Welch's ANOVA with blocking My understanding is that Welch's ANOVA is often a good option for a one-way comparison of treatment groups where the assumption of homogeneity of variance between treatment groups is violated.
The implementations of Welch's ANOVA that I've found so far (in R) are all for a one-way comparison.
I found this post, where one of the answers discusses factorial designs and Welch's ANOVA. So my understanding from this that it is theoretically possible to extend Welch's ANOVA beyond a one-way comparison...

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*Is it possible to implement a Welch's ANOVA with a treatment factor and a blocking factor, assuming that the blocking factor does not affect the variance?

*Is this any simpler than doing this for a factorial design?

*If yes to question 1, can anyone recommend an R package that is able to do this?

*On the other hand, if you don't think that this is possible, is there a mathematical explanation for why Welch's ANOVA only works for one-way comparisons (I can cope with equations but accompanying words of explanation would be very welcome)?

*Does anyone recommend an alternative method for comparing treatment group means, where there is a blocking factor and the standard ANOVA assumption of homogeneity of variance between treatment groups is violated?

 A: It depends on which aspect you want to extend.
Let's start with Welch's 2-sample $t$-test, the simplest case of one-way ANOVA.  Just as Student's $t$-statistic is what you get using a single binary predictor in a standard linear regression, Welch's 2-sample $t$-statistic is what you get using single binary predictor in a standard linear regression but using model-robust 'sandwich' standard errors (at least up to factors like $(n-1)/n$). This is shown by straightforward linear algebra, but it always takes me a while to reconstruct the proof, so I'm not giving it here (perhaps someone will ask a question about it)
So you can extend Welch's 2-sample $t$-test to one-way ANOVA, two-way fixed-effects ANOVA, and general fixed-effects linear regression by doing them as regression. The standard errors are available with ,robust in Stata and with the sandwich package in R. This is a lot easier than finding new ANOVA formulas.
What you don't get this way is an extension of Welch's rules for the $t$-test degrees of freedom (if you're working in a setting where the degrees of freedom are small enough to matter).  On the other hand, the paper given in the linked question doesn't seem to extend those rules, either: it just uses the ordinary residual df. You also don't (as far as I know) get random-effects ANOVA this way.
