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I have a fitted linear regression of a Box-Cox transformed dependent variable, using an indicator variable as one of the two predictors :

$$ g(Q, \lambda) = \hat{\beta_0} + \hat{\beta_1}P+ \hat{\beta_2}I +\epsilon = f(P,I) +\epsilon $$

This generates two parallel lines of slope $\beta_1$ and intercepts $\beta_0$, when $I=0$, and $\beta_0+\beta_2$, when $I=1$. $g(Q, \lambda)$ would be the Box-Cox transformation of $Q$ by parameter $\lambda$

I am interested in estimating how $Q$ (untransformed) changes, given a value of $P = \bar{P_0}$ when $I$ changes from $0$ to $1$. Note that $\bar{P_0}$ is actually the average of the values of $P$ where $I=0$, eventually I need to do the same for $\bar{P_1}$, i.e. the average of $P$ when $I=1$, but I guess that's just repeating the same procedure.

Since, I need $Q$ in the untransformed space, It is my understanding that I need to correct for bias in the back transformation process, for this I am fiddling with Duan's smearing, using the following formula to find $Q$ as a function of $I$:

$$ Q(I) = \frac{1}{n}\sum_{i=1}^nh(f(\bar{P_0},I) + \hat{\epsilon}_i, \lambda) $$

where $f$ is the linear regression model above, $h = g^{-1}$ is the inverse of the Box-Cox transformation and $\hat{\epsilon}_i$ are the residuals of the fitted linear regression.

Hoping to be using the right approach, there are a few things that I am finding unclear:

  1. The domain of the inverse transformation of Box-Cox is not the entire number line. in my formula above, it requires $f(P_0,I) + \hat{\epsilon}_i > \frac{-1}{\lambda}$. I have a few points where this condition doesn't hold, and my reverse transformation is impossible. This doesn't seem to be an issue that is considered in the (admittedly sparse) literature I have found on the topic (e.g. this article), so I'm not sure how to deal with it. At the moment I am substituting the entire $h(\cdot, \lambda)$ term with $0$ in these cases because logically that makes the most sense to me given the data ($Q$ and $P$ are volumes, so cannot be negative) and the shape of the Box-Cox transformation. Is there a better approach?

  2. How should I deal with the indicator variable when taking the $\hat{\epsilon}_i$ to add to the sum? Should I use all the residuals regardless of the value of $I$ that I am testing for, or only the residuals of the data points with $I$ matching the one I am testing at the specific moment? I guess that mostly means asking if I should consider the two lines that come out of the regression as two separate models or as a single one. I can see benefits and pitfalls in both approaches and admittedly the results I get are not that different. Still would be good to understand what the best practice is.

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