Every "adjusted" number in every paper you read is a regression coefficient
Strip away the jargon and regression does one simple thing: it draws the best line through a cloud of data, describing how an outcome changes as a predictor changes. When a paper reports that operative time increased "by 4 minutes per unit of BMI, adjusted for age and approach," that single sentence is a regression coefficient, its precision, and its adjustment all at once.
If you can read a regression table, you can read the results section of almost any clinical study. If you cannot, you are trusting the authors' interpretation of their own model.
Linear regression models a continuous outcome as a straight-line function of one or more predictors. It is the foundation that logistic regression, Cox regression, and nearly every "multivariable adjustment" in the literature are built upon.
Linear regression has exactly one equation. Learn it once and every coefficient table makes sense.
The outcome is what you are predicting (a continuous number: length of stay, operative time, blood loss). β0 (the intercept) is the predicted outcome when every predictor equals zero. β1 (a slope coefficient) is the heart of the model: for each 1-unit increase in X, the outcome changes by β1, holding all other variables in the model constant. The error term captures everything the predictors do not explain.
The key phrase: "holding other variables constant" is what makes a coefficient adjusted. It is the model's answer to "what is the effect of this one predictor if the others did not change?"
Here is the output from a model predicting length of stay (days) after elective colorectal resection:
| Variable | Coefficient (β) | 95% CI | p-value |
|---|---|---|---|
| Intercept | 2.10 | 0.90 – 3.30 | <0.001 |
| Age (per year) | 0.06 | 0.03 – 0.09 | <0.001 |
| BMI (per kg/m²) | 0.04 | -0.01 – 0.09 | 0.12 |
| Diabetes (yes vs no) | 1.30 | 0.55 – 2.05 | 0.001 |
| Open (vs laparoscopic) | 2.40 | 1.60 – 3.20 | <0.001 |
Direction: Positive β raises the outcome; negative β lowers it.
Magnitude: The size of the effect, in the outcome's own units (here, days).
Significance: If the 95% CI excludes 0 (and p < 0.05), the predictor is associated with the outcome.
Note: For a continuous outcome the null value is 0, not 1. (For ratios like hazard or odds ratios the null is 1 — do not confuse them.)
The line is a prediction. The distance from each patient to the line is what the model could not explain.
For every patient, the model produces a fitted value — the outcome the equation predicts from that patient's predictors. The residual is the gap between what actually happened and that prediction (observed minus fitted). Regression literally finds the line that makes the squared residuals as small as possible ("least squares").
Residuals are not nuisance — they are diagnostic. Patterns in the residuals (a curve, a fan shape, outliers) are how you discover that a straight line was the wrong model.
R² ranges from 0 to 1 and answers: "what fraction of the variation in the outcome do the predictors account for?" An R² of 0.30 means the model explains 30% of the variability in length of stay; the other 70% is driven by things not in the model.
High R² ≠ good model: you can get a high R² by overfitting, by including a predictor that is really a proxy for the outcome, or in data with little noise. Low R² ≠ useless model: a coefficient can be precisely estimated, clinically important, and trustworthy even when the model explains only a small share of total variance. Biology is noisy; a real effect lives inside a lot of unexplained scatter.
A coefficient is meaningless without its units. "β = 0.06" is trivial or enormous depending on whether it is per year, per decade, or per standard deviation. The coefficient is always expressed in outcome units per one unit of the predictor. Change the predictor's units (years to decades) and the coefficient scales accordingly (0.06/year becomes 0.6/decade).
By default the intercept is the outcome when every predictor is 0 — often nonsensical (a patient of age 0, BMI 0). Centering a predictor (subtracting its mean) makes the intercept the predicted outcome for an average patient, which is interpretable. Scaling (dividing by a standard deviation, or rescaling to per-decade) makes coefficients comparable and clinically natural.
Centering and scaling change the intercept and the numeric coefficients but do not change the model's fit, predictions, or p-values. They are reparameterizations for interpretability, not different models.
Read a regression in this order: direction and size of each coefficient (in its units), then the CI to judge precision, then R² for context — never R² first. A model can be both informative and far from a complete explanation.
Four ways a perfectly fitted model leads a careful reader astray.
"The intercept was 2.1 days — that's the baseline length of stay for our patients."
The intercept is the outcome when all predictors are zero, not the "average" or "baseline" patient. If the model includes uncentered age, the intercept describes a hypothetical patient of age 0, BMI 0, no diabetes, laparoscopic approach — a patient who does not exist. Unless the predictors were centered, the intercept is a mathematical anchor, not a clinical quantity. Do not interpret it as a typical stay.
"BMI was a significant predictor of operative time (β = 0.3 minutes/unit, p = 0.004)."
Statistical significance is not clinical importance. In a large sample, a coefficient of 0.3 minutes per BMI unit can be highly significant yet trivial: even a 10-point BMI difference predicts 3 extra minutes. Significance tells you the effect is probably not zero; the coefficient's magnitude tells you whether anyone should care. Always read the effect size, not just the p-value.
"Our model predicts length of stay increases 0.06 days per year, so a 95-year-old will stay ~6 days from age alone."
This is extrapolation beyond the data. If the study enrolled patients aged 40 to 80, the linear relationship is only supported within that range. The line may bend, flatten, or reverse outside it — the model cannot know. Applying a coefficient to predictor values not represented in the sample assumes a linearity you never observed.
"R² was 0.62, confirming that these risk factors cause prolonged stay."
R² measures explained variance, not causation. A high R² means the predictors track the outcome well in this dataset; it says nothing about whether they cause it, whether the association would replicate, or whether a confounder drives both. Regression on observational data describes association. Causal claims require design (randomization, instrumental variables, careful confounder control), not a large R².
When you read a regression: confirm the intercept was made interpretable (or ignore it), separate significance from effect size, check that conclusions stay within the observed range of predictors, and never let R² be quoted as evidence of cause.
Interpret coefficients, intercepts, R², and the limits of a model.
Module 5 - Lesson 1 complete
Next lesson: Logistic Regression and Odds Ratios — what changes when the outcome is yes/no instead of a number, and why the coefficients suddenly become ratios you exponentiate.