Follow this article and the cross references therein:
Harden, M., Friede, T. Sample size calculation in multi-center clinical trials. BMC Med Res Methodol 18, 156 (2018). https://doi.org/10.1186/s12874-018-0602-y
Calculating sample size for a multi-center prevalence study can be a bit more complex than for a single-center study, as there are a few additional factors to consider. Here are the general steps you can follow to calculate sample size for a multi-center prevalence study:
Define the population: Identify the target population you will be studying, and determine the total number of individuals in that population.
Determine the desired level of precision: Decide how precise you want your estimates of prevalence to be. This will depend on the research question, available resources and time, as well as the budget.
Determine the desired level of confidence: Decide how confident you want to be that your estimates of prevalence are accurate. This is usually set at 95% or 99%.
Estimate the prevalence: If possible, use any existing data or pilot data to estimate the prevalence of the condition in the population. If no data is available, use the best estimate based on expert opinion.
Calculate the required sample size: Use a sample size calculator or formula, taking into account the population size, desired level of precision, desired level of confidence, and estimated prevalence.
Multiply the sample size by the number of centers: Since the study will be conducted in multiple centers, you'll need to multiply the sample size by the number of centers to get the total sample size needed.
Add an extra sample size for non-response: Add an additional 10-20% of sample size to account for possible non-response, as not all individuals will agree to participate in the study.
Adjust for clustering: If the study is cluster sampling, meaning sampling individuals within a larger unit, such as a school or hospital, you'll need to adjust the sample size accordingly.
It's important to note that, even though sample size calculation is a crucial step in the study design, sometimes it may not be possible to achieve the desired sample size due to budget, time or other constraints. In these cases, a sensitivity analysis of the sample size may be performed to understand the potential impact of sample size on the precision of the estimates.