# Does it make sense to impute missing covariate data when the imputed value is a function of other covariates in the regression model?

We are building a model that adjusts for standard covariates (e.g., age, gender) and for the outcome at baseline. It would be ideal to adjust for each subject's baseline value like so: $$Y = \beta_0 + \beta_1*{\rm age} + \beta_2*{\rm gender} + \beta_3*{\rm baseline}$$ However, we don't have baseline measures for everyone. There is a standard clinical estimation of $Y$ (current and baseline) as a nonlinear function of age and gender and also a measure $X$ (which we do have on everyone), i.e., $\text{Estimated Baseline} = f(X,\ {\rm age},\ {\rm gender})$.

Does it make more sense to model $Y$ by including the raw value $X$ or the function of $X$? For example:

\begin{align} Y &= \beta_0 + \beta_1*{\rm age} + \beta_2*{\rm gender} + \beta_3*X \\ Y &= \beta_0 + \beta_1*{\rm age} + \beta_2*{\rm gender} + \beta_3*f(X,\ {\rm age},\ {\rm gender}) \end{align}

The estimated function is not a great estimate, but it is closer to the baseline measure.

• By "... and it is also a (nonlinear) function of age and gender...", I assume you mean that the best estimate of the estimated baseline value is a function of age & gender, not that X is a function of age & gender. Is that right? Feb 25, 2014 at 20:25
• Correct -- X is a different measure that is associated to our outcome, and cheaper to collect. Previous studies have published an estimating equation of the form: $$est(Y) = (X^a)(age^b)(cmale + dfemale) = f(X,age,gender) \\$$ Since X is easy and cheap to collect, most people just use this. So we could adjust our model for someone's estimated baseline, but that's just a function of other variables already in the model. Feb 25, 2014 at 20:45
• I tweaked that phrasing; please make sure it still says what you want. I think this is a great question, I'm interested in seeing what people will say. It seems to me that it taps several issues, including missing data, imputation schemes, measurement error, multicollinearity, & the connection b/t the modeling decisions made & the nature of the question you are trying to ask. Feb 25, 2014 at 20:59
• Huh -- I didn't actually think of this as an imputation question, but I guess it is the same idea. Using the estimated baseline is like doing a single imputation, but based on an external dataset. Feb 25, 2014 at 21:33