A zeta-diversity estimate can use the same recorded sites and still have different uncertainty under different sampling rules. Drawing pairs independently, drawing distinct pairs, and selecting sites before forming all their pairs are three different experiments.
The short answer is to define what is random before calculating a standard error. Overlapping site names do not, by themselves, make independently drawn pair labels dependent when the incidence table is fixed. This guide compares complete sampling distributions for an original four-site table, then changes the table and sampling design. It concerns uncertainty from selecting combinations within a recorded table, rather than uncertainty about an unobserved ecosystem.
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ROUTE 1 · CHAPTERS 1–6
Define the fixed-table experiment
Build the original incidence table and replacement sampling law.
ROUTE 2 · CHAPTERS 7–13
Compare the selection designs
Derive finite-population correction and exact site-induced estimates.
ROUTE 3 · CHAPTERS 14–20
Preserve the uncertainty model
Check joint inclusion, conditioning, software output and effective sample size.
ROUTE 4 · CHAPTERS 21–27
Transfer and retain the evidence
Change the table, audit resampling and preserve design information.
ROUTE 5 · CHAPTERS 28–33
Reproduce and continue
Repair misconceptions and check estimator variance and cross-order covariance.
Chapter contents
Define the fixed-table experiment · Chapters 1–6
1. Name the quantity being estimated
2. Keep three uncertainty questions separate
3. Build the original recorded incidence table
4. Calculate the complete pair population
5. Define independent pair-label sampling
6. Enumerate the first design rather than trusting a formula
Compare the selection designs · Chapters 7–13
7. Define distinct-pair sampling without replacement
8. Derive the finite-population correction
10. Change the sampling unit to sites
11. Work every three-site outcome
Preserve the uncertainty model · Chapters 14–20
14. Inspect joint inclusion for overlapping pairs
15. Inspect joint inclusion for disjoint pairs
16. Write the indicator variance account
17. Recognise what is conditional on the table
18. Distinguish spread from standard error
19. Read software documentation before importing an interval
Transfer and retain the evidence · Chapters 21–27
21. Change one incidence rule and predict the result
22. Try a changed table with no triangle differences
23. Check the changed-table solution independently
24. Explain why a bootstrap must match the intended experiment
25. Keep sampling weights tied to the target
Reproduce and continue · Chapters 28–33
28. Repair three common interpretation errors
29. Questions readers often ask
30. Reproduce the comparison in a short calculation
31. Estimate a variance when the pair census is unavailable
CHAPTER 1 OF 33 · Define the fixed-table experiment
1. Name the quantity being estimated
For N recorded sites, order-two zeta diversity is the average number of species shared by every unordered pair of distinct sites. There are C(N,2)=N(N−1)/2 pairs. Each pair contributes one shared-species count to that finite average.
We use unnormalised intersection counts throughout. A Jaccard ratio, a spatially weighted comparison or a nearest-neighbour statistic is a different target and needs a matching calculation. The word pair does not specify whether a pair can be selected again during the sampling experiment.
The target in this article is a fixed-table mean. Once the incidence table is supplied, its complete pair counts are fixed numbers. Randomness enters when we choose which of those numbers to inspect. This distinction makes it possible to calculate sampling uncertainty exactly without claiming that recorded absences are biological absences or that the selected sites represent a wider region.
CHAPTER 2 OF 33 · Define the fixed-table experiment
2. Keep three uncertainty questions separate
First, a computer might sample pair labels to approximate the complete mean of an existing incidence table. Increasing that computational sample can reduce this approximation error without collecting any new field observations.
Second, a field team might sample sites from a larger population. Its uncertainty concerns which sites entered the record, and its sampling unit is a site rather than a pre-existing pair label. Third, imperfect detection can change the incidence table itself, assigning species to the wrong recorded lists.
These questions can coexist, but their variance calculations are not interchangeable. A very precise Monte Carlo approximation to a biased or incomplete table remains an approximation to that table. Begin an analysis by stating which source of uncertainty the reported interval addresses. Here we deliberately hold the four incidence lists fixed, compare three declared selection designs, and leave field representativeness and detection inference outside the calculation.
CHAPTER 3 OF 33 · Define the fixed-table experiment
3. Build the original recorded incidence table
Use four fictional sites. Site A records species a, b, c, d, e and f. Site B records a, b, c and g. Site C records a, d and h. Site D records b, e and i. Each letter denotes one taxon under a consistent recording rule.
The lists are teaching constructions, not measurements from a named habitat. Their site richnesses are six, four, three and three. Species such as f or g occur on one list, while a and b occur on three lists.
Write out each intersection before discussing uncertainty. A and B share a, b and c. A and C share a and d. A and D share b and e. B and C share only a; B and D share only b. C and D share no species. Those six intersection counts provide the complete finite population that the first two designs will sample.
CHAPTER 4 OF 33 · Define the fixed-table experiment
4. Calculate the complete pair population
Label the pair counts AB=3, AC=2, AD=2, BC=1, BD=1 and CD=0. Their sum is nine, so the complete order-two zeta mean is μ=9/6=3/2.
Their squared values sum to nineteen: 9+4+4+1+1+0=19. With the population divisor six, the pair-count variance is σ²=19/6−(3/2)²=11/12. The variance with sample-style divisor five is S²=6σ²/5=11/10.
Both variances describe the same six fixed numbers but use different denominators. We will state which version appears in each formula. Confusing them changes a finite-population correction even when every intersection was counted correctly. This is also a useful hand-checking point: preserve the six values and their square sum beside any software result so a reader can reconstruct the arithmetic independently.
| Pair | Shared recorded taxa | Intersection count |
|---|---|---|
| AB | a, b, c | 3 |
| AC | a, d | 2 |
| AD | b, e | 2 |
| BC | a | 1 |
| BD | b | 1 |
| CD | None | 0 |
CHAPTER 5 OF 33 · Define the fixed-table experiment
5. Define independent pair-label sampling
In the first design, select one of the six pair labels uniformly, record its count, replace that label, and repeat independently. A pair can be drawn more than once. Each draw has expected count 3/2 and variance 11/12.
For m draws, the estimate is their arithmetic mean. Since the label draws are independent, its expectation is μ and its variance is σ²/m. At m=2 that variance is 11/24.
The word replacement concerns the sampling list, not an ecological event at a site. We are not removing species, resurveying habitats or changing the incidence matrix between draws. The draw AB followed by AC contains a shared site name, but the second label was still selected independently from the same six-label list. This fixed-table example is an important counterexample to the blanket claim that any overlap of sites automatically makes computational pair draws dependent.
CHAPTER 6 OF 33 · Define the fixed-table experiment
6. Enumerate the first design rather than trusting a formula
With two replacement draws, there are six choices for the first label and six for the second: thirty-six equally likely ordered outcomes. The estimate for AB then CD is (3+0)/2=3/2. The estimate for AB then AB is three. The estimate for CD then CD is zero.
Average all thirty-six estimates and the result is 3/2. Average their squared deviations from 3/2 and the result is 11/24. This enumeration checks both the independence assumption and the variance formula.
Repeated numerical counts are not the same as repeated labels. AC and AD both have value two but are distinct population members. Collapsing them into one equally likely category would change the selection probabilities. If enumeration is performed from the multiset of values, both entries must remain present so that the original uniform-label experiment is preserved.
CHAPTER 7 OF 33 · Compare the selection designs
7. Define distinct-pair sampling without replacement
In the second design, choose m distinct pair labels uniformly from the six possibilities. The selected pair counts are averaged. AB and AC can still appear together, but AB cannot be selected twice in the same sample.
For m=2 there are C(6,2)=15 equally likely unordered samples. This is simple random sampling without replacement from the pair population. The estimate remains unbiased for the complete pair mean.
Its variance is (1−m/M)S²/m, where M=6 is the number of pair labels and S² uses divisor M−1. At m=2 it is (1−2/6)(11/10)/2=11/30. It is smaller than 11/24 because a repeated draw provides no opportunity to miss a different label. This comparison concerns these specified designs and the same fixed table; it is not a universal statement that every design described informally as more diverse must be more precise.
CHAPTER 8 OF 33 · Compare the selection designs
8. Derive the finite-population correction
Let V₁ and V₂ be two ordered draws without replacement from M fixed values, whose population variance is σ² with divisor M. Each draw marginally has variance σ², but their covariance is −σ²/(M−1).
To see the sign, centre each population value around μ. Their centred sum is zero. The sum of products over all distinct ordered population members equals the square of that zero sum minus the sum of squared centred values, namely −Mσ². Dividing by M(M−1) gives the covariance.
For m draws, the variance of their mean is [mσ²+m(m−1)(−σ²/(M−1))]/m². Simplifying gives σ²(M−m)/[m(M−1)], equivalent to (1−m/M)S²/m. The negative dependence arises from excluding previously selected labels. It does not follow from whether the labels share a site name.
If all six pair labels are selected without replacement, their mean is the complete mean exactly. The finite-population formula gives variance zero when m=M, as it must.
If six labels are selected with replacement, some labels can be repeated and others omitted. Its variance remains σ²/6=11/72. The same number of computational draws does not guarantee a census.
This boundary check helps catch a wrong interpretation of sample count. A field or software report saying “six comparisons” has not yet specified whether every comparison was different or whether every possible pair was included. Record the actual selected labels when practical. A reproducible census claim requires coverage of the full label population, whereas a replacement Monte Carlo calculation can remain uncertain even when its draw count equals the number of distinct population members.
CHAPTER 10 OF 33 · Compare the selection designs
10. Change the sampling unit to sites
In the third design, choose three of the four site labels uniformly without replacement. Then compute the mean intersection count of all three pairs among those selected sites.
There are only four possible site samples: ABC, ABD, ACD and BCD. Their induced pair sets are not a uniform selection of three labels from all six pairs. Each selected set must form a triangle among its three sites.
For example, the three-pair set {AB, AC, AD} cannot arise from a three-site sample because it needs four sites. The set {AB, AC, BC} can arise. Treating the induced pair collection as an arbitrary simple random sample of three pair labels therefore describes a different experiment. The distinction is structural and can be checked before calculating any variance: list which sets of pair labels the design can actually produce.
For ABC, the pair counts are three, two and one, giving mean two. ABD also gives mean two from three, two and one. ACD gives (2+2+0)/3=4/3. BCD gives (1+1+0)/3=2/3.
The four site samples are equally likely. Their estimate averages to [2+2+4/3+2/3]/4=3/2, so this design is unbiased for the complete pair mean.
Its variance is the average of squared deviations: [(1/2)²+(1/2)²+(−1/6)²+(−5/6)²]/4=11/36. This is an exact design variance, not an estimate from four repeated field surveys. The four outcomes exhaust the hypothetical randomisation of the stated fixed table. No asymptotic approximation or normality assumption was needed to obtain the result.
| Selected sites | Induced pair counts | Estimate |
|---|---|---|
| ABC | 3, 2, 1 | 2 |
| ABD | 3, 2, 1 | 2 |
| ACD | 2, 2, 0 | 4/3 |
| BCD | 1, 1, 0 | 2/3 |
CHAPTER 12 OF 33 · Compare the selection designs
12. Compare three pairs with three induced pairs
A uniform sample of three distinct pair labels from the six-pair population has variance (1−3/6)(11/10)/3=11/60. A uniform sample of three sites followed by all induced pairs has variance 11/36.
Both estimates average three pair counts and both are unbiased for 3/2. Nevertheless their sampling distributions differ because their allowed joint selections differ. Using 11/60 for the site-sample estimate would understate its variance in this particular example.
The comparison is not a general ranking between site and pair sampling under every table or budget. A pair observation and a site survey can have different practical costs, and another incidence pattern can change their relative precision. The lesson is to match the variance calculation to the randomisation actually used. Equal numbers of pairwise calculations are not enough to establish equal information.
CHAPTER 13 OF 33 · Compare the selection designs
13. Derive why the site estimator is unbiased
For a general fixed population of N sites, select s sites uniformly without replacement, with 2≤s≤N. A given pair is included when both of its sites are selected. Its inclusion probability is C(N−2,s−2)/C(N,s)=s(s−1)/[N(N−1)].
The estimator averages C(s,2) induced pair counts. Taking its expectation adds each population pair count times that inclusion probability, then divides by C(s,2). The result is the sum of all pair counts divided by C(N,2), which is the complete pair mean.
This proof depends on uniform site selection and using all induced pairs with equal weight. It does not automatically cover convenience sites, unequal inclusion probabilities or selective removal of inconvenient comparisons. An unbiasedness claim must retain the design that justified it. The variance requires further joint-inclusion information, because single-pair inclusion probabilities alone do not specify which pairs tend to enter together.
CHAPTER 14 OF 33 · Preserve the uncertainty model
14. Inspect joint inclusion for overlapping pairs
Consider two distinct population pairs that share one site, such as AB and AC. Including both requires selecting three distinct sites. Under uniform s-site selection from N sites, their joint inclusion probability is (s)₃/(N)₃, using falling factorials.
For N=4 and s=3, this probability is 1/4. Each individual pair has inclusion probability 1/2, so the product of individual inclusion probabilities is also 1/4. In this particular small design, these overlapping-pair inclusion indicators have zero covariance.
That result is another warning against deciding the sign from the word overlap alone. Other N and s values can produce a different relationship. The correct comparison is joint inclusion probability minus the product of marginal inclusion probabilities. Then the pair values determine how those indicator covariances contribute to the estimator variance. A label-based slogan cannot replace that account.
CHAPTER 15 OF 33 · Preserve the uncertainty model
15. Inspect joint inclusion for disjoint pairs
For two disjoint pairs, such as AB and CD, selecting both requires all four sites. Under the three-of-four design this is impossible. Their joint inclusion probability is zero, while each marginal inclusion probability is 1/2.
Their inclusion-indicator covariance is therefore −1/4. By contrast, a uniform selection of three pair labels from six has joint pair-label inclusion probability 3×2/(6×5)=1/5 for any two distinct pair labels, regardless of whether their site names overlap.
These different joint probabilities explain why the two designs can have different variances even though their marginal means agree. They also show how to generalise beyond four sites: classify pair-pair relationships by the number of distinct sites required, calculate the matching joint inclusion probabilities, and retain the actual pair values. Merely counting pairwise comparisons loses the selection structure needed for uncertainty.
CHAPTER 16 OF 33 · Preserve the uncertainty model
16. Write the indicator variance account
Let I_e indicate that population pair e enters the site sample, and let v_e be its fixed intersection count. The site estimator is the sum of I_e v_e divided by C(s,2).
Its variance is the sum of v_e² Var(I_e) plus twice the sum over distinct unordered pair labels of v_e v_f Cov(I_e,I_f), all divided by C(s,2)². The inclusion probabilities from the preceding chapters supply those variances and covariances.
This expression is useful because it separates the fixed incidence information from the random sampling design. The values v_e belong to the table; the joint law of the indicators belongs to the selection rule. A changed table changes the former, and a changed selection procedure changes the latter. Reusing a standard-error formula after changing either part requires checking whether its assumptions still hold.
CHAPTER 17 OF 33 · Preserve the uncertainty model
17. Recognise what is conditional on the table
Our replacement draws are independent conditional on the fixed six-label table. If the incidence table itself is random under a larger model, two sampled pair values can share uncertainty about that common underlying table.
For example, different hypothetical regions could have different complete zeta means. Drawing several pairs within one realised region does not replace sampling several independent regions. Conditional computational variance and between-region variance answer different questions.
A layered analysis can use total variance: average the conditional sampling variance and add the variance of the conditional mean. That identity does not supply the missing ecological model; it states how the levels would combine once specified. Be explicit about which quantities were conditioned on. Otherwise a correct independence claim about random pair labels can be mistaken for a claim about independent ecological evidence, or a valid ecological dependence concern can be incorrectly imposed on a fixed-table Monte Carlo calculation.
CHAPTER 18 OF 33 · Preserve the uncertainty model
18. Distinguish spread from standard error
The spread of pair counts describes how different intersections are across the fixed table. The standard error of an estimator describes how its estimate varies under a specified sampling rule.
In the replacement design, pair-count variance is 11/12, whereas the variance of the mean of m independent label draws is 11/(12m). Taking the square root gives the corresponding standard deviations. They have the same count units but different meanings.
A software output labelled standard deviation might concern the sampled intersection counts, replicate estimates, a corrected population spread or another documented quantity. Do not divide or multiply it by a sample-size factor without reading that definition. The sampling unit and any correction options matter. A smaller reported spread is not automatically evidence that the complete zeta mean has been estimated more precisely under every design.
CHAPTER 19 OF 33 · Preserve the uncertainty model
19. Read software documentation before importing an interval
The zetadiv documentation for Zeta.order.mc describes a Monte Carlo calculation for a specified zeta order and exposes sampling, normalisation and standard-deviation correction options. Those options are part of the computational contract.
The three designs in this article are explicitly defined teaching experiments. We are not asserting that every version or option of a package implements one particular design or returns the standard error derived here. A reproducible software report should record the function, version, arguments, seed where relevant and interpretation of each output field.
Test the implemented procedure against a small incidence table whose complete combinations can be enumerated. Compare selected labels, means and repeated estimates with the declared selection rule. This is stronger than recognising the word Monte Carlo and assuming all resulting uncertainty measures have the same definition.
CHAPTER 20 OF 33 · Preserve the uncertainty model
20. Avoid a false effective-sample-size shortcut
It is tempting to replace the number of pairs by the number of sites, or by some adjusted number, and divide a variance by that quantity. That can be a useful approximation only when derived or validated for the actual design and target.
In our original table, the exact three-site variance is 11/36. Neither the mere presence of three pair counts nor the presence of three sites produces that value without considering their joint selections. The four possible outcomes provide a direct benchmark against which any approximation can be assessed.
An effective sample size is a summary of a specified dependence calculation, not an independent explanation of it. If a report uses one, retain the model and criterion that produced it. A single adjusted count cannot generally encode every weighting, spatial restriction, detection issue and selection relationship in a biodiversity study.
CHAPTER 21 OF 33 · Transfer and retain the evidence
21. Change one incidence rule and predict the result
Add one fictional species j to every site in the original table. Every pair intersection count increases by one. The complete pair mean increases from 3/2 to 5/2.
Each estimate under each of our designs also increases by one, because every selected count receives that same addition. Their sampling variances remain unchanged. A constant translation moves the sampling distribution without changing its centred deviations.
This is a useful transfer test because it changes the ecological-looking table while preserving a mathematical invariant. Recompute one replacement outcome, one distinct-pair sample and each induced-site outcome to confirm the prediction. A method that reports changed variance solely because every intersection increased by the same constant has mixed up a spread measure, a mean-dependent model or a programming convention with the exact fixed-table design variance considered here.
CHAPTER 22 OF 33 · Transfer and retain the evidence
22. Try a changed table with no triangle differences
Construct four new fictional sites with pair counts AB=2, AC=1, AD=0, BC=0, BD=1 and CD=2. One valid incidence construction uses two species restricted to AB, one restricted to AC, one restricted to BD and two restricted to CD. No species occurs in three sites.
The complete pair mean is one. The four induced three-site pair sums are all three, so every three-site estimate equals one. The site-sample estimator has variance zero, despite observing only three of the four sites.
The pair population still has nonzero spread: its squared values sum to ten, giving population variance 10/6−1=2/3. A random distinct-three-pair mean therefore has positive variance. This changed table reverses the precision ordering of the original example. It proves why the original comparison cannot be promoted into a universal ranking of site-first and pair-first designs.
CHAPTER 23 OF 33 · Transfer and retain the evidence
23. Check the changed-table solution independently
For ABC the changed counts are 2,1,0; for ABD they are 2,0,1; for ACD they are 1,0,2; and for BCD they are 0,1,2. Every mean is one.
For a simple random sample of three distinct labels from the six pair labels, S²=(6/5)(2/3)=4/5. Its estimator variance is (1−3/6)(4/5)/3=2/15. Two independent replacement pair draws have mean variance (2/3)/2=1/3.
A single draw can still be zero, one or two. Zero site-sample variance does not mean that all intersections are identical or that the entire incidence matrix has been recovered. It means this particular design’s triangle averages happen to equal the target for this particular table. Additional ecological questions and other zeta orders can remain unresolved.
CHAPTER 24 OF 33 · Transfer and retain the evidence
24. Explain why a bootstrap must match the intended experiment
A resampling procedure creates its own randomisation. Resampling pair-count entries independently mimics a pair-level empirical distribution. Resampling sites and rebuilding intersections changes several pair entries together. The resulting uncertainty need not agree.
Neither choice is automatically correct for every scientific target. If the question concerns computational error conditional on a known finite table, direct design enumeration or repeated implementation runs can be appropriate checks. If it concerns sampling sites from a population, the site-selection process and any clustering must be represented.
Resampling also cannot create missing detection information by itself. A bootstrap repeatedly rearranges the supplied record under a declared rule; it does not establish that a recorded zero was a true absence. Before using the word bootstrap as reassurance, state the unit resampled, the statistic recalculated, the assumptions about exchangeability and the inferential population. Our exact examples can reveal when a proposed resampling rule is reproducing the wrong experiment.
CHAPTER 25 OF 33 · Transfer and retain the evidence
25. Keep sampling weights tied to the target
Uniform pair selection estimates the equally weighted complete pair mean. If some pairs are selected more frequently because they are close together or easier to access, their ordinary sample mean generally targets a different weighted quantity.
Known inclusion probabilities can support a suitably weighted estimator under a specified design, but the weights and variance must be derived together. An arbitrary correction applied after seeing the counts is not a substitute for a documented selection mechanism.
The same issue arises when site inclusion probabilities differ. A site-induced pair is then included according to both of its sites and the joint design, not necessarily the product of two independent site probabilities. Preserve the selection records. A precise estimate of a weighted local statistic can be useful, but it should not be presented as the complete equally weighted zeta mean unless the estimator and design justify that interpretation.
CHAPTER 26 OF 33 · Transfer and retain the evidence
26. Separate computational effort from field coverage
A million replacement pair draws from four recorded sites can make the computational estimate of that table’s pair mean extremely precise. It does not turn four surveyed sites into a million independent field locations.
Conversely, fully enumerating six pairs removes combination-sampling error for the supplied table, while leaving uncertainty about detection, taxonomic consistency and regional representation. Zero Monte Carlo error is not zero scientific uncertainty.
A practical report can state these layers separately: how sites were sampled, how species were recorded, whether combinations were enumerated or sampled, and which uncertainty measure is attached to the reported estimate. This makes it possible to decide what more effort would improve. More computational draws repair approximation error; additional representative observations address a different information gap. The appropriate next step depends on which layer limits the intended claim.
CHAPTER 27 OF 33 · Transfer and retain the evidence
27. Build a reproducible evidence record
Keep the incidence matrix, site labels, taxonomic rule, empty-site handling and definition of the zeta statistic. Store the pair-label population or the code that reconstructs it. Record replacement, uniformity, sample size and any restrictions.
For simulated checks, retain the random seed and the actual sampled labels where practical. For a small table, enumerate the full sampling law and compare its mean and variance with repeated random estimates. Preserve exact rational calculations until the final display when the inputs are integer counts.
The record should also identify the scientific target: the complete mean of this table, a spatially restricted comparison or a population-level ecological quantity. Without that target, a standard error can be numerically correct but attached to the wrong question. A concise reproducible ledger is more informative than a large number of simulations whose selection mechanism and population remain unspecified.
CHAPTER 28 OF 33 · Reproduce and continue
28. Repair three common interpretation errors
The first error is that shared site names always imply dependent computational draws. Our independent replacement-label experiment disproves that conditional on a fixed table. The second is that every set of three observed pair counts is a simple random sample of three pair labels. A triangle induced by three sampled sites disproves that claim.
The third error is that enumerating all combinations removes every source of uncertainty. It removes the particular combination-selection uncertainty for the recorded incidence table, while leaving other observation and sampling questions.
To repair an analysis, return to its first random choice and list the possible outcomes. If that choice was a site, a pair label, a region or a detection event, the relevant probability law begins at a different level. This concrete reconstruction usually reveals the assumption more clearly than attaching a generic warning about dependence to the final estimate.
Can two pair draws share a site and still be independent?
Yes, under independent replacement draws from a fixed pair-label list. Their values are deterministic functions of independently sampled labels. A separate random-table or field-sampling model can add other dependence, so state the conditioning.
Is the sample standard deviation the standard error?
Not generally. One describes the spread of selected counts; the other describes the variability of an estimator under its sampling design. Check the software definition and selection rule.
Does site-first sampling always have larger variance?
No. The changed table has zero variance for the three-site mean while a distinct-three-pair sample has positive variance. The ordering depends on the incidence values and design.
Does a complete pair census establish ecological completeness?
No. It completes the pair calculation for the recorded sites. It does not establish complete detection or representative coverage of a larger ecological population.
CHAPTER 30 OF 33 · Reproduce and continue
30. Reproduce the comparison in a short calculation
Start with the six original values and enumerate the thirty-six ordered replacement pairs, fifteen unordered distinct-label pairs and four three-site subsets. For each outcome, calculate the mean and its probability under the declared design.
Then calculate the expectation and the average squared deviation from that expectation. Check 11/24 for two replacement draws, 11/30 for two distinct labels, 11/60 for three distinct labels, and 11/36 for three sampled sites followed by all their pairs.
Finally repeat with the changed six-value table and verify its zero site-sample variance. This sequence tests the full reasoning chain: incidence intersections, sampling unit, allowed outcomes, probabilities and estimator variance. It also provides a compact independent reference for a software implementation without assuming that a smooth simulation histogram certifies the correct sampling design.
CHAPTER 31 OF 33 · Reproduce and continue
31. Estimate a variance when the pair census is unavailable
The exact comparisons above use the complete six-value population because this is a teaching experiment. In a larger table, an analyst may know only the sampled values. Under independent replacement draws, the ordinary sample variance with divisor m−1 is an unbiased estimator of the pair-value variance when m≥2. Dividing that estimate by m estimates the variance of the sample mean under that specific design.
For a uniform sample without replacement, the sample variance estimates the finite-population S² with divisor M−1. Multiplying it by (1−m/M)/m gives the usual design-based variance estimate. These formulas require the corresponding random sampling assumptions; they do not automatically apply to site-induced triangles.
A variance estimate is itself random. With very few samples, it can be zero because selected values happen to agree even though the population varies. For example, selecting AC and AD gives two values of two. That observed zero spread does not establish that every original pair has count two. Keep the sampling uncertainty of the uncertainty estimate in view.
CHAPTER 32 OF 33 · Reproduce and continue
32. Preserve covariance when comparing several zeta orders
Suppose one selected site subset is used to calculate both its mean pair sharing and its triple sharing. Those two estimates are produced by the same random subset, so their joint variation matters when a later calculation combines them.
In the original three-site experiment, the pair means are 2, 2, 4/3 and 2/3, while the corresponding triple intersection counts are 1, 1, 0 and 0. Their expected product is one, their means are 3/2 and 1/2, and their covariance is therefore 1−3/4=1/4.
A difference or an inverse reconstruction using these estimates must retain that covariance rather than propagate each variance as though the estimates were independent. This example does not prescribe a universal covariance for zeta orders. It shows how a common random selection creates a joint law that can be enumerated and checked. Retain the pairing of replicate outputs when estimating that law from simulations.
CHAPTER 33 OF 33 · Reproduce and continue
33. Connect the calculation to the existing guides
Return to the Zeta Diversity subject guide for incidence definitions, sharing orders and ecological limits. The complete-spectrum occupancy reconstruction guide answers a different question: what an exact full set of complete-combination means contains about recorded occupancy classes.
Read The Gold Standard of Education: Learning Transfers into Life for the wider value of original examples, changed-context checks and explicit limits. The How X Works hub connects this method to other disciplines.
Keep a short account of why each sampling probability applies. That explanation is what lets another reader distinguish a computational approximation from a claim about new ecological observations.
A defensible final statement names the fixed record, the target statistic, the randomisation and the uncertainty included. It then identifies what additional evidence would be needed before extending the result beyond that mathematical experiment.