Abstract
At the end of the previous chapter, we spent some time considering the difference between statistical significance and the strength of an association. In this chapter, we spend some time on a measure that combines elements of both: the odds ratio, which for 2 ×2 tables, is the recommended test of both the strength and the statistical significance of an association. As with many SAS statistical procedures we have seen, the actual execution of the procedure is fairly simple. The challenge is in understanding the underlying statistical process and interpreting the results.
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Notes
- 1.
Note that this does not mean that the probability is 1/3 for or against.
- 2.
We used to teach the “rare disease assumption” as the reason that a case-control study can approximate a cohort study. There are now more fundamentally sound reasons supporting the validity of case-control studies. Rothman, in particular, has a very nice discussion. Still, the reason the odds ratio is a valid approximation to the relative risk is because the outcomes we study are, in fact, rare and do not when removed detract appreciably from the bottom number of the ratios themselves when compared to the denominator of the analogous probability. (See the formula for converting probabilities to odds.) Odds ratios will overestimate risk, when the outcomes are common. As a general rule of thumb, I find outcomes in much beyond 10 to 15 percent of the study population to be problematic.
- 3.
Smoking is so frequently a confounder that you might want to at least consider including it in almost any study in which it is not the actual exposure of interest.
- 4.
You also get the same two types of estimates for cohort data.
- 5.
You can also request / CMH rather than / all. If there is a 2 ×2 table, SAS will return the Mantel–Haenszel statistic in addition to the Cochrane–Mantel–Haenszel statistics.
- 6.
As you start to stratify by more than one variable, you may find that you will start having problems with small numbers. SAS has options to request exact tests for odds ratios. You request them with an “exact” statement on its own line.
- 7.
In fact, you wont go too far wrong just using type three routinely.
- 8.
There are no hard and fast rules for what constitutes an “appreciable” or “meaningful” difference between a crude and an adjusted estimate, nor is the comparison amenable to statistical testing. This is one of those areas (again) where training, experience and substance matter knowledge combine to guide our efforts. A 15% or 20% difference is an acceptable rule of thumb.
- 9.
Of course, the sample size was pretty small, and even with Tarone adjustment we may have to question these results. Also, see chapter 11 for a discussion of an epidemiological approach to interaction. You would want to consider using the Darroch/Rothman approach.
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DiMaggio, C. (2013). Categorical Data Analysis II. In: SAS for Epidemiologists. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4854-9_9
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