When the outcome is yes-or-no, linear regression breaks -- and the odds ratio takes over
Many surgical outcomes are binary: complication or no complication, alive or dead, readmitted or not. You might be tempted to fit a linear regression to predict the probability of the event. But a straight line keeps going -- it will happily predict a probability of 1.4 for a high-risk patient or -0.2 for a low-risk one. Probabilities cannot live outside 0 to 1.
Logistic regression solves this by modeling the log-odds of the outcome instead of the raw probability. The result is squeezed onto an S-shaped curve that can never escape 0 or 1, and the coefficients translate into odds ratios.
Logistic regression is the most common multivariable model in surgical papers with a binary outcome. Almost every "adjusted odds ratio" you read in a results table comes from one.
To read logistic output you need three related but distinct quantities.
Probability: the chance the event happens, from 0 to 1. If 20 of 100 patients get a surgical site infection, the probability is 0.20 (20%).
Odds: the ratio of the event happening to it not happening. Odds = p / (1 - p). For p = 0.20, odds = 0.20 / 0.80 = 0.25 (often written "1 to 4"). Odds run from 0 to infinity.
Log-odds (the logit): the natural log of the odds, ln(0.25) = -1.39. Log-odds run from minus infinity to plus infinity -- which is exactly what a linear equation needs to predict safely.
The logit link: logistic regression fits a straight line to the log-odds: logit(p) = b0 + b1x1 + b2x2 + ... Each coefficient is a change in log-odds. Exponentiate it (e^b) and you get an odds ratio, which is how results are reported.
Reading a logistic regression table predicting surgical site infection (SSI):
| Variable | OR | 95% CI | p-value |
|---|---|---|---|
| Diabetes (vs none) | 2.10 | 1.45 -- 3.04 | <0.001 |
| BMI (per 1 unit) | 1.08 | 1.04 -- 1.12 | <0.001 |
| Operative time (per 30 min) | 1.25 | 1.10 -- 1.42 | 0.001 |
| Laparoscopic (vs open) | 0.62 | 0.44 -- 0.87 | 0.006 |
| Male sex | 1.12 | 0.83 -- 1.51 | 0.46 |
OR > 1: The predictor increases the odds of the event.
OR < 1: The predictor decreases the odds -- it is protective (e.g., laparoscopic OR = 0.62).
OR = 1: No association.
Significance rule: if the 95% CI crosses 1.0, the result is not statistically significant -- the odds could be higher or lower. Male sex (OR 1.12, CI 0.83 -- 1.51) is an example: the CI spans 1.0, so there is no evidence of an effect.
This is the single most misread feature of logistic regression in surgical papers.
The odds ratio and the risk (relative) ratio answer different questions. When the outcome is rare (< ~10%), they are nearly identical. When the outcome is common (> ~10%), the OR overstates the RR -- pushing it further from 1.0.
Worked example. Suppose 50% of exposed patients develop a wound complication versus 25% of unexposed patients.
Risk ratio = 0.50 / 0.25 = 2.0 (exposed patients have twice the risk).
Odds (exposed) = 0.50 / 0.50 = 1.0. Odds (unexposed) = 0.25 / 0.75 = 0.333.
Odds ratio = 1.0 / 0.333 = 3.0.
The true relative risk is 2.0, but the logistic model reports OR = 3.0. A reader who treats the OR as a risk ratio overstates the effect by 50%.
Rule of thumb: the more common the outcome, the more the OR exaggerates the RR. With a 30% complication rate, an OR of 2.0 might correspond to an RR of only ~1.5. Never read "OR = 2.0" as "twice the risk" unless the outcome is rare.
If the OR is so easily misread, why is it everywhere? Because logistic regression produces odds ratios directly -- exponentiating a coefficient gives an OR with no extra assumptions. Estimating adjusted risk ratios requires different models (log-binomial or modified Poisson) that often fail to converge. The OR is the path of least resistance, so it dominates the literature. Your job as a reader is to keep its limitations in mind.
The OR for a continuous variable is tied to its unit. The underlying coefficient is fixed, but how it is reported depends on the scale.
If the per-1-year OR for age is 1.03, then the per-10-year OR is 1.03^10 ~ 1.34. Same data, same model -- a per-1-unit OR of 1.03 looks trivial, while the per-decade OR of 1.34 looks substantial.
This is why a per-1-unit OR on a wide-range variable (e.g., per 1 mL of blood loss) can look deceptively close to 1.0, and why authors often rescale (per 100 mL, per decade) to make the effect interpretable. Always check the units before judging the size of an OR.
Logistic output sometimes shows the raw coefficient (log-odds) rather than the OR. To convert, exponentiate: OR = e^b. A coefficient of 0 gives OR = 1 (no effect). A positive coefficient gives OR > 1; a negative coefficient gives OR < 1. Example: b = 0.74 gives OR = e^0.74 ~ 2.10 (the diabetes row above).
Mistakes that survive peer review but mislead readers.
"The OR for the new technique was 2.0, so patients were twice as likely to develop a complication."
Only true if the outcome is rare. The complication rate here was 35%. With a common outcome, OR = 2.0 overstates the risk ratio -- the actual relative risk might be closer to 1.5. "Twice the odds" is correct; "twice as likely" (twice the risk) is not. Report the absolute risks alongside the OR.
"Estimated blood loss was a strong predictor of SSI (OR 1.002, p < 0.001)."
The OR looks tiny because it is per 1 mL. Blood loss ranges over hundreds of millilitres. An OR of 1.002 per mL is 1.002^100 ~ 1.22 per 100 mL and 1.002^500 ~ 2.7 over a 500 mL difference. The effect is substantial; the per-1-unit scaling just hides it. Rescaling to a clinically meaningful unit reveals the true magnitude.
"Prior mesh infection was associated with recurrence (OR 47.3, 95% CI 6.1 -- 368.0)."
This is a separation / zero-cell problem. An enormous OR with an absurdly wide CI usually means that in the data, nearly everyone (or no one) in one cell had the event -- a category that perfectly or near-perfectly predicts the outcome. The model cannot estimate the coefficient stably, so it inflates toward infinity. The fix is penalized (Firth) logistic regression or collapsing sparse categories, not reporting OR = 47.3 at face value.
"Smoking was not associated with leak (OR 1.30, 95% CI 0.85 -- 1.99, p = 0.22), proving smoking has no effect."
A non-significant OR is not proof of no effect. The CI (0.85 -- 1.99) crosses 1.0, so the result is not significant -- but the interval is also wide and includes a 99% increase in odds. Absence of evidence is not evidence of absence; this study was simply underpowered to rule out a meaningful effect. "No significant association" is the most you can say.
When reviewing a logistic model: check the outcome frequency (is OR ~ RR?), check the units behind any continuous OR, watch for impossibly large ORs with huge CIs (separation), and never read a CI crossing 1.0 as proof of no effect.
Evaluate logistic regression results and interpretation.
Module 5 - Lesson 2 complete
Next lesson: Crude vs. Adjusted -- what "adjusting for" actually means, why crude and adjusted odds ratios differ, and how confounding shifts your estimates.