So I have a general multiple linear regression model as the following:

$$ y_i = \alpha_0 + \alpha_1z_i + \alpha_3w_i + \alpha_{13}z_iw_i + \epsilon_i,\quad \epsilon_i\sim N(0,\tau^2). $$

where $z_i =\left\{\begin{matrix} 0\text{ if subject $i$ is young} \\ 1\text{ if subject $i$ is old}\end{matrix}\right. $ and $w_i = \left\{\begin{matrix} 0\text{ if subject $i$ healthy} \\ 1\text{ if subject $i$ diseased}\end{matrix}\right.$ and $\alpha_j's$ are just coefficients.

My question is asking to consider imposing the belief: The difference in $E(y)$ between healthy and diseased groups is the same for young and old subjects.

My attempt was to consider the following indicators:

$$ x_i = \left\{\begin{matrix} 1\text{ if subject $i$ is healthy and diseased } \\ 0 \text{ otherwise}\end{matrix}\right. $$


$$ z_i = \left\{\begin{matrix} 1\text{ if subject $i$ is young and old} \\ 0\text{ otherwise}\end{matrix}\right. $$

The possible model I could have for this:

$$ y_i = \beta_0 + \alpha\times(x_i + z_i) + \epsilon_i, \quad \epsilon_i \sim N(0, \sigma^2) $$

Is this a correct way to do or any other possible ways to do this?


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