Why is treating three repeated blood-pressure readings per patient as three independent observations wrong?

## Rows are not units Every standard-error formula in ordinary least squares rests on one assumption that has nothing to do with normality or straight lines: the errors are independent of each other. Independence is what licenses the arithmetic `SE = s / sqrt(n)` and, more generally, the whole `(X'X)^-1 * sigma^2` variance formula. When a dataset has several rows per patient, per user, per store or per classroom, that assumption is usually false, and the sample size the software believes in is not the sample size you actually have. Call the readings on patient `j` at visits 1, 2 and 3 `y_j1`, `y_j2`, `y_j3`. Patients differ from each other in baseline blood pressure for reasons the model does not contain — age, weight, medication history, the anxiety of being in a clinic. Write that persistent patient-specific offset as `u_j` and the visit-to-visit noise as `e_jt`, so the error on any row is `u_j + e_jt`. Two rows from the same patient share `u_j`, so their errors are positively correlated; two rows from different patients share nothing and stay independent. The strength of the sharing is the intraclass correlation, `ICC = var(u) / (var(u) + var(e))`. ## What actually breaks Three things are worth separating, because interviewers probe exactly this distinction. **The coefficients are usually fine.** Under within-patient correlation, ordinary least squares remains unbiased for the coefficients as long as the mean model is correctly specified and the clustering is not itself confounded with the predictor. It is no longer the minimum-variance estimator — it is inefficient — but it is not systematically wrong. **The standard errors are not fine.** The reported variance assumes `n` independent errors. With positive within-patient correlation there is less independent information than `n`, so the reported standard error is too small. Confidence intervals are too narrow, t-statistics too large, p-values too small. A test you believe rejects 5% of the time under the null can easily reject 20% or 40% of the time. This is a false-positive machine, and it is silent: nothing in the output looks wrong. **The degrees of freedom are not fine either.** Software counts residual degrees of freedom from rows. With 300 rows from 100 patients, the reference distribution is far too generous. The severity is not uniform across predictors. A predictor that is constant within a patient — a patient-level treatment assignment, sex, a chronic diagnosis — suffers the worst inflation, because each patient really contributes just one observation of that predictor while the software counts three. A predictor that varies freely from visit to visit and is uncorrelated within the patient suffers much less. The damage grows with both the correlation of the errors within a cluster and the correlation of the predictor within a cluster. ## Recognising it before you model The diagnostic is structural, not graphical: look at the identifier columns. If a patient id, user id, device id or session id repeats down the rows, you have clustered data, and the burden of proof is on the analyst to show it does not matter. Counting distinct ids against total rows takes one line and tells you the worst case immediately: 300 rows and 100 patients means your true sample size is somewhere between 100 and 300, never above 300, and closer to 100 the more the patient dominates the variation. ## The three honest analyses **Aggregate to the unit.** Average each patient's readings into one number and analyse 100 rows. Simple, transparent, hard to argue with when cluster sizes are equal and you only care about patient-level predictors. It costs you any within-patient effect and some precision when cluster sizes are unequal. **Keep the rows, fix the inference.** Fit on all 300 rows but compute standard errors clustered by patient. This allows arbitrary correlation inside a patient while assuming independence across patients, and it changes only the uncertainty, never the estimates. **Model the structure.** Fit a random intercept per patient, which estimates `var(u)` explicitly, weights patients sensibly and recovers efficiency. It asks more of you: the random effect must be uncorrelated with the predictors, or the coefficients themselves become biased. ## The interview point The sentence that lands is that a p-value is only as trustworthy as the independence claim underneath it, and adding visits to the same patients buys measurement precision, not statistical power over patients. If you want more power over patients, recruit more patients.