Relation of parameters of the logistic link function with linear predictor parameters

Question

I am trying to self study Generlized Linear Models using the book "Doing Bayesian Data Analysis" by Kruschke. In the chapter on link functions, the author explains how the linear predictor y = β0 + β1 can be transformed using the logistic function to better fit the central tendency represented by the data. He then goes on to say that the logistic regression is more conveniently represented by parameters θ and γ, instead of β0 and β1

What I don't really understand is how we got from equation 15.13 to equation 15.14.

It doesnt seem to be a simple rearranging of the terms of the parameters, so β0 doesnt equal γ, and β1 doesnt equal θ. Could somebody exaplain it to me? Please note that my background is in biology, not mathematics, so the questions might be trivial, but nethertheless the answer would be greatly appreciated.

Thank you in advance, Jacek

Answer 1

You just let $\gamma=\beta_1$ and $\theta=-\beta_0 / \beta_1$. This is just a reparameterization, as long as $\beta_1\neq 0$ then there is a $\gamma$ and $\theta$ so that the model in 15.14 is identical to the model in 15.13 with a particular $\beta_1$ and $\beta_0$.