Statistical & Clinical Analysis
Hurlbert’s PIE Formula: Unbiased Probability of Interspecific Encounter & S_PIE
Compute Stuart H. Hurlbert’s (1971) finite-sample-unbiased Probability of Interspecific Encounter PIE = (N / (N - 1)) * (1 - SUM((n_i / N)^2)) and Effective Common Species Count S_PIE = 1 / (1 - PIE).
Why Hurlbert’s finite-sample N / (N - 1) factor fixes Gini-Simpson’s small-quadrat downward bias!
In his landmark 1971 Ecology critique ("The Nonconcept of Species Diversity"), Stuart H. Hurlbert replaced abstract "information bits" with a concrete biological probability: if two individuals are drawn at random WITHOUT replacement from a finite community sample of N = SUM(n_i) organisms, what is the exact probability PIE that they belong to DIFFERENT species?
Because the second draw comes from the remaining N - 1 individuals, exact combinatorics gives PIE = SUM_{i=1..S} (n_i / N) * ((N - n_i) / (N - 1)) = (N / (N - 1)) * (1 - SUM_{i=1..S} (n_i / N)^2)! Unlike naive plug-in 1 - SUM(p_i^2) (which is biased downward by (N - 1) / N in small quadrats), Hurlbert’s PIE is an exact minimum-variance unbiased estimator (MVUE) of true population interspecific encounter probability across ANY sample size N >= 2!
| Small-Sample Quadrat (Counts [6, 5, 5, 4]; N = 20, S = 4) | Naive Plug-In 1 - SUM(p_i^2) | Unbiased Hurlbert PIE | Effective Species S_PIE = 1 / (1 - PIE) |
|---|---|---|---|
| Observed 4-Species Quadrat (N = 20) | 0.7450 (Downward Biased) | 0.7842 | 4.6341 |
| Equal-Count Small Quadrat [5, 5, 5, 5] (N = 20) | 0.7500 | 0.7895 | 4.7500 |
| Monoculture Single Species [20, 0, 0, 0] | 0.0000 | 0.0000 | 1.0000 |
Unbiased Hurlbert Probability of Interspecific Encounter PIE (Without-Replacement)
Evaluates (N / (N - 1)) * (1 - SUMPRODUCT((n_i / N)^2)) across integer counts B2:B5 with N = SUM(B2:B5) >= 2.
=IF(OR(MIN(B2:B5) < 0, SUM(B2:B5) < 2), "Requires Non-Negative Integer Counts & N >= 2", ROUND((SUM(B2:B5) / (SUM(B2:B5) - 1)) * (1 - SUMPRODUCT((B2:B5 / SUM(B2:B5)) ^ 2)), 4))Chase-Knight Effective Number of Common Species S_PIE = 1 / (1 - PIE)
Converts PIE into effective species richness (guarding the pure-singleton case PIE = 1 where every individual is a different species so 1 - PIE = 0).
=IF(OR(MIN(B2:B5) < 0, SUM(B2:B5) < 2), "Requires Non-Negative Integer Counts & N >= 2", IF(MAX(B2:B5) = 1, "All Singletons: Infinite Asymptotic S_PIE", ROUND(1 / (1 - (SUM(B2:B5) / (SUM(B2:B5) - 1)) * (1 - SUMPRODUCT((B2:B5 / SUM(B2:B5)) ^ 2))), 4)))Try the example
Use the original values or change them to test your understanding.