Covariate

A covariate is an observed explanatory variable used to model the conditional distribution of a response. A vector of covariates can enter a linear predictor, a treatment-effect adjustment or a hazard multiplier. Conditioning on a covariate does not itself establish that its effect is causal.

A baseline covariate is measured before treatment assignment or the exposure being studied. A prognostic baseline covariate predicts the outcome. In a randomized controlled trial, adjusting for such a covariate can improve statistical efficiency without conditioning on a consequence of treatment. For instance, a measurement taken before treatment can explain persistent between-subject variation in a later measurement.^[ref]

Here is a linear regression example with explicit noise assumptions. Letwhere treatment $Z$ is independent of the baseline covariate $X$, all displayed variances are finite, $\mathbb{E}[\epsilon \mid X,Z]=0$ and $\mathrm{Var}(\epsilon \mid X,Z)={\sigma }^2$. ThenThe first identity uses the law of total variance. It isolates the unexplained population variation removed by adjustment for $X$. For $\gamma =2$, $\mathrm{Var}(X)=1$ and ${\sigma }^2=1$, this variation falls from $5$ to $1$. Estimating additional coefficients still uses data, so this population calculation is not a guarantee that every finite-sample reported standard error becomes smaller. Measurement before treatment also does not by itself make a covariate prognostic or establish a causal effect.