I would like to model data where the outcomes are produced, jointly, by a pair of indistinguishable subjects. As an example[*], consider the length of two-participant conversations. These data have the following properties
- Outcomes can only be measured at the pair level: for example, we only know the total length of the conversation $Y_{i,j}$, rather than how much $i$ and $j$ individually spoke during their interaction.
- The outcomes are undirected/symmetric $Y_{i,j} \triangleq Y_{j,i}$
- I have data--and equal amounts of it--for all possible pairs $\{i,j\}$ except for when $i=j$, where the outcome can't be measured.
- The pairs are sets: they're indistinguishable/unordered. This is somewhat different from many situations, where each member of the pair has a fixed role (e.g., buyer/seller) that could be modeled separately.
- I'm predominately interested in predictors that are also defined on the pairs (e.g. are $i$ and $j$ from the same organization? How far apart are they standing?), but would ideally like to include some subject-level and fixed effects.
When fitting a mixed-effect model, it seems wrong to treat the $Y_{i,j}$ as independent because there could be subject-specific effects: if Uncle Bob tends to ramble, conversations that include him may tend to be longer than average, regardless of the other participant. If the subjects were distinguishable, I would be tempted to account for this by adding a (random) effect for each subject: y_ij ~ X + (1 | i) + (1 | j)
. However, the $i$ and $j$ here are indistinguishable. Providing lmer()
with a pair of subject id vectors doesn't seem like it would account for the dependence between observations $<y_{i,j}, \textrm{ }i, j>$ and $<y_{j,k}, j, k>$ due to the possible contribution of $j$ to both.
How can I account for subject-specific effects when the subjects are indistinguishable?
I had considered adding in a ton of dummy variables (pair contains 1, pair contains 2, ...). However, this would add ~100 fixed effects that I don't actually care about, which doesn't seem great.
I also had a quick look at the dyadic literature, but nothing seems like quite the right fit. The actor-partner models seem to require subject-by-subject responses, ditto for social relations models, which always seem to have directed rating ($i$ likes $j$ is obviously different from $j$ likes $i$).
Would a better way to address this be changing the model's correlation structure, so that the covariance of observations from pair ${i,j}$ are some combination of $\sigma_i$ and $\sigma_j$? This is a little beyond my experience with mixed models, so I would like to know if this is both theoretically sound and if/how it's practically doable.
[*] The actual data is not conversation length--and is not even from humans--so don't worry too much about sociolinguistic-specific confounds. It seems like a very similar situation though, and the conversation example is much clearer.