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Presume we would like to model the dynamics of two related variables as a bi-variate Normal, while also accounting for the effect of other covariates (via regression).

E.g. I would like to model heights of various pairs of siblings, while also accounting for their gender. Then, let

  • $Y_1, Y_2$ - heights of two siblings (at an adult age, when they can't grow taller)
  • $x_1, x_2$ - their respective genders

I would like to both account for dependence between $Y_1$ and $Y_2$, while including $x$ as a regression covariate. Assuming normality for the time-being, there are two approaches of capturing that dependence I'm thinking about:

  1. Introducing appropriate covariance structure inside the bivariate normal framework: \begin{equation} \begin{pmatrix} Y_{1,i} \\ Y_{2,i} \end{pmatrix} \sim N(\begin{pmatrix} \alpha + \beta x_{1,i} \\ \alpha + \beta x_{2,i} \end{pmatrix}, \sigma^2 \begin{pmatrix} 1 & \rho \\ \rho & 1\end{pmatrix}), \ \ i =1,\dots, n , \end{equation} $i$ indicates $i^{th}$ pair of siblings, $\beta$ - gender effect coefficient (same for both siblings)

  2. Just explicitly include sibling's height as a covariate with a coefficient, without having to specify correlation parameter in the covariance structure (hence, no $\rho$), as follows: $$ \begin{pmatrix} Y_{1,i} \\ Y_{2,i} \end{pmatrix} \sim N(\begin{pmatrix} \alpha + \beta x_{1,i} + \gamma y_{2,i} \\ \alpha + \beta x_{2,i} + \gamma y_{1,i} \end{pmatrix}, \sigma^2 \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}), \ \ i =1,\dots, n,$$ where $y_{1,k}, y_{2,k}$ are the observed values of a "sibling's height" variable, which we explicitly use as fixed covariates (!! see Question #2 below !!) on the right-hand side.

Questions:

  1. Could someone please explain both theoretical and practical differences between these two attempts to capture bi-variate dependence, while including covariate info? Most importantly: how would my interpretations of covariate effect $\beta$ differ (e.g. how would dependence between $Y_1 \& Y_2$ play into interpretation of gender effect $\beta$ in approach #1)?

  2. Is approach # 2 even legitimate, given that I'm using inherently random responses $Y_1,Y_2$, as fixed covariates on the right-hand side? What issues/peculiarities does that aspect introduce theoretically?

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