Odds Ratio Calculator Formula

Understand the math behind the odds ratio calculator. Each variable explained with a worked example.

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Formulas Used

Odds Ratio (OR)

odds_ratio = (a * d) / (b * c)

ln(OR)

log_or = log((a * d) / (b * c))

Odds Group 1

odds_group1 = a / b

Odds Group 2

odds_group2 = c / d

Variables

VariableDescriptionDefault
aGroup 1 - Event (a)30
bGroup 1 - No Event (b)70
cGroup 2 - Event (c)15
dGroup 2 - No Event (d)85

How It Works

How to Calculate the Odds Ratio

Formula

OR = (a * d) / (b * c)

where a, b, c, d are cells of the 2x2 table:

  • a = exposed cases, b = exposed non-cases
  • c = unexposed cases, d = unexposed non-cases

    • OR = 1 means no association. OR > 1 means the exposure increases the odds of the outcome. OR < 1 means the exposure is protective.

    Worked Example

    Group 1: 30 events, 70 non-events. Group 2: 15 events, 85 non-events.

    a = 30b = 70c = 15d = 85
    1. 01Odds Group 1 = 30/70 = 0.4286
    2. 02Odds Group 2 = 15/85 = 0.1765
    3. 03OR = (30*85) / (70*15) = 2550 / 1050 = 2.4286
    4. 04ln(OR) = ln(2.4286) = 0.887
    5. 05The odds of the event are 2.43 times higher in Group 1

    Frequently Asked Questions

    How does odds ratio differ from relative risk?

    The odds ratio compares odds (event/non-event), while relative risk compares probabilities (event/total). For rare events (< 10%), OR approximates RR. For common events, OR exaggerates the association.

    Why is the log of OR used?

    The log odds ratio has better statistical properties: it is symmetric around 0 (OR=1), approximately normally distributed, and is used in logistic regression. Confidence intervals are computed on the log scale then exponentiated.

    Can the odds ratio be zero or negative?

    The OR is always positive (since all cells are non-negative counts). It can be zero only if a = 0 or d = 0 (no events in one cell). It can never be negative.

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