# Disease Association - Risk Ratio

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## Slide 1: Risk Ratio

Probability of disease, ate salad = 54/70 = 0.77

Probability of disease, no salad = 2/28 = 0.07 Risk ratio = 0.77/0.07 = 11

Illustrates risk ratio in cohort with complete follow-up Here is a example of a risk ratio from cohort data from a cohort with equal follow-up on everyone. It is the example of the outbreak of gastrointestinal illness we looked at earlier. Because the follow-up is short and identical for everyone, the risk ratio is just the ratio of the proportion with disease in the exposed group (those who ate the potato salad) and in the unexposed group (those who didn't eat the salad). Eleven is a large value for a risk ratio but that might be expected in a study such as this looking for a single likely food source for the outbreak. So the RR=11 is taken as strong evidence for assigning causality to eating the potato salad. It is highly likely that the potato salad caused the outbreak of gastroenteritis.

## Slide 2: Risk Ratio in a Cohort with Censoring

Choose a time point for comparing two cumulative incidences:

At 6 years, % dead in low CD4 group = 0.70 and in high CD4 group = 0.26. Risk ratio at 6 years = 0.70/0.26 = 2.69 As we have pointed out frequently, however, follow-up in most cohorts that are not very short term outbreak investigations, have differing amounts of follow-up time on the subjects and the risk of the event has to be estimated in the exposed and unexposed group using a method like the Kaplan-Meier or the life table. So in forming a risk ratio from a Kaplan-Meier analysis of the survival in two groups, we have to choose a point in time. As you can see from inspecting the curve, the risk ratio will be different for different points in time. If one point in time is selected, then the risk ratio becomes the ratio of the two proportions failing (or surviving, if you prefer) at that point in time. But, once again, in reporting Kaplan-Meier results you must always specify at what amount of follow-up time. This applies to the risk ratio as well.

When a Kaplan-Meier analysis is presented in the medical literature, a p-value that summarizes the probability that the two curves differ over their entire lengths is usually given. This is a more complex statistic than just comparing two proportions with a chi-square test as it compares proportions all along the curves whenever an event occurs. The most commonly used statistic is called the log rank test; an alternative test is called the Wilcoxon.

## Slide 3: OR compared to Risk Ratio

This graphic illustrates the point that, unless both the risk ratio and the odds ratio are 1.0 (no difference), the odds ratio is always farther from 1.0 than the risk ratio, larger if the risk ratio is greater than 1.0 and smaller if the risk ratio is less than 1.0. If Risk Ratio > 1, then OR farther from 1 than Risk Ratio:

When the ratio of two probabilities, a risk ratio, is > 1.0, the OR will be larger than the RR. The OR is dividing each probability by a quantity forced to be < 1.0 (unless probability = 1.0), so each probability increases and the ratio between them also increases. The only exception occurs when the risk ratio is exactly 1.0. In that case the OR will also be 1.0. This can easily be seen by modifying the example above to RR = [0.4/0.4] = 1.0. The OR would then be [0.4/0.6]/[0.4/0.6] = 1.0.

If Risk Ratio < 1, then OR farther from 1 than Risk Ratio:

The same phenomenon occurs when the ratio of two probabilities, a risk ratio, is less than 1.0 Since the OR is dividing each probability by a quantity forced to be < 1.0 unless probability = 1.0, each probability increases and the ratio between them also increases, which in this case moves the value farther away from 1.0. Values for both risk ratios and odds ratios less than 1 are bounded by 0. In that respect they differ from ratios with values greater than 1, which can be infinitely large.

## Slide 4: Important Property of Odds Ratio #2

• OR approximates Risk Ratio only if disease incidence is low in both the exposed and the unexposed group

## Slide 5: Risk of Disease Is Low

If risk of disease is low in both exposed and unexposed, RR and OR approximately equal.

Example: incidence of MI risk in high blood pressure group = 0.018

and incidence of MI risk in low bp group is 0.003:

Risk Ratio = 0.018/0.003 = 6.0 OR = 0.01833/0.00301 = 6.09

When incidence is very low in both groups, the risk and the odds ratio are very close. This is due to the fact that the odds, the probability of the event divided by the probability of the non-event, is dividing the probability of the event by a value close to 1.0. In this example the probability of the non-event in the group with the highest incidence (the exposed group) is 0.982 (1 – 0.018). The probability of the event in the unexposed is even closer to 1.0: 0.997 (1 – 0.003). So dividing the two probabilities by values close to 1.0 is going to have very little effect on their ratio. If either probability of the non-event had been far from 1.0, then there would have been a substantial effect on the ratio.

## Slide 6: Risk of Disease Is High

If risk of disease is high in either or both exposed and unexposed, Risk Ratio and OR differ

Example, if risk in exposed is 0.6 and 0.1 in unexposed:

RR = [0.6/0.1] = 6.0 OR = [0.6/0.4]/[0.1/0.9] = 13.5

OR approximates Risk Ratio only if incidence is low in both exposed and unexposed group

It is important to understand this feature of the odds ratio because you will see it referred to as an approximation of the risk ratio. As we have been stressing, it is no less valid as a mathematical expression than the risk ratio, but its less intuitive nature leads many investigators to prefer to talk about the risk ratio, a more intuitive concept. This results in measures which are in fact OR’s being presented in some papers as RR’s or relative risks. In many instances this is not appropriate even as an approximation because the incidence is not low in both the exposed and unexposed group and the OR and the risk ratio are quite different.

## Slide 7: “Bias” in OR as Estimate of RR

Some individuals refer to “bias” in OR as estimate of RR [OR = RR x (1-incid.unexp)/(1-incid.exp)] not “bias” in usual sense because both OR and RR are mathematically valid and use the same numbers

Simply that OR cannot be thought of as a surrogate for the RR unless incidence is low

It is preferable to reserve the word “bias” to situations where invalid or inaccurate results are being obtained because of some factor or factors that are affecting the study, the sources of bias. Since OR and RR are mathematically valid measures using the same numbers, it is a bit misleading to talk about the bias of the OR. But the point that the OR and the RR are not the same thing is important to note.