I have a question about terminology.
Suppose I have data categorized by three factors A, B, and C, with cell means $\bar{y}_{ijk}$ and cell frequencies $n_{ijk}$, where $i$, $j$, and $k$ index A, B, and C respectively. And suppose I want to summarize the results for factor A by computing some sort of weighted means $WM_i$, averaging over indices $j$ and $k$ with weights $w_{jk}$. Consider these four weighting schemes:
- Use equal weights, $w_{jk} = 1$.
- Use weights of $w_{jk} = n_{+jk}$ (where "+" denotes summing over that index).
- Use weights $w_{jk} = n_{+j+}n_{++k} $ (outer product of the one-factor marginal frequencies).
- Use weights $w_{ijk} = n_{ijk}$ (the only one where we use a different set of weights for each $i$).
Scheme 1 yields the "unweighted" or "least-squares" means, and scheme 4 yields the ordinary means for A, ignoring B and C altogether. Scheme 3 yields weighted averages over $k$ of weighted averages over $j$ (or vice versa).
My question is what to call these schemes; e.g., as a character argument in an R function. Preliminarily, I am calling them "equal", "proportional", "outer", and "actual". The first one is pretty obvious. But maybe there is some existing standard terminology for some or all of the others that I am unaware of (or have forgotten). Any suggestions?
