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Complex Samples Crosstabs
Risk Estimate
Figure
Risk estimate for newspaper subscription by response
The relative risk is a ratio of event probabilities. The relative risk of a response to the mailing is the ratio of the probability that a newspaper subscriber responds to the probability that a nonsubscriber responds. Thus, the estimate of the relative risk is simply 17.2%/10.3% = 1.673. Likewise, the relative risk of nonresponse is the ratio of the probability that a subscriber does not respond to the probability that a nonsubscriber does not respond. Your estimate of this relative risk is 0.923. Given these results, you can estimate that a newspaper subscriber is 1.673 times as likely to respond to the mailing as a nonsubscriber, or 0.923 times as likely as a nonsubscriber not to respond.
The odds ratio is a ratio of event odds. The odds of an event is the ratio of the probability that the event occurs to the probability that the event does not occur. Thus, the estimate of the odds that a newspaper subscriber responds to the mailing is 17.2%/82.8% = 0.208. Likewise, the estimate of the odds that a nonsubscriber responds is 10.3%/89.7% = 0.115. The estimate of the odds ratio is therefore 0.208/0.115 = 1.812 (note there is some rounding error in the intervening steps). The odds ratio is also the ratio of the relative risk of responding to the relative risk of not responding, or 1.673/0.923 = 1.812.
Odds Ratio versus Relative Risk
Since it is a ratio of ratios, the odds ratio is very difficult to interpret. The relative risk is easier to interpret, so the odds ratio alone is not very helpful. However, there are certain commonly occurring situations in which the estimate of the relative risk is not very good, and the odds ratio can be used to approximate the relative risk of the event of interest. The odds ratio should be used as an approximation of the relative risk of the event of interest when both of the following conditions are met:
The probability of the event of interest is small (< 0.1). This condition guarantees that the odds ratio will make a good approximation to the relative risk. In this example, the event of interest is a response to the mailing.
The design of the study is case control. This condition signals that the usual estimate of the relative risk will likely not be good. A
Neither condition is met in this example, since the overall proportion of respondents was 12.8% and the design of the study was not case control, so it’s safer to report 1.673 as the relative risk, rather than the value of the odds ratio.
