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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.

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A couple of points:

  • If you work with the differences $y_{1i} - y_{2i}$ you have univariate data, i.e., a single difference per subject. Hence, you would not need to work with a mixed model unless I missed something.
  • A potential benefit of working with a mixed model approach is if you have missing data for some subjects (i.e., you either have $y_{1i}$ or $y_{2i}$ but not both). In this case, you cannot compute differences for these subjects, but you could use all available data in a mixed model approach. For example, $$y_{mi} = \beta_0 + \beta_1 \texttt{Method}_{mi} + \beta_2 \texttt{Age}_i + \beta_3 \texttt{Sex}_i + b_i + \varepsilon_{mi},$$ where $\texttt{Method}_{mi}$ is zero for Method 2 and one for Method 1, and $b_i$ is a random intercept for the subjects with mean zero and variance $\sigma_b^2$. The coefficient $\beta_1$ will denote the difference in the expected outcome $y_i$ between the two methods controlled for age and sex.
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