Formula Used
Continuous endpoint:
N control = (1 + 1 / r) × (Z alpha + Z power)² × SD² / Difference²
N treatment = r × N control
Binary endpoint:
N control = (Z alpha + Z power)² × [p0(1 - p0) + p1(1 - p1) / r] / (p1 - p0)²
N treatment = r × N control
Longitudinal cluster design effect:
DE = 1 + (M effective - 1)ICC + (T - 1)R + (M effective - 1)(T - 1)ICC × R
Cluster size adjustment:
M effective = M × (1 + CV²)
Final adjusted units:
Adjusted units = N × DE × (1 - covariate R²) / (1 - dropout rate)
Clusters per arm:
Clusters = ceiling(adjusted units / average repeated observations per cluster)
How to Use This Calculator
- Select the endpoint type.
- Enter alpha, power, and test side.
- Add the expected effect size.
- Enter cluster size, visits, ICC, and repeated measure correlation.
- Add dropout, cluster size variation, and covariate adjustment.
- Press calculate to show results below the header.
- Use CSV for spreadsheet work.
- Use PDF for reports and planning notes.
Example Data Table
| Scenario |
Outcome |
Effect |
ICC |
Visits |
Participants per Cluster |
Power |
| Physics lab cohort |
Continuous score |
0.50 mean units |
0.02 |
4 |
30 |
0.80 |
| Education trial |
Binary success |
40% to 52% |
0.03 |
3 |
25 |
0.90 |
| Field measurement program |
Continuous index |
0.35 mean units |
0.01 |
5 |
40 |
0.80 |
Understanding Longitudinal Cluster Trial Sizing
Longitudinal cluster randomization trials assign groups, not single people. Schools, wards, laboratories, or clinics may become clusters. Repeated visits then measure each participant over time. This structure is useful in physics education, field studies, and applied research programs. It also creates dependence between observations. People within the same cluster tend to resemble one another. Repeated measures from one person also resemble earlier measures. A simple independent sample formula can understate the needed size.
Why Correlation Matters
The calculator uses a design effect. This factor inflates the independent sample count for cluster and time correlation. The intracluster correlation measures similarity inside a cluster. The within participant correlation measures stability across visits. More visits can improve information, but they also add correlation. Large clusters can be efficient when the ICC is small. They can become wasteful when the ICC is high. Unequal cluster sizes add another penalty through the cluster size coefficient of variation.
Planning Inputs
Start with the smallest effect worth detecting. For a continuous endpoint, enter the expected mean difference and standard deviation. For a binary endpoint, enter control and treatment rates. Select alpha, power, test side, and allocation ratio. Then add the average participants per cluster, number of visits, ICC, repeated measure correlation, dropout, and adjustment from covariates. The result estimates clusters per arm, total participants, total observations, and key inflation factors.
Using Results Wisely
The output is a planning guide, not a regulatory guarantee. Real trials may need simulation, mixed model review, or expert statistical design. Still, the estimate is valuable early. It reveals whether power is limited by clusters, participants, visits, or attrition. It also shows how much correlation changes the budget. Run several scenarios before approval. Compare optimistic, expected, and conservative assumptions. Save the CSV for spreadsheets. Export the PDF for notes, protocols, and stakeholder review.
Practical Review
Check assumptions before recruitment begins. ICC values from past studies are especially helpful. When none exist, use conservative values. Review cluster count feasibility first. More participants inside the same cluster cannot fully replace missing clusters. Balance matters too. A severe allocation ratio can raise the total size. Clear planning avoids weak evidence, wasted effort, and expensive redesigns. It supports team decisions early too.
FAQs
What is a longitudinal cluster randomization trial?
It is a study where groups are randomized, and participants are measured repeatedly over time. The grouped design and repeated visits both create correlation that affects sample size.
Why does ICC increase the required sample size?
ICC shows similarity within the same cluster. When similarity is high, each added participant gives less independent information. More clusters may be needed to protect power.
What does within participant correlation mean?
It measures how strongly repeated outcomes from the same participant resemble each other. Higher values reduce the independence of repeated measurements.
Should I use continuous or binary outcome mode?
Use continuous mode for mean scores, lab values, and measured scales. Use binary mode for success, failure, response, completion, or any yes or no endpoint.
What is the allocation ratio?
It is the treatment arm size divided by the control arm size. A value of 1 means equal allocation. Higher values assign more clusters to treatment.
Why include dropout?
Dropout reduces complete data. The calculator inflates the planned size so the expected completed observations remain closer to the needed amount.
What does covariate R squared do?
It represents variance explained by baseline covariates. Higher values can reduce required sample size because adjusted analysis may improve precision.
Is this enough for a final protocol?
This calculator is useful for planning. Complex studies should also be reviewed by a statistician, especially when using mixed models, unequal follow up, or unusual correlation structures.