bootstrapcorrelationhypothesis testing
I want to test the hypothesis that correlation coefficient between X and Y is 0 with a bootstrap, however I don't know which is a correct way to construct bootstrap samples. I have several ideas, which may be wrong:
1) randomly permute Y among X's as in permutation test, but sample Y with replacement
2) permute Y among X's without replacement and sample pairs (X,Y) with replacement
Could you please tell me the right way to do bootstrapping in this case and justify it?
Thank a lot!
I don't think use bootstrap to artificially increase your sample size would be a good idea. Any violation of the assumption of the independence of the observations dramatically increase the odds od a spurious result (which would be the case when n1 is significantly greater than n0).
I would estimate the confidence interval of the effect size (the strength of the difference / relationship you are trying to use) and assume the lower bound as the true effect. Then it would be easy to estimate the power.
[note: I assume you already have a significan result with n0 observations. Otherwise your data is compatible* with the null hypothesis and there is no way to have a conservative estimate of the powere - unless you are using the wrong test. Power analysis assume the knowledge of the "real" effect size so there is no way to use theme to "bypass" inferential statistics]
*likely to be observed if the null hypotesis is true
The short answer is that - at least in the simple cases - the observations are sampled with replacement. Imagine writing each of the data values on an n sided die and rolling the die n times.
If you're trying to bootstrap a correlation, you resample the data in pairs $(x_i,y_i)$. If you think of your data as two columns, each row is an observation, and you resample the observations (rows).
Here's an example:
More generally, think of a matrix of data where the observations (rows) are resampled.
(This is not a suitable resampling scheme for every situation, though. There are a plethora of bootstrap schemes.)
Best Answer
Just sample pairs (X,Y) with replacement. You should not be permuting anything.