Dear Celso,

I would only add a few comments to Martin's explanation below:

- tau-tests were primarily meant for general nested hypotheses, whereas

for a hypothesis of the form "beta_j = 0" for a single index "j" one can

also use (as it's done for glm estimators, say) *approximate* p-values

based on a normal approximation to the distribution of the ratio

"estimator / standard error" -- these are the p-values that

"summary.lmrob" currently reports, based on the *robust standard errors*

of Croux et al. 2003 (full reference in Martin's e-mail below) that

remain valid even when the data may contain asymmetric outliers;

- the asymptotic distribution of tau-tests is known for symmetrically

distributed errors and, furthermore, it involves a weighted sum of

independent chi-squared distributions, with weights depending on the

eigenvalues of the (asymptotic) covariance matrix of the explanatory

variables. Not surprisingly, their p-values are rather difficult to

calculate in practice (although approximations do exist: see Alfio

Marazzi's ROBETH and S-PLUS's robust libraries);

- for nested linear hypotheses, the tests in Markatou and Hettmansperger

(1990, "Robust bounded-influence tests in linear models", JASA, 85,

187-190) provide an alternative to the tau-tests with the "usual"

asymptotic chi-squared distribution, although this asymptotic

approximation is also known to hold for symmetrically distributed

errors, and moreover, seems to be rather sensitive to the presence of

outliers (see my paper in JSPI, 2005, 128, 241-257), while the Robust

Bootstrap performs quite well in estimating the p-values for these

robust tests.

Summarizing:

- the standard errors and p-values for individual hypotheses of the form

"beta_j=0" reported by summary.lmrob (in robustbase) are (robust)

asymptotic approximations, which should be interpreted and used accordingly;

- if you're interested in nested linear hypotheses, there are some

proposals in the literature to obtain robust p-values for robust tests

although they have not been implemented in robustbase yet (hopefully

they will be in the near future).

Best,

Matias

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______________________________________________________________

Matias Salibian-Barrera - Department of Statistics

University of British Columbia -

[hidden email]
Phone: (604) 822-3410 - Fax: (604) 822-6960

Martin Maechler wrote:

> [Oops! Written 6 hours ago, the following was accidentally not sent.]

>

>>>>>> "Celso" == Celso Barros <

[hidden email]>

>>>>>> on Wed, 5 Jul 2006 04:09:17 -0300 writes:

>

> Celso> When I run rlm to obtain robust standard errors, my output does not include

> Celso> p-values. Is there any reason p-values should not be used in this case?

>

> yes (see also below).

>

> Celso> Is there an argument I could use in rlm so that the output does

> Celso> include p-values?

> no.

>

> What are the reasons?

>

> How to properly do hypothesis testing in the context of robust

> regression has partly been an open research problem. Whereas

> or has been solved in Elvezio Ronchetti's PhD thesis (1982)

> by tau-tests, see chapter 7 of Hampel, Rousseeuw, Ronchetti,

> Stahel (1986), these are not (directly) related to standard

> errors, and t-tests with some degrees of freedom.

> Hence they are not so intuitively explainable, and not entirely

> trivial to implement. Probably this is one of reasons, why they

> (tau-tests) haven't been programmed for MASS (the book and the R package).

>

> Recent research, namely,

> Croux, C., Dhaene, G. and Hoorelbeke, D. (2003) _Robust standard

> errors for robust estimators_, Discussion Papers Series 03.16,

> K.U. Leuven, CES.

> has been made use of by Matias Salibian-Barrera's roblm()

> function now available as lmrob() from package 'robustbase'.

> There, mod <- lmrob(........); summary( mod )

> does provide you with P-values.

> But we still recommend *not* to ``believe in the P-values''

> blindly, but rather base your data analysis on serious analysis

> of residuals and other model checking.

>

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