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Re: st: types of standard error


From   Phil Schumm <pschumm@uchicago.edu>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: types of standard error
Date   Tue, 23 May 2006 11:57:09 -0500

On May 23, 2006, at 9:03 AM, Timothy.Mak@iop.kcl.ac.uk wrote:
For example, in glm, we have the option of using:

OIM, EIM, OPG, HAC, jacknife, one-stepped jacknife, unbiased sandwich...

If you really want to learn more about the differences between these estimators, that's great; be forewarned, however, you're not going to get very far without delving into some of the math. The distinction between the expected information (EIM) and observed information (OIM) is an important one, and comes out of basic likelihood theory (any basic book on statistical theory will discuss this). In some cases (e.g., generalized linear models with canonical link), they are equivalent. A classic paper comparing the two is "Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information" (Efron and Hinkley, Biometrika (65) 1978). This paper argues that the observed information is in general more appropriate, which (I presume) is why -glm- produces this by default.

The robust (sandwich) estimator is an attempt to relax the model assumptions on which the standard calculation is based. For a simple introduction (with examples), see "Model Robust Confidence Intervals Using Maximum Likelihood Estimators" (Royall, International Statistical Review (54) 1986). Also, as I said before, see [U] 20.14 and the references cited therein.

Efron and Tibshirani's book on the bootstrap (the one I suggested earlier) actually treats both the bootstrap and the jackknife in detail, and discusses the relationship between them and several other estimates of variance (e.g., expected and observed information, sandwich, etc.). In this respect, it is one of the better (and more accessible) treatments of the differences among the various estimators.

Finally, if you're primarily interested in this issue WRT -glm-, I believe Hardin and Hilbe's book (http://www.stata.com/bookstore/ glmext.html) has a good discussion of the differences between the various estimators available.

Keep in mind, more choice (and the ease with which the different estimates may be obtained and compared) is a very good thing. You can never go wrong by trying the different estimators to verify that they give similar results (which they often do). In cases where the results are substantially different, however, you need to be careful picking one over the others. And as always, they justifications for many of these estimators rely on asymptotic arguments, and therefore you should always be careful when applying them to data from small samples.


-- Phil

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