KIMRA

Research toolsStudy design

Sample size for comparing two proportions

Calculates how many participants per group are needed to detect a difference between two independent proportions, at a chosen significance level and statistical power.

These are the values under the alternative hypothesis — the difference you want the study to be able to detect.

1 for equal groups. 2 means twice as many in group 2.

Results

Total participants
186
Group 1
93
Group 2
93
Per group, before rounding
92.999
Critical values
z₁₋α⁄₂ = 1.9600z₁₋β = 0.8416

This is the sample for the primary outcome. Secondary outcomes and subgroups will be under-powered at this size.

When to use it

  • Powering a two-arm trial with a binary outcome.
  • Planning a cohort or case–control study comparing two groups.
  • Checking whether a proposed study is large enough to answer its question at all.

Formula

           [ z₁₋α⁄₂ · √( (1 + 1/k) · p̄(1 − p̄) )  +  z₁₋β · √( p₁(1−p₁) + p₂(1−p₂)/k ) ]²
  n₁  =  ──────────────────────────────────────────────────────────────────────────────
                                     ( p₁ − p₂ )²

  n₂  =  k · n₁

  p̄ = (p₁ + k·p₂) / (1 + k)     the weighted pooled proportion
  k = allocation ratio, n₂ : n₁  (k = 1 for equal groups)
  z₁₋α⁄₂ = 1.959964 at α = 0.05 two-sided
  z₁₋β   = 0.841621 at 80% power, 1.281552 at 90% power

Worked example

A two-arm trial expecting 50% response on the new treatment and 30% on the control.

Proportion, group 1
0.50
Proportion, group 2
0.30
Significance level
0.05, two-sided
Power
80%
Allocation
1 : 1

n = 92.998 per group, so 93 per group, 186 in total.

With 93 participants in each arm, a true difference of 20 percentage points would be detected about 80% of the time.

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