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

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?

• (+1) (should be the correct link now, sorry about that!) Ex-colleagues of mine recently published this:Depicting deterministic variables within directed acyclic graphs (DAGs): An aid for identifying and interpreting causal effects involving tautological associations, compositional data, and composite variables where they discuss this and related issues in some depth. Commented Nov 26, 2023 at 14:58
• I'm not sure I follow your example. Conditioning on Height, BMI, and Weight there are no observed variables left other than the outcome, no? What effect are you trying to estimate? Commented Nov 27, 2023 at 11:36
• @Scriddie Typically, I will see investigators try to estimate the effect of the composite variable (here, BMI) while conditioning on all the other variables which comprise the composite variable (here, height and weight). It seems silly to do because BMI should have no effect on the outcome except through height and weight. Here, I'm wondering if there may be some sort of bias that results. The paper linked by Robert Long answers "Yes" in some detail. Commented Nov 27, 2023 at 15:49
• It seems like the question you are asking is whether it is ok to include highly correlated predictors in a regression model. DAG or no DAG it's not going to work very well. It looks like the referenced paper (along with Dr. Harrell's answer below) offers alternatives for choosing amongst the best predictors / adjusting for lack of capture of the constituent variables in the composites, but including BMI along with height and weight in the same regression model just seems [IMO] silly. Commented Nov 30, 2023 at 18:18

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.

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.

• (+1) Nothing to forgive :) Commented Nov 26, 2023 at 18:03
• To be pedantic, I don't think BMI can be taken to model "weighted average effect" of weight & height for any common definition of weighted average. You could just say 'combined', for example. Commented Nov 27, 2023 at 9:46
• @SakariCajanus I have no problem with pedancy, but those are the author's words and not my own. Commented Nov 27, 2023 at 15:50

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.

• Granted, Bayes is more flexible, but this doesn't really solve the interpretation problem in my opinion. Even if I had good priors, the interpretation of the coefficients is difficult because BMI only changes when height and weight change. How then does Bayes help me interpret the coefficient of BMI in the model? I think the approach, Bayesian or otherwise, is not justifiable for this reason. However, leaving this point aside, I'm more interested in the causal aspect. Commented Nov 26, 2023 at 16:38
• I beg to disagree. The Bayesian approach recognizes that BMI is a function of height and weight and thus does not need BMI to be dealt with in any way save using the structure of BMI as the model’s starting point. A Bayesian model would not consider BMI as a separate entity other than the prior placing some favoritism on the BMI’s assumption about how height and weight related to outcome. Commented Nov 26, 2023 at 17:06

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?)

• For better formatted output, first set options(scipen=5) and then do ols_eigen_cindex(model1) |> round(4) Commented Nov 26, 2023 at 17:23