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I am doing a retrospective cohort study with sample size 600 and disease prevalence rate greater than 10%. I am leaning more towards using risk ratio because it is easier to interpret and because disease prevalence is over 10%. Do you have any thoughts on this matter?

Also, if possible can someone briefly explain the differences between odds ratio and risk ratio? A specific and brief example would be helpful.

Thanks in advance.

Jay
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2 Answers2

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In a two by two table of exposure $X$ versus response $Y$,

\begin{array}{c|cc} & Y & \bar{Y} \\ \hline X & a & b \\ \bar{X} & c & d \\ \end{array}

The odds ratio is calculated as $ad/bc$ whereas the risk ratio is $a(c+d)/(c(a+b))$.

  1. For hypothesis testing, if the OR > 1 then the RR > 1 and the likelihood ratio test for the logistic model is the uniformly most powerful test.
  2. The prevalence of the outcome only matters for using the OR as an approximation to the RR - the approximation is good when the outcome is rare, VERY rare. It's best just to report an OR as an OR always.
  3. Either modeling approach provides the chance to predict response probabilities across a range of exposures. Without this, I argue an OR or an RR on their own mean very little. Some reviewers demand also to see the risk difference as yet another contrast.
AdamO
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Odds ratio and relative risk are two measures used to describe the likelihood of an event happening. The odds ratio is defined as the ratio of the odds of an event or disease occurring in one group to the odds occurring in another group. The standard formula is $[X/(1-X)]/[Y/(1-Y)]$, where $X$ and $Y$ are the probability of that event in the two groups, respectively. In contrast, the relative risk is the risk of an event or disease relative to exposure. The standard formula is $A/B$.

For your study, I would strongly suggest to use the risk ratio. A general rule is that when the prevalence of the disease is <10%, the relative risk and the odds ratio tend to be very similar, so that for rare diseases both indexes can be used nearly in an interchangeable manner. On the other hand, for diseases with relatively high prevalences, the odds ratio is considerably different from the relative risk. In particular, it overstates the real effect: the odds ratio is smaller than the relative risk for odds ratios <1 (for example, it could lead to an overestimation of the beneficial effect of a pharmacological agent for the prevention of a disase), and larger than the relative risk for odds ratios >1 (e.g. it could lead to an overestimation of the real effect of a factor in increasing the risk of a disease). Also note that the magnitude of this overestimation increases as both the initial risk (prevalence of the disease) increases and the odds ratio departs from 1.

Lastly, we have to take into account that odds (unlike risks) are rather difficult to understand, particularly for readers that are not particularly trained in statistics. The risk of an event is a very simple concept: it is the ratio between the number of subjects who experience the event and the total number of subjects at risk. It is classically expressed as a proportion ranging between 0 and 1 or between 0 and 100%. In contrast, the odds of an event is the ratio between the number of subjects who experience the event and those who do not. It ranges from zero to infinity, and is clearly less intuitive. For instance, a relative risk of 70% corresponds to an odds ratio of 0.7/(1-0.7)=2.33: however, it is clearer to say to the layman that a certain risk factor "increases the probability of a disease by 70%" (relative risk) rather than that it "increases the probability of the disease by an odds ratio of 2.33".

Anatoly
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  • OR and RR do not describe the likelihood of an event any more than the slope of a linear regression describes the mean response. It's a second moment estimator, a contrast, between groups. – AdamO Oct 26 '22 at 16:26