bootstrapping in regression

Thomas Mang wrote:

Hi,

Please apologize if my questions sounds somewhat 'stupid' to the 
trained and experienced statisticians of you. Also I am not sure if I 
used all terms correctly, if not then corrections are welcome.

I have asked myself the following question regarding bootstrapping in 
regression:
Say for whatever reason one does not want to take the p-values for 
regression coefficients from the established test statistics 
distributions (t-distr for individual coefficients, F-values for 
whole-model-comparisons), but instead apply a more robust approach by 
bootstrapping.

In the simple linear regression case, one possibility is to randomly 
rearrange the X/Y data pairs, estimate the model and take the 
beta1-coefficient. Do this many many times, and so derive the null 
distribution for beta1. Finally compare beta1 for the observed data 
against this null-distribution.

There is a very basic difference between bootstrapping and random 
permutations.  What you are suggesting is to shuffle values between 
cases or rows in the frame.  That amounts to a variant of a permutation 
test, not a bootstrap.

What you do in a bootstrap test is different, you regard your sample as 
a population and then sample from that population (with replacement), 
normally by extracting a large number of random samples of the same size 
as the original sample and do the computations for whatever you are 
interested in for each sample.

In other words, with bootstrapping, the pattern of values within each 
case or row is unchanged, and you sample complete cases or rows.  With a 
permutation test you keep the original sample of cases or rows, but 
shuffle the observations on the same variable between cases or rows.

Have a look at the 'boot' package.

Tom

What I now wonder is how the situation looks like in the multiple 
regression case. Assume there are two predictors, X1 and X2. Is it 
then possible to do the same, but just only rearranging the values of 
one predictor (the one of interest) at a time? Say I want again to 
test beta1. Is it then valid to many times randomly rearrange the X1 
data (and keeping Y and X2 as observed), fit the model, take the beta1 
coefficient, and finally compare the beta1 of the observed data 
against the distributions of these beta1s ?
For X2, do the same, randomly rearrange X2 all the time while keeping 
Y and X1 as observed etc.
Is this valid ?

Second, if this is valid for the 'normal', fixed-effects only 
regression, is it also valid to derive null distributions for the 
regression coefficients of the fixed effects in a mixed model this 
way? Or does the quite different parameters estimation calculation 
forbid this approach (Forbid in the sense of bogus outcome) ?

Thanks, Thomas