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I have a model that is roughly: $y \approx \beta_0 + \beta_1^x$.

I'm trying to figure out if there is a transformation I can do on either y or x to make it linear.

The only thing I can think of is making $x' = ln(\beta_1^x)$ -- but that's not a transformation on x solely... eg (if I don't know $\beta_1$ ahead of time, I'm not sure how to proceed...

Thanks!

user1357015
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  • Are $y$ and $x$ centered? We need further details on the variables. – Firebug Apr 27 '17 at 21:29
  • @Firebug, does it matter. – user1357015 Apr 27 '17 at 21:29
  • Of course it does. – Firebug Apr 27 '17 at 21:30
  • @Firebug, can you provide more information? – user1357015 Apr 27 '17 at 21:30
  • [What is the relationship between the function $\mathbb{E}(Y \mid X = x)$ and linear regression?](https://stats.stackexchange.com/questions/221977/what-is-the-relationship-between-the-function-mathbbey-mid-x-x-and-lin?rq=1) – Firebug Apr 27 '17 at 21:31
  • That should just be $X\beta$ where $\beta$ is the vector of covariates, etc. – user1357015 Apr 27 '17 at 21:32
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    It's a mathematical fact, relatively straightforward to prove, that there exists no differentiable monotonic transformation of $y$ that will linearize this formula, regardless of how $x$ is transformed. Therefore any hope will rest on finding an *approximately* linearizing transform for the specific data you have. But why do you need to linearize? That's going to alter the implicit probability model for the errors and, besides, this is an easy model to fit: it's identical to $y = \beta_0 + \exp(\kappa x)$, an (unscaled!) exponential plus a constant. – whuber Apr 27 '17 at 22:14
  • this question might need the self-study tag – Taylor Apr 27 '17 at 22:52
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    @whuber: that was my suspicion as I couldn't find one-- but I was hoping to see a proof. The request came from a a non-stats colleague for his drug toxicity model. I – user1357015 Apr 27 '17 at 23:05
  • A somewhat loose argument goes - the model is presently linear in $\beta_0$ but not in $\beta_1$. If you applied some nonlinear transformation (which is your only hope of changing the situation with $\beta_1$, it would no longer linear in $\beta_0$. However, it's very simple in form and should give little trouble to estimate. What's more important in modelling this is what is understood to be the situation with the noise term. Even if the model for the expectation were linearizable, you have a model for the observations here -- an additive error would make it not linearizable – Glen_b Apr 28 '17 at 00:24

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