Use this **odds ratio calculator** to easily calculate the ratio of odds, confidence intervals and p-values for the odds ratio (OR) between an exposed and control group. One and two-sided confidence intervals are reported, as well as Z-scores.

Quick navigation:

- Using the odds ratio calculator
- What are odds and what is an odds ratio?
- Odds ratio vs. Risk Ratio (Relative Risk)
- Standard error and confidence interval formula for odds ratios
- What is an odds ratio confidence interval and "confidence level"
- One-sided vs. two-sided intervals

## Using the odds ratio calculator

This odds ratio calculator allows you to perform a post-hoc statistical evaluation of odds data when the outcome of interest is the change in the odds (the odds ratio) between an exposed/treatment group and a control group. To use the tool you need to simply enter the number of events and non-events (e.g. disease and no disease) for each of the two groups. You can select any level of significance you require for the confidence intervals.

The **odds ratio calculator will output**: odds ratio, two-sided confidence interval, left-sided and right-sided confidence interval, one-sided p-value and z-score. If the test was two-sided, you need to multiply the p-value by 2 to get the two-sided p-value.

## What are odds and what is an odds ratio?

Odds are the probability of an event occurring (e.g. developing a disease or condition, being injured, dying, etc.) versus the event not occurring (e.g. staying disease-free, symptom-free, staying alive, etc.), usually between an exposed group and a control group, or a treatment group and a control group, depending on context (though connected, betting odds are a different breed). An odds ratio (OR) expresses the ratio of two odds: OR = (Events_{treatment} / Non-events_{treatment}) / (Events_{control} / Non-events_{control}). If the odds ratio equals 1 there is no effect of the treatment or exposure.

Here is a **practical example**. If we take smokers and risk of lung cancer as an example, if we know that from the exposed group (smokers) 20 developed some kind of lung cancer and 80 remained cancer free, while in the non-smokers 1 person developed lung cancer and 99 remained cancer-free, what are the relative odds of smokers versus non-smokers?

If we denoted the smokers who developed cancer with **a**, those who did not with **b**, the non-smokers who developed cancer with **c** and those who did not with **d** the formula and solution to calculate the odds ratio will look like so:

This is the equation used in our odds ratio calculator. So a smoker will have 25 higher odds to develop lung cancer compared to a non-smoker. The odds ratio should not be confused with relative risk or hazard ratios which might be close in certain cases, but are completely different measures.

## Odds ratio vs. Risk Ratio (Relative Risk)

Odds ratios are not very intuitive to understand, but are sometimes used due to convenience in plugging them in other statistics. Where possible relative risk (risk ratio) should be reported due to it being much more a intuitive measure of effectiveness. Still, odds ratios are widely used in fields like epidemiology, clinical research, including randomized control trials, as well as cohort analysis and longitudal observational studies.

One possible advantage of odds ratios is that they are invariant to the variable of interest. Odds ratios calculated using our tool will vary proportionally in both effect directions while a risk ratio is skewed and can produce very different results when looking at the complimentary proportion instead. As an extreme example of the difference between risk ratio and odds ratio, if action A carries a risk of a negative outcome of 99.9% while action B has a risk of 99.0% the relative risk is approximately 1 while the odds ratio between A and B is 10 (1% = 0.1% x 10), more than 10 times higher.

The highly disparate results in RR vs OR are due to the definition of risk based on the negative events. If we define risk by using the positive outcome instead, we get a relative risk of 0.10 which has a much better correspondence with the odds ratio. However, this is also a disadvantage given the variable of interest is properly defined, which is why risk ratios are generally preferred. If we want to talk about risk reduction we should use the relative risk defined via the risk event (odds ratio can easily be misinterpreted), but if we are interested in the increase in non-events in the above example then the reverse relative risk should be reported (odds ratio now corresponds closely to relative risk).

## Standard error and confidence interval formula for odds ratios

The standard error of the log risk ratio is known to be:

Accordingly, confidence intervals are calculated using the formula:

where **OR** is the calculated odds ratio (relative odds), **SE _{lnOR}** is the standard error for the log odds ratio and

**Z**is the score statistic, corresponding to the desired confidence level. For reference, this is the formula used for CI limit calculations in this odds ratio calculator. The Z-score corresponding to a two-sided interval at level α (e.g. 0.90) is calculated for

**Z**, revealing that a two-sided interval, similarly to a two-sided p-value, is calculated by conjoining two one-sided intervals with half the error rate. E.g. a Z-score of 1.6448 is used for a 0.95 (95%) one-sided confidence interval and a 90% two-sided interval, while 1.956 is used for a 0.975 (97.5%) one-sided confidence interval and a 0.95 (95%) two-sided interval. Therefore it is important to use the right kind of interval: more on one-tailed vs. two-tailed intervals.

_{1-α/2}## What is an odds ratio confidence interval and "confidence level"

A confidence interval is defined by an upper and lower limit for the value of a variable of interest and it aims to aid in assessing the uncertainty associated with a measurement, usually in experimental context. The wider an interval is, the more uncertainty there is in the odds ratio estimate.

Every confidence interval is constructed based on a particular required confidence level, e.g. 0.09, 0.95, 0.99 (90%, 95%, 99%) which is also the coverage probability of the interval. A 95% confidence interval (CI), for example, **will contain the true value of interest** 95% of the time (in 95 out of 5 similar experiments).

Simple two-sided confidence intervals are symmetrical around the observed odds ratio, but in certain scenarios asymmetrical intervals may be produced (complex cases, not supported by our calculator). In any particular case the true odds ratio may lie anywhere within the interval, or it might not be contained within it, no matter how high the confidence level is. Raising the confidence level widens the interval, while decreasing it makes it narrower, as you can **verify simply by setting a different level in our odds ratio calculator**. Similarly, larger sample sizes result in narrower intervals, since the interval's asymptotic behavior is to be reduced to a single point.

## One-sided vs. two-sided intervals

While odds ratio confidence intervals are customarily given in their two-sided form, this can often be misleading if we are interested if a particular value below or above the interval can be excluded at a given significance level. A one-sided interval in which one side is plus or minus infinity is appropriate when we have a null / want to make statements about an odds ratio value lying **either above or below** the top / bottom bound ^{[1]}. By design a two-sided interval is constructed as the overlap between two one-sided intervals at 1/2 the error rate ^{2}.

For example, if we have the two-sided 90% odds ratio interval covering (2.5, 10), we can actually say that odds ratios less than 2.5 are excluded with 95% confidence precisely because a 90% two-sided interval is nothing more than two conjoined 95% one-sided intervals:

Therefore, to make directional statements about relative odds based on two-sided intervals, one needs to increase the significance level for the statement. In such cases it is better to **use the appropriate one-sided odds ratio interval** instead, to avoid confusion. Our free odds ratio calculator conveniently produces both one-sided intervals for you.

#### References

[1] Georgiev G.Z. (2017) "One-tailed vs Two-tailed Tests of Significance in A/B Testing", [online] https://blog.analytics-toolkit.com/2017/one-tailed-two-tailed-tests-significance-ab-testing/ (accessed Apr 28, 2018)

Our statistical calculators **have been featured** in scientific papers and articles published in high-profile science journals by:

## FAQs

### How to calculate p value from odds ratio and confidence interval? ›

**Steps to obtain the P value from the CI for an estimate of effect (Est)**

- If the upper and lower limits of a 95% CI are u and l respectively:
- 1 calculate the standard error: SE = (u − l)/(2×1.96)
- 2 calculate the test statistic: z = Est/SE.
- 3 calculate the P value2: P = exp(−0.717×z − 0.416×z
^{2}).

**How do you find the odds ratio and confidence interval? ›**

**ci = exp(log(or) ± Zα/2*√1/a + 1/b + 1/c + 1/d)**, where Zα/2 is the critical value of the Normal distribution at α/2 (e.g. for a confidence level of 95%, α is 0.05 and the critical value is 1.96).

**What is the p value for odds ratio? ›**

This probability is called the “p-value.” The p-value is **calculated using the same numbers that are used to calculate the odds ratio**. The larger the p-value, the higher the probability that you might observe such an association as a result of chance alone and that the exposure is probably not related to the disease.

**How do you find the 95% confidence interval for an odds ratio? ›**

**The following formula is used for a 95% confidence interval (CI).**

- Upper 95% CI = e ^ [ln(OR) + 1.96 sqrt(1/a + 1/b + 1/c + 1/d)]
- Lower 95% CI = e ^ [ln(OR) - 1.96 sqrt(1/a + 1/b + 1/c + 1/d)]

**What is the p-value of a 95% confidence interval? ›**

The uncorrected p-value associated with a 95 percent confidence level is **0.05**.

**How to calculate the odds ratio? ›**

To calculate the odds ratio, you take the number of exposures and divide it by the non-exposures for both the case and control groups. Case-control studies use this arrangement because they start with the disease outcome as the basis for sample selection, and then the researchers need to identify risk factors.

**How do you find the p value of an odds ratio in R? ›**

**Following the steps in the box we calculate P as follows:**

- Est = log(0.81) = −0.211.
- l = log(0.70) = −0.357, u = log (0.94) = −0.062.
- SE = [−0.062 − (−0.357)]/(2×1.96) = 0.0753.
- z = −0.211/0.0753 = −2.802. We take the positive value of z, 2.802.
- P = exp(−0.717×2.802 − 0.416×2.8022) = 0.005.

**How to calculate the confidence interval? ›**

**To calculate the confidence interval, use the following formula:**

- Confidence interval (CI) = ‾X ± Z(S ÷ √n)
- Confidence interval = 4.5 ± 0.97(2.5 ÷ √25) = 4.5 ± 0.97(2.5 ÷ 5) = 4.5 ± 0.97(0.5) = 4.5 ± 0.485 = 4.985, 4.015.

**How do you find the p-value? ›**

The p-value is calculated using the sampling distribution of the test statistic under the null hypothesis, the sample data, and the type of test being done (lower-tailed test, upper-tailed test, or two-sided test). The p-value for: a lower-tailed test is specified by: **p-value = P(TS ts | H _{0} is true) = cdf(ts)**

**What does p-value tell you? ›**

The p-value is **the probability that the null hypothesis is true**. (1 – the p-value) is the probability that the alternative hypothesis is true. A low p-value shows that the results are replicable. A low p-value shows that the effect is large or that the result is of major theoretical, clinical or practical importance.

### Why do we calculate odds ratio? ›

Odds ratios are used **to compare the relative odds of the occurrence of the outcome of interest** (e.g. disease or disorder), given exposure to the variable of interest (e.g. health characteristic, aspect of medical history).

**How do you construct a 95 confidence interval for p? ›**

Since 95% of values fall within two standard deviations of the mean according to the 68-95-99.7 Rule, simply **add and subtract two standard deviations from the mean** in order to obtain the 95% confidence interval.

**How do you calculate odds ratio with example? ›**

In other words, it's a ratio of successes (or wins) to losses (or failures). As an example, if a racehorse runs 100 races and wins 20 times, the odds of the horse winning a race is 20/80 = 1/4. The above odds definition is the odds in favor of an event.

**Is p-value 0.05 the same as 95 confidence interval? ›**

In accordance with the conventional acceptance of statistical significance at a P-value of 0.05 or 5%, **CI are frequently calculated at a confidence level of 95%**. In general, if an observed result is statistically significant at a P-value of 0.05, then the null hypothesis should not fall within the 95% CI.

**What is p-value and confidence interval? ›**

In exploratory studies, p-values enable the recognition of any statistically noteworthy findings. Confidence intervals provide information about a range in which the true value lies with a certain degree of probability, as well as about the direction and strength of the demonstrated effect.

**How do you calculate p-value from confidence? ›**

**(a) CI for a difference**

- 1 calculate the test statistic for a normal distribution test, z, from P3: z = −0.862 + √[0.743 − 2.404×log(P)]
- 2 calculate the standard error: SE = Est/z (ignoring minus signs)
- 3 calculate the 95% CI: Est –1.96×SE to Est + 1.96×SE.

**How do you calculate odds ratio and risk ratio? ›**

For example, when the odds are 1:10, or 0.1, one person will have the event for every 10 who do not, and, using the formula, the risk of the event is **0.1/(1+0.1) = 0.091**. In a sample of 100, about 9 individuals will have the event and 91 will not.

**What are odds and odds ratios? ›**

**Odds are the probability of an event occurring divided by the probability of the event not occurring.** **An odds ratio is the odds of the event in one group**, for example, those exposed to a drug, divided by the odds in another group not exposed. Odds ratios always exaggerate the true relative risk to some degree.

**How do you find the p-value and R value? ›**

The following describes the calculations to compute the test statistics and the p-value: The p-value is calculated using a t-distribution with n – 2 degrees of freedom. The formula for the test statistic is **t=r√n−2√1−r2** t = r n − 2 1 − r 2 .

**What is the p-value for 99% confidence interval? ›**

It turns out that the p value is 0.0057. There is a similar relationship between the 99% confidence interval and significance at the 0.01 level. Whenever an effect is significant, all values in the confidence interval will be on the same side of zero (either all positive or all negative).

### How do you find p-value from z-score in R? ›

In the statistics program R, the conversion of a z-score into a p-value uses the command **pnorm(z, mean, sd)**. For significance testing we want to know how extreme the observed z-score is relative to the null-hypothesis, which is defined by a standard normal distribution with mean = 0, and sd = 1).

**How do you calculate odds ratio and p value in Excel? ›**

The formula **=EXP(SUMPRODUCT(I28:I30,LN(H28:H30))/I31)** is used to calculate the alternative common odds ratio in cell H31. The value of p (cell I31) is calculated by =SUM(I28:I30) where p_{1} (in cell I28) is calculated by the formula =1/SUMPRODUCT(1/B4:E4).

**How do you calculate confidence interval using Excel data analysis? ›**

**How to calculate confidence interval in Excel**

- Calculate the sample mean. Arrange your data in ascending order in your spreadsheet. ...
- Find the standard deviation. Apply the =STDEV. ...
- Input the alpha value. ...
- Type in the confidence function. ...
- Calculate the confidence interval.

**What are P values and confidence intervals for dummies? ›**

p-values simply provide a cut-off beyond which we assert that the findings are 'statistically significant' (by convention, this is p<0.05). **A confidence interval that embraces the value of no difference between treatments indicates that the treatment under investigation is not significantly different from the control.**

**What is 95% confidence interval example? ›**

For example, if you are estimating a 95% confidence interval around the mean proportion of female babies born every year based on a random sample of babies, you might find **an upper bound of 0.56 and a lower bound of 0.48**. These are the upper and lower bounds of the confidence interval. The confidence level is 95%.

**Why do we calculate confidence intervals? ›**

Why have confidence intervals? Confidence intervals are **one way to represent how "good" an estimate is**; the larger a 90% confidence interval for a particular estimate, the more caution is required when using the estimate. Confidence intervals are an important reminder of the limitations of the estimates.

**What is the formula of probability p? ›**

...

Basic Probability Formulas.

All Probability Formulas List in Maths | |
---|---|

Independent Events | P(A∩B) = P(A) ⋅ P(B) |

Conditional Probability | P(A | B) = P(A∩B) / P(B) |

**What does p-value 0.05 mean? ›**

**P > 0.05 is the probability that the null hypothesis is true**. 1 minus the P value is the probability that the alternative hypothesis is true. A statistically significant test result (P ≤ 0.05) means that the test hypothesis is false or should be rejected. A P value greater than 0.05 means that no effect was observed.

**What is an example of p-value? ›**

P-values are expressed as decimals and can be converted into percentage. For example, **a p-value of 0.0237 is 2.37%**, which means there's a 2.37% chance of your results being random or having happened by chance. The smaller the P-value, the more significant your results are.

**What is p-value and why is it important? ›**

The P value means **the probability, for a given statistical model that, when the null hypothesis is true, the statistical summary would be equal to or more extreme than the actual observed results** [2].

### Where is odds ratio used? ›

It is standard in the medical literature to calculate the odds ratio and then use the rare-disease assumption (which is usually reasonable) to claim that the relative risk is approximately equal to it.

**How do you find the p-value from a confidence interval? ›**

For the p-value, we just take the effect estimate and divide it by the standard error of the effect estimate to get a z score from which we can calculate the p-value.

**How do you calculate p-value from confidence level? ›**

You can use either P values or confidence intervals to determine whether your results are statistically significant. If a hypothesis test produces both, these results will agree. **The confidence level is equivalent to 1 – the alpha level**. So, if your significance level is 0.05, the corresponding confidence level is 95%.

**Can you tell p-value from confidence interval? ›**

The width of the confidence interval and the size of the p value are related, **the narrower the interval, the smaller the p value**. However the confidence interval gives valuable information about the likely magnitude of the effect being investigated and the reliability of the estimate.

**Where do I find p-value? ›**

Graphically, the p value is **the area in the tail of a probability distribution**. It's calculated when you run hypothesis test and is the area to the right of the test statistic (if you're running a two-tailed test, it's the area to the left and to the right).

**Why use confidence intervals instead of p-values? ›**

Interpretation of trial results based solely on P values can be misleading. Confidence intervals are preferred because they give the plausible range of the treatment effects and a direct indication of the clinical importance of the trial results.