The best-fitting advertising response function in Example 7.5 fits the observed data. This is because we rigged the observed data to fall close to a curve of the form in Equation (7.4). See what happens when one of the observed points is an outlier—that is, it doesn’t fit the pattern of the others.
a. Specifically, change the number of exposures corresponding to 50 ads from 90.3 to 125, and then rerun Solver. Do you get essentially the same response function as before, or does this one outlier exert a large influence on the estimated response function?
b. Repeat part a, but now change the number of exposures corresponding to 50 ads from 90.3 to 55.
EXAMPLE 7.5 ESTIMATING AN ADVERTISING RESPONSE FUNCTION
Recall that the General Flakes Company from Example 4.1 of Chapter 4 sells a brand of low-fat breakfast cereal that appeals to people of all age groups and both genders. The company has advertised this product in various media for a number of years and has accumulated data on its advertising effectiveness. For example, the company has tracked the number of exposures to young men from ads placed on a particular television show for five different time periods. In each of these time periods, a different number of ads was used. Specifically, the numbers of ads were 1, 8, 20, 50, and 100. The corresponding numbers of exposures (in millions) were 4.7, 22.1, 48.7, 90.3, and 130.5. What type of nonlinear response function might fit these data well?
Objective To use nonlinear optimization to find the response function (from a given class of functions) that best fits the historical data.
WHERE DO THE NUMBERS COME FROM?
The question here is how the company measures the number of exposures a given number of ads has achieved. In particular, what does the company mean by “exposures”? If one person sees the same ad 10 times, does this mean 10 exposures? Is it the same thing as 10 people seeing the same ad once each? Although we defer to the marketing experts here, we suggest that one person seeing the same ad 10 times results in fewer exposures than 10 people seeing the same ad once each. However, the marketing experts decide to count exposures, it should lead to the decreasing marginal effects built into this example.