Odds ratio: Definition & Examples

An odds ratio compares the odds of the same event in two groups. If a credit analyst reports a default odds ratio of 0.444 for group A relative to group B, A has about 44.4% of B's default odds. That statement alone does not say that A has 44.4% of B's default probability.

Write

p_A=\mathbb{P}(D\mid A)

and

p_B=\mathbb{P}(D\mid B)

for the conditional probabilities of default over the same horizon. For

0<p_A,p_B<1

\mathrm{OR}_{A:B} =\frac{p_A/(1-p_A)}{p_B/(1-p_B)}.

It is dimensionless. An odds ratio of 1 means equal event probabilities, above 1 means higher odds in A, and below 1 means lower odds in A. The order of the groups and the definition of the event belong in the statement of the result.

Two loan cohorts, three answers

Consider two synthetic complete cohorts, each containing 100 loans followed for one year, with every loan's default outcome known. These invented counts illustrate arithmetic; they are not market estimates.

One-year outcome	Cohort A	Cohort B
Default	10	20
No default	90	80
Total loans	100	100

A's default probability is $10/100=10\%$ , but its default odds are $10/90=1/9$ . B's default probability is $20\%$ , and its default odds are $20/80=1/4$ . Consequently,

\mathrm{OR}_{A:B}=\frac{10/90}{20/80} =\frac{10\times80}{90\times20} =\frac49\approx0.444.

A has about 55.6% lower default odds than B. For the same table, the risk ratio, which compares probabilities, is

0.10/0.20=0.50

: A's default probability is half B's. The risk difference is

0.10-0.20=-0.10

, or 10 percentage points lower. These measures answer different questions; 55.6% lower odds is not a 55.6% reduction in default probability.

Synthetic one-year loan cohorts: A has 10 defaults and 90 nondefaults; B has 20 defaults and 80 nondefaults. A versus B odds ratio is 4/9, risk ratio is 1/2 and risk difference is minus 10 percentage points.

Original synthetic complete-cohort comparison, with 100 loans in each group and the same one-year default definition. Odds use nondefaults as the denominator; risks use all loans.

Reversing the reference group gives

\mathrm{OR}_{B:A}=9/4=2.25

. Keeping the groups fixed but changing the event to no default also takes the reciprocal. An unlabelled odds ratio is therefore easy to misread.

Reading a two-by-two table

Let the counts be $a$ defaults and $b$ nondefaults in A, and $c$ defaults and $d$ nondefaults in B. With all four cells positive,

\mathrm{OR}_{A:B}=\frac{a/b}{c/d}=\frac{ad}{bc}.

This is the cross-product ratio. In a fully observed cohort, the corresponding risks are

a/(a+b)

and

c/(c+d)

; using those denominators produces the risk ratio instead.

The sampling design matters. If a study deliberately selects defaulted loans and nondefaulted controls, its chosen mix of cases and controls generally cannot reveal population default probabilities. In a simple unmatched design, if selection probabilities depend only on default status, the population odds ratio is preserved in the selected distribution: the two outcome-specific selection factors cancel. The sample ratio then estimates that parameter. That algebra does not validate a biased control sample. Matched studies and other sampling schemes require their corresponding analysis; a selected sample's raw default fraction is not automatically a population default risk. CDC's comparison of association measures explains the standard cohort versus case-control distinction.

For observed data, the computed ratio is an estimate. A small table can be very uncertain even when its ratio looks large. Zero cells can give zero, an infinite boundary value or an undefined ratio, depending on which cells vanish. Do not silently add a pseudocount to every cell: any correction, exact procedure or penalised model changes the analysis and needs an explicit justification.

Why a logistic coefficient becomes an odds ratio

A simple logistic regression writes

\log\frac{p(x)}{1-p(x)}=\beta_0+\beta_1x.

Increasing

x

by one unit changes log-odds by

\beta_1

, so it multiplies odds by

e^{\beta_1}

. With an indicator

x=1

for A and

x=0

for B,

e^{\beta_1}

is the model's A-to-B odds ratio. With additional covariates, this interpretation holds at fixed values of those covariates, provided the model has the stated additive form without interactions involving

x

. Adjusted and raw table odds ratios need not agree. UCLA's coefficient interpretation derives this exponentiation.

An odds ratio still needs a baseline probability to become a probability forecast. If the odds ratio is $\theta$ and B's probability is $p_B$ , then

p_A=\frac{\theta p_B}{1-p_B+\theta p_B}.

Substituting $\theta=4/9$ and $p_B=0.20$ recovers $p_A=0.10$ . The distinction from a risk ratio can also be seen directly:

\mathrm{OR}_{A:B} =\frac{p_A}{p_B}\frac{1-p_B}{1-p_A}.

When both event probabilities are small, the second factor is close to 1, so the odds ratio approximates the risk ratio. This is an approximation with a baseline-dependent error, not a permission to relabel every odds ratio as relative risk.

Finally, an odds ratio measures association. Different leverage, collateral, underwriting or macroeconomic exposure could explain the cohort contrast. Neither the table nor exponentiating a fitted coefficient establishes that changing group membership would cause the reported change in default risk.

To build the model behind the coefficient, continue to Logistic Regression and Maximum Likelihood (public preview; full lesson requires membership).

Sources

James, G., Witten, D., Hastie, T., Tibshirani, R. and Taylor, J. An Introduction to Statistical Learning: with Applications in Python (2023), §4.3.1, “The Logistic Model,” pp. 139–140: odds, log-odds and multiplicative coefficient interpretation.
CDC. Principles of Epidemiology in Public Health Practice, third edition, Lesson 3, §5, “Measures of Association”: risk ratios, cross-product odds ratios and standard case-control sampling.
UCLA OARC. “Logistic Regression in Stata,” sections on the logistic model, coefficient interpretation and odds ratios.

Two loan cohorts, three answers

Reading a two-by-two table

Why a logistic coefficient becomes an odds ratio

Sources

Related terms

Conditional probability

Odds