Marketing and Research Consulting for a Brave New World
Subscribe via RSS

The Movable Middle can be sized by using the Beta probability distribution…you just need to know who the category buyers are, and the share and repeat rate of the brand (maps to mean and variance of the Beta). In fact, the share of buyers is the first moment and the repeat rate from a markov matrix is literally the second moment of the distribution. Wikiedia has a good description of the Beta. You will find that the Beta distribution for most brands is U-Shaped (mode is 0 and there is an uptick in the curve towards the 90-100% probability of buying the brand.) The Beta then gives the percent of category buyers who have a given probability of buying the brand, and that allows us to directly calculate the size of The Movable Middle. This can be broken down and sized for any targetable segment to guide ad delivery. If you are in CPG, you can also use frequent shopper data as probabilities of buying can be directly estimated from such data. The accuracy of the Beta distribution at tracing out the full curve of % category buyers having a given probability of buying the brand is amazingly at r = 99% across 46 brands from 4 CPG product categories with shares ranging from 1% to 35%.

There is every reason to believe that the model works just as well for non-CPG although we must realize that the probability of buying is a latent variable that generates outcomes. Such outcomes as strings of purchases are observable in CPG but not observable with autos, financial services, HD TVs, etc. but the model is the same. I have found that a constant sum question in a survey works really well at tracing out the Beta distribution for such products and services.

The breakthrough was in linking the probability of purchase, p (ij) of a given consumer(i) towards a given brand (j) to their likely ad responsiveness to j.  The logit model based on the log odds ratio [ln (p/(1-p)] is often used in MTA because it constrains outcomes to be between 0 and 1 (an important property when modeling a probability.)  When you exponentiate and move terms around, you get that p (probability of buying) is equal to e to the power of a starting probability of buying plus some function of marketing activity divided by 1+ e to the same power. This produces an S-shaped curve where you can differentiate and note that the slope is different at different starting points on the curve.  The first derivative is maximized at a probability of purchase of 50% (the mid-point of the Movable Middle) and the second derivative goes to 0 (inflection point)…that tells us where the ad response is highest to a small increment of media weight. Also, the first derivative is completely symmetric around 50%, falling off at the same rate to its LOWEST near 0 and near 100%.  Obviously, those with a probability of purchase close to 100% have nowhere to increase to but seeing that the curve is symmetric, tells us WHY non-buyers who mostly have probabilities of purchase close to 0 must also have low ad response. All of this was confirmed by the data on our project.

So, the logit curve and the Beta distribution have a multiple variable relationship in marketing application that we discovered on this project that our academic advisors (UCLA, Oxford) tell us they believe had never before been discovered (as per peer reviewed publications.)


One Response to “Math behind the Movable Middle and why it produces 50% improvement in ROAS”

  1. […] Math behind the Movable Middle and why it produces 50% improvement in ROAS […]