PUBLIC LIBRARY / CALCULATOR

Relative risk and odds ratio

Measures exposure–outcome association in cohorts and trials (relative risk), and case-control studies (odds ratio).

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Check the population, units and version. The result shows the formula or classification; it does not make an automatic treatment decision.

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Method · Limits of application

RR/Katz 1978 log CI; OR/Woolf 1955 log CI; Haldane 0.5 for zero; 95% z=1.96

Use counts from two independent groups and the same binary outcome; this table does not model matched pairs or person-time rates. Odds ratio and relative risk are not the same measure and must not be presented as interchangeable. Logarithmic intervals are approximations; Woolf 1955 notes the limitation when any frequency is small. When any cell is zero, this implementation adds 0.5 to all four cells; this choice does not establish validity in small samples. Association does not establish causation.

Conditions of use

Check the population, units, inclusion and exclusion criteria, and version in the original source. A result alone does not establish a diagnosis, discharge decision or prescription.

Documented parameters

Study design
Exposed with the outcome (a)
Exposed without the outcome (b)
Unexposed with the outcome (c)
Unexposed without the outcome (d)

4/4 reference cases checked. Numerical tests are not clinical validation.

Sources and documentation

Documented formula

Transcribed from the source or a recorded revision; this is not a clinical recommendation.

RR = [a/(a+b)] / [c/(c+d)], SE(ln RR) = √(1/a − 1/(a+b) + 1/c − 1/(c+d)) (Katz, 1978).

OR = (a × d) / (b × c), SE(ln OR) = √(1/a + 1/b + 1/c + 1/d) (Woolf, 1955).

CI 95% = exp(ln measure ± 1.96 × SE). If any cell is zero, add 0.5 to all cells (Haldane correction).

References

Identifier: risco-relativo-e-odds-ratio

Original clinical content: Brazilian Portuguese.

Method version

RR/Katz 1978 log CI; OR/Woolf 1955 log CI; Haldane 0.5 for zero; 95% z=1.96

risco-relativo-e-odds-ratio@native-2026-10-05+ebd85183dcf0

Katz D, Baptista J, Azen SP, Pike MC. Obtaining confidence intervals for the risk ratio in cohort studies. Biometrics, 1978.

The bibliographic edition identifies the source. By itself, it proves neither full conformity of the formula nor clinical approval.

Synthetic cases checked (4)

Synthetic cases for checking the implementation, with no patient data.

Case 1

reference-001

Inputs

Study design: Cohort or clinical trial
Exposed with the outcome (a): 20
Exposed without the outcome (b): 80
Unexposed with the outcome (c): 10
Unexposed without the outcome (d): 90

{
  "desenho": "coorte",
  "a": 20,
  "b": 80,
  "c": 10,
  "d": 90
}

Expected result

{
  "rr": 2,
  "rrLo": 0.987,
  "rrHi": 4.054,
  "or": 2.25,
  "orLo": 0.994,
  "orHi": 5.092
}

Actual result

{
  "rr": 2,
  "rrLo": 0.9865632012325182,
  "rrHi": 4.0544792214049545,
  "or": 2.25,
  "orLo": 0.9942948858766125,
  "orHi": 5.091547861615204
}

Case 2

reference-002

Inputs

Study design: Cohort or clinical trial
Exposed with the outcome (a): 30
Exposed without the outcome (b): 70
Unexposed with the outcome (c): 10
Unexposed without the outcome (d): 90

{
  "desenho": "coorte",
  "a": 30,
  "b": 70,
  "c": 10,
  "d": 90
}

Expected result

{
  "rr": 3,
  "rrLo": 1.551,
  "rrHi": 5.803
}

Actual result

{
  "rr": 2.9999999999999996,
  "rrLo": 1.5508297114721117,
  "rrHi": 5.803345095482356,
  "or": 3.857142857142857,
  "orLo": 1.7666027787432017,
  "orHi": 8.421559843233334
}

Case 3

reference-003

Inputs

Study design: Case-control
Exposed with the outcome (a): 30
Exposed without the outcome (b): 70
Unexposed with the outcome (c): 10
Unexposed without the outcome (d): 90

{
  "desenho": "caso",
  "a": 30,
  "b": 70,
  "c": 10,
  "d": 90
}

Expected result

{
  "or": 3.857,
  "orLo": 1.767,
  "orHi": 8.421
}

Actual result

{
  "rr": 2.9999999999999996,
  "rrLo": 1.5508297114721117,
  "rrHi": 5.803345095482356,
  "or": 3.857142857142857,
  "orLo": 1.7666027787432017,
  "orHi": 8.421559843233334
}

Case 4

reference-004

Inputs

Study design: Cohort or clinical trial
Exposed with the outcome (a): 10
Exposed without the outcome (b): 90
Unexposed with the outcome (c): 30
Unexposed without the outcome (d): 70

{
  "desenho": "coorte",
  "a": 10,
  "b": 90,
  "c": 30,
  "d": 70
}

Expected result

{
  "rr": 0.3333,
  "rrLo": 0.1723,
  "rrHi": 0.6448
}

Actual result

{
  "rr": 0.33333333333333337,
  "rrLo": 0.17231441238579018,
  "rrHi": 0.6448161217202621,
  "or": 0.25925925925925924,
  "orLo": 0.11874284795393257,
  "orHi": 0.5660582061981252
}

Code and reproducibility

This tool’s code on GitHub

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Sources, version and reproduction instructions

README

Local installation

Code license: Apache-2.0

Guidelines and conditions

This repository’s license applies to the code. Third-party instruments, questionnaires, data and models remain subject to their owners’ conditions.

The code tests are technical checks. Clinical approval and professional translation review are recorded separately.

Relationship with cancer research

Support for research — context required

Measures of association for 2×2 tables. Application to oncology research depends on the design, outcome and selected data.

Applications: Clinical trials and methods · Epidemiology