the structure of M permits; this static algorithm may be more representative of default MCMC algorithms provided by software packages, and is denoted as “Default.” Finally, for several example models we include an informed blocking of the model parameters, based upon expert or prior knowl- edge, which is referred to as “Informed Blocking.” Results for the random effects model also include the “Informed Cross-Level” MCMC algorithm which makes use of cross-level sampling, which is not in Ψ$_{M}$ . The left pane of Figure 3 displays the Efficiency performance for the model structures containing varying sized blocks of fixed correlation. For ρ = 0 . 2, the Auto Blocking algorithm selects cut height h = 0, which corresponds to re-selecting the algorithm All Scalar. Since this MCMC algorithm is identical to the initial state, the automated procedure terminates there. The All Blocked scheme actually runs faster, but the algorithmic efficiency loss inherent to large block sampling dominates, resulting in Efficiency approximately four times lower. For larger values of ρ , the All Scalar algo- rithm suffers progressively more since it fails to institute any blocking in the presence of increasing correlations. For ρ = 0 . 5 and 0.8, Auto Blocking algorithm selects cut heights h = 0 . 6 and h = 0 . 3, respectively, which each exactly place all correlated terms into sampling blocks. In every case, the slowest mixing parameter is from among the largest correlated group of 32 parameters. The right pane of Figure 3 presents results for the model structure containing fixed size parame- ter groupings with correlations between 0 and 0.9. For each size model, the automated blocking procedure selects a particular cut height (and hence, MCMC algorithm) twice consecutively, thus terminating on the third iteration. The cut heights selected for models N = 20, 50, and 100 are h = 0.5, 0.8, and 0.9, respectively (not shown), progressively pushing more of the correlated parameter groupings into sampling blocks. The Auto Blocking algorithm produces increases in Efficiency by factors of 4.5, 7, and 21 in the three models, over the static All Scalar and All Blocked algorithms. Varying Size Blocks of Fixed Correlation Fixed Size Blocks of Varying Correlation 22