Founder Equivalents:
Founder equivalents estimates the expected number of equally contributing founders that would be required to produce the observed genetic diversity in the current population.

f_e = 1 / Σ(p_i²)

Where p_i is the proportion of the genes of the living, descendant population contributed by founder i.

Founder Genome Equivalents:
Founder genome equivalents estimates the expected number of equally contributing founders that would be required to produce the observed genetic diversity in the current population if no alleles had been lost to drift.

f_g = 1 / Σ(p_i²/r_i)

Where p_i is the proportion of the genes of the living, descendant population contributed by founder i, and r_i is the allelic retention for founder i (estimated by gene drop).

Because the retention values r_i are estimated by gene drop, f_g is a Monte Carlo estimate reported with a sampling standard error (shown as the +/- next to FG); that standard error shrinks roughly in proportion to one over the square root of the number of gene-drop iterations. At small iteration counts f_g also carries a slight finite-sample bias that the standard error does not capture.

Gene Diversity (GD):
Gene diversity is the expected heterozygosity retained relative to the founding gene pool -- the fraction of the founders' allelic variation that still survives in the current population.

GD = 1 − 1 / (2 × f_g)

Where f_g is the founder genome equivalents above. GD is a proportion, not a count of individuals: 0 means none of the founding diversity is retained, and it approaches (but never reaches) 1 as f_g grows. It is computed over the same analysis set as the founder statistics above.

Gene diversity is derived here from the founder genome equivalents above, following Lacy RC. 1989. Analysis of founder representation in pedigrees: founder equivalents and founder genome equivalents. Zoo Biol 8:111-123.

Sex-Ratio Effective Population Size (Sex-Ratio Ne):
The sex-ratio effective size is the effective population size implied by an unequal breeding sex ratio -- the diversity lost when many of one sex are bred to few of the other, as in a harem colony.

N_e = 4 × N_m × N_f / (N_m + N_f)

Where N_m and N_f are the numbers of current living breeders known to be male and female. It equals the census count when the sexes are balanced and falls toward four times the rarer sex as the ratio skews; it is 0 when either breeding sex is absent. This estimate is taken over the current living breeders -- the living animals that appear as a sire or dam -- which is a different (usually smaller) population than the analysis set summarized by the founder statistics above.

Crow JF, Kimura M. 1970. An Introduction to Population Genetics Theory. Harper and Row, New York.

Variance Effective Population Size (Variance Ne):
The variance effective size measures the diversity lost to unequal family sizes -- usually the dominant reducer of effective size in a managed colony, where a few breeders produce most of the offspring.

N_e = (N × k − 1) / (k − 1 + V / k)

Where N is the number of current living breeders (of every sex), k is their mean number of lifetime offspring, and V is the variance of those offspring counts. This is the general Crow & Kimura (1970) form; it makes no constant-size assumption and reduces to the classic (4N − 2) / (V + 2) when the mean family size is at replacement (k = 2). It is undefined -- shown as N/A -- when fewer than two living breeders are present.

Crow JF, Kimura M. 1970. An Introduction to Population Genetics Theory. Harper and Row, New York.

Both effective-size estimates idealize a Wright-Fisher population (constant size, discrete generations, random union of gametes), so read each as an index of one source of diversity loss rather than a literal head count. They are two of several ways to estimate effective size; an estimate based on the rate of increase in coancestry may be added in a future version.

Genome Uniqueness (GU):
Genome uniqueness estimates, for each animal, the percentage of its simulated allele copies that are rare founder alleles (carried by at most a chosen threshold number of animals in all, counting the animal itself; the application offers 1-5, default 4) -- animals with high genome uniqueness are more likely to carry genetic material that would otherwise be lost from the colony.

GU_i = (rare_i / (2 × iterations)) × 100

Where rare_i is the number of simulated allele copies for animal i that are rare, summed across all gene-drop iterations, and iterations is the number of gene-drop simulations run. Because it is a Monte Carlo estimate, GU is reported with a sampling standard error (the guSE column) that shrinks roughly in proportion to one over the square root of the number of iterations.

Genome uniqueness is calculated using a gene-drop simulation according to MacCluer JW, et al. (1986) and Ballou JD, Lacy RC. (1995).

The distribution table also reports GU's skewness and kurtosis (excess) across the colony -- skewness indicates whether a minority of animals with unusually high GU (positive) or low GU (negative) stretch one tail of the distribution; kurtosis (excess) indicates whether extreme values are more common (positive) or less common (negative) than a normal distribution, which reads 0. Both are the bias-adjusted Fisher-Pearson estimators of Joanes DN, Gill CA. (1998).

Mean Kinship (MK):
Mean kinship summarizes how related an animal is to the rest of the living colony -- animals with low mean kinship carry genetic material that is comparatively underrepresented, making them of higher value for breeding.

MK_i = Σf_ij / N

Where f_ij is the pairwise kinship coefficient between animal i and every animal j (including i itself), and N is the number of animals in the kinship matrix. Ballou JD, Lacy RC. 1995. Identifying genetically important individuals for management of genetic variation in pedigreed populations, p 77-111. In: Ballou JD, Gilpin M, Foose TJ, editors. Population management for survival and recovery. New York (NY): Columbia University Press.

The distribution table also reports MK's skewness and kurtosis (excess) across the colony -- skewness indicates whether a minority of animals with unusually high MK (positive) or low MK (negative) stretch one tail of the distribution; kurtosis (excess) indicates whether extreme values are more common (positive) or less common (negative) than a normal distribution, which reads 0. Both are the bias-adjusted Fisher-Pearson estimators of Joanes DN, Gill CA. (1998).

Marker-Based Kinship (KING-robust):
Marker-based kinship estimates relatedness directly from a marker genotype panel, independent of the recorded pedigree -- an independent check on the pedigree-implied Mean Kinship (MK) above, not a replacement for it. It uses the "KING-robust" estimator, which is robust to population substructure and requires no assumed reference population.

φ_ij = 0.5 + (2N_AaAa - 4N_AAaa - N_Aa⁽ⁱ⁾ - N_Aa^(j)) / (4 min(N_Aa⁽ⁱ⁾, N_Aa^(j)))

Where, over the marker loci genotyped in both animals i and j, N_AaAa is the number of loci at which both are heterozygous, N_AAaa is the number of loci at which they have opposite homozygous genotypes, and N_Aa⁽ⁱ⁾/ N_Aa^(j) are each animal's own heterozygous-locus count. The estimator requires biallelic markers and is not bounded below by zero -- a negative value indicates more divergent ancestry than the reference sample, not an error. Manichaikul A, Mychaleckyj JC, Rich SS, Daly K, Sale M, Chen WM. 2010. Robust relationship inference in genome-wide association studies. Bioinformatics 26(22):2867-2873.

Heterozygosity (Observed vs. Expected):
The heterozygosity diagnostic compares each animal's observed heterozygosity (H_o) -- the fraction of its genotyped marker loci at which it carries two different alleles -- against the population's expected heterozygosity (H_e), Nei's gene diversity computed from marker allele frequencies. A population-wide H_o below H_e suggests inbreeding or population substructure; H_o above H_e suggests outbreeding or negative assortative mating.

H_e = 1 - Σp_i²

Where p_i is the population frequency of allele i at a marker locus, computed from all genotyped individuals; H_e is computed per locus and averaged across loci for a population-wide summary. Nei M. 1973. Analysis of gene diversity in subdivided populations. Proceedings of the National Academy of Sciences USA 70(12):3321-3323.

Mendelian-Exclusion Parentage Verification:
This diagnostic cross-references the pedigree's recorded dam and sire against an animal's marker genotype, locus by locus, and flags a recorded parent whose genotype evidence contradicts simple Mendelian inheritance -- a direct check on the pedigree's own recorded parentage, targeting real-world dam/sire misidentification. A locus is an "exclusion" when the animal and the candidate parent are both homozygous for a different allele (no shared allele is possible); the pair's exclusion count is compared to a tunable tolerance (default: flag only at 3 or more inconsistent loci, tolerating up to 2 -- ordinary genotyping error or mutation can produce an isolated mismatch even for a true parent-offspring pair). Cifuentes LO, Martinez EH, Acuna MP, Jonquera HG. 2006. Probability of exclusion in paternity testing: time to reassess. Journal of Forensic Sciences 51(2):349-350. de Groot NG, de Vos-Rouweler AJM, Heijmans CMC, et al. 2025. Genetic Conservation and Population Management of Non-Human Primates: Parentage Determination Using Seven Microsatellite-Based Multiplexes. Ecology and Evolution 15(4):e71216.

Cross-Center Differentiation (Hudson's Fst):
This statistic answers a different question than the marker-based kinship above: not how related two animals are, but how differentiated two centers' allele frequencies are at the same marker loci -- a population-level, between-dataset comparison, not a per-animal one. Hudson's estimator was chosen over the more commonly-cited Weir & Cockerham (1984) estimator because it is not biased by an imbalance between the two centers' sample sizes, which real per-center marker panels are expected to have.

F_ST = [(p₁ - p₂)² - p₁(1-p₁)/(n₁-1) - p₂(1-p₂)/(n₂-1)] / [p₁(1-p₂) + p₂(1-p₁)]

Where p₁/p₂ are the two centers' sample allele frequencies at a shared marker locus and n₁/n₂ are their allele counts. Per-locus values are pooled into a single summary as a ratio of sums, not a mean of per-locus ratios -- averaging per-locus ratios directly is a documented source of bias for this statistic. A value near zero indicates negligible differentiation at that locus; the statistic can range up to 1, and can come out slightly negative due to sampling noise (not clamped to zero). Hudson RR, Slatkin M, Maddison WP. 1992. Estimation of levels of gene flow from DNA sequence data. Genetics 132(2):583-589. Bhatia G, Patterson N, Sankararaman S, Price AL. 2013. Estimating and interpreting FST: The impact of rare variants. Genome Research 23(9):1514-1521.

Candidate Parent Assignment (Likelihood-Based, LOD):
For each recorded parent the Mendelian-Exclusion Parentage Verification diagnostic above flags as inconsistent, this ranks candidate replacement parents by a multilocus likelihood-ratio (LOD) score: for each shared marker locus, the log-ratio of the probability of the animal's genotype under "this candidate is the true parent" versus "this candidate is an unrelated individual drawn from the colony," summed across all jointly-genotyped loci. Population allele frequencies are computed colony-wide from the uploaded genotype file. This is report-only -- it never rewrites the pedigree's recorded dam/sire -- and requires curator review before acting on any result. A candidate scored on fewer than a minimum number of loci is flagged as low-power rather than ranked as if fully informative, and no simulation-calibrated percentage confidence is reported, since this package's realistic marker-panel sizes sit inside the range the underlying literature itself documents as underpowered for that kind of claim. Meagher TR, Thompson E. 1986. The relationship between single parent and parent pair genetic likelihoods in genealogy reconstruction. Theoretical Population Biology 29(1):87-106. Marshall TC, Slate J, Kruuk LEB, Pemberton JM. 1998. Statistical confidence for likelihood-based paternity inference in natural populations. Molecular Ecology 7(5):639-655. Kalinowski ST, Taper ML, Marshall TC. 2007. Revising how the computer program CERVUS accommodates genotyping error increases success in paternity assignment. Molecular Ecology 16(5):1099-1106.

Realized Relatedness Variance:
Pedigree kinship gives the expected relatedness between two animals, but the actual fraction of genome they share identical-by-descent (IBD) varies around that expectation because of Mendelian sampling and linkage -- a finite number of chromosomes and a finite genetic map length. For Parent-Offspring, Full-Sibling, and Half-Sibling pairs, this estimates the variance of that realized relatedness around the pedigree-expected value, so two pairs in the same relationship category are not assumed equally related just because their pedigree relationship coefficient is the same.

Var_FS = 2φ₂(l) - φ₁(l)    Var_HS = φ₂(l) - ½φ₁(l)

Where l is a chromosome's genetic map length in Morgans and φ_n is Hill & Weir's closed-form integration function; multiple chromosomes are combined via the equal-length approximation to their exact weighted-sum combination rule. Parent-Offspring pairs have zero variance -- exactly one allele is always shared IBD, regardless of map length. Hill WG, Weir BS. 2011. Variation in actual relationship as a consequence of Mendelian sampling and linkage. Genetics Research 93(1):47-64.

LD Block Statistic:
A descriptive, exploratory-use linkage-disequilibrium (LD) statistic for pairs of loci on the same chromosome -- deliberately the secondary statistic alongside Realized Relatedness Variance above, not a rigorous, pedigree-aware LD-block method. Classical LD theory assumes a sample drawn at random from a population in linkage equilibrium; a captive pedigreed colony genuinely violates that assumption, and no method exists that is both pedigree-aware and capable of handling multiallelic marker panels (e.g. microsatellites). Interpret with caution.

D′ = ∑_i∑_j p_iq_j |D′_ij|    r² = ∑_i∑_j D_ij² / (p_iq_j)

Where the sums run over every pair of alleles (i at the first locus, j at the second), p_i/q_j are their frequencies, and D_ij is estimated by a maximum-likelihood two-locus phase-frequency estimator applied to unphased genotype data (a double-heterozygote's phase cannot otherwise be resolved). D′ is Hedrick's (1987) frequency-weighted average of the standardized per-allele-pair disequilibrium; r² reduces exactly to the classic biallelic r² when both loci happen to be biallelic. Excoffier L, Slatkin M. 1995. Maximum-likelihood estimation of molecular haplotype frequencies in a diploid population. Molecular Biology and Evolution 12(5):921-927. Hedrick PW. 1987. Gametic disequilibrium measures: proceed with caution. Genetics 117(2):331-341. Weir BS. 1996. Genetic Data Analysis II: Methods for Discrete Population Genetic Data. Sinauer Associates.

Genomic Runs of Homozygosity (F_ROH):
Unlike the pedigree-based kinship/founder statistics above, this estimates an animal's own inbreeding directly from its sequence-scale marker genotypes, independent of (and often more complete than) its recorded pedigree. A "run of homozygosity" is a long, unbroken stretch of homozygous genotypes along one chromosome -- evidence that both chromosome copies in that stretch descend from a shared recent ancestor. F_ROH summarizes an animal's genomic inbreeding as the fraction of its genotyped genome covered by such runs.

F_ROH = ΣL_ROH / L_genotyped

Where L_ROH is the summed length of ROH segments -- consecutive homozygous genotyped loci, ordered by chromosome position, spanning at least a minimum number of loci and a minimum base-pair length (a field-standard dual threshold; too short a run reflects only background linkage disequilibrium, not real relatedness) -- and L_genotyped is the total genomic span covered by the marker panel, a single fixed value shared across every animal so F_ROH stays comparable across a cohort. Ceballos FC, Joshi PK, Clark DW, Ramsay M, Wilson JF. 2018. Runs of homozygosity: windows into population history and trait architecture. Nature Reviews Genetics 19(4):220-234.

MHC Haplotype and Rare-Haplotype Flag:
An MHC haplotype is the set of major histocompatibility complex genes an animal inherited together, on one chromosome, from one parent; each animal carries two. Colonies record them as short labels (for example A004_B002) following a community naming convention. This report treats each label as a name and never reads meaning into its parts. A trailing ? marks a provisional designation: it is counted and shown, but left out of the frequencies.

frequency = copies / certain designations

Where copies is how many times the haplotype appears (a homozygous animal adds two copies but counts as one carrier), and certain designations is two per genotyped animal, minus the missing and provisional ones. A haplotype is flagged rare when its frequency is at or below the frequency threshold (default 0.01) or it is carried by no more than the carrier threshold (default 2 animals). The second rule matters in small colonies: with 30 animals no haplotype can have a frequency as low as 0.01, yet a haplotype carried by only two animals is lost if those two are removed. Both thresholds are adjustable. Wiseman RW, Karl JA, Bohn PS, Nimityongskul FA, Starrett GJ, O'Connor DH. 2013. Haplessly hoping: macaque major histocompatibility complex made easy. ILAR Journal 54(2):196-210. Kanthaswamy S, et al. 2026. Next-generation short-read sequencing reveals impacts on major histocompatibility complex diversity resulting from differences in captive rhesus macaque (Macaca mulatta) colony expansion strategies. American Journal of Primatology 88(1):e70108. Hurley CK, et al. 2020. Common, intermediate and well-documented HLA alleles in world populations: CIWD version 3.0.0. HLA 95(6):516-531. Allendorf FW. 1986. Genetic drift and the loss of alleles versus heterozygosity. Zoo Biology 5(2):181-190.