Can anyone enlighten me with the equation structure for the following basic theoretical (although common) scenario:
Sales, $S$, are related to "Advertising", $A$, such that when $A$ is small $S$ grows exponentially, and when $A$ is large, $S$ grows more slowly until at some point any increase in $A$ produces zero increase in $S$. So, a typical S-shaped response to advertising.
The total number of sales made in any period in the market is $T$. So our average market share works out as $\Sigma S/\Sigma T$. When advertising is zero, our sales are static at some level $s$ (i.e. the effect some other supporting influence other than advertising).
I know this can be solved through linear regression by transforming the variables, but I'm struggling to get my head around the most basic version of this - essentially $f(S) =\gamma+ \beta g(A)+\epsilon $, (with $\gamma$ being some intercept (possibly $0$) and $\epsilon$ being residuals) but what do $f$ and $g$ look like, and therefore what does the equation look like that I need to solve to estimate $S$?
To show how the final equation would look, I have the logit transform in mind, so I'm looking for how the logit transform is applied using the parameters in the question, then what the final equation would look like with the transformations in place.
In addition, I'm specifically looking for a form to solve via linear regression rater than anything non-linear.