Module 3 · Odds Ratios
Two studies, same drug, both report "0.5." They don't agree.
One trial reports a relative risk of 0.5 — the drug halves your risk. Another reports an odds ratio of 0.5 — and it gets quoted in the press as "halves your risk" too.
If the event is rare, those two "0.5"s really do mean almost the same thing. But if the event is common, they can describe meaningfully different benefits — and the odds ratio will always be the one that sounds better.
An odds ratio of 0.5 — does that mean the same as "risk halved"?
Odds are not risk
You met this briefly before; now make it solid, because everything here turns on it.
Risk = events ÷ everyone. Out of 1,000 people, 100 have the event → risk = 100/1,000 = 10%.
Odds = events ÷ non-events. Those same 100 events versus 900 non-events → odds = 100/900 ≈ 0.11, or "1 to 9."
It's the gambler's way of counting: not "what fraction had it" but "how many had it for each one who didn't." When the event is rare, the two are almost identical (dividing by 1,000 vs 900 barely differs). But as the event gets common, they pull apart — at a 50% risk, the odds are 1-to-1, a number that no longer looks anything like the risk.
Hold that: risk and odds agree when events are rare, and diverge when events are common. Every quirk of the odds ratio flows from this one fact.
The odds ratio
The odds ratio (OR) does for odds what relative risk does for risk: it compares the two groups. From the 2×2 table, OR = (a/b) ÷ (c/d) — the treated group's odds divided by the control group's odds.
Let's compute one, to make odds concrete — they're the least intuitive thing in this block.
Odds of the event = events ÷ non-events = 100 ÷ 900 = ? (give 2 decimals)
OR = treated odds ÷ control odds = 0.05 ÷ 0.10 = ?
In last lesson's table (treated: 10 events / 990 not; control: 20 / 980), the odds ratio was about 0.49 — and the relative risk was 0.50. Almost identical, because the event there was rare (1–2%).
So why use this slightly awkward measure instead of the intuitive risk ratio? Two reasons we'll come to: sometimes OR is all you can compute (case-control studies), and it's what logistic regression naturally produces. But first, the trap — what happens to that reassuring "OR ≈ RR" when the event stops being rare.
Watch them diverge
Here's a drug with a fixed odds ratio of 0.50 — that never changes. All you'll move is how common the event is (the baseline risk). Watch what happens to the actual risk ratio.
Control risk: 1% · OR: 0.50 (fixed) · RR ≈ 0.50
OR and RR nearly identical — 'halving the odds' ≈ 'halving the risk' here.
Try this: start with a rare event (baseline 1%) — the OR and RR markers sit right on top of each other. Now slide the event up toward common (40–50%) and watch the risk ratio drift away from the odds ratio, toward "no effect."
At rare events, OR and RR sit on top of each other — reporting the OR is harmless. But as the event becomes common, the OR stays put at 0.50 while the real risk ratio drifts toward 1 — the true reduction in risk is much smaller than "half." Read that 0.50 as "risk halved," and you've overstated the benefit, sometimes badly.
OR always exaggerates
There's a rule worth burning in, because it's always true and always in the same direction:
So an OR of 0.5 is at best "risk halved" (when events are rare) and otherwise something less impressive dressed up to look like it. An OR of 2.0 is at best "risk doubled" and often less. Whenever you see an odds ratio reported for a common outcome and discussed as though it were a risk ratio, a benefit (or a harm) is being made to sound larger than it is.
This isn't the odds ratio being "wrong" — the number is correct. It's the interpretation that goes wrong, when OR is read as if it were RR.
So why use odds ratios at all?
If OR is so easy to misread, why does it exist everywhere? Because sometimes it's the only measure you can compute — and sometimes it's what the maths hands you.
The key case is the case-control study you met back in M2. There, researchers start by picking people based on the outcome — say, 200 people with the disease and 200 without — and look backwards. Because the proportion with the disease was fixed by the study design, not observed in a real population, you simply cannot calculate a risk (events ÷ everyone has no meaning here). But you can compare odds. The odds ratio is the only effect measure a case-control study can give.
OR isn't a trick — it's often a necessity. Case-control studies and logistic regression (which you'll meet in M11) both speak in odds ratios. The skill isn't to reject OR; it's to know what it is, and to refuse to read it as a risk ratio when the event is common.
Can you trust this OR as risk?
For each result, decide how to handle the odds ratio.
The reflex for odds ratios
Two quick questions whenever you meet an odds ratio:
- How common is the event? If it's rare, OR ≈ RR and you can read it loosely as a risk ratio. If it's common, the OR overstates the effect — get the risk ratio or the absolute risks.
- What kind of study is it? A case-control study can only give an OR — accept it, but read it as an association in odds, never as a risk reduction.
The danger is never the odds ratio itself — it's an odds ratio for a common event, quoted as though it were a risk ratio. Spot that, and you've caught one of the most common quiet exaggerations in the literature.
Why this matters for HTA
Odds ratios turn up constantly in submissions — from case-control evidence, meta-analyses, and regression models — and each is a chance for a benefit to look bigger than it is.
- An OR reported for a common outcome should prompt you to ask for the risk ratio or absolute risks — the OR will flatter the effect.
- An OR from a case-control study is legitimate, but it's an association in odds; it can't be turned into "X% lower risk" or an NNT without external information about the actual baseline risk.
- An OR quietly described as though it were a risk reduction is a red flag — the same words ("halves the risk") mean something weaker in odds than in risk when events are common.
- As always, check the confidence interval around the OR, and remember a ratio with no baseline still tells you nothing about absolute benefit.
An odds ratio is a correct answer to a question most readers aren't asking. Your job is to know when it's standing in for a risk ratio — and to refuse to let it overstate the effect.
Odds, odds ratios & when OR misleads, in one breath
- Odds = events ÷ non-events (not ÷ everyone); they match risk when events are rare and diverge when events are common.
- The odds ratio compares odds between groups: (a/b) ÷ (c/d).
- OR ≈ RR for rare events, but OR is always further from 1 than RR — it overstates the effect, more so the commoner the event.
- Case-control studies and logistic regression can only give odds ratios — by design, risk can't be computed there.
- The trap is reading an OR for a common event as if it were a risk ratio; the fix is to ask the event rate and the study type.
An odds ratio isn't wrong — but read as a risk ratio for a common event, it quietly inflates the benefit. Know which one you're holding.
That completes the measures of effect — every way of asking "how big?" when the outcome is a simple yes/no event. But many of the most important outcomes in medicine aren't "did it happen?" — they're "how long until it happened?" Survival, time to relapse, time to progression. Counting events alone throws away when they occurred, and that timing is often the whole story. The next block opens the dimension of time: survival analysis.