δ 0.025 0.05 0.075 0.1 π π π π N   0.025 0.05 0.075 0.1 0.025 0.05 0.075 0.1 0.025 0.05 0.075 0.1 0.025 0.05 0.075 0.1 10 RB  0.02947 0.02258 0.04016 0.03863 0.13102 0.07173 0.09012 0.06763 0.24288 0.12071 0.11051 0.10479 0.33570 0.18675 0.14564 0.15245 RBE 0.03021 0.02474 0.04221 0.04513 0.12475 0.07008 0.09192 0.07777 0.22789 0.11552 0.10892 0.11093 0.31133 0.16957 0.13704 0.15162 RMSE        0.99578 0.98579 0.99114 0.90046 1.01075 1.02702 1.00010 0.92797 1.04519 1.01203 1.04029 0.95725 1.04125 1.06464 1.04087 1.01525 30 RB  0.00802 0.01036 0.01316 0.00871 0.03736 0.01877 0.01972 0.01997 0.06229 0.03655 0.02350 0.03446 0.08924 0.06558 0.06124 0.03677 RBE 0.00778 0.01194 0.01684 0.01474 0.03303 0.01777 0.02252 0.02682 0.05223 0.03117 0.02224 0.03797 0.07198 0.05321 0.05455 0.03376 RMSE        1.00309 0.98850 0.96432 0.93459 1.02808 1.00170 0.99492 0.95885 1.04456 1.04185 1.02271 1.00198 1.08761 1.08077 1.03345 1.05222 50 RB  0.01008 0.00247 0.00309 0.00439 0.01118 0.01067 0.01058 0.00521 0.03868 0.01866 0.02416 0.01035 0.06565 0.03432 0.03297 0.03320 RBE 0.00997 0.00401 0.00688 0.01069 0.00796 0.01036 0.01337 0.01225 0.03002 0.01382 0.02270 0.01336 0.04995 0.02394 0.02592 0.03031 RMSE        1.00058 0.98683 0.96670 0.93616 1.01788 1.00520 0.99666 0.96695 1.04735 1.03331 1.01907 1.01246 1.07330 1.05838 1.05647 1.04896 100 RB  0.00395 0.00342 -0.00092 -0.00203 0.00787 0.00838 0.00384 0.00221 0.01871 0.01858 0.01445 0.00792 0.04271 0.02876 0.01196 0.01248 RBE 0.00383 0.00502 0.00295 0.00423 0.00494 0.00790 0.00697 0.00957 0.01080 0.01356 0.01322 0.01130 0.02889 0.01841 0.00537 0.01005 RMSE        1.00121 0.98672 0.96740 0.94398 1.02159 1.00838 0.98715 0.96012 1.04804 1.03945 1.01994 1.00181 1.08931 1.06980 1.05148 1.04374 200 RB  -0.00134 -0.00086 -0.00197 -0.00718 0.00898 -0.00057 -0.00191 -0.00822 0.01747 0.01051 0.00419 0.00133 0.02724 0.01607 0.00991 0.00703 RBE -0.00140 0.00076 0.00195 -0.00099 0.00610 -0.00095 0.00124 -0.00086 0.01031 0.00597 0.00307 0.00454 0.01496 0.00642 0.00388 0.00483 RMSE        1.00082 0.98763 0.96873 0.95629 1.02283 1.00717 0.99031 0.97972 1.05062 1.03692 1.02093 1.00423 1.08130 1.06762 1.05158 1.03280

Table 2: Relative bias (RB) and relative mean squared error (RMSE) for the beta-binomial model and Relative bias (RBE) for the exact distribution (Equation 3), using various combinations of π, δ and N with k=25 and nl=10, l=1,2,…,N.