 # Relationships among Three Popular Measures of Differential Risks: Relative Risk, Risk Difference, and Odds Ratio

## Article excerpt

1. Introduction

The relative risk, risk difference, and odds ratio are three major measures used to assess differences in the risk of diseases between different groups. These measures --which play important roles in research and practice in the biomedical, behavioral, and social sciences --have been extensively discussed in statistics, [1,2] epidemiology, [3-7] and biomedical  literature. Although straightforward to interpret when used independently, there is considerable confusion and frequent misinterpretation of the measures when they are used together. [9,10] As popular as they are, relationships among the three measures have never been made clear and remain elusive. For example, here is an excerpt from Kraft and colleagues: 

"... genetic profiles based on sets of risk markers can potentially identify rare highrisk and low-risk subgroups with large relative differences in risk (that is, with odds ratios greater than 10). These profiles can also have a high population attributable risk (PAR; also known as population attributable fraction)." (p.264)

In this excerpt the authors incorrectly assumed that a larger odds ratio would imply a higher PAR, which, in turn, would give rise to a larger relative risk. Their line of thinking is apparently logical, since all three measures have traditionally been viewed as equivalent measures of differential risks such that a larger value in any one measure would naturally imply larger values in the other two measures. This paper discusses the properties of the three measures, systematically assesses relationships between the three pairs of measures (i.e., relative risk and odds ratio, relative risk and risk difference, and risk difference and odds ratio), and presents examples to clarify the misconception in the above statement as well as other pitfalls when interpreting the relationships between the different measures.

2. Relationship between relative risk and odds ratio

Let pi and [p.sub.2] denote disease prevalence in two groups of interest. For simplicity, we assume that 0<[p.sub.1], [p.sub.2]<1, that is, there exist two subgroups with potentially different prevalence rates for the disease of interest. The relative risk (r), risk difference (d), and odds ratio (9) between the groups are defined as:

r = [p.sub.1]/[p.sub.2], d = [p.sub.1] - [p.sub.2], [theta] = [p.sub.1](1 - [p.sub.2])/(1 - [p.sub.1]) [p.sub.2]. (1)

Note that the relative risk is also called the risk ratio in the literature,  but for convenience we use the term 'relative risk' hereafter.

From the definitions above, we immediately see that the three measures have quite different ranges; both the relative risk and odds ratio vary between 0 and ~, while the risk difference is limited to a much smaller interval between -1 and 1. Despite the fact that the relative risk and odds ratio have the same range, they represent totally different measures of differential risks and, therefore, have quite different interpretations. For example, if [p.sub.1]=0.40 and [p.sub.2]=0.25, then the relative risk is r=1.60, but the odds ratio is [tjheta]=2.00.

Given two prevalence rates [p.sub.1] and [p.sub.2], we can calculate both the relative risk and odds ratio. However, there is generally more than one pair of prevalence rates ([p.sub.1], ([p.sub.2]) that yields any pre-specified relative risk (or odds ratio). For example, ([cp.sub.1], [cp.sub.2]) yields the same relative risk (r=Pi/([p.sub.2]) for any value of c when 0

Theorem 1. The relative risk r and the odds ratio 9 satisfy one of the following conditions:

r = [theta] = 1, [theta]r>1. …

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