Chapter 6 — Probability foundations: normal, binomial, Poisson

The “p = 0.02” trap

It’s a Tuesday morning. You’re reviewing a study that reports ICU mortality dropped from 30% to 18% (p = 0.02). Your fellow jumps: “Significant!” You hesitate — what did the p actually come from? Was it a comparison of two proportions, assuming a normal approximation to a binomial? Was the sample size small enough that the approximation broke down?

If you don’t know which distribution underlies the test, you may trust a result that was never meant to be there. This chapter will give you the three distributions you see most often in ICU statistics, what each assumes, and how to appraise whether the assumption was checked.

The three probability distributions you must recognise

Why each distribution matters for your clinical thinking

• Normal underlies many common tests (t‑tests, linear regression, ANOVA). If the underlying data are not normal — but the sample size is large enough — the Central Limit Theorem lets you use normal‑based tests for proportions or means anyway. But small samples (n < 30) violate this, and you need exact methods.
• Binomial governs every binary outcome you read: mortality, shock reversal, ICU‑acquired infection. If the number of events is small, or the proportion is near 0 or 1, the exact binomial test (Clopper‑Pearson) is safer than a chi‑square or z‑test.
• Poisson is the engine of every ICU “rate” — VAP, CLABSI, CAUTI, even “number of admissions per week.” Poisson regression lets you adjust rates for time at risk and confounders.

The normal distribution: symmetry, tails, and the 68‑95‑99.7 rule

A normal distribution with mean μ and standard deviation σ has these properties:

• ~68% of values fall within μ ± 1σ
• ~95% fall within μ ± 2σ
• ~99.7% fall within μ ± 3σ

In the ICU, a mean arterial pressure of 70 mmHg with SD 15 tells you that about 95% of your ventilated patients have a MAP between 40 and 100. If a colleague says “MAP 65 ± 5 in the new protocol group,” and you see that the normal range is 60–70, you know the distribution is tight — but you still should ask whether the data are truly symmetric, especially if there are a few very hypotensive patients at the tail.

When the normal approximation fails — the binomial example

Suppose an ICU study reports: “1 patient out of 50 developed a catheter‑related bloodstream infection (2% rate).” A p‑value for comparing two arms might be calculated using a chi‑square test, which assumes the normal approximation to the binomial. But with 1 event, np = 1 and n(1−p) = 49 — the rule of thumb (both ≥ 5) is violated.

Under these conditions, the chi‑square p‑value can be inflated or deflated. The correct alternative is the Fisher’s exact test, which computes the exact probability from the binomial distribution. A paper that uses chi‑square on sparse data will report an unreliable p‑value.

Appraisal checklist for normal use:

1. Check for normality — Was a histogram, Q‑Q plot, Shapiro‑Wilk, or Kolmogorov‑Smirnov test reported? If not, normality is an unverified assumption.
2. Check sample size — For binomial outcomes, ensure both np and n(1‑p) ≥ 5 for a valid chi‑square approximation. For continuous data, small samples (n < 30) may be non‑normal even after log transformation.
3. Look for “exact” wording — Did the authors mention Fisher’s exact, Clopper‑Pearson, or “exact binomial” for small samples? That is generally preferable when counts are sparse.

The binomial distribution: discrete successes out of fixed trials

The binomial distribution applies when you have exactly n independent trials, each with two possible outcomes, and the probability of each outcome is constant.

ICU example: “Of 200 patients on veno‑venous ECMO, 46 (23%) died during extracorporeal support.” Here:

• Each patient is a “trial.”
• The outcome is binary (death / survival).
• The probability of death is assumed constant across 200 patients (this is a key assumption — but ECMO patients vary widely in severity).

If you want a confidence interval for that 23%, the simplest method is the normal approximation (the Wald interval):

This is valid when n is large and p is not near 0 or 1. For 46/200, the Wald CI is 0.23 ± 0.087 = [0.14, 0.32]. However, when events are rare or n is small, the Wilson score interval (or the exact Clopper‑Pearson) is preferred.

When researchers misuse the binomial model

• Non‑independent events: Patients clustered in a single ICU, ventilated consecutively by one team. The independence assumption is broken. Look for cluster‑robust standard errors or generalized estimating equations (GEE).
• Changing probability: A study enrolling over 2 years during which practice changes (e.g., new sedation protocols). Pooling events from early vs late periods into one binomial count may be misleading.
• Rare events: Treating a handful of seizures as a binomial proportion without accounting for time‑dependent exposure. Some patients had 6 months of monitoring; others, 10 days.

The Poisson distribution: rare events in time or space

Poisson models the number of events occurring in a fixed interval, assuming a constant rate. It is ideal for:

• Infection rates: “2.5 CLABSI per 1,000 catheter‑days”
• ICU admissions: “12 emergency admissions per week”
• Cardiac arrests: “1 cardiac arrest per 850 patient‑days”

The key feature of Poisson is that the mean equals the variance. If the variance is far larger than the mean (overdispersion), a negative binomial model is required instead. This is common in ICU data: infections cluster (some ICUs have outbreaks), and patient heterogeneity inflates variance.

A useful rule: if the Poisson rate λ has 0 or 1 events in most time intervals, Poisson is appropriate. Once you see intervals with 5, 8, 12 events, the variance likely exceeds the mean.

Overdispersion: the Poisson’s most common failure

A Poisson model that underestimates variance will produce standard errors that are too small, leading to p‑values that are spuriously significant. Look for:

1. Dispersion parameter > 1 reported in regression output.
2. Deviance / residual degrees of freedom >> 1
3. Authors using negative binomial or robust standard errors when overdispersion detected

If a paper reports a highly significant ICU infection rate (p < 0.001) using Poisson regression but does not check for overdispersion, treat that p with caution.

Your appraisal checklist for distributional assumptions

Go deeper

• StatPearls – “Normal Distribution” (NBK551995): free, concise explanation of the 68‑95‑99.7 rule and clinical relevance. https://www.ncbi.nlm.nih.gov/books/NBK551995/
• OpenIntro Statistics – Section 3.1 “The binomial distribution” (free PDF): worked examples of when to use normal vs exact binomial tests. https://www.openintro.org/stat/textbook.php
• BMJ Statistics Notes – “The Poisson distribution” (Altman & Bland): explains Poisson rates, when to suspect overdispersion. https://www.bmj.com/content/bmj_stats_notes
• PMC8329901 – “Choosing between Poisson and negative binomial models for longitudinal neonatal mortality counts”: free full‑text paper demonstrating how to test for overdispersion and select the correct model. https://doi.org/10.1186/s12874-021‑01306-4

Next: Chapter 7 — Confidence intervals & statistical significance