The advertising response function in Equation (7.4) is only one of several nonlinear functions that could be used to get the same “increasing at a decreasing rate” behavior in Example 7.5. Another possibility is the function f (n) = anb, where a and b are again constants to be determined. Using the same data as in Example 7.5 and RMSE as the fitting criterion, find the best fit to this type of function. In terms of RMSE, which function appears to fit the data better, the one here or the one in the example? Can you spot any qualitative difference between the two types of functions?
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.