Medicine has many examples where some score is derived from other clinical measures. Examples include BMI (which is a function of weight and height) and MELD (which appears to be based on blood tests).
Often, researchers will want to describe the relationship between these scores and some clinical outcome. Regression is the main tool to do this, and I often see studies adjust for these scores and their constituent variables.
This seems wrong to me. On one hand, the interpretation of the regression coefficient as the expected change in the outcome per unit change in the covariate holding all others constant doesn't make sense (e.g. how can BMI change by one unit when height and weight remain constant). In the extreme case where the composite variable is a linear combination of the constituents, the design matrix is rank defficient and no unique maximum likelihood estimate exists.
On the other, I suspect this introduces confounding into the estimates but I can't demonstrate this in a DAG. Using height, weight, and BMI as an example, one possible DAG might look like this.
Here, conditioning on BMI, height, and weight closes all back doors. This remains true if BMI has a direct effect on the outcome.
Is the interpretation of the regression coefficient the only objection to adjusting for constituent variables and some composite score? Are there confounding issues as well? If so, under what assumptions?