Cross Validated Asked by whuber on December 16, 2020

I would be grateful for any reference, such as a textbook or peer-reviewed journal article (of any vintage), that contains explicit formulas for conducting a weighted ANOVA with arbitrary weights.

There are plenty of sources presenting the usual sample-size weights. This is different. Think of the situation, somewhat abstractly, as an observational study in which a triple of variables is observed: a value (such as the price of an economic transaction), a weight (such as the quantity of materials transacted), and a category (such as the purchaser). It is straightforward to work out the theory and the formulas for weighted regression of value vs category. They are obvious generalizations of the standard decompositions into components of variance; of pooled variances; and of effect sizes. But who has published this and where?

Ordinarily there should be no need of such a reference: the mathematics ought to suffice. And although one can tease out the relevant formulas from, say, standard accounts of weighted least squares regression, once again that relies on mathematical derivations. But this arises in the context of a larger dispute where the decision maker is going to rely on citations to authorities and will be unwilling (and likely unable) to follow any mathematical argument. Thus, the more elementary the reference, the better.

I have also posted this request in the ASA’s Consulting group and plan to share any answers that come through that channel.

BTW, among the sources I have investigated, to no avail, are Google (periodically over many months), Kendall & Stuart, Snedechor & Cochran, Johnson & Kotz’s *The Encyclopedia of Statistical Sciences,* various regression textbooks, Hanson Hurwitz & Madow (in the survey sampling literature), and several of Jacob Cohen’s books.

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