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Assume we have the following model: $y = \beta_0 + \alpha_1 * x_1 ^{\beta_1} + \alpha_2 * x_2^{\beta_2} + \alpha_3 * x_1^{\beta_1} * x_2^{\beta_2}$ where as we have the following priors for our IV's $\beta_1 \sim Beta(1, 1), \beta_2 \sim Beta(1, 1)$ thus $0 <= \beta <= 1$

I wonder how I can deal with the multicollinearity issues of the interaction term, notice that we cannot mean center $x_1, x_2$ in our interaction-term due to the exponent(we cannot have $x < 0$ since $\beta <= 1$ and $\beta$ is not an integer), does anyone have some ideas?

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  • $\begingroup$ Perhaps I am not reading your notation correctly, but your beta coefficients here are power terms? As in for each $x$, it gets raised to the power of the beta weight? Perhaps I am naive but I have never seen linear regressions defined this way, only power terms for $x$ (e.g. $\beta_1 x_1 + \beta_2 x_1^2)$. $\endgroup$ Commented Nov 16, 2023 at 8:48
  • $\begingroup$ yes they are powerterms aiming to capture diminishing marginal returns hence $0 <= \beta <= 1$ $\endgroup$ Commented Nov 16, 2023 at 11:54
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    $\begingroup$ Welcome to CV, Richard. Your remark about mean centering is puzzling (because that procedure is equivalent to including the intercept term $\beta_0,$ which is explicitly present) and it's not clear there will be much multicollinearity. But even if there is, so what? What problems do you anticipate it might create and what specifically do you mean by "dealing" with it? Also, consider re-expressing your model as $E[y]=(\beta_0-1)+(1+x_1^{\beta_1})(1+x_2^{\beta_2})$ for further analysis, because it reveals the underlying simplicity. $\endgroup$
    – whuber
    Commented Nov 16, 2023 at 14:03
  • $\begingroup$ @whuber edited the question a bit to make it clear that mean-centering refers to the interaction terms, thus mean-centering $x_1, x_2$ solely in the interaction-term, not the main effects $\beta_1*x_1 ,\beta_2*x_2$. I also edited the actual equation since i missed out on the coefficients $\alpha$ $\endgroup$ Commented Nov 16, 2023 at 15:34
  • $\begingroup$ The point about mean centering remains obscure. Your introduction of the $\alpha_i$ introduces another twist: what priors are you assuming for them? And, again, why are you concerned about multicollinearity in the first place? $\endgroup$
    – whuber
    Commented Nov 16, 2023 at 16:40

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Multicollinearity gets a lot of undue attention. As the Wikipedia entry states:

Multicollinearity does not affect the accuracy of the model or its predictions. It is a numerical problem, not a statistical one.

There is nevertheless a subsequent inferential problem, seen in your scenario with only two predictor variables. If $x_1$ and $x_2$ are correlated, then you won't be able to distinguish their separate associations with outcome very cleanly. But that will be the same problem whether or not $x_1$ and $x_2$ are mean-centered. See this answer, for example, with respect to regression that is linear in the parameters.

From what you describe, mean centering isn't possible in your situation with its nonlinearity in the parameters, anyway. So leave the predictors alone, and recognize the limitations of working with correlated predictors.

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