INTERACTIVE COMPANION · METHODS · SAMPLING

The Sampling Bench

A live model accompanying “How Many Samples Before Your Species Count Means Anything”

Two benches. The first lets you sample a community one individual at a time and watch the species count climb towards a true richness you can see on the screen, with the estimators trying to guess it from below. The second runs hundreds of replicate surveys and reports how far off each estimator was, on average and in the worst case.

Everything runs in your browser with no libraries and no network. The random number generator is seeded, so the same seed gives the same run every time. It is a different generator from the one numpy uses in the article's analysis, which means the two agree statistically and never digit for digit. The difference is shown rather than hidden.

Model 1. One individual at a time

The community below holds a number of species you chose. Nothing about the sampling is clever: an individual is drawn in proportion to its species' abundance, added to the running count, and dealt into one of ten quadrats in rotation. Watch the accumulation curve bend over long before it reaches the dashed line.

True richness 100 species The dashed line. Nobody in a real survey knows this number, which is the entire reason we are doing this on a computer. Push it to 200 and watch how much further the curve has to climb before it flattens. Unevenness σ 1.00 Log-normal spread. At 0.50 the community is nearly even and the curve saturates quickly. At 2.00 a handful of species dominate and the tail never arrives. Set it to 0 for MacArthur's broken stick instead. Try 0.50 and then 2.00 and compare how far the estimators get. Speed 4096 per second Drag to the left to watch individual draws land. The interesting part is the first two hundred, where the estimators swing about wildly before settling. Seed Change it and the whole community is redrawn. The estimator ordering usually survives; the exact numbers never do. Show Chao1 and Chao2 The two singleton-and-doubleton estimators. Both approach the truth from below and stay there. Show ACE The coverage-based estimator. Usually the smoothest of the three that approach from below. Show the jackknives These overshoot. Turn them on and find the crossing point, somewhere in the low hundreds at the default settings.

individuals: 0 species seen: 0 still missing: 100 singletons / doubletons: 0 / 0 Chao1: – Jack2: – expected S at this n: –

The last readout is the closed form, the thing the curve is a noisy realisation of:

E[S_obs] = sum over species of ( 1 − (1 − p_i)^n )

It uses the abundances the model knows and you do not. If the observed count and that number drift apart by more than a species or two, the run has been unlucky rather than wrong.

Model 2. What each estimator is worth

One survey tells you nothing about an estimator. This bench runs the whole thing again and again on fresh communities drawn from the same family, and reports the average miss and the root mean square miss. Bias is what the textbooks quote. Root mean square error is what you care about when you get one afternoon and one survey.

True richness 100 species Richer communities are harder at the same effort, because the same individuals are divided among more species. Unevenness σ 1.00 The lever that matters most. Run the defaults, then set σ to 2.00 and watch every estimator lose twenty species at once. Set it to 0 for the broken stick. Effort n 500 individuals Try the sequence 200, 500, 2000, 5000 at σ = 1.00. Somewhere past 2000 the plain count starts beating Chao1 on root mean square error, which is the one result in the article that nobody at the club table predicted. Quadrats T 10 Only Chao2 reads this, and it reads it hard. The same individuals cut into 4 units instead of 50 move the Chao2 bias by more than half a species. Nothing else on this page changes at all. Replicate communities 800 More replicates shrink the standard errors as one over the square root, and nothing else. The ordering of the estimators settles long before the third decimal place does. Seed Every run with the same seed and settings gives the same table.

configuration: – lowest RMSE: – at: –

estimator	mean ± SE	bias	RMSE	article	difference

–

Three numbers you can check with a calculator

The formulas are short enough to do by hand, which is the point of printing them.

Suppose a survey of 500 individuals turns up 86 species, of which 12 were seen exactly once and 5 exactly twice. Then the bias-corrected Chao1 is 86 + 12×11 / (2×6) = 86 + 11 = 97.00. The second-order jackknife is 86 + 12×997/500 − 5×498²/(500×499) = 86 + 23.928 − 4.970 = 104.958. And if those 86 species had been perfectly even at one percent each, the expected count from 500 draws would have been 100 × (1 − 0.99⁵⁰⁰) = 99.343 species, which is the number this page prints as the expected S. All three are checked against the running code before the page is published.

What the model leaves out

Everything spatial. A quadrat here is an independent draw from the whole community, which no real quadrat has ever been. Every individual is identified correctly, which no real survey manages. Detection is uniform across species, so nothing is cryptic, nocturnal or too small to key. The community does not change while you sample it. Those four assumptions are what make the truth knowable, and they are also why an estimator that looks acceptable here can be much worse in a wood.

The article's effort table is the summary of all of this. The analysis code is the version that produced it, at 6,000 replicates per cell rather than the few hundred this page can manage while you wait.