We explore the use of instrumental variables (IV) analysis with a multi-site randomized trial to estimate the effect of a mediating variable on an outcome in cases where it can be assumed that the observed mediator is the only mechanism linking treatment assignment to outcomes, as assumption known in the instrumental variables literature as the exclusion restriction. We use a random-coefficient IV model that allows both the impact of program assignment on the mediator (compliance with assignment) and the impact of the mediator on the outcome (the mediator effect) to vary across sites and to co-vary with one another. This extension of conventional fixed-coefficient IV analysis illuminates a potential bias in IV analysis which Reardon and Raudenbush (forthcoming) refer to as "compliance-effect covariance bias." We first derive an expression for this bias and then use simulations to investigate the sampling variance of the conventional fixed-coefficient two-stage least squares (2SLS) estimator in the presence of varying (and co-varying) compliance and treatment effects. We next develop two alternate IV estimators that are less susceptible to compliance-effect covariance bias. We compare the bias, sampling variance, and root mean squared error of these "bias-corrected IV estimators" to those of 2SLS and OLS. We find that, when the first stage -statistic exceeds 10 (a commonly-used threshold for instrument strength), the bias-corrected estimators typically perform better than 2SLS or OLS. In the last part of the paper we use both the new estimators and 2SLS to reanalyze data from two large multi-site studies. [This article was published in "Journal of Educational and Behavioral Statistics," v39 n1 p53-86 Feb 2014 (EJ1020034).]