I have two independent variables $y_{mi},z_{mi}$ ($z$ is measured in fasting, so it is the basal state), measured with two different methods $m$ (m=2 is the reference method) in the same subjects $i$, with two predictors (age and sex). One regression model is needed for each variable, and the question is how age, sex, and first-method value influence the differences between methods.
What approach would be recommendable?
A linear model (lm) in which $y_{1i}-y_{2i}$ is the independent variable, and age, sex and $y_{1i}$ is a predictor. The caveats are: A mixed model is recommended for paired-measures data; the residuals~fitted plot reveals a small amount of homocedasticity (a linear model of (abs(residuals)~fitted.values has a slope of 0.1). A linear model of $residuals = f(age+sex+y_{1i})$ reveals no association between the residuals and the predictors (Gauss Markov requirement).
A robust regression (ltsreg/lmrob) with the same model.
A mixed model (lme4) in which $y_{1i}-y_{2i}$ is the independent variable; age, sex and $y_{1i}$ are fixed effects and $y_{1i}$ is a random effect. The obtained coefficients are identical to those in the linear model! (I included $y_{1i}$ as a fixed effect because it's the most relevant predictor and we need its coefficient).
The results/coefficients obtained with the different methods are roughly equivalent.
My questions are:
Would using the differences as dependent variable and one of the values as a predictor confront the Gauss-Markov theorem? Is this worrying, considering all the models give roughly similar estimates?
I think I'm going to choose the easily-understandable linear model as the "solution" to this problem. Any suggestion/criticism?
Help is appreciated, thank you.