Odds ratios: calculation and interpretation

Odds are the ratio of something happening to it not happening; probability is the ratio of it happening to all outcomes. They describe the same underlying fact and are easy to confuse.

odds = p / (1 − p), and to go back, p = odds / (1 + odds). Note that odds are unbounded above while probability stops at 1 — which is precisely why odds are the quantity modelled in logistic regression.

An odds ratio divides the odds in one group by the odds in another. It answers “by what factor are the odds different?”, which is a narrower question than “by what factor is the outcome more likely?”

The asymmetry of odds is worth dwelling on, because it is the source of most confusion. Probability is bounded at both ends: it cannot fall below 0 or rise above 1. Odds are bounded only below, at zero, and rise without limit — a probability of 0.99 gives odds of 99, and 0.999 gives 999. This is why doubling the odds is a coherent operation at any starting point while doubling a probability is not: a probability of 0.6 cannot be doubled. That single property is what makes odds the natural scale for modelling, and it is also why an odds ratio and a risk ratio can differ so much for common outcomes.

A study of 100 participants compares an intervention against a control, with improvement as the outcome.

Step 1 — the odds in each group

Intervention: 34 improved, 16 did not, so odds = 34 / 16 = 2.125.

Control: 22 improved, 28 did not, so odds = 22 / 28 = 0.786.

Step 2 — the ratio

OR = 2.125 / 0.786 = 2.705

Equivalently, and more quickly, as the cross-product: OR = (a × d) / (b × c) = (34 × 28) / (16 × 22) = 952 / 352 = 2.705.

Step 3 — what it means

The odds ratio is symmetric in a way the risk ratio is not. Swapping the rows — treating the control group as the reference — gives 1 / 2.705 = 0.370, and modelling non-improvement instead of improvement gives the same reciprocal. That relationship holds exactly, which is part of why the measure is convenient to work with and why its logarithm is symmetric around zero.

The odds of improvement were about 2.7 times higher in the intervention group. Note the wording carefully: the odds, not the probability.

A third measure is worth knowing and is often the most useful of the three for a practical decision: the absolute risk difference. Here it is 0.68 − 0.44 = 0.24, or 24 percentage points. Its reciprocal gives the number needed to treat: 1 / 0.24 = 4.2, meaning roughly four people must receive the intervention for one additional person to improve. Ratios describe the strength of an association; absolute measures describe what it is worth doing about it, and clinical and policy audiences generally want the second.

For comparison, the risk ratio

RR = (34/50) / (22/50) = 0.68 / 0.44 = 1.545. Participants were about 1.5 times as likely to improve — a considerably more modest statement than 2.7.

An odds ratio without an interval is uninterpretable, and the interval cannot be computed directly on the ratio scale because ratios are skewed. It is calculated on the log scale and then converted back.

Step 1 — the standard error of the log odds ratio

SE(ln OR) = √(1/a + 1/b + 1/c + 1/d)

= √(1/34 + 1/16 + 1/22 + 1/28) = √(0.0294 + 0.0625 + 0.0455 + 0.0357) = √0.1731 = 0.4160

Step 2 — the interval on the log scale

ln(2.705) = 0.9949, so the interval is 0.9949 ± (1.96 × 0.4160) = [0.1795, 1.8104].

Step 3 — convert back

95% CI = [e⁰·¹⁸₀, e¹·⁸¹₁] = [1.20, 6.11]

The interval excludes 1, so the association is significant at the .05 level. But it runs from 1.20 to 6.11 — the data are consistent with a modest effect and with a very large one, which is what a sample of 100 buys you.

The width of that interval is driven by the smallest cell. Look again at the standard error formula: it sums the reciprocals of all four counts, so a cell containing 5 contributes 0.20 while a cell containing 50 contributes only 0.02. One sparse cell therefore dominates the uncertainty regardless of how large the rest of the table is. This is why a study with 500 participants but only 8 events can produce an interval as wide as one with 60 participants evenly spread, and why total sample size alone is a poor guide to how precise an odds ratio will be.

The pattern is consistent: the odds ratio always overstates the risk ratio when the outcome is more likely in the exposed group, and the overstatement grows with the baseline risk. For a rare outcome the two are practically interchangeable; for a common one, treating them as equivalent can double the apparent size of an effect.

There is a further reason the distinction is enforced so strictly in health research. Odds ratios from different studies with different baseline risks are not directly comparable as measures of practical effect, even when the underlying association is identical. Two trials of the same intervention in populations with 5% and 50% baseline risk will report similar odds ratios and very different risk ratios — and it is the risk ratios that tell a clinician what to expect in their own patients.

The choice is not purely stylistic. Some designs make it impossible to calculate a risk ratio at all.

The case-control row explains why odds ratios became standard in epidemiology. In a case-control study you decide how many cases and how many controls to recruit, so the proportion with the outcome in your sample is an artefact of your sampling, not a property of the population. A risk ratio computed from it would be meaningless. The odds ratio, remarkably, is unaffected by this and estimates the population odds ratio correctly.

Sparse cells are the other design issue worth anticipating. If any of the four counts is zero, the cross-product is either zero or undefined and the standard error formula fails, since it requires dividing by each cell. The conventional workaround is to add 0.5 to every cell — the Haldane-Anscombe correction — which produces a usable estimate but introduces a known bias towards the null. A better answer where the software allows it is Firth's penalised likelihood, which handles zero cells properly rather than by patching them.

An odds ratio from a 2×2 table is unadjusted, or crude. An adjusted odds ratio comes from a logistic regression that includes other variables, and describes the association with those variables held constant.

Report both where possible. A crude OR of 2.70 that falls to 1.45 after adjustment tells the reader that much of the apparent association was attributable to the covariates — which is itself an informative finding, and one that disappears if only the adjusted figure is shown.

A word on how many covariates to include. Each variable entered consumes degrees of freedom and, in logistic regression, requires roughly ten events in the rarer outcome category to be estimated stably. Adding covariates because they are available rather than because they are plausible confounders inflates the standard errors, widens every interval, and can produce an adjusted odds ratio less trustworthy than the crude one it replaced. Choose the adjustment set from subject knowledge before fitting, not from which variables turn out significant.

1. Saying 'times more likely' for an odds ratio

That describes a risk ratio. Say “the odds were 2.7 times higher”.

2. Reporting the odds ratio without a confidence interval

The point estimate alone conveys nothing about precision, and small studies produce very wide intervals.

3. Computing a symmetric interval

The interval must be built on the log scale and exponentiated, giving an asymmetric result.

4. Calculating a risk ratio from a case-control study

The proportion with the outcome is fixed by your sampling, so it is not a risk. Only the odds ratio is valid.

5. Comparing crude and adjusted odds ratios as if interchangeable

Label them separately and state which covariates the adjusted figure includes.

6. Interpreting an odds ratio below 1 loosely

An OR of 0.65 means the odds are 35% lower, not that the outcome is 35% less likely. The same care applies in both directions.

Forest plots of odds ratios are conventionally drawn on a logarithmic axis for the same reason the interval is computed on one: it places an odds ratio of 0.5 and one of 2.0 at equal distances either side of the null line, which is the correct visual representation of two effects of equal size in opposite directions. On a linear axis the protective effect would appear compressed and the harmful one exaggerated. If you produce your own figure, check the axis before interpreting the spread.