Analysis of Covariance: Measurement Errors in Covariates

In his excellent book Statistical Issues in Drug Development (2021), Professor Stephen Senn provides a comprehensive overview of statistical issues related to covariate adjustment. Many of these issues highlight widespread misconceptions, particularly regarding analysis of covariance (ANCOVA) in randomized clinical trials (RCTs). I will explore some of the key issues in a series of posts from my own perspective. The covariates under discussion will be restricted to continuous variables measured before randomized treatment.

Some statisticians believe that statistical inferences obtained using ANCOVA may be biased or invalid

if covariates are measured with errors. Such a perspective, however, overlooks an important fact that covariates are random variables before they are observed and, therefore, measurement errors are always an inherent component of any observed covariates. Contrary to this misconception, ANCOVA is valid precisely because of the presence of measurement errors in the covariates.

Suppose the response Υ and covariate Ⅹ are joint normal random variables with means 𝜇Υ and 𝜇Ⅹ, variances 𝜎Υ² and 𝜎Ⅹ², and a correlation ρ. It is well known that the distribution of Υ given Ⅹ = 𝑥, denoted by Υ|Ⅹ = 𝑥, is normally distributed with mean

and variance

The conditional distribution of Υ|Ⅹ = 𝑥, as determined by Equations (1) and (2), can be interpreted as an ANCOVA model corresponding to the analysis of variance (ANOVA) model represented by the marginal distribution of Υ. Equation (1) implies that the ANCOVA model remains unbiased if the ANOVA model is unbiased. Equation (2) indicates that ANCOVA is generally more efficient than ANOVA when ρ ≠ 0, due to the reduced error variance. Note that the ANCOVA model uses the observed covariate Ⅹ = 𝑥, which is expected to be measured with some random error. The measurement errors in Ⅹ also play an important role (through the variance 𝜎Ⅹ²) in the covariate adjustment in Equation (1). However, if the covariate were measured without any error (a scenario with a probability of zero), meaning 𝑥 = 𝜇Ⅹ, then Equation (1) suggests that no adjustment would be necessary.

Finally, I'd like to conclude with a statement from Professor Senn's book: "Analysis of covariance using observed baselines is not only adequate but better than attempts to improve it."

References

Senn, S. (2021). Statistical Issues in Drug Development (3rd Edition), Statistics in Practice. John Wiley and Sons, Ltd.