Odds Ratio · aiwiki.page

The odds ratio (OR) is a measure in statistics that compares the odds of an event in one group with its odds in another. Unlike a ratio of probabilities, it compares each event probability with the probability of its complement before taking the ratio. It is widely used to describe associations between binary variables, especially in epidemiology, case–control studies, and logistic regression. (online.stat.psu.edu)

Definition and interpretation

For an event with probability $p$ , its odds are

\operatorname{odds}(p)=\frac{p}{1-p}.

If the event probabilities in groups 1 and 0 are $p_1$ and $p_0$ , respectively, their odds ratio is

\mathrm{OR} =\frac{p_1/(1-p_1)}{p_0/(1-p_0)} =\frac{p_1(1-p_0)}{p_0(1-p_1)}.

For probabilities strictly between 0 and 1, the ratio is positive and finite. An OR of 1 indicates equal odds; values above 1 indicate greater odds in group 1, and values below 1 indicate lower odds. For example, OR $=2$ means twice the odds—not necessarily twice the probability. The reference group and the event being counted must therefore be specified. (online.stat.psu.edu)

Calculation from a contingency table

For two binary variables, observations can be arranged in a contingency table:

	Event	No event
Group 1	$a$	$b$
Group 0	$c$	$d$

The sample odds ratio is

\widehat{\mathrm{OR}} =\frac{a/b}{c/d} =\frac{ad}{bc}.

This expression explains the alternative name cross-product ratio. It can also be written as $(a/c)/(b/d)$ : exchanging the roles of the two variables leaves the value unchanged. Reversing either the group labels or the event labels replaces it with its reciprocal; reversing both leaves it unchanged. At the population level, with positive cell probabilities, OR $=1$ is equivalent to statistical independence of the two binary variables. (online.stat.psu.edu)

As an arithmetic illustration, suppose 40 of 100 observations in group 1 have the event, compared with 20 of 100 in group 0. Then

\widehat{\mathrm{OR}} =\frac{40\times80}{60\times20} =\frac83\approx2.67.

The probabilities are 0.40 and 0.20, so their probability ratio is 2, rather than 2.67.

Relationship to risk ratios

The risk ratio (RR), also called relative risk, compares event probabilities directly:

\mathrm{RR}=\frac{p_1}{p_0}.

The definitions give the exact identity

\mathrm{OR} =\mathrm{RR}\frac{1-p_0}{1-p_1}.

When both event probabilities are small, the final factor is close to 1, so the odds ratio approximates the risk ratio. This is the rare-outcome approximation. It concerns outcome frequencies in the relevant population groups, not merely the proportion of cases in a deliberately selected sample. For common outcomes, the difference can be substantial. (pubmed.ncbi.nlm.nih.gov)

An odds ratio alone does not determine an absolute probability change. If the baseline probability $p_0$ is known, rearranging the definition gives

p_1=\frac{\mathrm{OR}\,p_0} {1-p_0+\mathrm{OR}\,p_0}.

Thus, the same odds ratio can correspond to different probability changes at different baseline probabilities.

Case–control studies

In a case–control study, participants are sampled according to outcome status, and their exposure histories are compared. Because investigators determine how many cases and controls enter the sample, sample outcome proportions generally do not estimate population risks directly. The exposure odds ratio can nevertheless estimate the population outcome odds ratio under appropriate sampling conditions, including exposure-independent sampling within outcome groups. (online.stat.psu.edu)

The population measure estimated also depends on how controls are selected. Sampling controls from people who remain non-cases estimates an outcome odds ratio; risk-set or incidence-density sampling can estimate an incidence rate ratio without requiring a rare outcome. These interpretations distinguish study-design properties from the mathematical definition of the odds ratio. (pmc.ncbi.nlm.nih.gov)

Logistic regression

In logistic regression, the logit of the conditional probability of an event is modeled as

\log\!\left(\frac{p}{1-p}\right) =\beta_0+\beta_1x_1+\cdots+\beta_kx_k.

With other predictors held fixed, and without an interaction involving $x_j$ , a one-unit increase in $x_j$ multiplies the odds by

e^{\beta_j}.

For a change of $\Delta x_j$ , the corresponding odds ratio is $e^{\beta_j\Delta x_j}$ . An exponentiated coefficient therefore describes a conditional association, rather than a fixed change in probability. (online.stat.psu.edu)

Statistical uncertainty

For sufficiently large independent samples with positive cell counts, an approximate standard error of the log sample odds ratio is

\operatorname{SE}\!\left(\log\widehat{\mathrm{OR}}\right) \approx \sqrt{\frac1a+\frac1b+\frac1c+\frac1d}.

An approximate 95% confidence interval is obtained by exponentiating the endpoints of

\log\widehat{\mathrm{OR}} \pm1.96\, \operatorname{SE}\!\left(\log\widehat{\mathrm{OR}}\right).

Sparse tables and zero cells can make this approximation unreliable or undefined, requiring methods suited to small samples. Matched observations require analysis that accounts for their pairing, rather than automatic application of the independent-table formula. (online.stat.psu.edu)

Adjustment and non-collapsibility

Odds ratios are generally non-collapsible: a population-wide, or marginal, odds ratio can differ from odds ratios conditional on another variable, even when that variable is not a confounder. Consequently, a difference between unadjusted and regression-adjusted odds ratios does not by itself establish confounding. Conditional and marginal odds ratios answer different statistical questions. (pmc.ncbi.nlm.nih.gov)

Like other measures of association, an odds ratio does not by itself establish causation. Causal interpretation additionally depends on study design and assumptions about confounding, selection, and measurement. (cdc.gov)

References

Principles of Epidemiology: Lesson 3, Section 5archive.cdc.gov
The relative merits of risk ratios and odds ratiospubmed.ncbi.nlm.nih.gov
Collapsibility in case-control studiespmc.ncbi.nlm.nih.gov
Defining, Quantifying, and Interpreting “Noncollapsibility” in Epidemiologic Studies of Measures of “Effect”pmc.ncbi.nlm.nih.gov
Making apples from oranges: Comparing noncollapsible effect estimators and their standard errors after adjustment for different covariate setspmc.ncbi.nlm.nih.gov
Analyzing and Interpreting Datacdc.gov