Power for Logistic Regression

Sample size for logistic regression: events per variable, odds-ratio detection, and simulation

Introduction

Power analysis for logistic regression is less standardised than for linear models. Three approaches: (1) events-per-variable rule of thumb, (2) closed-form formulas for a single predictor, (3) simulation for complex models.

Prerequisites

Logistic regression, odds ratio, events-per-variable (EPV).

Theory

EPV rule (Peduzzi et al.): at least 10 events per predictor coefficient is a working minimum. Harrell’s 20 EPV is safer.

Single-predictor formula (Demidenko 2007) uses \(\log(OR)\) and the baseline event rate:

\[n = \frac{(z_{1-\alpha/2} + z_{1-\beta})^2}{p_0(1 - p_0) \log^2(OR) \cdot \sigma_X^2 / \pi(1 - \pi)},\]

with simplifications for dichotomous \(X\).

Simulation: specify the full data-generating process, fit the intended model, compute power across many replicates. More reliable than formulas for multiple predictors.

Assumptions

• Binary outcome.
• Logit-linear predictors.
• Sufficient separation of groups.

R Implementation

library(WebPower)

# Single dichotomous predictor: baseline event rate p0 = 0.2, OR = 1.5
wp.logistic(n = NULL, p0 = 0.2, p1 = 0.2 * 1.5 / (1 + 0.2 * (1.5 - 1)),
            alpha = 0.05, power = 0.80,
            family = "Bernoulli", parameter = NULL, alternative = "two.sided")

# Continuous predictor: detect OR = 1.5 per 1-SD increase
wp.logistic(n = NULL, p0 = 0.2, p1 = 0.2,
            alpha = 0.05, power = 0.80,
            family = "normal",
            alternative = "two.sided",
            parameter = log(1.5))

# EPV rule: 15 predictors, min events = 10*15 = 150
# If baseline event rate is 0.20: n = 150 / 0.20 = 750

Output & Results

Detecting a modest OR of 1.5 typically requires hundreds of observations; continuous predictors need somewhat less \(n\) than dichotomous, given comparable effect magnitude.

Interpretation

“To detect an OR of 1.5 per unit of the standardised predictor at 80 % power and two-sided \(\alpha = 0.05\), 250 participants are required (assuming 20 % baseline event rate).”

Practical Tips

• Prefer simulation over formulas when predictors are correlated or the model is complex.
• Rare-event outcomes dramatically inflate required \(n\); power is approximately symmetric around \(p_0 = 0.5\).
• For ordinal or multinomial logistic, scale up by roughly \(k - 1\) (number of contrasts).
• Penalised logistic (ridge, Firth) handles small samples with separation.
• Missing data and measurement error further increase required \(n\).