What Are The Odds Questions

What is an Odds Ratio?

An odds ratio (OR) calculates the relationship between a variable and the likelihood of an event occurring. A mutual interpretation for odds ratios is identifying take a chance factors by assessing the relationship between exposure to a chance factor and a medical outcome. For example, is at that place an association between exposure to a chemic and a disease?

To calculate an odds ratio, you must have a binary event. And you'll need either a grouping variable or a continuous variable that you want to chronicle to your issue of interest. Then, utilise an OR to assess the relationship between your variable and the likelihood that an result occurs.

When you lot accept a grouping variable, an odds ratio interpretation answers the question, is an result more or less likely to occur in one status or another? It calculates the odds of an issue occurring in ane context relative to a baseline or control condition. For instance, your group variable can be a subject field's exposure to a risk factor—yes or no—to meet how that relates to affliction condition.

With a continuous variable, calculating an odds ratio can determine whether the odds of an event occurring change as the continuous variable changes.

In this post, learn nigh ORs, including how to utilise the odds ratio formula to calculate them, different ways to arrange them for several types of studies, and how to translate odds ratios and their confidence intervals and p-values.

What Are Odds in Statistics?

Before you lot can calculate and translate an odds ratio, yous must know what the odds of an event represents. In common usage, people tend to use odds and probability interchangeably. However, in statistics, it has an exact definition. It is a specific type of probability.

Odds relate to a binary event where the event either occurs or does not occur. For example, written report subjects were either infected or not infected. A person graduates or does not graduate from college. Y'all win a game, or you lot lose.

Odds definition: The probability of the event occurring divided past the probability of the consequence not occurring.

As you tin see from the formula, it tells you how likely an event is to occur relative to information technology not happening. For example, imagine playing a dice-rolling game where a six is very good. Your odds of rolling a six are the post-obit:

Your odds of rolling a six is 0.twenty or ane in 5. Because the number of dice outcomes is a constant 6, yous can replace 1/6 and five/6 in the formula with a 1 and 5 to derive the aforementioned answer (1/5 = 0.twenty). I'll utilize that format in the examples throughout this postal service.

Imagine yous're playing a game. If your odds of winning are 2 (or 2 wins to ane loss), that indicates you are twice every bit likely to win every bit to lose. On the other hand, if your odds of winning are 0.5 (or 1 win to 2 losses), you lot're half as likely to win as to lose.

As you tin see, the odds of an event occurring is a ratio itself. Therefore, an OR is a ratio of two ratios.

Related mail: Probability Fundamentals

Odds Ratios Interpretation for Two Weather

Odds ratios with groups quantify the force of the human relationship between 2 conditions. They indicate how likely an effect is to occur in 1 context relative to some other.

The odds ratio formula below shows how to summate it for weather condition A and B.

The denominator (condition B) in the odds ratio formula is the baseline or control grouping. Consequently, the OR tells you how much more than or less likely the numerator events (condition A) are likely to occur relative to the denominator events.

If you have a treatment and command grouping, the handling will be in the numerator while the control grouping is in the denominator of the formula. This adding of an odds ratio indicates how your handling group fares compared to the controls.

For instance, a study assesses infections in a treatment and control group. Infections are the events for the binary outcome. By calculating the following OR, analysts can determine how likely infections are in the handling group relative to the control group.

The interpretation of this odds ratio is that when the handling is effective, the odds of infections in the treatment group will be lower than the control group, producing an OR of less than one.

Permit'southward move on to more than interpretation details!

Related post: Control Groups in Experiments

How to Interpret Odds Ratios

Due to the odds ratio formula, the value of one becomes critical during interpretation because it indicates both weather accept equal odds. Consequently, analysts always compare their OR results to one when interpreting the results. As the OR moves abroad from one in either direction, the association between the condition and outcome becomes stronger.

Odds Ratio = 1: The ratio equals one when the numerator and denominator are equal. This equivalence occurs when the odds of the event occurring in one condition equal the odds of it happening in the other condition. There is no association betwixt condition and event occurrence.

Odds Ratio > 1: The numerator is greater than the denominator. Hence, the consequence'southward odds are higher for the group/condition in the numerator. This is frequently a risk factor.

Odds Ratio < 1: The numerator is less than the denominator. Hence, the probability of the outcome occurring is lower for the group/status in the numerator. This can be a protective factor.

In the hypothetical infection experiment, the researchers hope that the OR is less than i considering that indicates the treatment grouping has lower odds of becoming infected than the control group.

Caution: ORs are a type of correlation and practice not necessarily represent causal relationships!

Odds ratios are similar to relative risks and hazard ratios, but they are dissimilar statistics. Acquire more nigh Relative Risks and Hazard Ratios.

How to Calculate an Odds Ratio

The equation below expands the earlier odds ratio formula for computing an OR with two conditions (A and B). Again, it'southward the ratio of ii odds. Hence, the numerator and denominator are also ratios.

In the infection example above, nosotros assessed the relationship between treatment and the odds of being infected. Our two conditions were the treatment (status A) and the control group (B). On the correct-mitt side, we'd enter the numbers of infections (events) and non-infections (non-events) from our sample for both groups.

Example Odds Ratio Calculations for Two Groups

Let's apply data from an actual report to summate an odds ratio. The North Carolina Division of Public Health needed to identify risk factors associated with an East. Coli breakout. We'll summate the OR for i take a chance factor, but they assessed multiple possibilities in their study.

In this study, the event is an exposure to a risk gene for E. coli infection. Our ii conditions are those who are sick versus not sick. It's an example of a case-control report, which analysts utilise to identify candidate run a risk factors using odds ratios.

	Got Sick (Cases)	Did Not Become Sick (Controls)
Visited Petting Zoo	36	64
Did Non Visit	9	123

By plugging these numbers into the odds ratio formula, nosotros can calculate the odds ratio to appraise the relationship between visiting a petting zoo and condign infected by E. coli. In case-control studies, all infected cases go in the numerator while the uninfected controls go in the denominator. The side by side section explains why.

The interpretation of the odds ratio is that those who became infected with E. coli (cases) were 7.7 times more likely to have visited the petting zoo than those without symptoms (controls). That'due south a large red flag for the petting zoo being the E. coli source!

This report also assessed whether awareness of the affliction gamble from contacting livestock was a protective factor. For this factor, the study calculated the odds ratio:

For interpreting the odds ratio, the value of 0.1 indicates that those who became infected with East. coli were simply 1-tenth as likely to be aware of the illness take a chance from contacting livestock as those who were non infected. Knowledge is ability! Presumably, those who were enlightened of the run a risk took precautions!

Related post: Case-Command Studies

Different Arrangements

You might take noticed differences between the treatment and control group experiment and the case-command study's OR arrangements. Different types of studies require specific types of ORs.

For the experiment, we put the treatment group in the numerator and the control grouping in the denominator of the odds ratio formula. Both odds in the ratio relate to infections and split up the number of infections by the number of uninfected. This arrangement allows you to calculate the odds ratio of illness in the treatment group compare to the command group.

All the same, in case-control studies, you put only the cases (ill) in the numerator and the controls (good for you) in the denominator of the odds ratio formula. Both odds in the ratio relate to exposure rather than disease. To calculate the odds ratio, you lot take the number of exposures and carve up it by the not-exposures for both the example and control groups. Instance-command studies utilise this arrangement because they start with the disease outcome equally the ground for sample choice, so the researchers need to identify take a chance factors.

Odds Ratios for Continuous Variables

When you perform binary logistic regression using the logit transformation, y'all can obtain ORs for continuous variables. Those odds ratio formulas and calculations are more complex and go beyond the telescopic of this post. Even so, I will bear witness you lot how to interpret odds ratios for continuous variables.

Unlike the groups in the previous examples, a continuous variable can increase or subtract in value. Fortunately, the interpretation of an odds ratio for a continuous variable is similar and still centers around the value of ane. When an OR is:

Greater than 1: As the continuous variable increases, the event is more than likely to occur.
Less than 1: Every bit the variable increases, the consequence is less likely to occur.
Equals 1: Equally the variable increases, the likelihood of the event does non modify.

Interpreting Odds Ratios for Continuous Variables

In another mail, I performed binary logistic regression and obtained ORs for ii continuous independent variables. Let'south interpret those odds ratios!

In that mail service, I appraise whether measures of conservativeness and establishmentarianism predict membership in the Freedom Caucus within the U.Southward. House of Representatives in 2014.

Here's how you interpret these scores:

Conservativeness: College scores represent more conservative viewpoints.
Establishmentarianism: College scores stand for viewpoints that favor the political establishment.

For this post, I'll focus on the ORs for this binary logistic model. For more details, read the full mail service: Statistical Analysis of the Republican Establishment Split.

The odds ratio interpretation for conservativeness indicates that for every 0.1 increment (the unit of change) in the conservativeness score, a House fellow member is ~ii.seven times as probable to belong to the Freedom Caucus.

Conversely, the odds ratio estimation for establishmentness indicates that for every 0.one increment in the establishmentarianism score, a Firm fellow member is but ~73% as probable to belong to the Freedom Caucus.

Taking both results together, House members who are more conservative and less favorable towards the establishment make up the Freedom Caucus.

Interpreting Confidence Intervals and P-values for Odds Ratios

So far, we've only looked at the point estimates for odds ratios. Those are the sample estimates that are a single value. Still, sample estimates e'er have a margin of mistake cheers to sampling fault. Conviction intervals and hypothesis tests (p-values) can account for that margin of error when yous're using samples to draw conclusions most populations (i.e., inferential statistics). Sample statistics are e'er wrong to some extent!

Every bit with whatever hypothesis test, there is a null and alternative hypothesis. In the context of interpreting odds ratios, the value of one represents no effect. Hence, these hypotheses focus on that value.

Null Hypothesis: The OR equals 1 (no human relationship).
Culling Hypothesis: The OR does not equal 1 (relationship exists).

If the p-value for your odds ratio is less than your significance level (e.grand., 0.05), reject the nada hypothesis. The interpretation is that difference between your sample's odds ratio and one is statistically significant. Your data provide sufficient testify to conclude that a relationship between the variable and the consequence'southward probability exists in the population.

Alternatively, you can use the confidence interval to interpret an odds ratio and depict the same conclusions equally using the p-value. If your CI excludes one, your results are significant. All the same, if your CI includes ane, you can't rule out ane as a likely value. Consequently, your results are not statistically significant.

The confidence intervals for the two Liberty Caucus odds ratios both exclude 1. Hence, they are statistically significant.

Additionally, the width of the conviction interval indicates the precision of the gauge. Narrower intervals correspond more than precise estimates.

Related posts: Descriptive vs. Inferential Statistics and Hypothesis Testing Overview

Reference

Explaining Odds Ratios (NIH)