[R-meta] origin of "sandwich"

[Yefeng Yang]

This is an interesting question. According to my impression, as long as one does not use the most efficient estimator (BLUP or MVUE), the formula for the sampling variance or error for beta coefficients looks like a sandwich. If we use the empirical residual distribution to approximate error distribution when the cluster numbers are large, we can get the so-called robust sandwich estimator or robust error estimator. One can also use the small-sample adjusted matrices to adjust the empirical residual to account for the small-sample bias, as implemented in clubSandwich. So, when mentioning sandwich estimator, I would not neglect "robust" or more precisely "cluster robust" before"sandwich". If I am wrong, James please help clarify.

Best,
Yefeng
________________________________
From: R-sig-meta-analysis <r-sig-meta-analysis-bounces using r-project.org> on behalf of James Pustejovsky via R-sig-meta-analysis <r-sig-meta-analysis using r-project.org>
Sent: Saturday, 15 July 2023 12:03
To: R Special Interest Group for Meta-Analysis <r-sig-meta-analysis using r-project.org>
Cc: James Pustejovsky <jepusto using gmail.com>
Subject: Re: [R-meta] origin of "sandwich"

Hi Sicong,

Your impression is correct. It's called a sandwich estimator because the
formula looks like a sandwich. It has the same quantity, called the
"bread," on both sides, with a quantity called the "meat" in the middle.
See Kaureman & Carroll (2001) for a technical explainer:
https://www.jstor.org/stable/3085907

We can torture this analogy further by noting that the meat matrix consist
of a bunch of slices---one for each cluster---so it really is like slices
of baloney or salami or something. Furthermore, if you consider the slices
individually, they are poor quality estimators of what we're trying to
quantify. But if you stack enough of them together, they work well enough
(...to satisfy your grumbling stomach...).

My clubSandwich package gets its name from the fact that it implements
sandwich-type formulas, but with extra "filling" in the middle of the
sandwich---not just meat,  but also some sprouts, Swiss cheese, thousand
island dressing, etc. The extra filling is there to improve the performance
of the sandwich estimator when you only have a few independent slices of
meat. I've explained this analogy in a few talks I've given, e.g.:
https://www.jepusto.com/talk/oslorug-2022-clubsandwich/
The reactions have varied from polite chuckles to outright eye rolling,
depending on how hungry the audience is. Your mileage may vary.

James

On Fri, Jul 14, 2023, 3:15 PM Sicong Liu via R-sig-meta-analysis <
r-sig-meta-analysis using r-project.org> wrote:

> Dear All,
>
> I have a burning question regarding the origin of the word �sandwich� that
> frequently appears in meta-analysis (e.g., �Egger sandwich test� and the R
> package �clubSandwich�). Could some meta tycoon(s) share
> knowledge/discussion on this?
>
> My vague impression was that �sandwich� may relate to the looking of a
> linear algebra expression for computing uncertainties. However, I could no
> longer identify this source (a book I think) and feel much bothered by it.
> Thank you all in advance!
>
> Best regards,
> Sicong (Zone)
>
> ---------------------------------------------------------
> Sicong (Zone) Liu, Ph.D.
> Research Associate
> University of Pennsylvania
>
> 3620 Walnut Street,
> Philadelphia, PA 19104-6220
> Email: zone using upenn.edu<mailto:zone using upenn.edu>
> Cell: 850-345-5788
> ---------------------------------------------------------
>
>
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