# Favoring one regression line over another

Skip to the end to see the question, this is just background information.

In the industry I work in, it is often desirable to find any line that best models the data as a sort of worst-case scenario. This leads to some conflict in my analysis from one project to another as:

1. Project 1 has engineering defined slope, an inverse-square root slope, so we just find the intercept that minimizes RSS.
2. Project 2 has historically been a mixture of pure linear trends and logarithmic trends. - In working on Project 2, it has also been suggested to test an inverse-square root as well, even though this project does not follow the same engineering as Project 1.

The trends above were found in R using the following coding format:

Linear: lm(Y~X)

Logarithmic: lm(Y~log(X))

Inverse Square Root: lm(Y~sqrt(X)^-1)
OR
lm(Y~sqrt(X^-1))
OR
lm(Y~1/sqrt(X))

Note: These are all the same, obviously


## Question

My main question is focused on Project 2. Outside of engineering specified instructions like in the first, why should I favor/choose one over the other? Is it possible to lose information if a trend that was previously found with a linear trend is found to have an inverse square root that has a better AIC/BIC score than the linear?

Also, how do other factors help choose the line that best fits? Namely, one p-value is significantly stronger than another, or our ability to believe the trend is more true than the other, aka Power is higher for one than another.

I can not share example data for proprietary related reasons. I might try to generate a random example when I have a little more time.