# Apparent paradox about using a pilot study to determine sample size

I'm using some pilot data to determine the necessary sample size for an upcoming study.

The primary goal of this study is to measure the correlation between two variables, A and B, and to determine the confidence interval for this correlation statistic.

What I have been doing is looking at the distribution of differences between the two variables in the pilot data, and using the standard deviation of this distribution as a basis for a simulation. Let us call this standard deviation the ERROR.

The simulation does the following:

It generates a normally weighted random value for variable A, and then adds Gaussian noise to this value to generate a value for variable B (the standard deviation of which is equal to ERROR). It repeats this N number of times, and then measures the correlation and the confidence interval for this correlation.

At the end of the simulation, I can determine the confidence intervals as a function of N.

This is all fine, I've figured out how to do this, how to use the Monte Carlo method when the distributions in question aren't parametric, and how to test for and deal with nonindependent errors.

I am, however, having a mild crisis of faith here. The sample data is the "ground truth" from which all my simulation results are derived. Now, the goal of the actual study is to determine the correlation and confidence interval of this correlation. This correlation is directly proportional to the standard deviation of the distribution of the differences between A and B. Yet, it is this very error I am using that I am using, in my simulation, to determine how many samples of A and B I'll need to achieve a given confidence interval (I'm using the sample error, which I termed ERROR above).

Something seems circular here, and I feel that the answer is very simple and obvious. Can someone guide me back to the path of light? I think it's similar to the issue of using a sample mean to determine the distribution of sample means.