate and the dependent variable is linear.
• The regression slopes are homogeneous: the covariate-outcome relationship is similar across groups, unless this interaction is explicitly modeled.
• Observations and residuals are independent.
• Residuals are approximately normally distributed within groups.
• The residual variance is approximately equal across groups and covariate values.

The covariate should be measured reliably and, in an experiment, should generally be measured before the treatment. Adjusting for a variable that was itself affected by the treatment can bias the result. The groups should also have sufficient overlap in their covariate values; otherwise, adjusted comparisons may rely on extrapolation. In observational data, ANCOVA does not by itself remove confounding or establish causality.

In short: ANCOVA compares adjusted group means after accounting for the modeled relationship between the covariate and the outcome.

You can also run an ANCOVA online with numiqo. Feel free to try out our ANCOVA Calculator.