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Logistic regression and the odds ratio

Which ratio an OR actually is, why it necessarily overstates the effect when the outcome is not rare, how an adjusted OR differs from a crude one for reasons other than confounding, what separation looks like and how to fix it, and where the "at least ten events per variable" rule came from and why it is now disputed.

What this model is for

The second large family of clinical outcomes is binary: low birth weight or not, hospital-acquired infection or not, readmission within thirty days or not, a postoperative complication or not.

Fitting a 0/1 outcome with linear regression breaks in two places: fitted values leave the 0–1 range (a negative probability means nothing), and the residual variance changes with the fitted probability (the variance of a 0/1 variable is $p(1-p)$ , which is not constant by construction).

Logistic regression fixes this by changing the scale first. Turn the probability $p$ into the odds $p/(1-p)$ , then take the logarithm; the range opens from $[0,1]$ out to the whole real line, and the linear model is built there:

\log \frac{p}{1-p} = \beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k

So each $\beta$ means “with the other variables held constant, a one-unit increase in $x$ raises the log-odds by $\beta$ ”, and exponentiating gives $\exp(\beta)$ , the odds ratio (OR). This model produces an OR rather than an RR not because anyone chose it, but because of how the model is built.

The example on this page

We keep MASS::birthwt from the linear regression page and switch the outcome to the binary low: whether the newborn weighed under 2500 g. Of 189 mothers, 59 had a low-birth-weight baby, a prevalence of 31.2%.

That prevalence is the reason this example was chosen. It is nowhere near rare, so the OR and the RR will visibly part company in the same table — which is what the next section is about.

library(MASS)
data(birthwt, package = "MASS")

bw <- birthwt
bw$race_f  <- factor(bw$race,  levels = 1:3, labels = c("White", "Black", "Other"))
bw$smoke_f <- factor(bw$smoke, levels = 0:1, labels = c("No", "Yes"))
bw$ptl_any <- as.integer(bw$ptl > 0)

# Crude OR: one predictor
fit_c <- glm(low ~ smoke_f, data = bw, family = binomial)
exp(cbind(OR = coef(fit_c), confint.default(fit_c)))

# The RR from the same 2x2 -- you compute it yourself, glm will not give it
tb <- table(bw$smoke_f, bw$low)
(tb[2, 2] / sum(tb[2, ])) / (tb[1, 2] / sum(tb[1, ]))

# Adjusted OR
fit <- glm(low ~ smoke_f + age + lwt + race_f + ht + ui + ptl_any,
           data = bw, family = binomial)
summary(fit)
exp(cbind(OR = coef(fit), confint.default(fit)))   # Wald, the format papers use

# What separation looks like: treat ptl as a four-level factor
summary(glm(low ~ smoke_f + factor(ptl), data = bw, family = binomial))

Verified with R 4.6.0 and MASS 7.3.65. For a glm, confint() returns profile-likelihood intervals; the Wald intervals papers usually report come from confint.default().

import numpy as np
import statsmodels.api as sm
import statsmodels.formula.api as smf

bw = sm.datasets.get_rdataset("birthwt", "MASS").data
bw["race_f"]  = bw["race"].map({1: "White", 2: "Black", 3: "Other"})
bw["smoke_f"] = bw["smoke"].map({0: "No", 1: "Yes"})
bw["ptl_any"] = (bw["ptl"] > 0).astype(int)

fit = smf.logit(
    "low ~ C(smoke_f) + age + lwt + C(race_f) + ht + ui + ptl_any", data=bw).fit()
print(fit.summary())
print(np.exp(fit.params))          # OR
print(np.exp(fit.conf_int()))      # 95% CI, Wald

# On separation statsmodels raises PerfectSeparationWarning, which beats R's silence

statsmodels Logit and GLM(family=Binomial()) give the same fit; the Logit summary is the one that looks closest to R's summary().

Crude OR and RR: one 2×2 table, two answers

Before fitting anything, do it by hand. Smoking against low birth weight:

	Low birth weight	Normal	Total	Risk	Odds
Smoked	30	44	74	40.5%	0.682
Did not smoke	29	86	115	25.2%	0.337

Risk ratio (RR) = one risk divided by the other = 1.608 (1.06–2.44)
Odds ratio (OR) = one odds divided by the other = 2.022 (1.08–3.78)

Same table, same people, and the OR comes out about 26% larger than the RR. Read that OR aloud as “the risk doubled” and you have overstated the effect. What actually happened is that the risk went from 25.2% to 40.5%.

A curve plot. The horizontal axis is the unexposed group's risk in per cent; the vertical axis is the risk ratio implied by a fixed odds ratio. The curve falls from upper left to lower right, hugging the dashed horizontal line at OR = 2.02 when risk is low and pulling away from it as risk rises. A filled point marks this page's data: unexposed risk 25.2% with a risk ratio of 1.61, labelled birthwt: RR = 1.61. A vertical dashed line marks the overall low-birth-weight prevalence of 31.2%. — The OR is held fixed and only the unexposed group's baseline risk changes. When the outcome is rare, RR is nearly equal to OR; when the outcome is common, the OR is always further from 1 than the RR. The example on this page sits at the marked point.Plotting script `figures/scripts/B2-02-logistic.R`

This is algebra, not coincidence. Given an OR and an unexposed risk $p_0$ , the exposed risk is

p_1 = \frac{\mathrm{OR} \cdot p_0 / (1-p_0)}{1 + \mathrm{OR} \cdot p_0 / (1-p_0)} \quad\Longrightarrow\quad \mathrm{RR} = \frac{p_1}{p_0}

Holding this page’s OR fixed and varying the baseline risk:

Risk in the unexposed	RR implied by this OR
1%	2.001
5%	1.924
20%	1.679
40%	1.435

When an OR may be read as an RR: only when the outcome is rare.

The usual threshold quoted is an unexposed risk below 10%. The rare disease assumption is an approximation, not an identity: the baseline risk in these data is 25.2%, and the amount by which the OR overstates the RR is already plain.

Three practical consequences:

A case-control study can only report an OR (the denominators were chosen by the investigator), which is why the rare disease assumption matters so much there; see case-control studies, where the OR is this method applied directly.
Cohort studies and RCTs have denominators, so they can report an RR or a risk difference outright. Reporting an OR instead swaps a directly answerable question for a quantity that needs an extra assumption before it can be read. For an adjusted RR, use log-binomial regression, or Poisson regression with robust standard errors (see the adjusted risk ratio).
When you see “the risk went up N-fold”, check whether that number is an RR or an OR, and ask how common the outcome is. The more common it is, the larger the error from reading an OR as an RR.

The adjusted OR

A forest plot of adjusted odds ratios with 95% confidence intervals for eight variables. Five intervals do not cross 1: smoking, Black race, history of hypertension and previous preterm labour all have odds ratios above 1, with their whole line to the right of the dashed reference line; the mother's weight has an odds ratio of 0.985 with its whole line to the left, the only variable falling below 1. Three intervals cross 1: age, uterine irritability, and Other race. — Adjusted odds ratios from the eight-variable logistic model. The dashed line is OR = 1. An interval crossing that line means this study did not detect an association between that variable and low birth weight.Plotting script `figures/scripts/B2-02-logistic.R`

Variable	Adjusted OR	95% CI	p
Smoking during pregnancy	2.33	1.048–5.187	0.038
Age, per year	0.96	0.894–1.037	0.318
Mother's weight, per lb	0.99	0.972–0.999	0.034
Race: Black vs White	3.36	1.184–9.548	0.023
Race: Other vs White	2.23	0.928–5.382	0.073
History of hypertension	6.29	1.585–24.954	0.009
Uterine irritability	2.04	0.822–5.047	0.125
Previous preterm labour	3.39	1.369–8.408	0.008

The OR for smoking moves from a crude 2.022 to an adjusted 2.331.

Most textbooks will tell you the difference is confounding. Half of it is; the other half is not, and the other half is almost never mentioned:

The OR is non-collapsible.

Even with no confounding at all — exposure independent of every covariate — the conditional OR and the marginal OR still do not agree, and the conditional one is always further from 1. The reason is that the logit transformation is not linear, so averaging people across strata of different risk is not averaging things measured on one common scale.

The RR and the risk difference do not behave this way; the HR does, to a milder degree. Three consequences:

“The OR got bigger after adjustment” cannot automatically be read as “confounding was masking it”. At least part of the move is this algebraic property.
ORs from different models cannot be compared directly — different variables mean conditioning at a different level, so the two ORs define different contrasts. The same reasoning applies to the HR from a Cox model.
If you want an effect measure that can be interpreted at the population level (to compute an NNT, say), report a marginal estimate: standardisation / G-computation or weighting, rather than the model’s OR as it comes. For how that is done, see marginal estimates and G-computation.

The remaining columns read exactly as in linear regression: an OR on a continuous variable must carry its unit (the OR of 0.963 for age is per year); a variable whose confidence interval crosses 1 should be written up as “this study did not detect an association”, not as “no association”; and the global model test (likelihood ratio 37.8, df = 8, p < 0.001) asks whether all the coefficients are simultaneously 0.

Separation

Sometimes glm() returns a dramatic-looking coefficient attached to an absurd standard error. That is usually not strong evidence; it is separation — some variable splits the outcome perfectly, or nearly perfectly.

This dataset contains a ready-made example, no fabrication needed. Treat the number of previous preterm labours, ptl, as a four-level factor:

Previous preterm labours	Normal weight	Low birth weight
0	118	41
1	8	16
2	3	2
3	1	0

The last row contains 1 mother, and her baby was not low birth weight. That cell is 0, so within these data the level “three previous preterm labours” is synonymous with “low birth weight cannot happen”. The model responds like this:

Term	Coefficient	SE	p
(Intercept)	-1.28	0.23	< 0.001
smoke_fYes	0.58	0.34	0.087
ptl_f1	1.65	0.48	< 0.001
ptl_f2	0.52	0.94	0.582
ptl_f3	-13.86	882.74	0.987

The symptoms of separation, in the order they appear:

One coefficient is absurdly large in absolute value (here -13.86, an OR of about 9.5e-7 — on the order of one in a million) with a ridiculous standard error (883).
The confidence interval is so wide it carries no information, and the p-value approaches 1 — not because there is no effect, but because nothing can be estimated at all.
The decisive check: raise the iteration limit and refit; the coefficient keeps travelling. Here it moves from -13.86 to -22.86. A genuine maximum likelihood estimate does not change just because the optimiser was given more rounds. It runs away because the estimate does not exist — the likelihood only approaches its maximum out at infinity. R does not raise an error for this; at most you get fitted probabilities numerically 0 or 1 occurred, a line that is easy to dismiss as noise.

What to do, from simplest to most involved:

Collapse the sparse categories. Here, merging “2” and “3 or more” into “2 or more” is enough: the merged OR is 1.21 (0.21–7.03), an estimate that can actually be read. In most cases this is the most honest option — that level never had enough people in it to begin with.
Firth penalised likelihood (the logistf package) adds a penalty term to the likelihood, guaranteeing a finite estimate. It is the standard recommendation for small samples and rare events.
Exact logistic regression is feasible when the sample is very small; the computation grows quickly with the data.
Do not simply delete the variable without saying so, and do not report that OR as if it meant something.

EPV: at least ten events per variable

How many variables a logistic model can carry depends on the number of events, not the sample size. The model on this page has 59 events and 8 parameters, so EPV (events per variable) = 7.38.

Where the rule came from, and why it is disputed.

“At least ten events per variable” comes from simulation work by Peduzzi and colleagues in the 1990s: below an EPV of 10, coefficients drifted, standard errors were underestimated, and confidence interval coverage fell below its nominal level. The rule is easy to remember, which is how it became a fixture of peer review.

Later simulations, from the 2010s onward, found it wrong in both directions:

Sometimes too lenient. For prediction models an EPV of 10 is often not enough — out-of-sample performance still comes out optimistic. Current advice is to compute the required sample size from the amount of shrinkage you are willing to accept and the outcome prevalence, rather than applying a fixed multiplier.
Sometimes too strict. When covariates correlate weakly with the exposure and well-calibrated prediction is not the goal, bias around an EPV of 5 need not be severe.

So how should this page’s model (EPV 7.38) be described? Honestly: by the old rule it is below threshold (59 events supports only 5 parameters at an EPV of 10), so the coefficients and intervals should be treated as exploratory — while also noting that EPV 10 is a rule of thumb, not a theorem.

What you must not do is fit the model once, see which variables are significant, and drop the rest to manufacture an acceptable EPV. That invalidates the p-values, for the reasons given in the stepwise regression section of regression diagnostics.

Common misuses

Misuse	Why it is wrong
Reading an OR as an RR when the outcome is not rare	The OR is always further from 1, and the gap widens as the outcome becomes more common
Reporting only an OR from a cohort study or RCT that has denominators	An RR or risk difference is available directly; the OR demands an extra assumption
Attributing the whole crude-to-adjusted change in an OR to confounding	The OR is non-collapsible; a conditional OR sits further from 1 by construction
Comparing the size of ORs between two papers	Different adjustment sets mean the two ORs define different contrasts
Using a model’s OR to compute an NNT or a population-level impact	That needs a marginal estimate (standardisation / G-computation), not a conditional OR
Reporting an OR on a continuous variable without its unit	Per year and per decade are entirely different numbers
Reporting a huge OR produced by separation	The estimate does not exist; collapse categories or use Firth
Dismissing `fitted probabilities numerically 0 or 1` as noise	It is usually the only warning of separation you will get
Loading many covariates into a model with few events	Overfitting; neither the coefficients nor the intervals can be trusted
Choosing variables by looking at significance first	Post-selection p-values and confidence intervals are invalid
Ranking variables by “importance” using the size of their ORs	The scale depends on each variable’s unit and distribution, not on importance
Fitting a matched case-control study with ordinary logistic regression	Use conditional logistic regression; see case-control studies
Describing an effect size when the confidence interval crosses 1	Write “no significant association was detected” and give the interval

Reproducing every number on this page

/opt/homebrew/bin/Rscript figures/scripts/B2-02-logistic.R

Read the figure

The answer comes from the same statistical output that produced this page's figures, not from a number typed in beside them.

Low birth weight is not rare in these data, and the crude odds ratio is 2.02. Can you say smoking doubles the risk?

Show the answer and why

Correct answer: No - the risk ratio is 1.61, and when an outcome is common the OR sits further from one than the RR

The risk ratio is 1.61 against an odds ratio of 2.02, a wide gap because this outcome is anything but rare. The two converge only when events are uncommon; the higher the prevalence the further they separate, and the OR is always the one further from one. Reading an OR as "risk multiplied by" overstates the effect, which is how abstracts routinely phrase it. 3.78 is the OR's upper confidence bound, describing uncertainty rather than a second effect measure.

After adjusting for age, maternal weight and race, smoking's odds ratio moves away from the crude 2.02. What has adjustment done?

Show the answer and why

Correct answer: It becomes 2.33 - adjustment can move an estimate in any direction, and here it moves outward

The adjusted OR is 2.33, further from one than the crude 2.02. "Adjustment pulls estimates towards the null" is a common intuition and it is simply not true: the direction depends on how each covariate relates to the exposure and to the outcome, and an estimate can move outward, inward, or across. 3.36 is the race row of the same model, not smoking's. Faced with a crude and an adjusted figure, the question is which variables entered and why, not which way it should have gone.

30 of the 74 smokers had a low-birth-weight baby. What is the risk in the smoking group?

Show the answer and why

Correct answer: 0.405 - denominator is everyone in that group

Risk divides by everyone, giving 0.405. 0.682 is the same group's odds, whose denominator excludes those who had the event, which is why odds always exceed risk. When events are common the two diverge widely - and logistic regression hands you the odds side, which is why talking about risk from a logistic model requires converting first. 0.252 is the non-smokers' risk, read off the wrong row.

Chapters that use this method

Watch next

ENStatQuest with Josh Starmer· 9 minNine minutes to build the intuition — why probability gets converted to log-odds and where the S-shaped curve comes from. Watch before the first section.

ENStatQuest with Josh Starmer· 19 minSpecifically on how a coefficient becomes an OR, which is exactly the column the fourth and fifth sections here are reading.

繁中EDMAN MURMURS· 13 minIn Mandarin, from a clinician's point of view. A full episode, useful if you need the Chinese terms alongside the English ones.

繁中EDMAN MURMURS· 11 minIn Mandarin. The one episode that tackles the RR-versus-OR difference head on, matching the third section of this page.

繁中MeDA School（洪弘）· 45 minIn Mandarin, graduate-level hands-on R. Watch it if you want to type the whole workflow out yourself.

Sources and licences

This page is original writing