Modeling Spending on Internet Advertising You are the new director of Impact Advertisin...

Modeling Spending on Internet Advertising

You are the new director of Impact Advertising Inc.’s Internet division, which has enjoyed a steady 0.25% of the Internet advertising market. You have drawn up an ambitious proposal to expand your division in light of your anticipation that Internet advertising will continue to skyrocket. However, upper management sees things differently and, based on the following email, does not seem likely to approve the budget for your proposal.

Refusing to admit defeat, you contact the Market Research department and request the details of their projections on Internet advertising. They fax you the following information:

Now you see where the VP got that $2.7 billion figure: The slope of the regression equation is close to 2.7, indicating a rate of increase of about $2.7 billion per year. Also, the correlation coefficient is very high—an indication that the linear model fits the data well. In view of this strong evidence, it seems difficult to argue that revenues will increase by significantly more than the projected $2.7 billion per year. To get a better picture of what’s going on, you decide to graph the data together with the regression line in your spreadsheet. What you get is shown in Figure 32. You immediately notice that the data points seem to suggest a curve, and not a straight line. Then again, perhaps the suggestion of a curve is an illusion. Thus there are, you surmise, two possible interpretations of the data:

1. (Your first impression) As a function of time, Internet advertising revenue is nonlinear, and is in fact accelerating (the rate of change is increasing), so a linear model is inappropriate.

2. (Devil’s advocate) Internet advertising revenue is a linear function of time; the fact that the points do not lie on the regression line is simply a consequence of random factors that do not reflect a long-term trend, such as world events, mergers and acquisitions, short-term fluctuations in economy or the stock market, etc.

You suspect that the VP will probably opt for the second interpretation and discount the graphical evidence of accelerating growth by claiming that it is an illusion :a “statistical fluctuation.” That is, of course, a possibility, but you wonder how likely it really is.

For the sake of comparison, you decide to try a regression based on the simplest nonlinear model you can think of—a quadratic function.

which is $5.5 billion above the 2014 spending figure in the table above. Given Impact Advertising’s 0.25% market share, this translates into an increase in revenues of $13.75 million, which is about double the estimate predicted by the linear model!

You quickly draft an email to Lombardo, and are about to click “Send” when you decide, as a precaution, to check with a colleague who is knowledgeable in statistics. He tells you to be cautious: The value of r will always tend to increase if you pass from a linear model to a quadratic one because of the increase in “degrees of freedom.”† A good way to test whether a quadratic model is more appropriate than a linear one is to compute a statistic called the “p-value” associated with the coefficient of x². A low value of p indicates a high degree of confidence that the coefficient of x² cannot be zero (see below). Notice that if the coefficient of x² is zero, then you have a linear model.

You can, your colleague explains, obtain the p-value using your spreadsheet as follows (the method we describe here works on all the popular spreadsheets, including Excel, Google Docs, and Open Office Calc).

Suppose you are given the following data for the spending on Internet advertising in a hypothetical country in which Impact Advertising also has a 0.25% share of the market.

Perform a regression analysis using the quadratic model and find the associated p-value. What does it tell you about the appropriateness of a quadratic model?