Calculating the sample size for a complex multi-stage sampling design involves several steps. I will guide you through the process.

Step 1: Determine the required level of precision

The level of precision refers to how close you want your estimate to be to the true population value. This is usually expressed as a margin of error (MOE) or confidence interval. For example, if you want to estimate the proportion of girls facing challenges accessing menstrual health hygiene products within a specific margin of error (e.g., +/- 3%), you would need to determine the desired level of precision.

Step 2: Determine the desired level of confidence

The level of confidence refers to the probability that the true population parameter falls within the estimated interval. Common levels of confidence are 95% or 99%. Decide on the level of confidence you would like for your study.

Step 3: Adjust the sample size for the design effect

In complex multi-stage sampling designs, there is often a design effect that accounts for the clustering and stratification. The design effect is a measure of how much larger the required sample size is compared to a simple random sample. It accounts for the correlation among participants within the same cluster or stratum. The design effect varies depending on the specific design and analysis plan. For now, let's assume a design effect of 2 for your study.

Step 4: Calculate the initial sample size

To calculate the initial sample size, you will need to use a formula appropriate for estimating proportions. The most commonly used formula is the one for calculating sample size for a simple random sample:

n = [(Z^2 * p * (1-p)) / MOE^2]

where:

n is the required sample size

Z is the z-score corresponding to the desired level of confidence (e.g., 1.96 for 95% confidence)

p is the estimated proportion of the population with the characteristic of interest (use 0.54 as the estimated proportion)

MOE is the margin of error expressed as a decimal (e.g., 0.03 for +/- 3% MOE)

Using this formula will give you the required sample size for a simple random sample.

Step 5: Adjust the sample size for the design effect

To adjust the sample size for the design effect in multi-stage sampling, you will multiply the initial sample size by the design effect:

final_sample_size = initial_sample_size * design_effect

For example, if your initial sample size calculated in Step 4 is 400, and the design effect is 2, the final sample size would be 400 * 2 = 800.

Step 6: Determine the number of clusters or units at each stage

At each stage of the sampling design, you need to determine the number of clusters or units to select. For example, you mentioned selecting 12 counties, wards within each county, and schools within each ward. The number of clusters or units at each stage will depend on the population size and the desired precision. It is important to consult resources specific to your study area or consult with a statistician to determine the appropriate number of clusters or units at each stage.

Considering the complexity of your sampling design, it would be beneficial to consult with a statistician to properly determine the final sample size and the number of clusters or units at each stage based on the specific details of your study.