============================================================================== HOW MANY SAMPLES BEFORE YOUR SPECIES COUNT MEANS ANYTHING Science Journaling Club, Volume 1, Issue 1, Fall 2024 ============================================================================== This is a simulation study. No organism was counted. The club has no field site; the communities below exist only as arrays of relative abundances, and the surveys are multinomial draws from them. We do it this way because measuring the bias of a richness estimator requires knowing the true richness, which no fieldworker ever does. master seed 20241104 numpy 2.4.2 python 3.12.3 true richness S 100 species, every configuration quadrats per survey T 10 effort ladder n [50, 100, 200, 500, 1000, 2000, 5000, 10000, 20000, 50000] replicate communities 6000 per configuration configurations 5 total surveys 300,000 estimators S_obs, Chao1, Chao2, ACE, Jack1, Jack2 ============================================================================== PART 1. THE SWEEP ============================================================================== lognormal-0.5 Pielou J = 0.9733 +/- 0.0001 inverse Simpson = 78.44 mean rarest p = 2.592e-03 [ 25.8 s] lognormal-1.0 Pielou J = 0.8950 +/- 0.0003 inverse Simpson = 41.76 mean rarest p = 5.426e-04 [ 30.9 s] lognormal-1.5 Pielou J = 0.7803 +/- 0.0008 inverse Simpson = 20.85 mean rarest p = 9.802e-05 [ 32.4 s] lognormal-2.0 Pielou J = 0.6523 +/- 0.0013 inverse Simpson = 11.53 mean rarest p = 1.513e-05 [ 30.5 s] broken-stick Pielou J = 0.9093 +/- 0.0001 inverse Simpson = 50.93 mean rarest p = 9.946e-05 [ 30.1 s] Pielou's J is 1.000 for a perfectly even community and falls towards 0 as abundance concentrates in a few species. The inverse Simpson number is the effective count of common species: at sigma = 2.0 the community behaves like a handful of species plus noise, even though 100 species are genuinely present. ============================================================================== PART 2. VALIDATION 1 - RAREFACTION AGAINST THE ANALYTIC EXPECTATION ============================================================================== For multinomial sampling of n individuals from relative abundances p, the expected number of species observed is exactly E[S_obs] = sum_i ( 1 - (1 - p_i)^n ) We evaluate that expression on the same communities we sampled and compare it with the simulated mean. The comparison is paired, one community at a time, so the standard error is the standard error of the paired difference and the test is sharp. Two standard errors are printed. SE(emp) is the ordinary standard error of the paired difference across replicates. SE(null) is the exact standard error the difference should have if the sampler is correct, sqrt(mean_r Var(S_obs) / R), with Var(S_obs) evaluated in closed form including every pairwise covariance, on 250 of the replicate communities. They agree everywhere except in cells where every replicate saturates at all 100 species, where SE(emp) collapses to exactly zero and is useless. z is against SE(null). config n club sim analytic diff SE(emp) SE(null) z -------------------------------------------------------------------------------------------- lognormal-0.5 50 37.6780 37.6260 +0.051980 0.031728 0.032057 +1.62 lognormal-0.5 100 59.0417 59.1266 -0.084892 0.041824 0.042158 -2.01 lognormal-0.5 200 80.5733 80.5678 +0.005501 0.040902 0.041249 +0.13 lognormal-0.5 500 96.6520 96.6549 -0.002948 0.021699 0.021773 -0.14 lognormal-0.5 1000 99.5992 99.6134 -0.014210 0.008038 0.007873 -1.80 lognormal-0.5 2000 99.9835 99.9820 +0.001539 0.001631 0.001724 +0.89 lognormal-0.5 5000 100.0000 99.9999 +0.000062 0.000005 0.000127 +0.49 lognormal-0.5 10000 100.0000 100.0000 +0.000000 0.000000 0.000008 +0.02 lognormal-0.5 20000 100.0000 100.0000 +0.000000 0.000000 0.000000 +0.00 lognormal-0.5 50000 100.0000 100.0000 +0.000000 0.000000 0.000000 +0.00 -------------------------------------------------------------------------------------------- lognormal-1.0 50 32.4407 32.4088 +0.031835 0.035283 0.034898 +0.91 lognormal-1.0 100 49.1132 49.1219 -0.008695 0.043006 0.043390 -0.20 lognormal-1.0 200 67.0713 67.0593 +0.011994 0.045389 0.045336 +0.26 lognormal-1.0 500 86.2757 86.3030 -0.027355 0.036724 0.036448 -0.75 lognormal-1.0 1000 94.5983 94.6641 -0.065784 0.025153 0.025137 -2.62 lognormal-1.0 2000 98.4072 98.4223 -0.015105 0.014112 0.014567 -1.04 lognormal-1.0 5000 99.8022 99.8017 +0.000467 0.005416 0.005340 +0.09 lognormal-1.0 10000 99.9727 99.9719 +0.000742 0.002025 0.001955 +0.38 lognormal-1.0 20000 99.9973 99.9973 +0.000032 0.000659 0.000539 +0.06 lognormal-1.0 50000 100.0000 99.9999 +0.000061 0.000016 0.000061 +1.00 -------------------------------------------------------------------------------------------- lognormal-1.5 50 25.9572 25.9499 +0.007255 0.034853 0.034727 +0.21 lognormal-1.5 100 38.2810 38.2358 +0.045206 0.041069 0.041560 +1.09 lognormal-1.5 200 52.2478 52.2781 -0.030279 0.044348 0.044723 -0.68 lognormal-1.5 500 70.5863 70.6370 -0.050676 0.042362 0.042250 -1.20 lognormal-1.5 1000 82.1055 82.1173 -0.011786 0.036407 0.036246 -0.33 lognormal-1.5 2000 90.3723 90.3883 -0.015947 0.028659 0.028472 -0.56 lognormal-1.5 5000 96.5758 96.5674 +0.008447 0.018633 0.018159 +0.47 lognormal-1.5 10000 98.6645 98.6666 -0.002143 0.012216 0.011733 -0.18 lognormal-1.5 20000 99.5612 99.5535 +0.007667 0.007375 0.007068 +1.08 lognormal-1.5 50000 99.9205 99.9169 +0.003594 0.003130 0.003251 +1.11 -------------------------------------------------------------------------------------------- lognormal-2.0 50 19.9473 20.0013 -0.054014 0.031439 0.032122 -1.68 lognormal-2.0 100 28.7737 28.7941 -0.020480 0.037439 0.037651 -0.54 lognormal-2.0 200 39.2822 39.2406 +0.041555 0.041446 0.041340 +1.01 lognormal-2.0 500 54.3780 54.4280 -0.049996 0.041923 0.042634 -1.17 lognormal-2.0 1000 65.7952 65.7540 +0.041129 0.041015 0.040793 +1.01 lognormal-2.0 2000 75.9207 75.8981 +0.022561 0.036458 0.037009 +0.61 lognormal-2.0 5000 86.4233 86.4258 -0.002467 0.030085 0.030155 -0.08 lognormal-2.0 10000 91.9518 91.9588 -0.006977 0.024383 0.024391 -0.29 lognormal-2.0 20000 95.6255 95.6086 +0.016938 0.018787 0.018799 +0.90 lognormal-2.0 50000 98.2503 98.2673 -0.016933 0.012757 0.012320 -1.37 -------------------------------------------------------------------------------------------- broken-stick 50 33.4823 33.5537 -0.071400 0.033793 0.033600 -2.13 broken-stick 100 50.1597 50.2469 -0.087272 0.041538 0.041686 -2.09 broken-stick 200 66.9373 66.8907 +0.046603 0.041982 0.042572 +1.09 broken-stick 500 83.5177 83.4910 +0.026707 0.034870 0.034598 +0.77 broken-stick 1000 91.0028 91.0116 -0.008762 0.026230 0.026532 -0.33 broken-stick 2000 95.2973 95.2938 +0.003574 0.019900 0.019492 +0.18 broken-stick 5000 98.0628 98.0653 -0.002433 0.012613 0.012626 -0.19 broken-stick 10000 99.0245 99.0281 -0.003572 0.008949 0.009034 -0.40 broken-stick 20000 99.5232 99.5154 +0.007748 0.006297 0.006471 +1.20 broken-stick 50000 99.8105 99.8071 +0.003436 0.003927 0.004167 +0.82 -------------------------------------------------------------------------------------------- cells compared 50 largest |z| 2.62 (lognormal-1.0 at n = 1000) cells with |z| > 2 4 (expected about 2.3 by chance) cells with |z| > 3 0 (expected about 0.14 by chance) mean z -0.0875 sd of z 1.0125 (should be near 1.00) saturated cells 4 (SE(emp) far below SE(null); the empirical test is dead there) A note on the debugging, because the first version of this table was wrong and we would rather show the repair than pretend. Using SE(emp) alone, three cells came back past 3 standard errors and one at 12.4. All of them sat at efforts where the simulated richness was exactly 100.000 in every replicate and the analytic value was 99.99994. The difference was six hundred-thousandths of a species. The z score was enormous only because the empirical variance of a constant is zero. The null variance is the right denominator and it is computable in closed form, so we compute it. The largest real difference anywhere in the table is 0.08727 species out of 100. VERDICT: the rarefaction implementation agrees with the closed form. No cell disagrees by as much as 4 standard errors, the spread of z across cells is close to the standard normal, and the signed mean of z is near zero, so there is no systematic offset in either direction. The sampler is doing what the algebra says it should. ============================================================================== PART 3. VALIDATION 2 - HURLBERT'S WITHIN-SAMPLE RAREFACTION ============================================================================== The curve above rarefies the population. Ecologists usually rarefy the sample instead: given a collection of n individuals, how many species would a random subsample of k of them have contained? Hurlbert (1971) gives the hypergeometric answer E[S_k] = sum_i [ 1 - C(n - x_i, k) / C(n, k) ] We take one sample of 2000 individuals and subsample it 20000 times. reference sample: n = 2000, S_obs = 99, f1 = 2, f2 = 3 k subsampled SE Hurlbert diff z ------------------------------------------------------------------------------ 25 19.8867 0.0128 19.8904 -0.0037 -0.29 50 33.1213 0.0189 33.1095 +0.0118 +0.62 100 50.1751 0.0234 50.1623 +0.0128 +0.55 200 68.2184 0.0239 68.2355 -0.0171 -0.71 400 83.4781 0.0203 83.4745 +0.0036 +0.18 800 93.6237 0.0137 93.6262 -0.0025 -0.18 1600 98.4107 0.0050 98.4085 +0.0022 +0.44 2000 99.0000 0.0000 99.0000 +0.0000 +0.00 ------------------------------------------------------------------------------ largest |z| across 8 subsample sizes: 0.71 At k = n the subsample is the whole sample, both numbers must be the observed richness exactly, and they are. ============================================================================== PART 4. VALIDATION 3 - EVERY ESTIMATOR AT EXHAUSTIVE EFFORT ============================================================================== If sampling is so heavy that no species is left as a singleton or as a unique, every one of these estimators is algebraically forced back to S_obs. That is the sanity check: at exhaustive effort they must all return exactly 100.000, not approximately. community n f1 f2 Q1 S_obs Chao1 Chao2 ACE Jack1 Jack2 ------------------------------------------------------------------------------ lognormal-0.5 4,000,000 0 0 0 100.000 100.000 100.000 100.000 100.000 100.000 lognormal-1.0 4,000,000 0 0 0 100.000 100.000 100.000 100.000 100.000 100.000 lognormal-2.0 4,000,000 0 0 0 100.000 100.000 100.000 100.000 100.000 100.000 broken-stick 4,000,000 0 0 0 100.000 100.000 100.000 100.000 100.000 100.000 synthetic 7,000 0 0 0 100.000 100.000 100.000 100.000 100.000 100.000 ------------------------------------------------------------------------------ club value 100.000 vs true value 100.000, difference 0.000, for every estimator and every community. This is a weak test and we say so. It confirms the formulas are typed correctly and nothing more. An estimator can pass it and still be badly wrong at every effort a student could actually afford. ============================================================================== PART 5. BIAS AND ROOT MEAN SQUARE ERROR AGAINST KNOWN TRUTH ============================================================================== Truth is 100 species by construction. Bias is mean(estimate) - 100. RMSE is sqrt(mean((estimate - truth)^2)), so it charges an estimator for scatter as well as for being off-centre. Standard errors on the mean come from the 6000 replicates; standard errors on RMSE come from a 200-resample bootstrap over the same replicates. --- lognormal-0.5 (mean Pielou J = 0.9733) -------------------- n estimator mean SE bias rel.bias RMSE SE(RMSE) 50 S_obs 37.678 0.032 -62.322 -0.6232 62.371 0.032 50 Chao1 88.197 0.389 -11.803 -0.1180 32.344 0.567 50 Chao2 86.578 0.379 -13.422 -0.1342 32.267 0.483 50 ACE 98.647 0.424 -1.353 -0.0135 32.872 0.596 50 Jack1 65.091 0.082 -34.909 -0.3491 35.487 0.080 50 Jack2 84.794 0.147 -15.206 -0.1521 19.016 0.129 100 S_obs 59.042 0.043 -40.958 -0.4096 41.092 0.044 100 Chao1 92.268 0.208 -7.732 -0.0773 17.875 0.163 100 Chao2 91.388 0.200 -8.612 -0.0861 17.732 0.154 100 ACE 96.296 0.207 -3.704 -0.0370 16.444 0.191 100 Jack1 91.705 0.100 -8.295 -0.0830 11.337 0.084 100 Jack2 108.256 0.174 +8.256 +0.0826 15.782 0.152 200 S_obs 80.573 0.042 -19.427 -0.1943 19.700 0.040 200 Chao1 96.791 0.111 -3.209 -0.0321 9.154 0.097 200 Chao2 96.190 0.104 -3.810 -0.0381 8.920 0.079 200 ACE 98.825 0.100 -1.175 -0.0117 7.867 0.080 200 Jack1 107.358 0.086 +7.358 +0.0736 9.919 0.079 200 Jack2 112.063 0.154 +12.063 +0.1206 16.992 0.142 500 S_obs 96.652 0.023 -3.348 -0.0335 3.776 0.025 500 Chao1 99.530 0.038 -0.470 -0.0047 2.979 0.038 500 Chao2 99.365 0.035 -0.635 -0.0063 2.780 0.036 500 ACE 99.442 0.029 -0.558 -0.0056 2.323 0.023 500 Jack1 105.147 0.044 +5.147 +0.0515 6.157 0.038 500 Jack2 101.159 0.091 +1.159 +0.0116 7.122 0.065 1000 S_obs 99.599 0.008 -0.401 -0.0040 0.753 0.011 1000 Chao1 99.934 0.013 -0.066 -0.0007 1.045 0.046 1000 Chao2 99.922 0.012 -0.078 -0.0008 0.934 0.038 1000 ACE 99.856 0.009 -0.144 -0.0014 0.702 0.010 1000 Jack1 101.038 0.018 +1.038 +0.0104 1.713 0.018 1000 Jack2 99.439 0.040 -0.561 -0.0056 3.144 0.027 2000 S_obs 99.984 0.002 -0.016 -0.0002 0.128 0.006 2000 Chao1 99.987 0.002 -0.013 -0.0001 0.142 0.006 2000 Chao2 99.989 0.002 -0.011 -0.0001 0.145 0.006 2000 ACE 99.998 0.002 -0.002 -0.0000 0.137 0.006 2000 Jack1 100.075 0.004 +0.075 +0.0008 0.338 0.007 2000 Jack2 99.903 0.010 -0.097 -0.0010 0.806 0.013 5000 S_obs 100.000 0.000 +0.000 +0.0000 0.000 0.000 5000 Chao1 100.000 0.000 +0.000 +0.0000 0.000 0.000 5000 Chao2 100.000 0.000 +0.000 +0.0000 0.000 0.000 5000 ACE 100.000 0.000 +0.000 +0.0000 0.014 0.009 5000 Jack1 100.000 0.000 +0.000 +0.0000 0.018 0.008 5000 Jack2 99.998 0.001 -0.002 -0.0000 0.061 0.009 10000 S_obs 100.000 0.000 +0.000 +0.0000 0.000 0.000 10000 Chao1 100.000 0.000 +0.000 +0.0000 0.000 0.000 10000 Chao2 100.000 0.000 +0.000 +0.0000 0.000 0.000 10000 ACE 100.000 0.000 +0.000 +0.0000 0.000 0.000 10000 Jack1 100.000 0.000 +0.000 +0.0000 0.000 0.000 10000 Jack2 100.000 0.000 -0.000 -0.0000 0.013 0.008 20000 S_obs 100.000 0.000 +0.000 +0.0000 0.000 0.000 20000 Chao1 100.000 0.000 +0.000 +0.0000 0.000 0.000 20000 Chao2 100.000 0.000 +0.000 +0.0000 0.000 0.000 20000 ACE 100.000 0.000 +0.000 +0.0000 0.000 0.000 20000 Jack1 100.000 0.000 +0.000 +0.0000 0.000 0.000 20000 Jack2 100.000 0.000 +0.000 +0.0000 0.000 0.000 50000 S_obs 100.000 0.000 +0.000 +0.0000 0.000 0.000 50000 Chao1 100.000 0.000 +0.000 +0.0000 0.000 0.000 50000 Chao2 100.000 0.000 +0.000 +0.0000 0.000 0.000 50000 ACE 100.000 0.000 +0.000 +0.0000 0.000 0.000 50000 Jack1 100.000 0.000 +0.000 +0.0000 0.000 0.000 50000 Jack2 100.000 0.000 +0.000 +0.0000 0.000 0.000 --- lognormal-1.0 (mean Pielou J = 0.8950) -------------------- n estimator mean SE bias rel.bias RMSE SE(RMSE) 50 S_obs 32.441 0.040 -67.559 -0.6756 67.631 0.039 50 Chao1 68.582 0.343 -31.418 -0.3142 41.151 0.552 50 Chao2 66.306 0.306 -33.694 -0.3369 41.193 0.250 50 ACE 77.092 0.355 -22.908 -0.2291 35.800 0.287 50 Jack1 53.920 0.090 -46.080 -0.4608 46.599 0.091 50 Jack2 68.717 0.148 -31.283 -0.3128 33.306 0.147 100 S_obs 49.113 0.051 -50.887 -0.5089 51.042 0.049 100 Chao1 78.400 0.216 -21.600 -0.2160 27.317 0.155 100 Chao2 76.698 0.203 -23.302 -0.2330 28.105 0.147 100 ACE 84.864 0.219 -15.136 -0.1514 22.707 0.157 100 Jack1 74.881 0.102 -25.119 -0.2512 26.321 0.094 100 Jack2 89.189 0.162 -10.811 -0.1081 16.574 0.147 200 S_obs 67.071 0.057 -32.929 -0.3293 33.219 0.054 200 Chao1 87.468 0.151 -12.532 -0.1253 17.132 0.106 200 Chao2 86.113 0.139 -13.887 -0.1389 17.551 0.103 200 ACE 90.622 0.136 -9.378 -0.0938 14.110 0.097 200 Jack1 91.926 0.099 -8.074 -0.0807 11.124 0.079 200 Jack2 101.567 0.159 +1.567 +0.0157 12.435 0.114 500 S_obs 86.276 0.046 -13.724 -0.1372 14.174 0.047 500 Chao1 95.422 0.086 -4.578 -0.0458 8.083 0.070 500 Chao2 94.743 0.079 -5.257 -0.0526 8.040 0.061 500 ACE 95.493 0.068 -4.507 -0.0451 6.910 0.051 500 Jack1 102.277 0.071 +2.277 +0.0228 5.983 0.060 500 Jack2 104.257 0.124 +4.257 +0.0426 10.521 0.100 1000 S_obs 94.598 0.030 -5.402 -0.0540 5.892 0.033 1000 Chao1 98.448 0.054 -1.552 -0.0155 4.445 0.065 1000 Chao2 98.144 0.047 -1.856 -0.0186 4.097 0.042 1000 ACE 97.950 0.038 -2.050 -0.0205 3.562 0.032 1000 Jack1 102.906 0.049 +2.906 +0.0291 4.751 0.042 1000 Jack2 102.178 0.091 +2.178 +0.0218 7.384 0.068 2000 S_obs 98.407 0.016 -1.593 -0.0159 2.026 0.018 2000 Chao1 99.522 0.027 -0.478 -0.0048 2.114 0.042 2000 Chao2 99.444 0.025 -0.556 -0.0056 1.982 0.048 2000 ACE 99.297 0.018 -0.703 -0.0070 1.577 0.017 2000 Jack1 101.456 0.028 +1.456 +0.0146 2.595 0.025 2000 Jack2 100.365 0.055 +0.365 +0.0036 4.297 0.043 5000 S_obs 99.802 0.006 -0.198 -0.0020 0.490 0.008 5000 Chao1 99.882 0.007 -0.118 -0.0012 0.570 0.013 5000 Chao2 99.894 0.007 -0.106 -0.0011 0.585 0.020 5000 ACE 99.929 0.006 -0.071 -0.0007 0.504 0.008 5000 Jack1 100.306 0.011 +0.306 +0.0031 0.888 0.010 5000 Jack2 99.961 0.022 -0.039 -0.0004 1.734 0.018 10000 S_obs 99.973 0.002 -0.027 -0.0003 0.166 0.007 10000 Chao1 99.977 0.002 -0.023 -0.0002 0.179 0.008 10000 Chao2 99.977 0.002 -0.023 -0.0002 0.178 0.007 10000 ACE 100.010 0.003 +0.010 +0.0001 0.231 0.010 10000 Jack1 100.060 0.004 +0.060 +0.0006 0.344 0.008 10000 Jack2 99.974 0.010 -0.026 -0.0003 0.740 0.013 20000 S_obs 99.997 0.001 -0.003 -0.0000 0.052 0.006 20000 Chao1 99.997 0.001 -0.003 -0.0000 0.052 0.007 20000 Chao2 99.997 0.001 -0.003 -0.0000 0.053 0.007 20000 ACE 100.002 0.001 +0.002 +0.0000 0.083 0.006 20000 Jack1 100.006 0.001 +0.006 +0.0001 0.105 0.007 20000 Jack2 99.990 0.003 -0.010 -0.0001 0.245 0.011 50000 S_obs 100.000 0.000 +0.000 +0.0000 0.000 0.000 50000 Chao1 100.000 0.000 +0.000 +0.0000 0.000 0.000 50000 Chao2 100.000 0.000 +0.000 +0.0000 0.000 0.000 50000 ACE 100.000 0.000 +0.000 +0.0000 0.000 0.000 50000 Jack1 100.000 0.000 +0.000 +0.0000 0.013 0.009 50000 Jack2 100.000 0.000 -0.000 -0.0000 0.037 0.010 --- lognormal-1.5 (mean Pielou J = 0.7803) -------------------- n estimator mean SE bias rel.bias RMSE SE(RMSE) 50 S_obs 25.957 0.050 -74.043 -0.7404 74.145 0.052 50 Chao1 51.136 0.276 -48.864 -0.4886 53.339 0.193 50 Chao2 49.500 0.270 -50.500 -0.5050 54.645 0.271 50 ACE 58.578 0.293 -41.422 -0.4142 47.222 0.207 50 Jack1 41.642 0.095 -58.358 -0.5836 58.821 0.094 50 Jack2 52.276 0.141 -47.724 -0.4772 48.962 0.136 100 S_obs 38.281 0.064 -61.719 -0.6172 61.919 0.071 100 Chao1 62.754 0.242 -37.246 -0.3725 41.701 0.298 100 Chao2 60.776 0.204 -39.224 -0.3922 42.295 0.181 100 ACE 67.616 0.210 -32.384 -0.3238 36.239 0.165 100 Jack1 57.561 0.109 -42.439 -0.4244 43.267 0.102 100 Jack2 69.084 0.157 -30.916 -0.3092 33.214 0.149 200 S_obs 52.248 0.074 -47.752 -0.4775 48.098 0.073 200 Chao1 73.141 0.180 -26.859 -0.2686 30.274 0.157 200 Chao2 71.670 0.172 -28.330 -0.2833 31.310 0.204 200 ACE 75.984 0.156 -24.016 -0.2402 26.898 0.128 200 Jack1 72.921 0.111 -27.079 -0.2708 28.422 0.109 200 Jack2 83.327 0.157 -16.673 -0.1667 20.633 0.150 500 S_obs 70.586 0.074 -29.414 -0.2941 29.961 0.080 500 Chao1 85.337 0.135 -14.663 -0.1466 17.998 0.131 500 Chao2 84.081 0.122 -15.919 -0.1592 18.521 0.109 500 ACE 85.789 0.106 -14.211 -0.1421 16.427 0.096 500 Jack1 89.087 0.096 -10.913 -0.1091 13.198 0.089 500 Jack2 95.923 0.140 -4.077 -0.0408 11.588 0.098 1000 S_obs 82.106 0.062 -17.894 -0.1789 18.537 0.067 1000 Chao1 91.863 0.104 -8.137 -0.0814 11.433 0.105 1000 Chao2 91.029 0.094 -8.971 -0.0897 11.555 0.092 1000 ACE 91.433 0.078 -8.567 -0.0857 10.475 0.078 1000 Jack1 96.470 0.078 -3.530 -0.0353 6.966 0.072 1000 Jack2 100.235 0.121 +0.235 +0.0023 9.395 0.086 2000 S_obs 90.372 0.047 -9.628 -0.0963 10.306 0.056 2000 Chao1 95.957 0.073 -4.043 -0.0404 6.960 0.067 2000 Chao2 95.454 0.067 -4.546 -0.0455 6.884 0.064 2000 ACE 95.322 0.053 -4.678 -0.0468 6.229 0.051 2000 Jack1 99.877 0.058 -0.123 -0.0012 4.501 0.044 2000 Jack2 101.265 0.097 +1.265 +0.0126 7.631 0.067 5000 S_obs 96.576 0.027 -3.424 -0.0342 4.013 0.032 5000 Chao1 98.673 0.041 -1.327 -0.0133 3.465 0.054 5000 Chao2 98.502 0.039 -1.498 -0.0150 3.367 0.059 5000 ACE 98.349 0.029 -1.651 -0.0165 2.803 0.026 5000 Jack1 100.912 0.036 +0.912 +0.0091 2.906 0.027 5000 Jack2 100.870 0.064 +0.870 +0.0087 5.031 0.051 10000 S_obs 98.665 0.016 -1.335 -0.0134 1.841 0.021 10000 Chao1 99.419 0.025 -0.581 -0.0058 2.013 0.071 10000 Chao2 99.397 0.024 -0.603 -0.0060 1.948 0.054 10000 ACE 99.380 0.018 -0.620 -0.0062 1.516 0.020 10000 Jack1 100.623 0.023 +0.623 +0.0062 1.904 0.020 10000 Jack2 100.350 0.043 +0.350 +0.0035 3.363 0.035 20000 S_obs 99.561 0.009 -0.439 -0.0044 0.817 0.013 20000 Chao1 99.766 0.013 -0.234 -0.0023 1.019 0.045 20000 Chao2 99.772 0.012 -0.228 -0.0023 0.981 0.028 20000 ACE 99.849 0.011 -0.151 -0.0015 0.827 0.013 20000 Jack1 100.330 0.014 +0.330 +0.0033 1.169 0.016 20000 Jack2 100.157 0.028 +0.157 +0.0016 2.150 0.028 50000 S_obs 99.921 0.004 -0.079 -0.0008 0.293 0.008 50000 Chao1 99.937 0.004 -0.063 -0.0006 0.341 0.027 50000 Chao2 99.937 0.004 -0.063 -0.0006 0.321 0.010 50000 ACE 100.009 0.005 +0.009 +0.0001 0.412 0.018 50000 Jack1 100.079 0.006 +0.079 +0.0008 0.499 0.009 50000 Jack2 100.008 0.013 +0.008 +0.0001 0.986 0.019 --- lognormal-2.0 (mean Pielou J = 0.6523) -------------------- n estimator mean SE bias rel.bias RMSE SE(RMSE) 50 S_obs 19.947 0.057 -80.053 -0.8005 80.175 0.059 50 Chao1 37.037 0.232 -62.963 -0.6296 65.471 0.172 50 Chao2 35.566 0.209 -64.434 -0.6443 66.432 0.154 50 ACE 42.379 0.237 -57.621 -0.5762 60.473 0.179 50 Jack1 31.036 0.098 -68.964 -0.6896 69.378 0.098 50 Jack2 38.486 0.136 -61.514 -0.6151 62.404 0.139 100 S_obs 28.774 0.076 -71.226 -0.7123 71.468 0.079 100 Chao1 46.869 0.222 -53.131 -0.5313 55.832 0.175 100 Chao2 45.492 0.204 -54.508 -0.5451 56.760 0.158 100 ACE 50.841 0.203 -49.159 -0.4916 51.615 0.167 100 Jack1 42.574 0.118 -57.426 -0.5743 58.145 0.116 100 Jack2 51.037 0.156 -48.963 -0.4896 50.438 0.160 200 S_obs 39.282 0.091 -60.718 -0.6072 61.125 0.091 200 Chao1 57.727 0.209 -42.273 -0.4227 45.268 0.251 200 Chao2 56.129 0.185 -43.871 -0.4387 46.157 0.148 200 ACE 60.115 0.174 -39.885 -0.3989 42.111 0.156 200 Jack1 55.235 0.129 -44.765 -0.4476 45.867 0.128 200 Jack2 64.235 0.167 -35.765 -0.3576 38.030 0.183 500 S_obs 54.378 0.101 -45.622 -0.4562 46.282 0.098 500 Chao1 70.916 0.181 -29.084 -0.2908 32.296 0.198 500 Chao2 69.365 0.160 -30.635 -0.3063 33.054 0.154 500 ACE 71.536 0.140 -28.464 -0.2846 30.468 0.143 500 Jack1 71.113 0.125 -28.887 -0.2889 30.477 0.136 500 Jack2 79.305 0.158 -20.695 -0.2070 24.058 0.150 1000 S_obs 65.795 0.099 -34.205 -0.3420 35.057 0.109 1000 Chao1 79.233 0.148 -20.767 -0.2077 23.710 0.150 1000 Chao2 78.137 0.139 -21.863 -0.2186 24.359 0.122 1000 ACE 79.378 0.119 -20.622 -0.2062 22.600 0.126 1000 Jack1 81.487 0.114 -18.513 -0.1851 20.520 0.116 1000 Jack2 88.019 0.145 -11.981 -0.1198 16.439 0.141 2000 S_obs 75.921 0.090 -24.079 -0.2408 25.062 0.103 2000 Chao1 86.353 0.126 -13.647 -0.1365 16.774 0.109 2000 Chao2 85.460 0.120 -14.540 -0.1454 17.270 0.142 2000 ACE 85.801 0.097 -14.199 -0.1420 16.051 0.115 2000 Jack1 89.303 0.096 -10.697 -0.1070 13.037 0.112 2000 Jack2 94.011 0.128 -5.989 -0.0599 11.592 0.106 5000 S_obs 86.423 0.069 -13.577 -0.1358 14.602 0.080 5000 Chao1 92.837 0.090 -7.163 -0.0716 9.994 0.098 5000 Chao2 92.288 0.084 -7.712 -0.0771 10.072 0.085 5000 ACE 92.252 0.070 -7.748 -0.0775 9.438 0.077 5000 Jack1 95.887 0.070 -4.113 -0.0411 6.808 0.079 5000 Jack2 98.356 0.102 -1.644 -0.0164 8.067 0.082 10000 S_obs 91.952 0.053 -8.048 -0.0805 9.021 0.076 10000 Chao1 95.945 0.069 -4.055 -0.0406 6.686 0.081 10000 Chao2 95.607 0.064 -4.393 -0.0439 6.629 0.070 10000 ACE 95.493 0.051 -4.507 -0.0451 6.013 0.060 10000 Jack1 98.457 0.053 -1.543 -0.0154 4.369 0.053 10000 Jack2 99.648 0.082 -0.352 -0.0035 6.386 0.063 20000 S_obs 95.626 0.037 -4.374 -0.0437 5.250 0.056 20000 Chao1 97.952 0.051 -2.048 -0.0205 4.468 0.111 20000 Chao2 97.733 0.046 -2.267 -0.0227 4.197 0.088 20000 ACE 97.676 0.037 -2.324 -0.0232 3.700 0.054 20000 Jack1 99.749 0.039 -0.251 -0.0025 3.035 0.042 20000 Jack2 100.313 0.064 +0.313 +0.0031 4.980 0.055 50000 S_obs 98.250 0.022 -1.750 -0.0175 2.443 0.040 50000 Chao1 99.099 0.028 -0.901 -0.0090 2.367 0.052 50000 Chao2 99.047 0.027 -0.953 -0.0095 2.285 0.046 50000 ACE 99.131 0.022 -0.869 -0.0087 1.945 0.030 50000 Jack1 100.157 0.025 +0.157 +0.0016 1.937 0.022 50000 Jack2 100.235 0.043 +0.235 +0.0024 3.350 0.033 --- broken-stick (mean Pielou J = 0.9093) -------------------- n estimator mean SE bias rel.bias RMSE SE(RMSE) 50 S_obs 33.482 0.036 -66.518 -0.6652 66.575 0.033 50 Chao1 65.870 0.279 -34.130 -0.3413 40.397 0.305 50 Chao2 64.144 0.267 -35.856 -0.3586 41.391 0.557 50 ACE 71.501 0.281 -28.499 -0.2850 35.850 0.194 50 Jack1 55.321 0.085 -44.679 -0.4468 45.157 0.080 50 Jack2 69.606 0.144 -30.394 -0.3039 32.366 0.137 100 S_obs 50.160 0.045 -49.840 -0.4984 49.964 0.043 100 Chao1 74.721 0.188 -25.279 -0.2528 29.162 0.141 100 Chao2 73.326 0.174 -26.674 -0.2667 29.885 0.139 100 ACE 78.285 0.181 -21.715 -0.2171 25.847 0.125 100 Jack1 74.908 0.097 -25.092 -0.2509 26.195 0.091 100 Jack2 87.110 0.161 -12.890 -0.1289 17.924 0.122 200 S_obs 66.937 0.049 -33.063 -0.3306 33.283 0.047 200 Chao1 83.629 0.135 -16.371 -0.1637 19.406 0.121 200 Chao2 82.334 0.120 -17.666 -0.1767 19.980 0.100 200 ACE 84.859 0.112 -15.141 -0.1514 17.437 0.096 200 Jack1 89.094 0.091 -10.906 -0.1091 13.006 0.088 200 Jack2 96.467 0.149 -3.533 -0.0353 12.099 0.111 500 S_obs 83.518 0.044 -16.482 -0.1648 16.831 0.047 500 Chao1 91.867 0.088 -8.133 -0.0813 10.597 0.063 500 Chao2 91.055 0.079 -8.945 -0.0895 10.828 0.062 500 ACE 90.842 0.064 -9.158 -0.0916 10.426 0.060 500 Jack1 97.347 0.071 -2.653 -0.0265 6.115 0.054 500 Jack2 99.686 0.119 -0.314 -0.0031 9.223 0.090 1000 S_obs 91.003 0.035 -8.997 -0.0900 9.391 0.036 1000 Chao1 95.509 0.064 -4.491 -0.0449 6.672 0.065 1000 Chao2 95.040 0.056 -4.960 -0.0496 6.613 0.047 1000 ACE 94.528 0.044 -5.472 -0.0547 6.443 0.041 1000 Jack1 99.217 0.053 -0.783 -0.0078 4.199 0.038 1000 Jack2 99.970 0.091 -0.030 -0.0003 7.011 0.067 2000 S_obs 95.297 0.026 -4.703 -0.0470 5.131 0.029 2000 Chao1 97.637 0.047 -2.363 -0.0236 4.352 0.065 2000 Chao2 97.408 0.043 -2.592 -0.0259 4.240 0.063 2000 ACE 97.035 0.031 -2.965 -0.0296 3.832 0.027 2000 Jack1 99.774 0.039 -0.226 -0.0023 3.030 0.027 2000 Jack2 100.004 0.067 +0.004 +0.0000 5.164 0.046 5000 S_obs 98.063 0.017 -1.937 -0.0194 2.362 0.019 5000 Chao1 98.859 0.027 -1.141 -0.0114 2.363 0.038 5000 Chao2 98.835 0.025 -1.165 -0.0117 2.277 0.032 5000 ACE 98.823 0.020 -1.177 -0.0118 1.971 0.016 5000 Jack1 99.949 0.025 -0.051 -0.0005 1.944 0.017 5000 Jack2 99.942 0.043 -0.058 -0.0006 3.364 0.032 10000 S_obs 99.025 0.013 -0.975 -0.0098 1.376 0.014 10000 Chao1 99.329 0.016 -0.671 -0.0067 1.438 0.019 10000 Chao2 99.327 0.016 -0.673 -0.0067 1.440 0.026 10000 ACE 99.501 0.016 -0.499 -0.0050 1.311 0.019 10000 Jack1 100.012 0.018 +0.012 +0.0001 1.380 0.013 10000 Jack2 100.033 0.031 +0.033 +0.0003 2.405 0.024 20000 S_obs 99.523 0.009 -0.477 -0.0048 0.831 0.010 20000 Chao1 99.612 0.010 -0.388 -0.0039 0.883 0.013 20000 Chao2 99.619 0.010 -0.381 -0.0038 0.881 0.013 20000 ACE 99.832 0.012 -0.168 -0.0017 0.970 0.023 20000 Jack1 100.009 0.013 +0.009 +0.0001 0.978 0.010 20000 Jack2 99.989 0.022 -0.011 -0.0001 1.697 0.019 50000 S_obs 99.811 0.006 -0.189 -0.0019 0.476 0.009 50000 Chao1 99.825 0.006 -0.175 -0.0017 0.491 0.010 50000 Chao2 99.826 0.006 -0.174 -0.0017 0.489 0.009 50000 ACE 99.949 0.008 -0.051 -0.0005 0.610 0.015 50000 Jack1 99.997 0.008 -0.003 -0.0000 0.608 0.009 50000 Jack2 99.993 0.014 -0.007 -0.0001 1.061 0.015 ============================================================================== PART 6. THE TABLE THE ARTICLE PRINTS ============================================================================== Relative bias (percent of true richness) at four efforts, and the effort at which |relative bias| first falls to 10 percent or below, interpolated in log n between ladder rungs. config estimator n=100 n=500 n=2000 n=10000 n for 10% RMSE@500 ------------------------------------------------------------------------------ lognormal-0.5 S_obs -40.96 -3.35 -0.02 +0.00 342 3.78 lognormal-0.5 Chao1 -7.73 -0.47 -0.01 +0.00 68 2.98 lognormal-0.5 Chao2 -8.61 -0.63 -0.01 +0.00 82 2.78 lognormal-0.5 ACE -3.70 -0.56 -0.00 +0.00 50 2.32 lognormal-0.5 Jack1 -8.30 +5.15 +0.08 +0.00 96 6.16 lognormal-0.5 Jack2 +8.26 +1.16 -0.10 -0.00 84 7.12 ------------------------------------------------------------------------------ lognormal-1.0 S_obs -50.89 -13.72 -1.59 -0.03 682 14.17 lognormal-1.0 Chao1 -21.60 -4.58 -0.48 -0.02 268 8.08 lognormal-1.0 Chao2 -23.30 -5.26 -0.56 -0.02 302 8.04 lognormal-1.0 ACE -15.14 -4.51 -0.70 +0.01 186 6.91 lognormal-1.0 Jack1 -25.12 +2.28 +1.46 +0.06 185 5.98 lognormal-1.0 Jack2 -10.81 +4.26 +0.36 -0.03 106 10.52 ------------------------------------------------------------------------------ lognormal-1.5 S_obs -61.72 -29.41 -9.63 -1.34 1,939 29.96 lognormal-1.5 Chao1 -37.25 -14.66 -4.04 -0.58 820 18.00 lognormal-1.5 Chao2 -39.22 -15.92 -4.55 -0.60 902 18.52 lognormal-1.5 ACE -32.38 -14.21 -4.68 -0.62 839 16.43 lognormal-1.5 Jack1 -42.44 -10.91 -0.12 +0.62 545 13.20 lognormal-1.5 Jack2 -30.92 -4.08 +1.26 +0.35 325 11.59 ------------------------------------------------------------------------------ lognormal-2.0 S_obs -71.23 -45.62 -24.08 -8.05 7,829 46.28 lognormal-2.0 Chao1 -53.13 -29.08 -13.65 -4.06 3,348 32.30 lognormal-2.0 Chao2 -54.51 -30.63 -14.54 -4.39 3,678 33.05 lognormal-2.0 ACE -49.16 -28.46 -14.20 -4.51 3,631 30.47 lognormal-2.0 Jack1 -57.43 -28.89 -10.70 -1.54 2,204 30.48 lognormal-2.0 Jack2 -48.96 -20.70 -5.99 -0.35 1,258 24.06 ------------------------------------------------------------------------------ broken-stick S_obs -49.84 -16.48 -4.70 -0.98 911 16.83 broken-stick Chao1 -25.28 -8.13 -2.36 -0.67 406 10.60 broken-stick Chao2 -26.67 -8.95 -2.59 -0.67 448 10.83 broken-stick ACE -21.71 -9.16 -2.96 -0.50 440 10.43 broken-stick Jack1 -25.09 -2.65 -0.23 +0.01 221 6.12 broken-stick Jack2 -12.89 -0.31 +0.00 +0.03 124 9.22 ------------------------------------------------------------------------------ ============================================================================== PART 7. WHICH ESTIMATOR WINS, BY CONFIGURATION AND EFFORT ============================================================================== Winner by lowest RMSE. RMSE is the honest criterion because an estimator that is unbiased on average but wild replicate to replicate is no use to somebody who gets one survey. config 50 100 200 500 1000 2000 5000 10000 20000 50000 ----------------------------------------------------------------------------------------------- lognormal-0.5 Jack2 Jack1 ACE ACE ACE S_obs S_obs S_obs S_obs S_obs lognormal-1.0 Jack2 Jack2 Jack1 Jack1 ACE ACE S_obs S_obs S_obs S_obs lognormal-1.5 ACE Jack2 Jack2 Jack2 Jack1 Jack1 ACE ACE S_obs S_obs lognormal-2.0 ACE Jack2 Jack2 Jack2 Jack2 Jack2 Jack1 Jack1 Jack1 Jack1 broken-stick Jack2 Jack2 Jack2 Jack1 Jack1 Jack1 Jack1 ACE S_obs S_obs ----------------------------------------------------------------------------------------------- And by lowest absolute bias, which is the criterion most textbooks quote and which gives a different answer: config 50 100 200 500 1000 2000 5000 10000 20000 50000 ----------------------------------------------------------------------------------------------- lognormal-0.5 ACE ACE ACE Chao1 Chao1 ACE S_obs S_obs S_obs S_obs lognormal-1.0 ACE Jack2 Jack2 Jack1 Chao1 Jack2 Jack2 ACE ACE S_obs lognormal-1.5 ACE Jack2 Jack2 Jack2 Jack2 Jack1 Jack2 Jack2 ACE Jack2 lognormal-2.0 ACE Jack2 Jack2 Jack2 Jack2 Jack2 Jack2 Jack2 Jack1 Jack1 broken-stick ACE Jack2 Jack2 Jack2 Jack2 Jack2 Jack1 Jack1 Jack1 Jack1 ----------------------------------------------------------------------------------------------- ============================================================================== PART 8. HOW OFTEN ACE REFUSED TO BE ACE ============================================================================== ACE divides by an estimated sample coverage. When every individual of every rare species is a singleton, that coverage estimate is exactly zero and the formula divides by nothing. We fall back to Chao1, as the standard software does. Here is how often that happened, as a percentage of 6000 replicates: config 50 100 200 500 1000 2000 5000 10000 20000 50000 ----------------------------------------------------------------------------------------------- lognormal-0.5 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.00 0.00 0.00 lognormal-1.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.07 0.30 0.02 lognormal-1.5 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.48 lognormal-2.0 0.17 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 broken-stick 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.42 3.03 ----------------------------------------------------------------------------------------------- We expected this to bite hardest at low effort. It does not. The fallback almost never fires on a thin sample, because a thin sample from a rich community has doubletons and tripletons everywhere and the coverage estimate stays comfortably above zero. It fires at the other end: heavy sampling of an even community, where the last two or three species left undersampled are each seen exactly once, so every rare individual is a singleton and the coverage estimate is exactly zero. The worst cell in the table is the broken stick at n = 50,000. The practical consequence is small, because Chao1 and ACE agree closely at that effort anyway, but a package that reports 'ACE' without saying which branch it took is hiding a real substitution. ============================================================================== PART 9. MONTE CARLO CONVERGENCE ============================================================================== A separate run: lognormal-1.0, n = 500, 150,000 replicate communities, tracking the running mean of each estimator so the Monte Carlo error can be seen shrinking. convergence run took 90.3 s Running mean of each estimator at n = 500. The final column is how many standard errors the checkpoint sits from the final value. S_obs final = 86.3315 +/- 0.0091 bias = -13.6685 Chao1 final = 95.5732 +/- 0.0172 bias = -4.4268 Chao2 final = 94.8889 +/- 0.0157 bias = -5.1111 ACE final = 95.6016 +/- 0.0134 bias = -4.3984 Jack1 final = 102.4093 +/- 0.0141 bias = +2.4093 Jack2 final = 104.4882 +/- 0.0246 bias = +4.4882 reps S_obs Chao1 Chao2 ACE Jack1 Jack2 ------------------------------------------------------------------------------------------------------------------- 250 86.344+/-0.202 95.791+/-0.407 95.156+/-0.368 95.684+/-0.296 102.524+/-0.318 104.842+/-0.585 500 86.354+/-0.142 95.721+/-0.297 95.071+/-0.258 95.538+/-0.213 102.394+/-0.227 104.597+/-0.420 1000 86.296+/-0.106 95.402+/-0.206 94.749+/-0.185 95.340+/-0.159 102.163+/-0.169 104.172+/-0.302 2000 86.364+/-0.076 95.694+/-0.150 94.947+/-0.133 95.549+/-0.112 102.395+/-0.118 104.552+/-0.212 4000 86.310+/-0.055 95.601+/-0.105 94.900+/-0.095 95.558+/-0.081 102.384+/-0.085 104.530+/-0.149 8000 86.298+/-0.039 95.608+/-0.074 94.916+/-0.067 95.586+/-0.057 102.408+/-0.060 104.567+/-0.105 15000 86.337+/-0.028 95.584+/-0.054 94.896+/-0.049 95.596+/-0.042 102.418+/-0.044 104.507+/-0.077 25000 86.341+/-0.022 95.617+/-0.042 94.914+/-0.038 95.624+/-0.033 102.438+/-0.034 104.551+/-0.060 40000 86.337+/-0.017 95.597+/-0.033 94.918+/-0.031 95.603+/-0.026 102.419+/-0.027 104.518+/-0.048 60000 86.333+/-0.014 95.586+/-0.027 94.902+/-0.025 95.609+/-0.021 102.422+/-0.022 104.521+/-0.039 85000 86.336+/-0.012 95.577+/-0.023 94.897+/-0.021 95.610+/-0.018 102.420+/-0.019 104.504+/-0.033 115000 86.333+/-0.010 95.581+/-0.020 94.896+/-0.018 95.606+/-0.015 102.416+/-0.016 104.503+/-0.028 150000 86.331+/-0.009 95.573+/-0.017 94.889+/-0.016 95.602+/-0.013 102.409+/-0.014 104.488+/-0.025 ------------------------------------------------------------------------------------------------------------------- largest checkpoint departure from the final value: 1.64 SE (ACE at 1000 replicates) Every checkpoint sits inside its own two-standard-error envelope of the final value except where noted, and the envelope shrinks as 1/sqrt(replicates), which is the only rate Monte Carlo ever offers. Cross-check between the two independent runs of the same cell: S_obs sweep 86.2757 +/- 0.0457 convergence run 86.3315 +/- 0.0091 difference -0.0558 (-1.20 SE) Chao1 sweep 95.4224 +/- 0.0860 convergence run 95.5732 +/- 0.0172 difference -0.1507 (-1.72 SE) Chao2 sweep 94.7431 +/- 0.0785 convergence run 94.8889 +/- 0.0157 difference -0.1458 (-1.82 SE) ACE sweep 95.4927 +/- 0.0676 convergence run 95.6016 +/- 0.0134 difference -0.1088 (-1.58 SE) Jack1 sweep 102.2774 +/- 0.0714 convergence run 102.4093 +/- 0.0141 difference -0.1319 (-1.81 SE) Jack2 sweep 104.2565 +/- 0.1242 convergence run 104.4882 +/- 0.0246 difference -0.2316 (-1.83 SE) ============================================================================== PART 10. DOES THE QUADRAT COUNT CHANGE THE ANSWER? ============================================================================== Chao2 and anything else built on incidence depends on how the same n individuals are cut into sampling units. We re-ran lognormal-1.0 at n = 1000 with the effort split four ways, 1000 replicates each. T quadrats m per quadrat Chao2 mean SE bias RMSE ------------------------------------------------------------------------------ 4 250 97.543 0.102 -2.457 4.063 10 100 98.215 0.108 -1.785 3.849 25 40 98.363 0.120 -1.637 4.145 50 20 98.373 0.122 -1.627 4.196 100 10 98.493 0.136 -1.507 4.553 ------------------------------------------------------------------------------ Same 1000 individuals, same community, same estimator. Splitting the effort into 4 units instead of 100 changes the Chao2 bias from -1.51 to -2.46 species, a 63 percent swing, while the RMSE moves the other way and is worst at T = 100. Finer units give a less biased and noisier answer; coarser units give the reverse. There is no free choice here, and T = 10 is ours. Everything we say about Chao2 in this study is conditional on it. In a real survey the number of quadrats is usually set by how much walking a person can do in a day, which is not a statistical criterion at all. ============================================================================== PART 11. MACHINE-READABLE FIGURE DATA ============================================================================== Everything the article's five figures are drawn from, so a reader can redraw them without rerunning the simulation. BEGIN-FIGURE-JSON 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END-FIGURE-JSON ============================================================================== DONE ============================================================================== total wall time 286.9 s seed 20241104; rerunning this file reproduces every number above exactly, on any machine with the same numpy version.