I have data with two different kinds of measurements at the same set of $S$ sites. One of these (call it $X$) returns m estimates at each site, which are not necessarily independent of one another. So $X=\{x_{1,1}, x_{1,2}, x_{1,3}, ..., x_{2,1}, x_{2,2}, x_{2,3}, ..., x_{S,1}, x_{S,2}, x_{S,3}, ...,x_{S,M}\}$ where the first subscript indexes site ($S$ of these) and the second (M of these per site) the measurement at each site. The second set (call it $Y$) has just one value at each site, so $Y=\{y_1, y_2, y_3, ..., y_S\}$. There are errors on these in the typical sense of the word (errors of measurement and variation in the underlying process creating the values), but I don't have multiple measurements at each site to estimate them.
So I am wondering if there is a regression/correlation technique that would allow me to compare the two sets across sites while accounting for the estimates of error in the $X$ and the unknown error in $Y$. I could take the means across the M estimates at each site in $X$ then apply Type II/major axis regression which accounts for error in $X$ (and $Y$), but it seems a shame to throw away information I actually have on the errors. This is related to, but different from the case where errors are known for both $X$ and $Y$.
