Sample size ( $n$ )

The sample size is the number of observational units included in a study or assigned to a specified study arm.

Table of contents
- Sample size for comparing two proportions Sample size
  - Rare-event collaboration size at half power Sample size for comparing two proportions
  - Endpoint-specific rare-event sample size Sample size for comparing two proportions
- Normal-mean sample size calculation Sample size

Sample size for comparing two proportions ( $n ≃ [z_{1 - α /2} 2 \overset{p}{ˉ} (1 - \overset{p}{ˉ}) + z_{1 - β} p_{1} (1 - p_{1}) + p_{0} (1 - p_{0})]^{2} / (p_{1} - p_{0})^{2}$ )

For equal independent binomial samples, the displayed normal-approximation size per group targets a two-sided significance level

α

and power

1 - β

at specified alternative rates, with

\overset{p}{ˉ} = (p_{1} + p_{0}) /2

. Its first variance term sets the null critical value and its second the alternative distribution. A numerical sample size is not determined by an effect ratio alone; power, sidedness, allocation and baseline rates must also be specified. Rare-event discreteness and clustered designs can materially change the achieved power.

Rare-event collaboration size at half power ( $m ≃ ⌈ \frac{2 p ˉ ( 1 - p ˉ ) z _{*}^{2}}{n _{0} ( p _{A} - p _{B} ) ^{2}} ⌉$ )

 0  0

Suppose each collaborating centre contributes

n_{0}

independently randomized participants per arm with specified event probabilities. At 50% statistical power, the usual normal-approximation alternative quantile is zero. The required size per arm is

2 \overset{p}{ˉ} (1 - \overset{p}{ˉ}) z_{*}^{2} / (p_{A} - p_{B})^{2}

, with

z_{*} = z_{1 - α /2}

for a two-sided test or

z_{1 - α}

for a one-sided test. Divide by the per-centre arm size and round up. Endpoints must use the same time origin and follow-up window, and cluster dependence requires an additional adjustment.

Endpoint-specific rare-event sample size

 0  0

A sample size calculation for a binary endpoint requires its own baseline probability, not merely the rate of a related outcome. A fatal-event probability equals the probability of the underlying event times its conditional fatality probability; knowing only that product does not identify the incidence of all events. Thus a mortality rate can size a mortality trial but cannot determine the size for all fatal and nonfatal events without further assumptions. The significance level, allocation, target statistical power and effect definition must also be specified.