Enter an odds ratio (OR) or a hazard ratio (HR) plus the baseline risk (event probability) in the reference group to estimate the missing measure. This is an approximation that assumes proportional hazards and a constant hazard over the follow-up period (exponential survival), and that the OR describes the event odds by the same time point as the baseline risk.

Odds Ratio To Hazard Ratio Calculator (Approx.)

Enter Baseline Risk and either Odds Ratio or Hazard Ratio (leave the other blank).

Related Calculators

Odds Ratio to Hazard Ratio Formula (Approximate)

There is no universally valid way to convert an odds ratio (OR) to a hazard ratio (HR) without additional information. One common approximation uses the baseline event risk in the reference group by a specific time point and assumes proportional hazards with an exponential survival model (constant hazards over time).

p_1=\frac{OR\cdot p_0}{(1-p_0)+OR\cdot p_0},\qquad HR \approx \frac{\ln(1-p_1)}{\ln(1-p_0)}

Variables:

  • HR is the hazard ratio
  • OR is the odds ratio
  • p0 is the baseline risk (event probability) in the reference group by a specified time point
  • p1 is the risk (event probability) in the comparison group by that same time point
  • ln is the natural logarithm function

To estimate HR from OR, first convert OR and the baseline risk p0 into the implied comparison-group risk p1, then compute HR using the log survival relationship. This is only an approximation and should not replace a proper time-to-event analysis when individual follow-up times or survival curves are available.

What is an Odds Ratio?

An odds ratio (OR) is a measure of association between an exposure and an outcome. It represents the odds that an outcome will occur given a particular exposure, compared to the odds of the outcome occurring without that exposure. An OR of 1 indicates no association, an OR greater than 1 indicates a positive association, and an OR less than 1 indicates a negative association.

What is a Hazard Ratio?

A hazard ratio (HR) compares the instantaneous event rate (hazard) in one group to the instantaneous event rate in another group over time. It is commonly used in survival analysis. An HR of 1 indicates no difference in hazard, an HR greater than 1 indicates a higher hazard, and an HR less than 1 indicates a lower hazard.

How to Calculate Odds Ratio to Hazard Ratio?

The following steps outline how to estimate the hazard ratio from an odds ratio when you also know the baseline risk by a specific time point and you are willing to use the exponential survival approximation.

  1. Determine the odds ratio (OR) for the event by a specific time point.
  2. Determine the baseline risk p0 (event probability) in the reference group by that same time point.
  3. Compute p1 from OR and p0: p1 = (OR × p0) / ((1 − p0) + OR × p0).
  4. Estimate HR: HR ≈ ln(1 − p1) / ln(1 − p0).
  5. After inserting the values and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Odds Ratio (OR) = 2.5, Baseline Risk (p0) = 0.20

First compute p1 = (2.5 × 0.20) / ((1 − 0.20) + (2.5 × 0.20)) = 0.3846. Then estimate HR ≈ ln(1 − 0.3846) / ln(1 − 0.20) ≈ 2.176.