Estimate exposed and unexposed cohort sample sizes from outcome rates, confidence, power, allocation ratio, and dropout, or reverse-solve detectable risk.
Related Calculators
- Cochranโs Sample Size Calculator
- P-hat Calculator
- Risk Difference Calculator
- Per 100 000 Population Calculator
- All Statistics Calculators
Cohort Study Sample Size Formula
The calculator uses an unpooled normal (Wald) approximation for two independent cohorts with a binary outcome and the same follow-up period. It reports exposed recruitment, unexposed recruitment and their total, with the selected allocation and loss allowance. Other planning methods can give different estimates.
- nE = raw exposed sample size; raw unexposed size is r ร nE. Recruit each group by rounding up its raw size divided by (1 โ dropout).
- Zฮฑ/2 = Z-value for the selected two-sided confidence level
- Zฮฒ = Z-value for the selected study power
- pE = exposed outcome proportion internally; enter a percentage in the widget.
- pU = unexposed outcome proportion internally; r is the unexposed-to-exposed allocation ratio.
The calculator computes normal quantiles from the confidence and power percentages. Smaller outcome differences generally require larger samples. The reverse mode searches only for an increased exposed outcome rate below 100% that fits within the available recruitment under this approximation. Sensitivity scenarios scale the signed risk difference by 75% or 125%; out-of-domain rates are reported rather than clamped.
Enter outcome percentages, for example 25 for 25%. Confidence and power are also percentages. The formula uses decimal probabilities internally. Counts are rounded up separately because recruitment cannot include a fraction of a participant.
Common Z-Values for Cohort Study Planning
These values are commonly used for confidence level and power inputs.
| Input | Common setting | Computed Z-value (approx.) |
|---|---|---|
| Zฮฑ/2 | 90% confidence, two-sided | 1.645 |
| Zฮฑ/2 | 95% confidence, two-sided | 1.96 |
| Zฮฑ/2 | 99% confidence, two-sided | 2.576 |
| Zฮฒ | 80% power | 0.84 |
| Zฮฒ | 90% power | 1.28 |
How Input Differences Affect Sample Size
| Input pattern | Effect on sample size | Reason |
|---|---|---|
| p1 and p2 are close together | Larger sample size | A smaller difference is harder to detect. |
| Higher confidence level | Larger sample size | A larger Zฮฑ/2 increases the numerator. |
| Higher power | Larger sample size | A larger Zฮฒ increases the numerator. |
| p1 equals p2 | Cannot calculate | The denominator becomes zero. |
Example Cohort Study Sample Size Calculations
Example 1: 95% confidence and 80% power
Suppose you expect the outcome proportion to be 0.30 in the exposed group and 0.20 in the unexposed group.
- Zฮฑ/2 = 1.96
- Zฮฒ = 0.84
- p1 = 0.30
- p2 = 0.20
Using the formula:
With r = 1 and dropout = 0%, the widget uses z โ 1.959963985 and 0.841621234: nE โ 290.40855, so recruit 291 per group, 582 total. The displayed 1.96 and 0.84 above are rounded references.
Example 2: Larger expected difference
Suppose the expected outcome proportion is 0.40 in the exposed group and 0.25 in the unexposed group, using 95% confidence and 80% power.
- Zฮฑ/2 = 1.96
- Zฮฒ = 0.84
- p1 = 0.40
- p2 = 0.25
Using the formula:
With r = 1 and dropout = 0%, the widgetโs computed quantiles give nE โ 149.128715, so recruit 150 per group, 300 total.
Cohort Study Sample Size FAQ
Is the sample size the total sample size or the sample size per group?
The result shows the total recruitment and each group separately. Set r = 1 for equal groups. With unequal allocation, raw unexposed size is r times raw exposed size; separate rounding may slightly change the achieved ratio. The entered dropout allowance is already included in the recruitment counts.
How should you enter percentages?
Enter 10 for 10%, 35 for 35%, and 60 for 60%. Outcome rates must be strictly between 0% and 100%; confidence and power must be strictly between 50% and 100%. Equal expected outcome rates cannot produce a finite sample-size estimate.
Should you increase the result for loss to follow-up?
Use the optional dropout field when participants may be lost or have missing outcomes. The widget divides each raw group size by the common retention rate. For 200 retained participants and 10% dropout, 200 / 0.90 = 222.22, rounded up to 223 recruited participants. This is about an 11.11% recruitment increase, not a 10% increase. Normal approximations may be unreliable with few expected events or non-events; this tool does not guarantee exact achieved power or account for matching, clustering, censoring or adjusted analyses.
