Calculate cohort study sample size from Zα/2, Zβ, and outcome proportions in exposed and unexposed groups for two-proportion comparison.
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Cohort Study Sample Size Formula
The calculator estimates the required sample size for a cohort study comparing the proportion of an outcome in an exposed group with the proportion in an unexposed group.
n = ((Z_{\alpha/2} + Z_{\beta})^2 * (p_1(1 - p_1) + p_2(1 - p_2))) / (p_1 - p_2)^2- n = required sample size, rounded up to the next whole participant
- Zα/2 = Z-value for the selected two-sided confidence level
- Zβ = Z-value for the selected study power
- p1 = expected outcome proportion in the exposed group
- p2 = expected outcome proportion in the unexposed group
The calculator uses the entered Z-values and expected proportions to calculate the sample size needed to detect a difference between two proportions. The difference between p1 and p2 is the effect size. Smaller differences require larger sample sizes.
Enter proportions as decimals. For example, enter 25% as 0.25. The calculator rounds the result up because a study 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 | Z-value to enter |
|---|---|---|
| Zα/2 | 90% confidence, two-sided | 1.645 |
| Zα/2 | 95% confidence, two-sided | 1.96 |
| Zα/2 | 99% confidence, two-sided | 2.576 |
| Zβ | 80% power | 0.84 |
| Zβ | 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 Zα/2 increases the numerator. |
| Higher power | Larger sample size | A larger Zβ 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.
- Zα/2 = 1.96
- Zβ = 0.84
- p1 = 0.30
- p2 = 0.20
Using the formula:
n = 290.08, so the rounded sample size is 291.
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.
- Zα/2 = 1.96
- Zβ = 0.84
- p1 = 0.40
- p2 = 0.25
Using the formula:
n = 151.36, so the rounded sample size is 152.
Cohort Study Sample Size FAQ
Is the sample size the total sample size or the sample size per group?
This formula is commonly used for equal-sized exposed and unexposed groups. In that setting, the result is the required sample size for each group. If you need the total sample size, multiply the result by 2. For example, if the result is 291, you would plan for about 291 exposed participants and 291 unexposed participants, before adding any allowance for loss to follow-up.
How should you enter percentages?
Enter percentages as proportions between 0 and 1. For example, enter 10% as 0.10, 35% as 0.35, and 60% as 0.60. Do not enter 35 for 35%, because that is outside the valid range for a proportion.
Should you increase the result for loss to follow-up?
Yes, if you expect some participants to drop out or have missing outcome data. To adjust for loss to follow-up, divide the calculated sample size by the expected retention rate. For example, if the calculated sample size is 200 and you expect 90% retention, use 200 / 0.90 = 222.22, then round up to 223.