As a student studying randomization methods, I am struggling with the discretization of continuous stratification factors (e.g., age, biomarker levels) in stratified blocked randomization.

My current understanding is that stratification aims to control for important covariates (e.g., age, biomarkers) to ensure comparability between treatment groups. Thus, the choice of cut-off values for continuous covariates should ideally preserve inter-stratum heterogeneity in outcomes. I'm not sure if my understanding is correct. And I am unsure if percentile-based splits (e.g., median/quartile divisions) are theoretically valid, as they focus on balance of sample size instead of true prognostic differences.