Overinvolvement Ratios

There are a number of situations where it is difficult to obtain desired conditional probabilities, but alternative conditional probabilities are available. It may be difficult to obtain probabilities because the costs of enumeration are high or because critical, ethical, or legal restriction prevents direct collection of probabilities. In some of those cases it may be possible to use basic probability relationships to derive desired probabilities from available probabilities.

Suppose that we know 60% of the purchasers of our product have seen our advertisement, but only 30% of the nonpurchasers have seen the advertisement. The ratio of 60% to 30% is the overinvolvement of the event “Seen our advertisement” in the purchasers group, compared to the nonpurchasers group. In the analysis to follow we show how an overinvolvment ratio greater than 1.0 provides evidence that, for example, advertising influences purchase behaviour.

The probability of event A1, conditional on event B1, divided by the probability of A1, conditional on event B2, is defined as the overinvolvement ratio

Consider a company that wishes to determine the effectiveness of a new advertisement. An experiment is conducted in which the advertisement is shown to one customer group and not to another, followed by observation of the purchase behaviour of both groups. Studies of this have high probability of error; they can be biased because people who are watched closely often behave differently than they do when not being observed. It is possible, however, to measure the percentage of buyers who have seen an ad and to measure the percentage of nonbuyers who have seen the ad. Let us consider how those study data can be analysed to determine the effectiveness of the new advertisement.

Advertising effectiveness is determined using the following analysis. The population is divided into

                   B1 : Buyers
                   B2 : Nonbuyers

And into
                    A1 : Those who have seen the advertisement
                    A2 : Those who have not seen the advertisement.

Similarly, we can define the conditional odds, in which we use the ratio of the probabilities that are both conditional on the same event. For this problem the odds of a buyer conditional on “Have seen an advertisement” are

If the conditional odds are greater than the unconditional odds, the conditioning event is said to have influence on the vent of interest. Thus, advertising would be considered effective if

This result shows that, if a larger percent of buyers have seen the advertisement, compared to nonbuyers, then the odds in favor of purchasing conditional on having seen the advertisement are greater than the unconditional odds. Therefore, we have evidence that the advertising is associated with an increased probability of purchase.