Fascination with S-curves

I have had this fascination with S-curves since I started in media planning.

My first bosses - Malu and Oyie for (then) Basic Advertising - introduced me to the idea of Nat-Log curves (which depict the usual reach-curves and the "point of diminishing returns") which appear to hold for all GRPs x Cumed Reach relationships.

They also told me that audiences' response to advertising - whether it is in the form of awareness/recall, message acceptance, or action - could potentially be modeled using an S-curve: Response starts off slow, but once you hit a 'critical point', the 'response-variable' accelerates - until it reaches another point where the effect tapes off.

It was a good argument for "minimum spends" or "minimum GRPs" ("below which don't even bother spending") and for "maximum spends/GRPs" ("above which you're just wasting money").

I recently went back to the books to understand more deeply how response-curves, modeled through S-curves could be implemented in media planning - or communications planning.

In the chart above, one can imagine the horizontal axis to be "GRPs" or "Marketing Spends" and the vertical axis to be Purchase Intention or Actual Past-Week Purchase.

The above curve uses three paramaters. [Formally speaking, it is a three-parameter sigmoid curve - but you really don't want the details that much, right?]

The first parameter is the "Maximum", which in this case I defined as 100. The beauty of a three-parameter sigmoid curve is that you can change this "maximum". It could very well be your total population potential or the number of items you need to sell. I chose 100 to make it a little easier for me.
The second parameter is the "Starting Point", which in this case I defined as 1. If you were modelling, example, awareness or recall, this would be the latest reading. Or it could be your 'baseline' sales - the number of sales you're going to get even if you didn't advertise (assuming that that is what you're trying to model).
The third parameter is the most important one - and is almost always the 'unknown': The slope "c". The slope actually changes as you go through the line - so "slope" is probably not the right term, but I am calling it slope anyway (to make it easier). The slope determines the "conversion" of the increments in the "horizontal axis" to increments in the "vertical axis". (OK, elasticity of Y with respect to variable X...)

Using certain techniques and with sufficient data, one can come up with a similar curve.

And I think it's a very useful curve.

GRPs to awareness, GRPs to action, marketing spends to conversion, marketing spends to actual purchase, frequency of exposures to actual comprehension... These are some of the few that I think can be described by S-curves.

S-curves have also been used to demonstrate the diffusion of technologies and adoption of new trends. It's not very similar to Bass Diffusion Models which relies on coefficients of innovation and coefficients of imitation - but it is generic enough and "theory-agnostic" (well, almost).

Once you've got the formulas right, you can then apply some techniques from differential calculus to derive the inflection points.

How do you arrive at this curve?

My experience showed that the minimum that I needed was three data-points. But of course, the more data-points you have (say, GRPs x Recall), the better.

Also, I have used this on differenced/logged data (that is, increments on increments) - and it seems to work.

What's the secret sauce? Basic statistics (hint: goodness of fit) and Microsoft Excel(R)'s Solver Add-in. If you are interested, I can send you a copy of the EXCEL(R) file I did up for this and a simple macro that tries to accesses Solver to make life easier.