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# How to show that the odds ratio is always further away from 1 than relative risk

How to Show that the Odds Ratio is Always Further Away from 1 than Relative Risk

When analyzing data in medical or statistical research, it is essential to understand the differences between odds ratio and relative risk. Both measures assess the association between two variables, but they have distinct interpretations. This guide aims to explain how to demonstrate that the odds ratio is consistently further away from 1 than relative risk. By understanding this concept, researchers can gain valuable insights into the strength of associations and make more informed decisions.

I. Understanding Odds Ratio and Relative Risk:

To comprehend why the odds ratio is always further away from 1 than relative risk, it is crucial to understand the definitions and interpretations of these measures.

A. Odds Ratio:

1. Definition: The odds ratio measures the odds of an event occurring in one group compared to another group.
2. Interpretation: An odds ratio of 1 indicates no association, while values greater or less than 1 suggest a positive or negative association, respectively.

B. Relative Risk:

1. Definition: The relative risk assesses the probability of an event occurring in one group compared to another group.
2. Interpretation: A relative risk of 1 indicates no association, while values greater or less than 1 suggest a
Odds ratios often are mistaken for relative risk ratios. 2,3 Although for rare outcomes odds ratios approximate relative risk ratios, when the outcomes are not rare, odds ratios always overestimate relative risk ratios, a problem that becomes more acute as the baseline prevalence of the outcome exceeds 10%.

## When can you not use risk ratio?

In retrospective (case-control) studies, where the total number of exposed people is not available, RR cannot be calculated and OR is used as a measure of the strength of association between exposure and outcome.

## Why would you use an odds ratio in a case-control study and not a relative risk ratio?

Firstly, RRs cannot be applied in all cases. Because risk in a sample is an estimate of risk in a population, the sample must be reasonably representative of the population. As such, case-control studies, by simple virtue of the fact that ratios of outcomes are controlled, cannot have a risk ratio reported.

## When can odds ratios mislead?

The discrepancy between a relative risk reduction and the equivalent relative odds reduction (100×(1−odds ratio)%) can be misleading. When event rates are high (commonly the case in trials and systematic reviews) the relative odds reduction can be many times larger than the equivalent relative risk reduction.

## Why use odds ratio and not relative risk?

When the outcome is not rare in the population, if the odds ratio is used to estimate the relative risk it will overstate the effect of the treatment on the outcome measure. The odds ratio will be greater than the relative risk if the relative risk is greater than one and less than the relative risk otherwise.

## Why use odds ratio instead of relative risk?

“Risk” refers to the probability of occurrence of an event or outcome. Statistically, risk = chance of the outcome of interest/all possible outcomes. The term “odds” is often used instead of risk. “Odds” refers to the probability of occurrence of an event/probability of the event not occurring.

## Does the odds ratio give a good approximation to the relative risk for these data why OR why not?

Odds ratios often are mistaken for relative risk ratios. 2,3 Although for rare outcomes odds ratios approximate relative risk ratios, when the outcomes are not rare, odds ratios always overestimate relative risk ratios, a problem that becomes more acute as the baseline prevalence of the outcome exceeds 10%.

#### In which kind of study should you use odds ratio instead of the relative risk?

In contrast to clinical trials and cohort studies, odds ratios are not only preferred in case-control studies, but are often the only measure of association that can be applied to this type of study design.

#### What is the relationship between odd ratio and relative risk?

The relative risk (also known as risk ratio [RR]) is the ratio of risk of an event in one group (e.g., exposed group) versus the risk of the event in the other group (e.g., nonexposed group). The odds ratio (OR) is the ratio of odds of an event in one group versus the odds of the event in the other group.

#### What if the odds ratio is close to 1?

An odds ratio of 1 indicates that the condition or event under study is equally likely to occur in both groups. An odds ratio greater than 1 indicates that the condition or event is more likely to occur in the first group.

#### What can odds tell us about risk?

RELATIVE RISK AND ODDS RATIO An RR (or OR) more than 1.0 indicates an increase in risk (or odds) among the exposed compared to the unexposed, whereas a RR (or OR) <1.0 indicates a decrease in risk (or odds) in the exposed group. As for other summary statistics, confidence intervals can be calculated for RR and OR.

#### How do you choose between odds ratio and relative risk?

The relative risk (RR) is the risk of the event in an experimental group relative to that in a control group. The odds ratio (OR) is the odds of an event in an experimental group relative to that in a control group. An RR or OR of 1.00 indicates that the risk is comparable in the two groups.

#### How do you calculate hazard ratio from odds ratio?

With the following equation, the probability of healing first can easily be derived from the odds of healing first, which is the probability of healing first divided by the probability of not healing first: hazard ratio (HR) = odds = P/(1 − P); P = HR/(1 + HR).

## FAQ

Can odds ratio be interpreted as risk?
For initial risks of 10% or less, even odds ratios of up to eight can reasonably be interpreted as relative risks; for initial risks up to 30% the approximation breaks down when the effect size gives odds ratios of more than about three.
Why are the relative risk and odds ratio approximately equal?
If the outcome is rare in both exposed and unexposed persons, the odds ratio ([A/B]/[C/D]) will approximate the risk ratio ([A/(A + B)]/[C/(C + D)]). However, when the study outcome is common and the risk ratio is not close to 1, the odds ratio will be further from 1 compared with the risk ratio.
Is relative risk the same as rate ratio?
Risk ratio: ratio of the risk of an event in one group (exposure or intervention) to that in another group (control). So it depends on your definitions of rate and risk. The term "relative risk" is sometimes used as a synonym for risk ratio, and rate ratio is one of the relative risk measures too.
How do you calculate relative risk from odds ratio?
Thus the odds ratio is (a/b) / (c/d) which simplifies to ad/bc. This is compared to the relative risk which is (a / (a+b)) / (c / (c+d)). If the disease condition (event) is rare, then the odds ratio and relative risk may be comparable, but the odds ratio will overestimate the risk if the disease is more common.
When interpreting a relative risk or odds ratio if the calculation is equal to OR less than 1.0 then which of the following is true?
The odds ratio is interpreted in the same manner as the risk ratio or rate ratio with an OR of 1.0 indicating no association, an OR greater than 1.0 indicating a positive association, and an OR less than 1.0 indicating a negative, or protective association.
What is relative risk equal to?
The relative risk (RR) or risk ratio is the ratio of the probability of an outcome in an exposed group to the probability of an outcome in an unexposed group.

## How to show that the odds ratio is always further away from 1 than relative risk

 Does odds ratio have to be greater than 1? Important points about Odds ratio: Calculated in case-control studies as the incidence of outcome is not known. OR >1 indicates increased occurrence of an event. OR <1 indicates decreased occurrence of an event (protective exposure) How do you interpret odds ratio equal to 1? If an odds ratio (OR) is 1, it means there is no association between the exposure and outcome. So, if the 95% confidence interval for an OR includes 1, it means the results are not statistically significant. What does an odds ratio of 0.2 mean? An odds of 0.2 however seems less intuitive: 0.2 people will experience the event for every one that does not. This translates to one event for every five non-events (a risk of one in six or 17%). What is the relationship between odds ratio and relative risk? The relative risk (also known as risk ratio [RR]) is the ratio of risk of an event in one group (e.g., exposed group) versus the risk of the event in the other group (e.g., nonexposed group). The odds ratio (OR) is the ratio of odds of an event in one group versus the odds of the event in the other group. Is odds ratio a relative measure of association? Odds ratios (OR) are commonly reported in the medical literature as the measure of association between exposure and outcome. However, it is relative risk that people more intuitively understand as a measure of association. Does odds ratio show correlation? Odds ratio and correlation don't measure the same thing- correlation looks at how much one variable is explained by another (for example, is how much is wieght explaine by height?). Odds ratio compare the odds of a specific outcome in two groups, one exposied an done unexposed to a specific exposure.
• How do you interpret odds ratio and adjusted odds ratio?
• Odds of an event happening is defined as the likelihood that an event will occur, expressed as a proportion of the likelihood that the event will not occur. Therefore, if A is the probability of subjects affected and B is the probability of subjects not affected, then odds = A /B.
• What is an example of odds ratio and relative risk?
• 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 p1=0.40 and p2=0.25, then the relative risk is r=1.60, but the odds ratio is θ =2.00.
• Why is odds ratio different than relative risk?
• The relative risk (also known as risk ratio [RR]) is the ratio of risk of an event in one group (e.g., exposed group) versus the risk of the event in the other group (e.g., nonexposed group). The odds ratio (OR) is the ratio of odds of an event in one group versus the odds of the event in the other group.
• When odds ratio overestimates risk ratio?
• Odds ratios often are mistaken for relative risk ratios. 2,3 Although for rare outcomes odds ratios approximate relative risk ratios, when the outcomes are not rare, odds ratios always overestimate relative risk ratios, a problem that becomes more acute as the baseline prevalence of the outcome exceeds 10%.
• How do you know if a relative risk is statistically significant?
• Any RR > 2 is statistically-significant when N*P1 is at least 10. Any RR > 1.6 is statistically-significant when N*P1 is at least 25. As the count in the smallest cell decreases, the Normal Approximation becomes less adequate.
• Should relative risk and odds ratio be the same?
• The relative risk (RR) is the risk of the event in an experimental group relative to that in a control group. The odds ratio (OR) is the odds of an event in an experimental group relative to that in a control group. An RR or OR of 1.00 indicates that the risk is comparable in the two groups.

February 8, 2024
February 8, 2024
February 8, 2024