[srean]

It might help to pose the problem as regression analysis with with categorical dependent variables. Perhaps with generalized linear model to account for the case that the predicted variable is positive.

Alternatively, if you have enough data, see if an orthogonal array design is feasible. It will not be very kosher because you would be selecting as opposed to assigning.

[kqr]

My current best attempt is a mixed-effects quantile regression (to capture the influence of the fixed effects on the tenth percentile while accounting for dependence between trials from the same person) but it is so compute hungry with this (apparently relatively large) data set that I'm looking into Bayesian methods for accomplishing similar things.

[srean]

Quantile regression is such an under-used tool. It really needs to be better known.