extend summary.lm for hccm?
On Mon, 25 Dec 2006, Achim Zeileis wrote:
On Sun, 24 Dec 2006, John Fox wrote:
If I remember Freedman's recent paper correctly, he argues that sandwich variance estimator, though problematic in general, is not problematic in the case that White described -- an otherwise correctly specified linear model with heteroscedasticity estimated by least-squares.
More generally, sandwich-type estimators are valid (i.e., estimate the right quantity, although not necessarily precisely, as Frank pointed out) in situations where the estimating functions are correctly specified but remaining aspets of the likelihood (not captured in the estimating functions) are potentially not.
Not exactly. The asymptotic properites are good, but in samples of moderate size the properties (including both biasedness and variance) can be surprisingly bad. And if you are trying to calculate a p-value, getting a too-small variance estimate gives you spuriously small p-values. FWIW, I've seen a case in which the nominal size of a test based on the sandwich estimator was several orders of magnitude smaller than a test with correct nominal size. There is a modest literature on this. Some refs: Background Papers: Drum M, McCullagh P. Comment. Statistical Science 1993; 8:300-301. Freedman DA. On the So-Called "Huber Sandwich Estimator" and "Robust" Standard Errors. The American Statistician, Volume 60, Number 4, November 2006, pp. 299-302(4) ----- Some Proposed Corrections: Fay MP and Graubard BI. Small-Sample Adjustments for Walt-Type Tests Using Sandwich Estimators. Biometrics 2001; 57: 1198-1206. Guo X, Pan W, Connett JE, Hannan PJ and French SA. Small-sample performance of the robust score test and its modifications in generalized estimating equations Statistics in Medicine 2005; 24:3479-3495 Mancl LA and DeRouen TA, A Covariance Estimator for GEE with Improved Small-Sample Properties. Biometrics 2001; 57:126-134. Morel JG, Bokossa MC, and Neerchal NK. Small Sample Correction for the Variance of GEE Estimators Biometrical Journal 2003; 45(4): 395-409. Pan W and Wall MM. Small-sample adjustments in using the sandwich variance estimator in generalized estimating equations. Statistics in Medicine 2002; 21:1429-1441. HTH, Chuck
In linear models, it is easy to see what this means: the mean function has to be correctly specified (i.e., the errors have zero mean) but the correlation structure of the errors (i.e., their (co-)variances) might differ from the usual assumptions. In GLMs, in particular logistic regression, it is much more difficult to see against which types of misspecification sandwich-based inference is robust. Freedman's paper stresses the point that many model misspecifications also imply misspecified estimating functions (and in his example this is rather obvious) so that consequently the sandwich-type estimators estimate the wrong quantity. Best wishes, Z
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Charles C. Berry (858) 534-2098
Dept of Family/Preventive Medicine
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