How to test for a main effect when dealing with bootstrapped distributions in a factorial design

Question

I have a 2 x 2 experimental design and have bootstrapped the distribution of a particular statistic for each cell. I would like to compute a p-value associated with the main effect of each factor. What is the proper way to do this?

To provide more context: the raw data going into this analysis are real numbers on the interval [0,1]. Each of these numbers is associated with a particular cell in the experimental design. The statistic I'm interested in is entropy. So, to create my bootstrapped distributions, I do the following

1) resample the data with replacement

2) for each cell, compute an estimate of entropy using the data corresponding to that cell

3) repeat the above many times

I've asked a similar question previously, and the recommendation I received was to perform an anova on each bootstrap iteration (i.e. bootstrap the F statistic). But this doesn't seem possible because each bootstrap iteration yields only 1 entropy estimate per cell.

My current solution is to average entropy estimates across one factor on each each iteration, yielding two distributions which can then be compared, but I'm wondering if their are other methods with more power.

Answer 1

Here's the solution I came up with.

For each bootstrap iteration 1) Fit a linear model isomorphic to an anova where I predict the 4 entropy values I get on each iteration from the two factors, their interaction, and a constant. 2) Save the fit coefficients.

At the end of the bootstrap I compare the distribution of each coefficient vs zero.