Adjusted Odds Ratio Calculator (AOR Formula, Logistic & MH)

Calculate an adjusted odds ratio (AOR) and confidence interval from a logistic regression coefficient and SE, or pool two strata by Mantel-Haenszel.

Related Calculators

Adjusted Odds Ratio Formula

The adjusted odds ratio depends on the input method. If you have logistic regression output, the calculator exponentiates the regression coefficient. If you have stratified 2×2 tables, it uses the Mantel-Haenszel pooled odds ratio.

From logistic regression coefficient and standard error

AOR = exp(beta)

CI = exp(beta ± z×SE)

zstat = beta/SE

AOR = adjusted odds ratio
beta = logistic regression coefficient on the log-odds scale
SE = standard error of the coefficient
z = critical value for the selected confidence level, such as 1.96 for 95%
CI = confidence interval for the adjusted odds ratio
z_stat = Wald test statistic used to estimate the p-value

From stratified 2×2 data using Mantel-Haenszel

ORMH = sum(ai×di/ni) / sum(bi×ci/ni)

CI = exp(ln(ORMH) ± z×SEln)

OR_MH = Mantel-Haenszel adjusted odds ratio
a_i = exposed cases in stratum i
b_i = exposed non-cases in stratum i
c_i = unexposed cases in stratum i
d_i = unexposed non-cases in stratum i
n_i = total observations in stratum i
SE_ln = standard error of the log Mantel-Haenszel odds ratio

The From β & SE mode is for logistic regression output. Use it when the model already includes the covariates or confounders you want to adjust for.

The Mantel-Haenszel mode is for stratified count data. It pools the stratum-specific 2×2 tables into one odds ratio adjusted for the stratifying variable.

The Mantel-Haenszel interval uses the Robins–Breslow–Greenland log-variance estimate and the selected confidence-level z value. The default zero-cell convention adds 0.5 to all eight cells if any cell is zero; you can choose no adjustment. Each stratum must contain observations. Odds ratios describe odds, not risks, and do not establish causation. A zero standard error produces a degenerate interval and cannot establish statistical significance.

Crude odds ratio from one 2×2 table

Select Crude odds ratio: one 2×2 table for unadjusted counts. Enter a = exposed cases, b = exposed non-cases, c = unexposed cases, and d = unexposed non-cases. These should be independent observations, with each observation in one cell. This mode does not control for confounders.

Crude OR = (a × d) / (b × c)
SE(ln OR) = √(1/a + 1/b + 1/c + 1/d)
Approximate CI = exp(ln OR ± z × SE(ln OR))

The result shows the crude odds ratio, the selected 90%, 95%, or 99% approximate confidence interval, and whether a correction was used. At 95%, z is approximately 1.96. If any cell is zero, the default adds 0.5 to all four cells before calculating the ratio and interval; select No adjustment to require positive cells instead. Both groups and both outcome categories must be represented. This log-normal (Woolf) interval can be unreliable for sparse data; specialist exact methods may be more appropriate. The odds ratio describes association, not a risk ratio or proof of causation.

Confidence Levels and Odds Ratio Interpretation

Confidence level	Critical value used	Effect on interval width
90%	1.6449	Narrower interval
95%	1.9600	Common default for published results
99%	2.5758	Wider interval

Adjusted odds ratio result	Typical interpretation
AOR = 1	No difference in odds after adjustment
AOR > 1	Higher adjusted odds in the exposed or comparison group
AOR < 1	Lower adjusted odds in the exposed or comparison group
CI includes 1	Not statistically significant at that confidence level
CI does not include 1	Statistically significant at that confidence level

Example

Example 1: Adjusted odds ratio from logistic regression output

Suppose a logistic regression model gives:

beta = 0.693
SE = 0.250
Confidence level = 95%

Calculate the adjusted odds ratio:

AOR = exp(0.693) \approx 2.00

Calculate the 95% confidence interval:

CI = exp(0.693 ± 1.96×0.250) ≈ 1.23 to 3.26

The adjusted odds ratio is about 2.00. Since the 95% CI does not include 1, the association is statistically significant at the 95% level.

Example 2: Mantel-Haenszel adjusted odds ratio

Suppose you have two strata:

Stratum 1: a = 30, b = 70, c = 20, d = 80
Stratum 2: a = 40, b = 60, c = 25, d = 75

For stratum 1, n = 200, so a*d/n = 12.00 and b*c/n = 7.00.

For stratum 2, n = 200, so a*d/n = 15.00 and b*c/n = 7.50.

ORMH = (12.00 + 15.00) / (7.00 + 7.50) = 1.86

The Mantel-Haenszel adjusted odds ratio is about 1.86, with a 95% CI of about 1.20 to 2.90.

FAQ

What is an adjusted odds ratio?

An adjusted odds ratio is an odds ratio that accounts for other variables. In logistic regression, this means the coefficient is estimated while other predictors are included in the model. In a stratified analysis, this means the odds ratio is pooled across strata, such as age groups or study sites.

How is an adjusted odds ratio different from an unadjusted odds ratio?

An unadjusted odds ratio compares two groups without controlling for other variables. An adjusted odds ratio attempts to remove the influence of specified confounders. For example, if age is related to both exposure and outcome, adjusting for age can give a more appropriate estimate of the exposure-outcome association.

What does it mean if the confidence interval includes 1?

If the confidence interval includes 1, the result is not statistically significant at that confidence level. An odds ratio of 1 means equal odds, so an interval that crosses 1 includes the possibility of no association.