The “adjust for everything” reflex
Your colleague shows you a paper: “Adjusted mortality 18% vs 28%, adjusted OR 0.63 (95% CI 0.44–0.90), p = 0.01.” The table lists 12 covariates: age, APACHE, SOFA, lactate, creatinine, vasopressor count, steroids, antibiotics, mechanical ventilation, renal replacement, cancer history, and sepsis source.
The paper says: “After adjustment, the intervention effect remained significant.” But you notice: 12 covariates, 80 events (deaths), events‑per‑variable (EPV) = 6.7. That is below the conventional EPV ≥ 10 rule. The model may be overfitted, and the OR may be unreliable.
This chapter covers linear and logistic regression — the workhorses of multivariable analysis in ICU research — and the checks you should apply before trusting the reported effect.
Linear regression: continuous outcomes
Linear regression models the mean of a continuous outcome as a linear combination of predictors:
[ E(Y) = _0 + _1 X_1 + _2 X_2 + + _p X_p ]
In ICU, Y could be ICU length of stay, SOFA change at day 3, or lactate clearance at 6 h.
Assumptions (and what to check)
- Linearity — the relationship between predictors and outcome is linear. Check with residual plots (fitted vs residuals should show no pattern).
- Independence — observations are independent (no clustering; if clustered, use mixed effects).
- Homoscedasticity — residual variance is constant across fitted values. Check with a scale‑location plot.
- Normality of residuals — residuals should be approximately normal (QQ‑plot, Shapiro‑Wilk on residuals).
- No multicollinearity — predictors should not be highly correlated (check VIF; VIF > 5 or 10 indicates problematic collinearity).
Common ICU pitfalls
- Non‑linear relationships — e.g., age and mortality may be J‑shaped; forcing a linear term misses the curve. Use splines or fractional polynomials.
- Missing not at random — if sicker patients have missing lactate values, complete‑case analysis is biased. Multiple imputation or sensitivity analysis needed.
- Time‑varying predictors — lactate measured at hour 6 and hour 24 are correlated; including both without accounting for time inflates collinearity. Use time‑varying coefficients or longitudinal models.
Logistic regression: binary outcomes
Logistic regression models the log‑odds of a binary outcome:
[ () = _0 + _1 X_1 + + _p X_p ]
In ICU: mortality, AKI, VAP, delirium — all binary.
The odds‑ratio interpretation
The reported OR for each predictor is ((_j)). A OR = 0.63 means: “a one‑unit increase in X is associated with 37% lower odds of the outcome, holding other covariates constant.”
Assumptions (and what to check)
- Linearity of log‑odds — continuous predictors should have a linear relationship with the log‑odds. Check with Box‑Tidwell test or residual plots.
- Independence — no clustering; if clustered, use GEE or mixed‑effects logistic regression.
- No separation — if a predictor perfectly predicts the outcome (e.g., all deaths in one arm), maximum likelihood fails; use Firth’s penalized logistic regression.
- Events per variable (EPV) — ≥ 10 events per predictor variable is the rule of thumb. 12 covariates need ≥ 120 events. If EPV < 10, the model is likely overfitted.
The calibration–discrimination trade‑off
- Discrimination — does the model separate events from non‑events? (C‑statistic or AUC; c‑statistic ≥ 0.70 is acceptable; ≥ 0.80 is good.)
- Calibration — do predicted probabilities match observed probabilities? (Hosmer‑Lemeshow test, calibration plot.)
A model with C‑statistic = 0.85 but poor calibration (Hosmer‑Lemeshow p < 0.05) may be useless for bedside prediction — the predicted risk is wrong even if it separates events from non‑events.
Your appraisal checklist for regression papers
# | Question | Why it matters |
1 | Is the outcome correctly specified? (Continuous → linear; binary → logistic) | Wrong link → wrong model. |
2 | Was the linearity of continuous predictors checked? (Residual plots, Box‑Tidwell, splines) | Non‑linearity → biased estimates. |
3 | Was multicollinearity assessed? (VIF > 5 is concerning) | Multicollinearity inflates standard errors. |
4 | Events per variable ≥ 10? (EPV rule) | EPV < 10 → overfitting; ORs unreliable. |
5 | Was calibration assessed? (Hosmer‑Lemeshow, calibration plot) | Poor calibration → predictions are wrong. |
6 | Was separation or complete prediction checked? (Firth’s if needed) | Perfect separation → infinite ORs; model fails. |
7 | Was clustering accounted for? (GEE, mixed effects) | Clustered data → underestimated standard errors. |
Common ICU examples at a glance
Outcome | Model type | Common predictors | EPV check |
28‑day mortality | Logistic | APACHE, SOFA, age, lactate, vasopressors | ≥ 10 events per predictor |
ICU length of stay | Linear (or log‑normal if skewed) | APACHE, SOFA, procedure count, sepsis | Check normality of residuals |
VAP (yes/no) | Logistic | mechanical ventilation days, sedation, H2 blocker, antibiotics | ≥ 10 VAP events per predictor |
AKI stage 3 | Logistic | fluid balance, norepinephrine, baseline creatinine | ≥ 10 AKI events per predictor |
Time to extubation | Cox (see Chapter 11) | treatment arm, SOFA, ARDS severity | ≥ 10 events per predictor |
Go deeper
- StatPearls – “Logistic Regression” (NBK539695): free, clear logistic‑regression explanation, odds‑ratio interpretation, EPV rule. https://www.ncbi.nlm.nih.gov/books/NBK539695/
- StatPearls – “Linear Regression” (NBK560123): free, covers assumptions, residual checks, multicollinearity. https://www.ncbi.nlm.nih.gov/books/NBK560123/
- OpenIntro Statistics – Chapter 9 “Multiple regression” (free PDF): worked examples, diagnostics, R code. https://www.openintro.org/stat/textbook.php
- PMC6620654 – “Regression analysis for the ICU clinician” (free full text): ICU‑focused practical guide to linear, logistic, Cox, and common pitfalls. https://doi.org/10.1186/s13054-019-2493-7 (PMC free)
Next: Chapter 13 — Confounding, interaction & model building