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Proportion · PROP-KI

Comparing Three or More Independent Proportions

Plan an omnibus comparison of a binary outcome across three or more independent groups.

Enter anticipated percentages for mutually independent groups. The result targets the global Pearson chi-square test, not pairwise comparisons.

1. Anticipated group proportions and allocation

Group 1

Group 2

Group 3

2. Researcher settings

3. Results

Enter the planning assumptions and select Calculate.

Formula, assumptions, limitations, and methodological note

Example clinical scenario

A pragmatic trial compares smoking-cessation success at six months across standard counselling, a mobile intervention, and combined counselling plus medication.

Formula and methodological note

For candidate group sizes ni, the allocation-weighted pooled probability is p̄ = Σnipi/N. The noncentrality parameter is λ = Σni(pi − p̄)²/[p̄(1 − p̄)]. Power is the upper-tail probability of a noncentral chi-square distribution with k − 1 degrees of freedom beyond the central chi-square critical value. Whole-number allocation ratios are preserved exactly, and the smallest valid multiplier reaching target power is reported.

The method assumes independent participants, one binary endpoint, prespecified group probabilities, and adequate expected event/non-event counts. It plans only the omnibus test. It does not control power or multiplicity for pairwise comparisons and does not cover ordered trends, covariate adjustment, paired/repeated data, clustering, exact methods, or sparse tables. Dropout is applied after statistical sizing and rounded upward by group.

This tool supports research planning and does not replace protocol-specific statistical consultation.

Technical results

Method references

  1. Fleiss JL, Levin B, Paik MC. Statistical Methods for Rates and Proportions. 3rd ed. Wiley; 2003.
  2. R Core Team. The (Non-central) Chi-Squared Distribution. R stats documentation.

Suggested citation

Yaacob, Najib Majdi. “Sample Size for Comparing Three or More Independent Proportions (PROP-KI).” Web calculator. NajibMY, 2026. Accessed August 19, 2026. https://najibmy.netlify.app/tools/sample-size/proportion/three-or-more-independent-proportions/.