# Seeking authoritative references on weighted ANOVA

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.

## Related Questions

### Fixed effects versus random effects in panel data for intervention group only? Change in dependent variable for each time period

0  Asked on December 11, 2021 by isobel-m

### How to calculate ARIMA(1,0,0)(1,0,1)12 prediction by hand

1  Asked on December 10, 2021 by code_diy

### Compare RMSE for the same model but varying sample size

3  Asked on December 8, 2021 by skoestlmeier

### ANOVA determining percentage of variation

2  Asked on December 8, 2021 by unistudent87

### Oscillating validation accuracy for a convolutional neural network?

3  Asked on December 8, 2021 by rockthestar

### Beta values for mixed models

1  Asked on December 8, 2021

### Comparing top level group effects using a 3-level hierarchical regression

1  Asked on December 8, 2021 by kev8484

### What are the worst (commonly adopted) ideas/principles in statistics?

32  Asked on December 8, 2021

### Statistical Analysis over different samples – Prediction for the number of objects

0  Asked on December 8, 2021

### Group level distribution for positive parameters in Bayesian multilevel models

0  Asked on December 8, 2021 by likao

### Learning more about glm parameters, how to dig deeper?

0  Asked on December 8, 2021

### OLS regression interpretation when sample means from t-test are insignificant

0  Asked on December 8, 2021 by thetagang

### Can we compare the effects of continuous covariate and categorical covariate on response variable in generalized linear regression?

0  Asked on December 8, 2021

### what is Multivariate Data

2  Asked on December 8, 2021

### Why LIME does not show the attribution for each features

0  Asked on December 8, 2021

### Is it useful or even necessary to standardize independent variables for linear regression?

1  Asked on December 8, 2021

### Why and under what conditions does Q learning converge?

0  Asked on December 8, 2021

### How to factor this conditional probability?

1  Asked on December 8, 2021 by user292136

### What are some good resources to learn Statistical Genetics?

1  Asked on December 6, 2021 by hulk