Statistics terminology: $n$-way and $m$-sample 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"?
 A: 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.
