In this example we will learn how to estimate sample size to compare two proportions. Researchers want to estimate sample size required for a randomised controlled trial designed to test if a newer treatment as compared to the standard treatment reduces the mortality rate among the patients with sepsis admitted in an ICU.

We need to take certain decisions before calculating sample size. First we need to set or fix Type-I and Type-II errors. Type-I error decides level confidence while Type-II error decides power of study. Conventionally Type-I and Type-II errors are fixed and 5% and 20% respectively which means study is designed to have 95% confidence level and 80% power in estimating the effect (difference) under investigation.

We also need some prior data (or a expert "guess" in case data is not available) about effect i.e. difference in outcome rates among controls and intervention. Let us assume outcome rate in control as 30% and in intervention as 20%. With these assumptions and information now let us proceed to sample size calculation

Navigate to [https://wnarifin.github.io/ssc_web.html](https://wnarifin.github.io/ssc_web.html)

Click "Two-proportion comparison (independent)"

Click on proportion in cases (intervention)

Leave other parameters to default, since we have also assumed same as defaults

We can see sample size as well as adjusted sample size for dropout rate.

See what happens when we increase power to 90%

If we increase the power, sample size increases. It means we require more sample size to correctly detect the true difference.

Now let us decrease the difference to be detected. It can be done by either increasing the outcome rate in intervention or cases or decreasing outcome rate in controls. 

We can see that, we require quite large sample size to detect smaller differences. Unfortunately smaller differences are reality in clinical research and that is the reason we need to conduct multi-centric studies.

2 proportions - Hypothesis Testing
Proportion in control (p0):
Proportion in case (p1):
Significance level (α): Two-tailed
Power (1 - β): %
Expected dropout rate: %
 
Sample size, n =
Sample size (with 10% dropout), ndrop =

How to write these calculations in sample size section of your protocol? "We have calculated sample size using web based sample size calculator (Arifin, W. N. (2022). Sample size calculator (web). Retrieved from [wnarifin.github.io](http://wnarifin.github.io) ). Assumptions in sample size calculation were based on study by XXX (cite the study). Sample size was estimated to detect the difference in outcome rate (mortality) among patients with sepsis admitted in an ICU who are randomised to newer treatment as compared to standard treatment assuming reduction from 30% (in standard treatment) to 20% (in newer treatment), with Type-I error of 5% (95% confidence) and Type-II error of 20% (80% power). Required sample size was found to be 327 per group and total of 652 participants would be needed to be enrolled."

Sample size for comparing two group proportion