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I'm trying to implement the "Free Step-Down Resampling Method" described by Westfall and Young in "Resampling-Based Multiple Testing" (algorithm ~2.8 in the text). My goal is to perform a multivariate linear regression.

So, I have an error estimate (from the original sample) like this (using OLS):

$\epsilon = Y - (\beta_0 + \beta_1X_1 + \beta_2X_2 + \dots \beta_pX_p)$

[BTW, $X_1 \dots X_p$ are dummy variables.]

In order to resample ($i$ times), I have to do:

$Y_i^* = \epsilon_i^*$

where $\epsilon_i^*$ is a with replacement sample from the original $\epsilon$.

Here is the problem:

In my dataset, responses (rows) are clustered (data come from related individuals); so, I would usually have applied Huber-White estimators to account for correlations in OLS-based linear regressions.

I don't know how to proceed here... Should Huber-White estimators be used? If so, how?

Apologies if my question is too simple, but I'm new in resampling methods... I guess the answer is simple, too. Suggestions are welcome.

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don't you mean $Y−(\beta_1 X_1+ \beta_2 X_2+ \beta_p X_p)$, and should there be an intercept ($\beta_0$) in there as well? Also fix the stray "epsilon" by adding a \. I'd have put the parentheses and \ in myself but that's not enough characters of change to successfully edit. – Glen_b Oct 30 '12 at 22:17
True! I'll add that. – Elabore Oct 30 '12 at 22:28
You still didn't put the parentheses around the $(\beta_0 + \beta_1 X_1 + ... + \beta_p X_p)$ which means you're subtracting only the first of those terms and adding the rest. a - b + c is NOT the same as a - (b+c), and you really don't mean what you wrote. – Glen_b Oct 30 '12 at 23:21
Ok, just a typo... ;) – Elabore Oct 31 '12 at 0:15
White considered heteroskedasticity... at least in his 1980 article. The clustered standard errors are usually named as such without trying to attribute them to a person. – StasK Oct 31 '12 at 4:10
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1 Answer

up vote 1 down vote accepted

Usually there is a reasonable correspondence between the bootstrap scheme and the sandwich estimator.

  1. Simple bootstrap with resampling $\Leftrightarrow$ White's heteroskedasticity robust estimator
  2. Block bootstrap with blocks of length $l \Leftrightarrow$ Newey-West estimator with $l$ lags
  3. Bootstrap of clusters $\Leftrightarrow$ cluster-corrected standard errors.

To read on clustered bootstraps, start from Rao and Wu (1988). Methods of this kind would be implemented in survey package (although I am not sure it does the Rao and Wu bootstrap, precisely; my understanding is that survey just resamples clusters without the small sample corrections that Rao and Wu introduced). To read about resampling methods in regression analysis, take a look at Wu (1986).

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Thanks for your fast responses. I'll read those Rao's and Wu's publications. I've put a better reference in the original post.. and I also have a question about the other comment you wrote. Please take a look at it... Regards, – Elabore Oct 31 '12 at 17:35
An implementation of the model I mentioned (even based on the same references) is in flip. [**Although I do not find flip too friendly, so I've prefered either running the original code that appears in the book (for SAS) or re-writting it for R.] But I still don't have a sense of where to include corrections for non-heteroskedasticity, if necessary... – Elabore Oct 31 '12 at 17:48
I don't have the book, and I don't know what flip is, so I can't help you figuring out what the algorithm does and how to modify it for your situation. – StasK Nov 1 '12 at 13:53
Dear StasK and dear All, Apologies for the delay in getting back to the topic. Finally I've realized how to get bootweights using survey and then create permutated samples: $\epsilon_i^*$. Nonetheless, I'm still unable to figure out whether the original $\epsilon = Y - (\beta_0 + \beta_1X_1 + \beta_2X_2 + \dots \beta_pX_p)$ must be obtained using Huber-White correction for correlated responses... As you said, Simple bootstrap with resampling ⇔ White's heteroskedasticity robust estimator, but I don't know how to proceed in this case. – Elabore Nov 15 '12 at 14:39
Similarly, I wonder whether in the next steps, when doing $Y_i^* = \epsilon_i^*$ I should, again, use some kind of corrections to account for heteroskedasticity of each case when performing new regressions. I hope not to bother you; thank you in advance for your help. – Elabore Nov 15 '12 at 14:40
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