For what reasons should regression models not adjust for variables comprising a composite score?

Question

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?

Answer 1

Forgive me for answering my own question, but this sparked some conversation on twitter (er, perhaps the site formerly known as twitter) with one of the authors from the paper linked by Robert Long.

You can read his answer in the linked thread, but I will distill it here for completeness. He writes:

Your problem is that a deterministic node (e.g. BMI) is technically barren. It cannot have a causal effect independent of its parents (e.g. weight and height) because it contains no additional information...

When interested in a deterministic variable, you hence have chose whether to focus on the causal effect of the parents (e.g. weight & height) or the weighted average effect implied by the child (e.g. BMI), acknowledging that the latter still technically acts through the parents.

This makes sense to me. One should either estimate the effects of the constituent variables on the outcome or the implied weighted average of effects of the composite score, but not both.

With regards to confounding, one needs to be careful not to introduce time varying confounding. The paper linked by Robert provides very interesting examples of this in figure 5.

Answer 2

Bayesian modeling opens up a vast array of possibilities. Bayesian models can more precisely reflect the data generating mechanism and prior knowledge. For example, when adjusting for a supposedly linear effect of log BMI but not wanting to fully trust that log height and low weight are optimally captured in log BMI, notice that BMI assumes the ratio of the coefficients of log weight and log height are -2. Put a prior on this ratio that is centered at -2 but allows for the possibility of other ratios. In this way you’ll have excellent adjustment for height in weight while not spending a full 2 parameters on them, thus reducing mean squared errors of estimates.

Answer 3

I'm not sure if this answers what you are asking, but leaving aside DAG and causality, if you include e.g weight, height, and BMI, you will have very high collinearity. I made up some data in R:

library(olsrr)

set.seed(1234) # Set a seed


ht <- rnorm(100, 1.6, .10)
wt <- ht*50 + rnorm(100, 0, 10)

bmi <- wt/(ht^2)


depvar <- bmi*10 + rnorm(100,0, 20)

where height and weight seem to be in at least the right ballpark. Then I made a model and checked for collinearity:

model1 <- lm(depvar~bmi+ht+wt)
ols_eigen_cindex(model1)

yielding:

    Eigenvalue Condition Index    intercept          bmi           ht           wt
    1 3.973092e+00         1.00000 3.617950e-06 0.0000132548 3.563885e-06 1.299638e-05
    2 1.871737e-02        14.56941 4.146123e-04 0.0052085840 1.005284e-03 7.847182e-04
    3 8.167689e-03        22.05539 1.566721e-03 0.0047716438 1.463788e-04 1.443944e-02
    4 2.247451e-05       420.45475 9.980150e-01 0.9900065174 9.988448e-01 9.847628e-01
    > 

Not surprisingly, the highest condition index was huge, with 98 or 99 percent of variance shared among all variables.

(As an aside question, does anyone know why R printed some of the numbers in scientific notation and some not?)