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In statistics, I see certain things described by "$n$-way" or "$m$-sample." For example, there is "$n$-way" ANOVA for any $n$ and "$m$-sample" t-tests for $m=1,2$. I want to get a handle on what these descriptors mean in general. It seems to me like "$n$-way" means that there are $n$ nominal variables, and that "$m$-sample" means that there is one nominal variable with $m$ possible values. Is this correct? Are there any other usages of the phrases "$n$-way" and "$m$-sample"?

Frans Rodenburg
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You are correct that $n$-way ANOVA implies a model with $n$ categorical explanatory variables with some unspecified number of categories, whereas an $m$-way $t$-test can be considered a model with one categorical explanatory variable consisting of $m$ categories.

To answer your other question, there are many uses of $n$-way, or $k$-way or whatsoever. The use of $n$ and $m$ is arbitrary here. For example, first order interactions between variables are also called one-way interactions; second order interaction are called two-way interactions, and so on. So one might call this $n$-way interaction.

I think the important thing to take from this type of nomenclature is generalization. It should suffice to understand that $n$-way ANOVA means an ANOVA-type model with any number of variables $n \geq 1$.

Edit
As per Nick Cox's suggestion, it is probably better to avoid using $n$ for anything other than sample size when teaching statistics to beginners. Multi-way ANOVA is a somewhat less ambiguous alternative.

Moreover, since there only exist one- and two-sample $t$-tests, these probably do not require their own symbol either.

Nick Cox
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Frans Rodenburg
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  • It's a matter of taste, but mine is that the symbol $n$ should be avoided for this purpose given its common use for sample size. While many people understand easily that choice of symbol is arbitrary, people new to the field can get confused by pointless notation; it's also true that some symbols are better than others given other associations! For that matter, "multi-way" is perhaps preferable here (and only trivially longer). But there are many clashes: while $p$ is a common notation for number of predictors, it could easily be objected that it is in common use for probability. And so on. – Nick Cox Oct 05 '17 at 08:42
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    I agree that $n$ is an unfortunate choice, but I interpreted the question as a notation the OP had read or heard somewhere before. Personally, I would not use $n$-way for that matter. I will add to my answer that "multi-way" is less likely to avoid confusion. – Frans Rodenburg Oct 06 '17 at 02:30