A shared species count can be wrong even when every site has been included. Missed presences remove real overlap; false positive records can create overlap. When observation sensitivity and specificity are independently known, a simple centered correction can recover a pair-sharing total in expectation. Multiplying corrected site indicators works only with an appropriate joint observation model.
This guide derives that statement on a fixed three-site, four-species teaching frame. It checks the bias of raw recorded sharing, the corrected estimator’s exact variance, and the covariance between pairs that reuse a site. The examples are fictional. Their purpose is to expose the assumptions needed before interpreting a corrected number as evidence about a real ecological community.
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Start with the observation model, work the fixed-table example, or audit what the correction can support.
ROUTE 1 · CHAPTERS 1–6
Define the observation correction
Separate true incidence, recorded values and joint error assumptions.
ROUTE 2 · CHAPTERS 7–12
Calculate bias and variance
Work the original species frame and corrected pair products.
ROUTE 3 · CHAPTERS 13–18
Check shared records and transfer
Derive covariance, enumerate outcomes and change calibration.
ROUTE 4 · CHAPTERS 19–24
Protect the evidence boundary
Inspect conditioning, calibration uncertainty, design and sources.
ROUTE 5 · CHAPTERS 25–30
Audit additional observations
Test shortcuts, aggregation, heterogeneous rates and reproduction.
Chapter contents
Define the observation correction · Chapters 1–6
1. Separate the incidence table from its recorded version
2. Define the observation probabilities precisely
3. Derive a corrected single-site indicator
4. State the pair-level independence requirement
Calculate bias and variance · Chapters 7–12
7. Calculate the bias of raw recorded sharing
8. Construct the corrected pair-sharing mean
9. See why an all-negative record still has a contribution
10. Derive each corrected indicator's variance
Check shared records and transfer · Chapters 13–18
13. Account for pairs that reuse a site
14. Finish the exact variance of the mean
15. Verify by enumerating all observation records
16. Show that higher-order raw sharing can be biased upward
Protect the evidence boundary · Chapters 19–24
19. Inspect weak and perfect observation boundaries
20. Keep calibration uncertainty in the analysis
21. Combine site sampling and observation correction only with a joint contract
22. Distinguish an unbiased estimator from a constrained ecological estimate
Audit additional observations · Chapters 25–30
25. See why combining more visits can eventually destroy contrast
26. Permit different rates at different sites
27. Bound a possible joint-error bias from its marginals
28. Distinguish repeated observation precision from ecological replication
29. Questions readers often ask
30. Reproduce the calculation and continue through verified routes
CHAPTER 1 OF 30 · Define the observation correction
1. Separate the incidence table from its recorded version
For a named species at site i, let T_i be its true presence indicator: one means present and zero means absent. Treat these true values as fixed throughout our observation experiment. Let O_i be the binary value recorded by a fallible survey or classification procedure.
The distinction is concrete. A species may be present but missed, producing T_i=1 and O_i=0. A species may be absent but wrongly recorded, producing T_i=0 and O_i=1. Neither kind of error is repaired by counting the same erroneous record more carefully.
We will average over repeated hypothetical observations of the same fixed table. We are not initially sampling sites from a landscape or modeling the ecological process that generated their presences. Those are additional layers of uncertainty and meaning. Keeping the table fixed makes it possible to identify exactly which expectation the correction addresses before adding those layers.
CHAPTER 2 OF 30 · Define the observation correction
2. Define the observation probabilities precisely
Write s_i=P(O_i=1 given T_i=1) for sensitivity and f_i=P(O_i=1 given T_i=0) for the false positive probability. Specificity is 1−f_i, not f_i. Both probabilities refer to a declared survey protocol and species-site observation unit.
Then E(O_i given T_i)=f_i+(s_i−f_i)T_i. If T_i=0 this gives f_i; if T_i=1 it gives s_i. This two-case check is the starting point of the calculation.
Initially regard s_i and f_i as known constants established independently of the current observation. We will later explain why substituting fitted or uncertain probabilities changes the analysis. Assume s_i>f_i for the examples, so a positive record is more likely under presence than absence.
A reported “accuracy” percentage does not supply both of these conditional probabilities. Its value can depend on how many present and absent cases entered a validation set. Preserve the two conditional rates and the validation design.
CHAPTER 3 OF 30 · Define the observation correction
3. Derive a corrected single-site indicator
Set d_i=s_i−f_i and define U_i=(O_i−f_i)/d_i. Substituting the observation mean gives E(U_i given T_i)=T_i whenever d_i is nonzero.
Thus U_i is unbiased for the fixed true indicator under the declared observation law. It is an estimating variable, not a posterior probability of presence. It may lie outside the interval from zero to one.
For the original rates s=4/5 and f=1/10, d=7/10. A negative record gives U=−1/7, while a positive record gives U=9/7. These values look unusual because the estimator must cancel false positive contributions across repeated outcomes.
Replacing the negative value by zero or the value above one by one would change the expectation. A constrained estimate may have a useful role, but its bias and uncertainty need separate treatment. The unbiasedness claim belongs to the unmodified formula and its explicitly stated averaging experiment.
CHAPTER 4 OF 30 · Define the observation correction
4. State the pair-level independence requirement
True sharing by a named species between sites i and j is T_iT_j. A natural corrected contribution is U_iU_j. If the two observation errors are independent conditional on the fixed true table and calibration probabilities, E(U_iU_j)=E(U_i)E(U_j)=T_iT_j.
The conditioning is essential. We are not assuming that ecological presences at different sites are independent; the whole true table is fixed. We are assuming a particular relationship between the observation errors after those presences have been specified.
A shared observer, laboratory batch, reference library or contamination pathway can make that assumption fail. Simply assigning each site a correct marginal sensitivity and specificity does not determine their joint error law.
The pair correction is therefore more than two single-site calibrations placed next to each other. It requires enough observation-process information to justify multiplying their expectations. We will calculate the additional bias term when that relationship does not hold.
CHAPTER 5 OF 30 · Define the observation correction
5. Work all four recorded pair outcomes
With the original common rates, the product U_iU_j has only four possible recorded outcomes. Keep their signs and scales visible before averaging.
| Recorded pair O_i,O_j | Corrected contribution U_iU_j |
|---|---|
| 0,0 | 1/49 |
| 1,0 | −9/49 |
| 0,1 | −9/49 |
| 1,1 | 81/49 |
If both true presences are one, the independent observation probabilities are 1/25 for 00, 4/25 for each discordant outcome, and 16/25 for 11. The weighted average is [1−36−36+1296]/1225=1.
If both true presences are zero, the probabilities are 81/100, 9/100, 9/100 and 1/100. The weighted average becomes [81−81−81+81]/4900=0.
A mixed true pair also has mean zero. It may yield a positive corrected contribution in an individual repetition, but the complete signed average cancels. Check the whole outcome law rather than interpreting a single corrected product as a verified shared presence.
CHAPTER 6 OF 30 · Define the observation correction
6. Define the original three-site species frame
Use four named teaching species r, s, t and u. Their true incidence sets are A={r,s}, B={r,t}, and C={s,t,u}. The candidate species list is fixed in advance and contains exactly these four taxa.
Pairs AB, AC and BC each truly share one species: respectively r, s and t. Their equally weighted mean pair-sharing count is therefore ζ₂=1. No species is truly present at all three sites, so ζ₃=0.
Every one of the twelve species-site observations uses sensitivity 4/5 and false positive probability 1/10. For the exact variance calculations, assume all twelve errors are independent conditional on the true table. This includes independence between species as well as between sites.
The independent-error model is deliberately simple enough for exhaustive verification. It is not an assertion that actual field surveys behave this way. A real analysis needs evidence about which errors share observers, conditions, classifications or samples before adopting the same covariance structure.
CHAPTER 7 OF 30 · Calculate bias and variance
7. Calculate the bias of raw recorded sharing
For a given species, the probability that both site records are positive is the product of their positive-record probabilities under our independence assumption. Those probabilities are 4/5 for a true presence and 1/10 for a true absence.
For pair AB, species r contributes 16/25, s contributes 2/25, t contributes 2/25, and u contributes 1/100. The expected raw recorded shared count is 81/100, although the true count is one.
For AC and BC the expected raw sharing is 22/25 each. Their different values arise because u is truly present at C and absent at A and B. Averaging all three raw counts gives 257/300, approximately 0.8567.
False negatives dominate this particular pair-sharing expectation, but that is a consequence of this frame and these rates. False positives can raise other sharing quantities, as the all-three-site calculation below demonstrates. There is no universal direction of bias implied by the phrase “imperfect detection.”
CHAPTER 8 OF 30 · Calculate bias and variance
8. Construct the corrected pair-sharing mean
For each pair ij, sum corrected species contributions: V̂_ij=Σ_a U_ai U_aj, where a ranges over the four predeclared taxa. Define ζ̂₂=(V̂_AB+V̂_AC+V̂_BC)/3.
Linearity of expectation and the pair calculation give E(V̂_ij)=Σ_a T_aiT_aj. Therefore each corrected pair count has mean one, and E(ζ̂₂)=1.
The denominator is the fixed number of site pairs, three. It is not the number of pairs whose records happened to show at least one species. Dropping apparently empty pairs would condition the data on an observation outcome and change the estimator law.
The numerator must also retain every species in the fixed candidate list, including taxa recorded negative everywhere in a particular repetition. Selecting the list after seeing detections would change the calculation. The species frame and site frame are part of the definition, not optional details to add after estimating the result.
CHAPTER 9 OF 30 · Calculate bias and variance
9. See why an all-negative record still has a contribution
If all twelve records happen to be zero, every U equals −1/7. Each species contributes 1/49 to every pair, so ζ̂₂=4/49 in that repetition.
This is not a claim that four species were observed sharing the sites. It is one value of an estimator whose expectation is evaluated over every possible record generated by the stipulated error process.
If the species list were instead constructed as the observed union, the all-negative record would generate an empty list and a different value, zero. Under our finite-frame calculation that post-observation exclusion removes contributions needed by the unbiasedness proof.
In a real biodiversity survey, a complete candidate list may be unavailable. The formula then does not magically discover every unrecorded species. Its finite-list target must be stated honestly, and any extrapolation to an unknown wider species pool requires a separate sampling or occupancy model. A simple algebraic correction cannot replace missing frame information.
CHAPTER 10 OF 30 · Calculate bias and variance
10. Derive each corrected indicator's variance
Conditional on a fixed true indicator, O is Bernoulli with probability p=s when present and p=f when absent. Since U=(O−f)/d, its variance is p(1−p)/d².
For the original rates, a true absence has variance 9/49 and a true presence has variance 16/49. These are variances of corrected estimating variables, not ecological variability in the fixed T values.
Their second moments follow from variance plus squared mean. For a true absence E(U²)=9/49; for a true presence E(U²)=1+16/49=65/49.
This distinction makes the pair variance easy to calculate. Independence allows the second moment of a product to factor into the two second moments. It does not allow a product variance to be computed merely by multiplying the two variances when either factor has nonzero mean. Keeping both means and second moments avoids that common shortcut error.
CHAPTER 11 OF 30 · Calculate bias and variance
11. Derive the three types of pair-product variance
If a species is truly present at both sites, U_iU_j has mean one and second moment (65/49)². Its variance is therefore 1824/2401.
For a mixed true pair, the mean product is zero. Its variance equals its second moment, (65/49)(9/49)=585/2401. For a true-absence pair, both means are zero and the variance is (9/49)²=81/2401.
These formulas depend on conditional observation independence. They account for a product of noisy corrected values, not a Bernoulli shared-presence indicator. The product can be negative or exceed one, so a Bernoulli variance formula would be inappropriate.
Under the additional independence between species, sum these variances to obtain a pair-count variance. If taxa are confused with one another, species-level independence may fail too. For example, incorrectly assigning one specimen to one taxon may remove its assignment to another, creating a covariance that the current sum would omit.
CHAPTER 12 OF 30 · Calculate bias and variance
12. Sum the pair-count variances on the original frame
For AB there is one true 11 species, two mixed species and one true 00 species. Thus Var(V̂_AB)=[1824+2×585+81]/2401=3075/2401.
For AC, species s is the true shared species; r, t and u are mixed pairs. Therefore Var(V̂_AC)=[1824+3×585]/2401=3579/2401. BC has the same classification counts and the same variance.
The three means are all one, but their variances are not all equal. True sharing alone does not determine observation-error variance: the surrounding pattern of species presences and absences matters as well.
This is one reason to retain the incidence table rather than only its final ζ₂ mean. Two ecological tables can have the same average sharing but different vulnerability to a specified observation process. A variance estimate copied from one table into another needs its own justification even when the point target agrees.
CHAPTER 13 OF 30 · Check shared records and transfer
13. Account for pairs that reuse a site
Pairs AB and AC both use the corrected observation U_A for the same species. Their covariance is not generally zero, even though all distinct species-site observation errors are independent.
For one species, independence gives Cov(U_AU_B,U_AU_C)=Var(U_A)T_BT_C. The shared factor appears squared in the product expectation, producing the extra variance term.
On our frame, the only species present at both B and C is t. It is absent at A, so the contribution is Var(U_At)=9/49. All other species have T_BT_C=0. Hence Cov(V̂_AB,V̂_AC)=9/49.
The other two pair-count covariances are also 9/49: species s supplies the AB–BC contribution through B, and species r supplies the AC–BC contribution through C. Reusing one noisy record creates dependence between derived pair counts even when the original individual records were generated independently.
CHAPTER 14 OF 30 · Check shared records and transfer
14. Finish the exact variance of the mean
The variance of a sum includes each covariance twice. Therefore Var(ζ̂₂)=[3075+3579+3579+2×3×441]/(9×2401)=1431/2401, approximately 0.5960.
Here 441/2401 equals 9/49, so the numerator places every term over the same denominator before division by the nine from averaging three pairs.
Ignoring pair covariances would give 1137/2401, approximately 0.4736. That understates the observation variance for this exact frame and model. Calling the three pair counts independent would introduce a new assumption contradicted by their shared data.
This variance is conditional on fixed true incidence and known calibration rates. It is not a confidence interval, and its square root is not a complete ecological uncertainty assessment. Unknown calibration, sampled sites, seasonal changes or dependence between species introduce further uncertainty. The calculation is useful because it tells us exactly what it has included and exposes the terms that a naive pair-only account would miss.
CHAPTER 15 OF 30 · Check shared records and transfer
15. Verify by enumerating all observation records
There are twelve binary observations, so the independent model has 2¹²=4096 possible recorded tables. Each receives a probability equal to the product of its twelve Bernoulli outcome probabilities, using 4/5 or 1/10 according to the corresponding true incidence.
For each record, compute the twelve U values, the three corrected pair counts and their mean. Sum probability times the estimator to verify mean one. Sum probability times its squared deviation from one to verify variance 1431/2401.
The same enumeration verifies the raw recorded mean 257/300. It supplies a separate route to the covariance-derived result and can reveal indexing or denominator mistakes in code.
This exhaustive calculation is feasible only because the teaching frame is small. A larger analysis may need analytic formulas or simulation, but its checking cases should still be small enough to reproduce independently. A program returning many decimal places is less informative than agreement between two differently organized derivations on an explicit frame.
CHAPTER 16 OF 30 · Check shared records and transfer
16. Show that higher-order raw sharing can be biased upward
No species in the original table occupies all three sites. The true all-three shared count is zero. Nevertheless, raw triple-positive records can occur through a false positive at a site where a species is absent.
Species r, s and t each have two true presences and one absence. Each contributes probability (4/5)²(1/10)=8/125. Species u has one presence and two absences, contributing (4/5)(1/10)²=1/125. Their sum is 25/125=1/5.
Thus raw ζ₂ has downward bias in this example while raw ζ₃ has upward bias. A single general assertion that imperfect detection makes the whole decline curve smaller would be wrong.
Under the stronger independence across all three observations, the corrected triple product U_AU_BU_C has expectation T_AT_BT_C. Its sum therefore has mean zero on this frame. Pairwise independence alone would not justify factoring a three-way product expectation; the observation assumptions must match the sharing order being estimated.
CHAPTER 17 OF 30 · Check shared records and transfer
17. Derive the bias when observation errors are correlated
Allow a conditional covariance c_ij=Cov(O_i,O_j given the fixed true table), while retaining the correct marginal rates. Centering and scaling give Cov(U_i,U_j)=c_ij/(d_id_j).
Therefore E(U_iU_j)=T_iT_j+c_ij/(d_id_j). The correction remains unbiased only when the additional term vanishes or is explicitly modeled and removed using justified joint information.
For the original common d=7/10, a conditional covariance of 1/100 would contribute 1/49 to the expected pair product. That covariance must be feasible under the relevant Bernoulli marginals; it is not an unconstrained parameter one can choose arbitrarily for every state.
The calculation identifies what extra validation is needed. Correct marginal sensitivity and specificity do not certify joint error independence. Shared contamination or a common observer effect may require paired validation, batch effects or another joint model. Increasing the number of sites does not automatically eliminate a systematic shared observation error.
CHAPTER 18 OF 30 · Check shared records and transfer
18. Transfer to a different calibrated observation rule
Change the calibration to sensitivity 9/10 and false positive probability 1/5. The new difference d is still 7/10, but the corrected values become U(0)=−2/7 and U(1)=8/7.
Pair contributions are now 4/49 for 00, −16/49 for either discordant outcome and 64/49 for 11. They are not the earlier 1/49, −9/49 and 81/49, despite the unchanged denominator d.
For a true shared pair, the outcome probabilities are 1/100, 9/100, 9/100 and 81/100. Weighting the new contributions gives [4−144−144+5184]/4900=1, as required.
This transfer check separates a reusable derivation from memorized weights. Sensitivity and false positive probability enter the centering as well as their difference. When either calibration changes, reconstruct both the allowed U values and the observation probabilities before recomputing bias or variance. A familiar denominator does not establish an unchanged estimator law.
CHAPTER 19 OF 30 · Protect the evidence boundary
19. Inspect weak and perfect observation boundaries
If s=f, the recorded indicator has the same distribution under true presence and absence. The denominator d is zero, and the proposed correction is undefined. The observation supplies no distinction between those two states under the specified model.
As s approaches f, the corrected values and their variances can become large. This is a conditioning problem: a small difference in observed behavior is being amplified to distinguish the latent states. More precise arithmetic cannot improve the informational contrast of the observation procedure.
At the perfect-observation boundary s=1 and f=0, U=O=T. Every variance in the fixed-table observation experiment is zero, and the corrected sharing equals the true sharing exactly.
These boundaries are useful checks on formulas. They do not imply that every apparently weak or perfect empirical calibration should be taken literally. Finite validation data carry uncertainty, particularly when a study observes no errors in a limited sample. The mathematical boundary and evidence for reaching it are separate questions.
CHAPTER 20 OF 30 · Protect the evidence boundary
20. Keep calibration uncertainty in the analysis
The unbiasedness proof treats s and f as known. In practice they may be estimated from a validation sample. Substituting estimates into (O−f)/(s−f) produces a ratio involving random quantities; its expectation need not equal the ratio obtained by substituting the true rates.
Even a separate validation study does not make the estimates exact. Its uncertainty must be propagated through the centering, denominators and product structure. If the same estimated rates are reused across many sites, they can induce dependence between corrected values that persists even when the raw records were independently collected.
A bootstrap or simulation must reproduce the validation and survey designs it is intended to approximate. Resampling only the ecological table while treating estimated calibration as fixed addresses a different conditional uncertainty question.
Report which quantities were validated, the cases included, how verification was selected and whether the protocol matches the present survey. A correction derived under one observer or habitat may fail when those rates are transferred into a different observation setting.
CHAPTER 21 OF 30 · Protect the evidence boundary
21. Combine site sampling and observation correction only with a joint contract
Suppose sites are additionally selected from a finite frame with known joint pair inclusion probabilities π_ij. If the selection design and observation law justify the required conditional expectations, one can estimate a frame pair total with contributions I_ij V̂_ij/π_ij, where I_ij indicates that the pair was observed.
Then divide by the fixed number of frame pairs if the target is their equally weighted mean. The selection weights address which pairs entered the data; the U corrections address errors in their incidence records. Neither repairs the other’s missing assumptions.
The unequal-inclusion pair-sharing guide derives the design-weighting layer separately. Combining the two requires conditional observation rates appropriate to selected sites and a joint variance account for both mechanisms.
Do not simply multiply two familiar correction factors and assume all covariance terms have been handled. The target, conditioning, denominators and dependence structure must remain explicit across the complete estimator.
CHAPTER 22 OF 30 · Protect the evidence boundary
22. Distinguish an unbiased estimator from a constrained ecological estimate
True shared-species counts are nonnegative integers. Corrected estimating variables can be fractional, negative or above a count that seems possible in one recorded table. That does not contradict unbiasedness over repeated observations under the specified model.
A constrained likelihood or Bayesian model may enforce nonnegative latent counts or incidence states. Such an approach answers its question through a full model and may be preferable for some decisions. Its result should not be described as the same unmodified estimator if its constraints or prior information alter the calculation.
Clipping a negative ζ̂₂ to zero can make a presentation look more natural while changing its expectation and sampling law. If a constrained summary is reported, retain the unconstrained calculation when useful for checking, explain the transformation and assess its uncertainty under the actual procedure.
The practical aim is an honest inference, not preserving one estimator at every cost. Match the method to the evidence and decision, while keeping its inferential guarantees narrower than the assumptions can support.
CHAPTER 23 OF 30 · Protect the evidence boundary
23. Link the correction to the broader evidence framework
Royle and Link’s original paper on false positive and false negative occupancy errors establishes why both observation-error directions belong in an occupancy model. The USGS author publication record provides its bibliographic and research context.
The finite-table centered-indicator derivation and numerical frame here are independent teaching calculations. They do not reproduce the paper’s fitted model or empirical study. Use the sources for the observation-model framework and this guide for the narrower conditional pair-sharing proof.
The existing multievent state-uncertainty guide develops hidden ecological states and observed events over time. It complements this fixed-incidence correction without making the assumptions interchangeable. Survival, state transitions and identification errors introduce additional structure absent from our deliberately fixed frame.
CHAPTER 24 OF 30 · Protect the evidence boundary
24. Check a false-negative-only shortcut on the same frame
Suppose an analyst divides every observed positive indicator by sensitivity 4/5 but ignores false positives. The resulting variable W=O/(4/5) has expectation one under true presence and 1/8 under true absence. Its expectation is not the true indicator for both states.
Under independence, the expected pair product is therefore one for a true shared species, 1/8 for a mixed pair and 1/64 for a true-absence pair. For AB, the expected shortcut count is 1+1/8+1/8+1/64=81/64. For AC and BC it is 1+3/8=11/8.
The expected shortcut mean across the three pairs is 257/192, approximately 1.3385. Raw recorded sharing was downward biased, but this incomplete correction overshoots the true target one. Applying a correction is not itself evidence that bias has been reduced.
The counterexample isolates the omitted mechanism. Dividing by sensitivity addresses missed presences only under a zero-false-positive model. When false positives occur, subtracting their baseline is part of the derivation. Always check a proposed correction under both true absence and true presence before multiplying it across sites.
CHAPTER 25 OF 30 · Audit additional observations
25. See why combining more visits can eventually destroy contrast
Suppose R visits have independent errors conditional on the same unchanged true state, and the site is recorded positive if any visit is positive. The effective sensitivity is s_R=1−(1−s)^R, while the effective false positive probability is f_R=1−(1−f)^R.
For the original rates and R=2, s_R=24/25 and f_R=19/100. Their difference is 77/100. The centered correction must use these combined-record rates, not the rates for one visit.
With positive f, however, both effective rates approach one as R grows. Their difference d_R=(1−f)^R−(1−s)^R approaches zero. An indefinitely long sequence summarized only as “any positive” eventually becomes nearly always positive under either true state. The summary loses contrast even though the underlying repeated records contain more observations.
This is an information-loss calculation for a particular aggregation rule. It does not say that repeated surveys are generally harmful. Retaining complete detection histories supports other models and checks. It says that more data compressed through an unsuitable rule do not automatically produce a better-conditioned binary correction. State whether rates describe a single visit, a combined rule or a fitted history model.
CHAPTER 26 OF 30 · Audit additional observations
26. Permit different rates at different sites
The correction does not require common calibration at every site. For example, let site A have sensitivity 9/10 and false positive probability 1/10. Then U_A(0)=−1/8 and U_A(1)=9/8.
Let site B instead have sensitivity 4/5 and false positive probability 1/5. Then U_B(0)=−1/3 and U_B(1)=4/3. Under true presence at both sites and conditional error independence, each corrected factor has mean one, so their product also has mean one.
The four product values now differ from both earlier tables. Their observation probabilities must also be calculated from the site-specific rates. Sharing one formula does not imply sharing one set of numerical weights.
This example helps audit implementation. Store rates with the correct species, site, protocol and observation occasion rather than joining one global calibration to every row without justification. A shuffled rate column can leave valid-looking corrected numbers while breaking the expectation identity. A small deliberately heterogeneous test case is more informative for that error than a test where every row uses the same probabilities.
CHAPTER 27 OF 30 · Audit additional observations
27. Bound a possible joint-error bias from its marginals
For two Bernoulli observations with fixed positive-record probabilities p and q, their joint positive probability lies between max(0,p+q−1) and min(p,q). Consequently their covariance lies between those bounds minus pq.
For a species truly absent at both original sites, p=q=1/10. The allowable covariance ranges from −1/100 to 9/100. Dividing by d²=49/100 gives a possible corrected-product expectation between −1/49 and 9/49 if no joint information beyond those marginals is supplied.
Independence selects zero within that interval. The correct single-site calibrations alone do not select it. This is a useful sensitivity calculation: it shows the amount of pair bias that a feasible joint law can introduce before any field data are analyzed.
The bounds are conditional on the stated true-state case. Since the true states are unknown in an actual survey, a complete inference must account for the corresponding cases and any shared dependence parameters. The interval is not a confidence interval for a measured community. Its role is to expose an assumption hidden by marginal validation, using only the elementary constraints that a joint probability must satisfy.
CHAPTER 28 OF 30 · Audit additional observations
28. Distinguish repeated observation precision from ecological replication
If m independent complete observation repetitions were obtained on exactly the same fixed frame under known rates, averaging their corrected ζ₂ estimates would have variance 1431/(2401m). For m=4 the variance would be 1431/9604, one quarter of the single-repetition value.
That result requires independence between complete repetitions as well as the within-repetition assumptions already stated. Reusing one uncertain calibration estimate, one observer-specific bias or one contaminated reference can leave a shared error component that does not decrease through the same division by m.
Repeated observations also do not create additional ecological sites. They improve information about the declared fixed table under an observation model; they do not establish that the table represents a wider landscape or another season. If ecological presences change between visits, the fixed-state target itself needs revision.
A clear study report therefore distinguishes observation replication, calibration replication and ecological sampling. Each can be valuable, but they support different precision and transfer claims. Increasing a count labeled “sample size” without identifying what was repeated does not establish the uncertainty reduction being claimed.
Is U a probability of presence?
No. Its expectation equals a fixed indicator under the stated repeated-observation law. Its possible values can be negative or above one.
Does knowing sensitivity alone repair false positives?
No. The centering uses the false positive probability too. A false-negative-only correction assumes a different observation model.
Do independent species-site observations make all pair estimates independent?
No. Derived pairs can reuse the same corrected observation. The explicit shared-site covariance is nonzero in our example.
Can this identify species never included in the candidate list?
No. The finite-list target is explicit. An unknown species pool requires further information and modeling.
Can an unbiased correction have an uncertain value?
Yes. Unbiasedness concerns an expectation. Our exact observation variance is substantial even though the estimator has the correct mean.
Can I clip a negative answer to zero?
You can define that new procedure, but its expectation and uncertainty must be assessed separately. It does not inherit the unmodified estimator’s proof automatically.
CHAPTER 30 OF 30 · Audit additional observations
30. Reproduce the calculation and continue through verified routes
Write the true four-species incidence table, then derive E(O|T), U and the four possible pair products. Verify all four true-state pair cases. Calculate the raw means for AB, AC and BC before constructing the corrected mean.
Next derive single-site second moments, pair-product variances and the three shared-site covariance terms. Check the final variance against exhaustive enumeration of 4096 observation records. Repeat the changed calibration without reusing the original product weights.
Return to the Zeta Diversity subject guide for incidence definitions and ecological interpretation. The complete-spectrum occupancy guide explains what an exact recorded spectrum contains; the present correction explains why recorded incidence and true incidence are different targets.
Connect those checks through the Gold Standard of Education parent and the How X Works hub. A useful final statement names the frame, the observation law, the conditional independence assumption, the estimator and the uncertainty that remains outside its calculation.
Before applying the method to a new dataset, ask another reader to reconstruct one corrected species-pair contribution from the original record and its calibration row. Then ask them to identify every other pair that reuses either observation. This short audit connects the numerical weight to its source and the covariance to the actual data reuse. Agreement on a final total alone can conceal a wrong join or an independence assumption applied at the wrong level.