To estimate treatment effects in multi-site trials, one might focus on a distribution of treatment effects defined over a population of sites; or a distribution defined over a population of participants. For each case, we can define unbiased and consistent estimators, but these estimators may be grossly inefficient. On the other hand, precision weighted estimators may produce non-trivial but commonly ignored large-sample bias. To study these problems, we derive large-sample approximations to the mean and variance of competing
estimators, including the site fixed effects estimator, an estimator with site fixed effects and random coefficients, and an estimator that adjusts for the natural logarithm of the site-specific sampling precision.