Risk, odds, and how to compare them
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Risk and odds both answer the same plain question: how often does the disease happen? They just package the answer differently. Risk is the share of people who get the disease. Odds is the ratio of getting it to not getting it. Clinicians usually think in risk. Most regression models for a yes or no outcome report odds. So you need to read both, convert between them, and know when they almost agree.
This lesson stays with one group at a time. Comparing two groups, where ratios and differences enter, comes next. Here the job is to pull a single risk and a single odds off a row of counts, and to say what each one means.
A binary outcome and its probability
A binary outcome is a result with two possible values for each person. Call them for diseased and for healthy. The labels are just bookkeeping. could mean died, tested positive, or quit smoking; is the other case. Nothing says has to be the bad outcome.
The probability of is the proportion of people in category over the long run. If you follow many similar people, it is the fraction who end up diseased. A probability sits between 0 and 1. Zero means it never happens, one means it always happens, 0.5 means an even chance. You can read it as a percent if that is clearer to your audience.
In a sample you estimate that probability with the observed proportion. If people out of have the disease, the estimate is .
Key term
Probability and risk are the same number when the outcome is disease over a fixed follow-up. We say "risk" when we mean the probability that a person becomes a case during a defined period.
Risk is cumulative incidence
Risk is the probability that a person who is free of the disease at the start becomes a case during a stated period. Another name for it is cumulative incidence: the share of an at-risk group that develops new disease over follow-up. The two phrases point to the same quantity. Risk is the clinical word, cumulative incidence is the epidemiology word.
Risk always needs a time frame attached. A 5-year risk of death after a cancer diagnosis and a 30-day risk are different numbers about the same disease. Quote risk without a period and the figure is meaningless. If 18 of 200 patients die within five years, the 5-year risk of death is , or 9%.
Prevalence is not incidence
Cumulative incidence counts new cases over a period among people who could become cases. Prevalence counts existing cases at a single point in time: the proportion of a population that has the disease right now. Incidence is a film of new events; prevalence is a photo of who currently has it.
The gap matters because prevalence depends on how long people stay cases. A disease people live with for years piles up prevalent cases even when few new ones appear, so prevalence runs high while incidence is low. A disease that resolves or kills quickly can have high incidence yet low prevalence, because cases leave the "currently ill" pool fast.
Watch out
Only incidence (risk) is a probability of developing disease. Do not feed a prevalence proportion into a risk calculation and call the result a risk. They answer different questions.
Odds of disease
The odds of disease are the probability the disease happens divided by the probability it does not happen. With probability of disease, the probability of staying healthy is , so:
Odds is the language of betting. Odds of 4 to 1 against a horse mean the bookmaker rates losing as four times as likely as winning, so the win probability is . The same algebra turns any odds back into a probability:
Probabilities are stuck between 0 and 1. Odds are not. As climbs toward 1, the odds run off toward infinity. That open-ended scale is the reason odds sit at the centre of logistic regression and case-control analysis, which you meet later.
How risk and odds relate
Odds is always larger than the risk it came from, because you divide by , a number below 1. The gap is tiny for small probabilities and large for big ones. At the odds are exactly 1. At the odds are 9. At the odds are 99. So odds and risk are two views of the same chance, and you can move between them with the two formulas above.
Key term
Odds of an event is its probability divided by one minus its probability. It runs from 0 to infinity, while a probability runs from 0 to 1.
When odds is close to risk
When the disease is rare, odds and risk nearly coincide. If is about 0.1 or less, then is close to 1, and dividing by something near 1 barely changes the value:
A risk of 0.01 gives odds of , a difference well under a percent. A risk of 0.05 gives odds of about 0.0526. This is the "rare disease assumption" that lets an odds ratio stand in for a risk ratio in case-control studies. The approximation breaks once the outcome is common: at a risk of 0.5 the odds is double the risk.
Reading risk and odds off a 2x2 row
Lay the counts for one group in a row. Let be the number who got the disease and the number who did not. The row total is .
| Group | Disease (D) | Healthy (H) | Row total |
|---|---|---|---|
| Exposed | a | b | a + b |
| Unexposed | c | d | c + d |
| Column total | a + c | b + d | a + b + c + d |
Risk for the exposed row uses the row total as its denominator. Odds for that row pits cases against non-cases in the same row:
Risk has the whole row underneath it. Odds has only the healthy count underneath it. That single change of denominator is the entire difference between the two measures. Read across one row at a time and you will not mix groups.
Worked example 1: a single risk and odds
A clinic follows 250 patients on a new regimen for one year. By the end, 40 have a relapse () and 210 do not ().
- One-year risk of relapse: , that is 16%.
- Odds of relapse: .
- Check by converting the odds back: . It matches the risk, so the arithmetic is sound.
The odds (0.190) is larger than the risk (0.16), as it must be. The gap is noticeable because a 16% outcome is not rare.
Compute the one-year risk and odds of relapse from the counts, then convert the odds back to a risk to check.
a <- 40 # relapsed b <- 210 # did not
risk <- # fill in odds <- # fill in
risk <- a / (a + b) odds <- a / b back <- odds / (1 + odds) c(risk = risk, odds = odds, back = back) # risk 0.160, odds 0.190, back 0.160
Compute the one-year risk and odds of relapse from the counts, then convert the odds back to a risk to check.
a = 40 # relapsed b = 210 # did not
risk = # fill in odds = # fill in
risk = a / (a + b) odds = a / b back = odds / (1 + odds) print(round(risk, 3), round(odds, 3), round(back, 3)) # 0.16 0.19 0.16
In worked example 1, why is the odds of relapse (0.190) bigger than the risk (0.16)?
Worked example 2: a rare outcome
A district screens 5,000 newborns for a rare metabolic disorder. Eight test positive (); 4,992 test negative.
- Risk (here, prevalence at screening): , or 0.16%.
- Odds: .
- The risk and the odds agree to four decimal places. The outcome is rare, so is almost 1 and the two measures collapse onto each other.
Note the label. This is a one-time screen, so the proportion positive is a prevalence, not an incidence. The risk-versus-odds arithmetic is identical, but call the quantity by its right name when you report it.
Example
Rare outcome means you can quote the odds and the risk almost interchangeably. At a positive rate of 0.16%, odds (0.0016026) and risk (0.0016000) differ in the fifth decimal. Nobody acting on the number would notice.
Build the screening row as a one-row matrix, then read the risk and odds of a positive test and show how close they are.
pos <- 8 neg <- 4992
risk <- # fill in odds <- # fill in
m <- matrix(c(pos, neg), nrow = 1,
dimnames = list("screened", c("D", "H")))
risk <- m[1, "D"] / sum(m)
odds <- m[1, "D"] / m[1, "H"]
c(risk = risk, odds = odds, gap = odds - risk)
# risk 0.0016000, odds 0.0016026, gap 0.0000026Build the screening row as a NumPy array, then read the risk and odds of a positive test and show how close they are.
import numpy as np row = np.array([8, 4992]) # [D, H]
risk = # fill in odds = # fill in
risk = row[0] / row.sum() odds = row[0] / row[1] print(round(risk, 7), round(odds, 7), round(odds - risk, 7)) # 0.0016 0.0016026 0.0000026
Tip
Before you compute anything, decide whether the proportion is incidence (new cases over follow-up) or prevalence (cases at one moment). The formula is the same; the interpretation and the word you use are not.
Common mistakes
- Quoting risk with no time window. "The risk of relapse is 16%" is incomplete. Risk is cumulative incidence over a stated period. Say "16% over one year" so the number can be compared and acted on.
- Treating prevalence as a risk of developing disease. A point-prevalence proportion tells you who is currently a case, not the chance a healthy person becomes one. Do not call it a risk.
- Putting the row total under the odds. Odds is cases over non-cases, , not cases over the row total. Cases over the total is the risk. Mixing the denominators silently inflates or deflates one of them.
- Reading down a column instead of across a row. Risk and odds for a group come from that group's own row. Slip into a column and you are blending exposed and unexposed people.
- Assuming odds equals risk when the outcome is common. The approximation only holds for rare outcomes. At a risk of 0.5 the odds is 1, double the risk. Check that is roughly 0.1 or less before you treat them as interchangeable.
Tips
- Sanity-check any odds by converting it back with . If it does not return your original risk, you mislabelled a cell.
- Report risk to clinicians and patients. It is the number they reason with. Carry odds when you move into logistic regression or case-control work, where the open 0-to-infinity scale is what the model needs.
- Write the 2x2 the same way every time: disease in the first column, healthy in the second, groups in the rows. A fixed layout stops , , , from drifting.
- For an outcome around 1% or below, you can quote odds and risk as the same number to a clinical audience, but keep the right label on the quantity.
A hospital reports that 30% of its current inpatients have hypertension. A colleague says "so the risk of developing hypertension is 30%." What is wrong?