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Statistical samplingWe can describe the dependence between pairs of uniting alleles in a single population without invoking an evolutionary model for the history of the population. In this “statistical sampling” framework (Weir, 1996) we do not consider the variation associated with evolutionary processes but we do consider the variation among samples from the same population. Although extensive sets of genetic data allow individual-level inbreeding coefficients to be estimated with high precision, we start with population-level estimation.Allelic dependencies can be quantified with the within-population inbreeding coefficient, written here as fW to emphasize it is a within-population quantity, defined by$${H}_{l}=2{p}_{l}(1-{p}_{l})(1-{f}_{W})$$
(1)
where Hl is the population proportion of heterozygotes for the reference allele at SNP l and pl is the population proportion of that allele. The same value of fW is assumed to apply for all SNPs. An immediate consequence of this definition is that the population proportions of homozygotes for the reference and alternative alleles are ({p}_{l}^{2}+{p}_{l}(1-{p}_{l}){f}_{W}) and ({(1-{p}_{l})}^{2}+{p}_{l}(1-{p}_{l}){f}_{W}) respectively. This formulation allows fW to be negative, with the maximum of −pl/(1 − pl) and −(1 − pl)/pl as lower bound. It is bounded above by 1. Hardy–Weinberg equilibrium, HWE, corresponds to fW = 0 and textbooks (e.g., (Hedrick, 2000)) point out that negative values of fW indicate more heterozygotes than expected under HWE.Observed heterozygote proportions ({tilde{H}}_{l}) have Hl as within-population expectation ({{{{{{mathcal{E}}}}}}}_{W}) over samples from the study population, ({{{{{{mathcal{E}}}}}}}_{W}({tilde{H}}_{l})={H}_{l}), and this would provide a simple estimator of fW if the population allele proportions were known. In practice, however, these proportions are unknown. Steele et al. (2014) suggested use of data external to the study sample to provide reference allele proportions in forensic applications where a reference database is used for making inferences about the population relevant for a particular crime. The more usual approach is to use study sample proportions ({tilde{p}}_{l}) in place of the true proportions pl, as in equation 1 of Li & Horvitz (1953):$${hat{f}}_{{W}_{l}}=1-frac{{tilde{H}}_{l}}{2{tilde{p}}_{l}(1-{tilde{p}}_{l})}$$
(2)
The moment estimator in Eq. (2) is also an MLE of fW when only one locus is considered, but it is biased (Robertson & Hill, 1984) since not only is it a ratio of statistics but also the expected value ({{{{{{mathcal{E}}}}}}}_{W}[2{tilde{p}}_{l}(1-{tilde{p}}_{l})]) over repeated samples of n from the population is 2pl(1 − pl)[1 − (1 + fW)/(2n)] (e.g., (Weir, 1996), p39).This approach can be used to estimate the within-population inbreeding coefficient fj for each individual j in a sample from one population. These are the “simple” estimators of Hall et al. (2012) and the ({hat{f}}_{{{{{{{rm{HOM}}}}}}}_{j}}) of Yengo et al. (2017):$${hat{f}}_{{{{{{{rm{HOM}}}}}}}_{jl}}=1-frac{{tilde{H}}_{jl}}{2{tilde{p}}_{l}(1-{tilde{p}}_{l})}$$
(3)
The sample heterozygosity indicator ({tilde{H}}_{jl}) is one if individual j is heterozygous at SNP l and is zero otherwise. Averaging Eq. (3) over individuals gives the estimator based on SNP l in Eq. (2).A single SNP provides estimates that are either 1 or a negative value depending on ({tilde{p}}_{l}), so many SNPs are used in practice. In both Hall et al. (2012) and Yengo et al. (2017) data were combined over loci as weighted or “ratio of averages” estimators:$${hat{f}}_{{{{{{{rm{Hom}}}}}}}_{j}}=1-frac{{sum }_{l}({tilde{H}}_{jl})}{{sum }_{l}[2{tilde{p}}_{l}(1-{tilde{p}}_{l})]}$$
(4)
Gazal et al. (2014) referred to this estimator as fPLINK as it is an option in PLINK. We show below the good performance of this weighted estimator for large sample sizes and large numbers of loci. We will consider throughout that a large number L of SNPs are used so that ratios of sums of statistics over loci, such as in Eq. (4), have expected values equal to the ratio of expected values of their numerators and denominators. Ochoa & Storey (2021) showed statistics of the form ({tilde{A}}_{L}/{tilde{B}}_{L}), where ({tilde{A}}_{L}=mathop{sum }nolimits_{l = 1}^{L}{a}_{l}/L) and ({tilde{B}}_{L}=mathop{sum }nolimits_{l = 1}^{L}{b}_{l}/L), have expected values that converge almost surely to the ratio A/B when ({{{{{{mathcal{E}}}}}}}_{W}({tilde{A}}_{L})=A{c}_{L}) and ({{{{{{mathcal{E}}}}}}}_{W}({tilde{B}}_{L})=B{c}_{L}). This result rests on the expectations ({{{{{{mathcal{E}}}}}}}_{W}({a}_{l})=A{c}_{l}) and ({{{{{{mathcal{E}}}}}}}_{W}({b}_{l})=B{c}_{l}) with ({c}_{L}=mathop{sum }nolimits_{l = 1}^{L}{c}_{l}/L). It requires ∣al∣, ∣bl∣ to both be no greater than some finite quantity C, cL to converge to a finite value c as L increases, and for Bc not to be zero. For the ratio in Eq. (4), ({a}_{l}={tilde{H}}_{jl}), ({b}_{l}=2{tilde{p}}_{l}(1-{tilde{p}}_{l})) so A = (1 − fj), B = 1 for large sample sizes n, and cL = ∑l2pl(1 − pl)/L ≤ 1/2. The conditions are satisfied providing at least one SNP is polymorphic. For an “average of ratios” estimator of the form (mathop{sum }nolimits_{l = 1}^{L}({a}_{l}/{b}_{l})/L), the denominators bl can be very small and convergence of its expected value is not assured.As an alternative to using sample allele frequencies, Hall et al. (2012) used maximum likelihood to estimate population allele proportions for multiple loci whereas Ayres & Balding (1998) used Markov chain Monte Carlo methods in a Bayesian approach that integrated out the allele proportion parameters. Neither of those papers considered data of the size we now face in sequence-based studies of many organisms, and we doubt the computational effort to estimate, or integrate over, hundreds of millions of allele proportions in Eqs. (2) or (4) adds much value to inferences about f. The allele-sharing estimators we describe below regard allele probabilities as unknown nuisance parameters and we show how to avoid estimating them or assigning them values.Hall et al. (2012) used an EM algorithm to find MLEs for fj when population allele proportions were regarded as being known and equal to sample proportions. Alternatively, a grid search can be conducted over the range of validity for the single parameter fj that maximizes the log-likelihood$${{{{mathrm{ln}}}}},[{{{{{rm{Lik}}}}}}({f}_{j})]={{{{{rm{Constant}}}}}}+mathop{sum }limits_{l=1}^{L}{{tilde{H}}_{jl}{{{{mathrm{ln}}}}},[(1-{f}_{j})]+(1-{tilde{H}}_{jl}){{{{mathrm{ln}}}}},[1-2{tilde{p}}_{l}(1-{tilde{p}}_{l})(1-{f}_{j})]}$$Estimation of the within-population inbreeding coefficients fW (FIS of (Wright, 1922)) and fj does not require any information beyond genotype proportions in samples from a study population, nor does it make any assumptions about that population or the evolutionary forces that shaped the population. The coefficients are simply measures of dependence of pairs of alleles within individuals.Genetic samplingInbreeding parameters of most interest in genetic studies are those that recognize the contribution of previous generations to inbreeding in the present study population. This requires accounting for “genetic sampling” (Weir, 1996) between generations, thereby leading to an ibd interpretation of inbreeding: ibd alleles descend from a single allele in a reference population. It also allows the prediction of inbreeding coefficients by path counting when pedigrees are known (Wright, 1922). If individual J is ancestral to both individuals (j^{prime}) and j″, and if there are n individuals in the pedigree path joining (j^{prime}) to j″ through J, then Fj = ∑(0.5)n(1 + FJ) where FJ is the inbreeding coefficient of ancestor J and Fj is the inbreeding coefficient of offspring j of parents (j^{prime}) and j″. The sum is over all ancestors J and all paths joining (j^{prime}) to j″ through J. The expression is also the coancestry ({theta }_{j^{prime} j^{primeprime} }) of (j^{prime}) and j″: the probability an allele drawn randomly from (j^{prime}) is ibd to an allele drawn randomly from j″.The allele proportion pl in a study population has expectation πl over evolutionary replicates of the population from an ancestral reference population to the present time. Sample allele proportions ({tilde{p}}_{l}) provide information about the population proportions pl, and their statistical sampling properties follow from the binomial distribution. We do not invoke a specific genetic sampling distribution for the pl about their expectations πl although we do assume the second moments of that distribution depend on probabilities of ibd for pairs of alleles. One consequence of the assumed moments is that the probability of individual j in the study sample being heterozygous, i.e., the total expected value ({{{{{{mathcal{E}}}}}}}_{T}) of the heterozygosity indicator over replicates of the history of that individual, is$${{{{{{mathcal{E}}}}}}}_{T}({tilde{H}}_{{j}_{l}})=2{pi }_{l}(1-{pi }_{l})(1-{F}_{j})$$
(5)
The quantity Fj is the individual-specific version of FIT of Wright (1922) and we can regard it as the probability the two alleles at any locus for individual j are ibd. There is an implicit assumption in Eq. (5) that the reference population needed to define ibd is infinite and in HWE: there is probability Fj that j has homologous alleles with a single ancestral allele in that population and probability (1 − Fj) of j having homologous alleles with distinct ancestral alleles there. In the first place, the single ancestral allele has probability π of being the reference allele for that locus and the implicit assumption is that two ancestral alleles are both the reference type with probability π2. This does not mean there is an actual ancestral population with those properties, any more than use of ({{{{{{mathcal{E}}}}}}}_{T}) means there are actual replicates of the history of any population or individual, and we note that Eq. (5) does not allow higher heterozygosity than predicted by HWE. Nonetheless, the concept of ibd allows theoretical constructions of great utility and we now present a framework for approaching empirical situations.Inbreeding, or ibd, implies a common ancestral origin for uniting alleles and statements about sample allele proportions ({tilde{p}}_{l}) require consideration of possible ibd for other pairs of alleles in the sample. The total expectation of (2{tilde{p}}_{l}(1-{tilde{p}}_{l})) over samples from the population and over evolutionary replicates of the study population is ((Weir, 1996), p176)$${{{{{{mathcal{E}}}}}}}_{T}[2{tilde{p}}_{l}(1-{tilde{p}}_{l})]=2{pi }_{l}(1-{pi }_{l})left[(1-{theta }_{S})-frac{1}{2n}left(1+{F}_{W}-2{theta }_{S}right)right]$$
(6)
where FW is the parametric inbreeding coefficient averaged over sample members, ({F}_{W}=mathop{sum }nolimits_{j = 1}^{n}{F}_{j}/n), and θS is the average parametric coancestry in the sample, ({theta }_{S}=mathop{sum }nolimits_{j = 1}^{n}{sum }_{j^{prime} ne j}{theta }_{jj^{prime} }/[n(n-1)]). Equivalent expressions were given by McPeek et al. (2004) and DeGiorgio and Rosenberg (2009). We note the relationship fW = (FW − θS)/(1 − θS) given by Wright (1922) and we showed in WG17 the equivalent expression fj = (Fj − θS)/(1 − θS) for individual-specific values (θS is Wright’s FST).For a large number of SNPs, the expectation of a ratio estimator of the type considered here is the ratio of expectations (Ochoa & Storey, 2021). Therefore, the total expectations of the ({hat{f}}_{{{{{{{rm{Hom}}}}}}}_{j}}), taking into account both statistical and genetic sampling, are$${{{{{{mathcal{E}}}}}}}_{T}({hat{f}}_{{{{{{{rm{HOM}}}}}}}_{j}})=1-frac{1-{F}_{j}}{(1-{theta }_{S})-frac{1}{2n}left(1+{F}_{W}-2{theta }_{S}right)}=frac{{f}_{j}-frac{1}{2n}(1+{f}_{W})}{1-frac{1}{2n}(1+{f}_{W})}$$
(7)
For all sample sizes, ({hat{f}}_{{{{{{{rm{HOM}}}}}}}_{j}}) has an expected value less than the true value fj, with the bias being of the order of 1/n. The ranking of ({{{{{{mathcal{E}}}}}}}_{T}({hat{f}}_{{{{{{{rm{HOM}}}}}}}_{j}})) values, however, is the same as the ranking of the fj and, therefore, of the Fj. For large sample sizes, Eq. (7) reduces to ({{{{{{mathcal{E}}}}}}}_{T}({hat{f}}_{{{{{{{rm{HOM}}}}}}}_{j}})={f}_{j}). Averaging over individuals shows that ({{{{{{mathcal{E}}}}}}}_{T}({hat{f}}_{{{{{{rm{HOM}}}}}}})={f}_{W}): the population-level estimator in Eq. (2) has total expectation of fW, not FW.A different outcome is found for the ({hat{f}}_{{{{{{{rm{UNI}}}}}}}_{j}}) estimator of Yengo et al. (2017) (i.e., ({hat{f}}^{III}) of Yang et al. (2011); ({hat{f}}_{{{{{{rm{GCTA}}}}}}3}) of (Gazal et al., 2014)). This estimator, with the weighted (w) ratio of averages over loci we recommend, as opposed to the unweighted (u) average of ratios over loci used in their papers, is$${hat{f}}_{{{{{{{rm{UNI}}}}}}}_{j}}^{w}=frac{mathop{sum }nolimits_{l = 1}^{L}[{X}_{jl}^{2}-(1+2{tilde{p}}_{l}){X}_{jl}+2{tilde{p}}_{l}^{2}]}{mathop{sum }nolimits_{l = 1}^{L}2{tilde{p}}_{l}(1-{tilde{p}}_{l})}$$
(8)
In this equation Xjl is the reference allele dosage, the number of copies of the reference allele, at SNP l for individual j. It is equivalent to the estimator given by (Ritland (1996), eq. 5) and attributed by him to Li & Horvitz (1953).Ochoa & Storey (2021) showed that ({hat{f}}_{{{{{{{rm{UNI}}}}}}}_{j}}^{w}) has expectation, for a large number of SNPs and a large sample size, of$${{{{{{mathcal{E}}}}}}}_{T}({hat{f}}_{{{{{{{rm{UNI}}}}}}}_{j}}^{w})=frac{{F}_{j}-2{{{Psi }}}_{j}+{theta }_{S}}{1-{theta }_{S}}={f}_{j}-2{psi }_{j}$$
(9)
where Ψj is the average coancestry of individual j with other members of the study sample: ({{{Psi }}}_{j}=mathop{sum }nolimits_{j^{prime} = 1,j^{prime} ne j}^{n}{theta }_{jj^{prime} }/(n-1)). We term ψj = (Ψj − θS)/(1 − θS) the within-population individual-specific average kinship coefficient. The Ψj have an average of θS over members of the sample, so the average of the ψj’s is zero and expected value of the average of the ({hat{f}}_{{{{{{{rm{UNI}}}}}}}_{j}}^{w}) is fW, as is the case for ({hat{f}}_{{{{{{{rm{AS}}}}}}}_{j}}) below.Equation (9) shows that the ({hat{f}}_{{{{{{{rm{UNI}}}}}}}_{j}}^{w}) have expected values with the same ranking as the Fj values only if every individual j in the sample has the same average kinship ψj with other sample members.Finally, we mention another common estimator described by VanRaden (2008), termed fGCTA1 by Gazal et al. (2014) and available from the GCTA software (Yang et al., 2011) with option –ibc. We referred to this as the “standard” estimator in WG17. The weighted version for multiple loci is$${hat{f}}_{{{{{{{rm{STD}}}}}}}_{j}}^{w}=frac{{sum }_{l}{({X}_{jl}-2{tilde{p}}_{l})}^{2}}{{sum }_{l}2{tilde{p}}_{l}(1-{tilde{p}}_{l})}-1$$
(10)
and it has the large-sample expectation of (fj − 4ψj) as is implied by WG17 (Eq. 13) and as was given by Ochoa & Storey (2021). We summarize the various measures of inbreeding and coancestry in Table 1, and we include sample sizes in the expectations shown in Table 2.Table 1 Measures of inbreeding and coancestry.Full size tableTable 2 Estimators of inbreeding.Full size tableThe ({hat{f}}_{{{{{{rm{HOM}}}}}}}), ({hat{f}}_{{{{{{rm{UNI}}}}}}},{hat{f}}_{{{{{{rm{STD}}}}}}}) and ({hat{f}}_{{{{{{rm{MLE}}}}}}}) estimators of individual or population inbreeding coefficients make explicit use of sample allele proportions. This means that all four have small-sample biases, and none of the four provide estimates of the ibd quantities F or Fj. We showed that ({hat{f}}_{{{{{{rm{HOM}}}}}}}) is actually estimating the within-population inbreeding coefficients: the total inbreeding coefficients relative to the average coancestry of pairs of individuals in the sample, but ({hat{f}}_{{{{{{rm{UNI}}}}}}}) and ({hat{f}}_{{{{{{rm{STD}}}}}}}) are estimating expressions that also involve average kinships ψ.Allele sharingIn a genetic sampling framework, and with the ibd viewpoint, we consider within-individual allele sharing proportions Ajl for SNP l in individual j (we wrote M rather than A in WG17 and in (Goudet et al., 2018)). These equal one for homozygotes and zero for heterozygotes and sample values can be expressed in terms of allele dosages, ({tilde{A}}_{jl}={({X}_{jl}-1)}^{2}). We also consider between-individual sharing proportions ({A}_{jj^{prime} l}) for SNP l and individuals j and (j^{prime}). These are equal to one for both individuals being the same homozygote, zero for different homozygotes, and 0.5 otherwise. Observed values can be written as ({tilde{A}}_{jj^{prime} l}=[1+({X}_{jl}-1)({X}_{j^{prime} l}-1)]/2), with an average over all pairs of distinct individuals in a sample of ({tilde{A}}_{Sl}). Astle & Balding (2009) introduced ({tilde{A}}_{jj^{prime} l}) as a measure of identity in state of alleles chosen randomly from individuals j and (j^{prime}), and Ochoa & Storey (2021) used a simple transformation of this quantity. The allele sharing for an individual with itself is Ajjl = (1 + Ajl)/2.The same logic that led to Eq. (5) provides total expectations for allele-sharing proportions for all (j,j^{prime}):$$begin{array}{lll}{{{{{{mathcal{E}}}}}}}_{T}({tilde{A}}_{jj^{prime} l})&=&1-2{pi }_{l}(1-{pi }_{l})(1-{theta }_{jj^{prime} })\ {{{{{{mathcal{E}}}}}}}_{T}({tilde{A}}_{Sl})&=&1-2{pi }_{l}(1-{pi }_{l})(1-{theta }_{S})end{array}$$Note that θjj = (1 + Fj)/2. The nuisance parameter 2πl(1 − πl) cancels out of the ratio ({{{{{{mathcal{E}}}}}}}_{T}({tilde{A}}_{jj^{prime} l}-{tilde{A}}_{Sl})/{{{{{{mathcal{E}}}}}}}_{T}(1-{tilde{A}}_{Sl})) and this motivates definitions of allele-sharing estimators of the inbreeding coefficient for individual j and the kinship coefficient for individuals (j,j^{prime}) as$${hat{f}}_{{{{{{{rm{AS}}}}}}}_{j}}=frac{{sum }_{l}({tilde{A}}_{{j}_{l}}-{tilde{A}}_{{S}_{l}})}{{sum }_{l}(1-{tilde{A}}_{Sl})},{hat{psi }}_{{{{{{{rm{AS}}}}}}}_{jj^{prime} }}=frac{{sum }_{l}({tilde{A}}_{jj^{prime} l}-{tilde{A}}_{{S}_{l}})}{{sum }_{l}(1-{tilde{A}}_{Sl})}$$
(11)
For a large number of SNPs, these are unbiased for fj and ({psi }_{jj^{prime} }) for all sample sizes. We showed in WG17 there is no need to filter on minor allele frequency to preserve the lack of bias. Note that ({hat{f}}_{{{{{{{rm{AS}}}}}}}_{j}}) is a linear function of the form ({a}_{S}+{b}_{S}{tilde{A}}_{j}) with ({tilde{A}}_{j}) being the total homozygosity for j and constants aS, bS being the same for all individuals j. Changing the scope of the study, from population to world for example, preserves linearity (with different values of aS, bS). The changed estimates are linear functions of the old estimates: old and new estimates are completely correlated and are rank invariant over all samples that include particular individuals, i.e., over all reference populations. Unlike the case for ({hat{f}}_{{{{{{rm{UNI}}}}}}}) or ({hat{f}}_{{{{{{rm{STD}}}}}}}), rank invariance is guaranteed for ({hat{f}}_{{{{{{{rm{AS}}}}}}}_{j}}) for any two individuals even if only one more individual is added to the study.For large sample sizes, ((1-{tilde{A}}_{Sl})approx 2{tilde{p}}_{l}(1-{tilde{p}}_{l})). Under that approximation, ({hat{f}}_{{{{{{{rm{AS}}}}}}}_{j}}) is the same as ({hat{f}}_{{{{{{{rm{Hom}}}}}}}_{j}}) but the approximation is not necessary in computer-based analyses. Summing the large-sample estimates over individuals not equal to j gives an estimator for the average individual kinship ψj:$${hat{psi }}_{{{{{{{rm{AS}}}}}}}_{j}}=-frac{{sum }_{l}({X}_{jl}-2{tilde{p}}_{l})(1-2{tilde{p}}_{l})}{{sum }_{l}4{tilde{p}}_{l}(1-{tilde{p}}_{l})}$$
(12)
Adding (2{hat{psi }}_{{{{{{{rm{AS}}}}}}}_{j}}) to ({hat{f}}_{{{{{{{rm{UNI}}}}}}}_{j}}^{w}) gives ({hat{f}}_{{{{{{{rm{AS}}}}}}}_{j}}), as expected, as does adding (4{hat{psi }}_{{{{{{{rm{AS}}}}}}}_{j}}) to ({hat{f}}_{{{{{{{rm{STD}}}}}}}_{j}}^{w}). Similarly, ({hat{psi }}_{{{{{{{rm{AS}}}}}}}_{jj^{prime} }}) is obtained by adding ({hat{psi }}_{{{{{{{rm{AS}}}}}}}_{j}}) and ({hat{psi }}_{{{{{{{rm{AS}}}}}}}_{j^{prime} }}) to ({hat{psi }}_{{{{{{{rm{STD}}}}}}}_{jj^{prime} }}), where (Yang et al., 2011)$${hat{psi }}_{{{{{{{rm{STD}}}}}}}_{jj^{prime} }}=frac{mathop{sum}nolimits_{l}({X}_{jl}-2{tilde{p}}_{l})({X}_{j^{prime} l}-2{tilde{p}}_{l})}{mathop{sum}nolimits_{l}4{tilde{p}}_{l}(1-{tilde{p}}_{l})}$$These are the elements of the first method for constructing the GRM given by VanRaden (2008).When inbreeding and coancestry coefficients are defined as ibd probabilities they are non-negative, but the within-population values f and ψ will be negative for individuals, or pairs of individuals, having smaller ibd allele probabilities than do pairs of individuals in the sample, on average. Individual-specific values of f always have the same ranking as the individual-specific F values, and they are estimable. Negative estimates can be avoided by the transformation to (({hat{f}}_{{{{{{{rm{AS}}}}}}}_{j}}-{hat{f}}_{{{{{{{rm{AS}}}}}}}_{j}}^{min })/(1-{hat{f}}_{{{{{{{rm{AS}}}}}}}_{j}}^{min })) where ({hat{f}}_{{{{{{{rm{AS}}}}}}}_{j}}^{min }) is the smallest value over individuals of the ({hat{f}}_{{{{{{{rm{AS}}}}}}}_{j}})’s. We don’t see the need for this transformation, and we noted above the recognition of the utility of negative values. Ochoa & Storey (2021) wished to estimate Fj rather than fj and, to overcome the lack of information about the ancestral population serving as a reference point for ibd, they assumed the least related pair of individuals in a sample have a coancestry of zero. We showed in WG17 that this brings estimates in line with path-counting predicted values when founders are assumed to be not inbred and unrelated, but we prefer to avoid the assumption. We stress that, absent external information or assumptions, F is not estimable. Instead, linear functions of F that describe ibd of target pairs of alleles relative to ibd in a specified set of alleles are estimable and have utility in empirical studies.Runs of homozygosityEach of the inbreeding estimators considered so far has been constructed for individual SNPs and then combined over SNPs. Observed values of allelic state are used to make inferences about the unobserved state of identity by descent. Estimators based on ROH, however, suppose that ibd for a region of the genome can be observed. Although F is the probability an individual has ibd alleles at any single SNP, in fact ibd occurs in blocks within which there has been no recombination in the paths of descent from common ancestor to the individual’s parents. Whereas a single SNP can be homozygous without the two alleles being ibd, if many adjacent SNPs are homozygous the most likely explanation is that they are in a block of ibd (Gibson et al., 2006). There can be exceptions, from mutation for example, and several publications give strategies for identifying runs of homozygotes for which ibd may be assumed (e.g., Gazal et al. (2014); (Joshi et al., 2015)). These strategies include adjusting the size of the blocks, the numbers of heterozygotes or missing values allowed per block, the minor allele frequency, and so on. These software parameters affect the size of the estimates (Meyermans et al., 2020). Some methods (e.g., Gazal et al. (2014); (Narasimhan et al., 2016)) use hidden Markov models where ibd is the hidden status of an observed homozygote. Model-based approaches necessarily have assumptions, such as HWE in the sampled population.We provide more details elsewhere, but we note here that ROH methods offer a useful alternative to SNP-by-SNP methods even though they cannot completely compensate for lack of information on the ibd reference population. We note also that shorter runs of ibd result from more distant relatedness of an individual’s parents, and ROH procedures can be set to distinguish recent (familial) ibd from distant (evolutionary) ibd. SNP-by-SNP estimators do not make a distinction between these two time scales. More

Reef life surveyReef fish communities were censused by a combination of experienced marine scientists and trained recreational SCUBA divers using globally standardized Reef Life Survey methods. All surveys were undertaken on 50 m long transects laid along a contour (at consistent depth) on predominantly hard substrate (usually rocky or coral reef) in shallow waters (depth range of transects 1 to 20 m, average ~7.2 m). Full details of fish census methods, data quality, and training of divers are provided in refs. 22,34,35 and in an online methods manual (www.reeflifesurvey.com). Fish abundance counts and size estimates per 500 m2 transect area (2 ×250 m2 blocks) were converted to biomass using length–weight relationships for each species obtained from Fishbase (www.fishbase.org). In cases where length–weight relationships were provided in Fishbase using standard length or fork length, rather than total length as estimated by divers, length–length relationships provided in Fishbase allowed conversion to the total length. For improved accuracy in biomass assessments, observed sizes were also adjusted to account for the bias in divers’ perception of fish size underwater using an empirical calibration36. Length–weight coefficients from similar-shaped close relatives were used for those species where length–weight relationships were not available in Fishbase. All transects were collapsed into a single average value of biomass for each species at a location to account for any differences in the total number of transect surveys performed.Decomposition of difference in ecosystem functioningOur equation was inspired by previous decompositions, principally the Price equation originally derived in the field of evolutionary biology as a means of separating genetic and environmental influences on phenotypic change over time37. Fox38 and later Fox and Kerr12 modified the Price equation to describe how the difference in the ecological function between two communities can be decomposed into components with different ecological interpretations. We follow a similar approach but use a different decomposition where the resulting components are similar to, but not the same as, the components proposed by Fox and Kerr12.We begin by assuming that the ecological function of the community, such as biomass, is a simple additive function of the contributions of its constituent species. We go on to compare two communities, one of which we consider the “reference” community and the other we refer to as the “comparison” community. The species present in the reference community can be classified into two types: species that are unique to the reference community (i.e., not present in the comparison community) and those that are in common with the comparison community. Let suB be the number of unique species in the reference community, and sc be the number in common between the two communities. Let ({bar{z}}_{{uB}}) be the average ecological function contributed per unique species to the reference community, and ({bar{z}}_{{cB}}) be the average ecological function contributed per shared species in the reference community. The total ecological function TB of the reference community can thus be decomposed as:$${T}_{B}={s}_{{uB}}{bar{z}}_{{uB}}+{s}_{c}{bar{z}}_{{cB}}$$
(1)
where the first term represents the ecological function contributed by species that are unique to the reference community (i.e., not present in the comparison community) and the latter term represents the contribution from species that are also found in the comparison community.Analogously, in the comparison community, the total ecological function can be decomposed as:$${T}_{F}={s}_{{uF}}{bar{z}}_{{uF}}+{s}_{c}{bar{z}}_{{cF}}$$
(2)
with a similar interpretation to Eq. (1). Though there are sc species in common between the two communities, the average per species contribution need not be the same in the two communities (i.e., ({bar{z}}_{{cB}}) may differ from ({bar{z}}_{{cF}})).The species in common between the two communities can serve as a reference point for comparison between communities. It is useful to define ({delta }_{B}={bar{z}}_{{uB}}-{bar{z}}_{{cB}}) and ({delta }_{F}={bar{z}}_{{uF}}-{bar{z}}_{{cF}}) as the difference in average ecological function per species of unique species versus shared species in reference and comparison communities, respectively. From this perspective, we consider the average ecological function of a species unique to the reference community as being equal to the average ecological function of shared species (as measured in the same community) plus the deviation from this value ({bar{z}}_{{uB}}={bar{z}}_{{cB}}+{delta }_{B}). Using this equality and the analogous one for ({bar{z}}_{{uF}}), along with Eqs. (1) and (2), the difference in the ecological function between communities can be decomposed as$$Delta T={T}_{F}-{T}_{B}={-s}_{{uB}}{bar{z}}_{{cB}}-{s}_{{uB}}{delta }_{B}+{s}_{{uF}}{bar{z}}_{{cF}}+{s}_{{uF}}{delta }_{F}+{s}_{c}left({bar{z}}_{{cF}}-{bar{z}}_{{cB}}right)$$
(3)
The first two terms represent the loss in ecological function in the comparison community due to the loss of species that are unique to the reference community. Specifically, the first term represents the loss in ecological function due to the absence of unique species if these species had the same average value of functioning as each of the shared species. In other words, it is the amount by which biomass is expected to decline if species were interchangeable. Therefore, we interpret this term as the “richness loss” or the loss in functioning due strictly to the loss of species: RICH-L ((={-s}_{{uB}}{bar{z}}_{{cB}})). It will always be negative, assuming there is at least one species unique to the reference population. In cases where ({bar{z}}_{{cB}} > {bar{z}}_{{uB}}), it is possible for RICH-L to exceed the total functioning observed at the reference site, which complicates interpretation of the raw values. In this case, it is useful to consider only the relative quantities (each component is scaled by the sum of the absolute values of all components). We note that this situation arises only 41 times out of 2867 comparisons in our analysis, and removing these cases has no effect on our findings. We advise future applications be aware of this potential issue and test for its influence.The second term accounts for the fact that the true loss in ecological function due to these lost species will often differ from the “richness expectation” because the lost species differ in value from the average value of shared species. In other words, this term reflects the deviation in the actual contributions of lost species from the average of shared species, which implies that not all species contribute equally (and that the identities of the species are important in determining differences in biomass between the two communities). We, therefore, interpret this term as indicating “compositional loss,” or the degree to which loss in biomass is due to loss of particular species: COMP-L ((= – {s}_{{uB}}{delta}_{B})). If the average lost species provide a higher contribution to the reference community than the average shared species (({bar{z}}_{{uB}} > {bar{z}}_{{cB}})), the COMP-L term will be negative. On the other hand, if the average lost species represent lower contributions, the COMP-L term will be positive (({bar{z}}_{{uB}} < {bar{z}}_{{cB}})).The next two terms are analogous to the first two terms but instead represent the increase in ecological function in the comparison community due to the “gain” of unique species that are lacking from the reference community. The third term represents the expected increase in ecological function due to an increase in species richness assuming these gained species had the same per species contribution as the shared species: RICH-G ((={+s}_{{uF}}{bar{z}}_{{cF}})). It is always positive, assuming the comparison community has at least one unique species. The fourth term, COMP-G ((=+{s}_{{uF}}{delta }_{F})), reflects the difference in composition (with respect to average value) of gained versus shared species. This term can be positive or negative, being positive if the gained species have a higher per species value than the shared species.The final term focuses on the changes in biomass considering only the species that are present in both communities. This can be thought of as holding richness and composition constant and considering changes in the community biomass that are controlled extrinsically, i.e., by underlying gradients in resource availability and other environmental factors. Historically, this term has been referred to as the “context-dependent effect,” or CDE, and is the number of shared species (({s}_{c})), multiplied by the difference in biomasses among shared species at both sites ((={s}_{c}({bar{z}}_{{cF}}-{bar{z}}_{{cB}}))). It can be of either sign: positive if shared species have a higher value in the comparison community than in the reference, negative if they have a higher value in the reference community. The number of shared species has the potential to bias away from the CDE term if it is very low. However, we note that, on average, 49.1 ± 0.003% of species are shared for each comparison at the 100-km scale, and this value is remarkably consistent regardless of spatial scale (51.3–50.0% for 15–50 km).Our decomposition is similar to, but not the same as, that of Fox and Kerr12, though both are mathematically sound. Only the CDE term is mathematically identical across the two decompositions and, thus, shares the same interpretation. By extension, the sum across the loss and gain terms (the total diversity effect, or DIV) must also be identical, because both equations partition the same total quantity. Thus, it is important to note that using either decomposition yields the same inference with respect to comparisons of DIV and CDE.Our decomposition differs from Fox and Kerr’s because the two approaches use different reference points. We take the perspective that the shared species form the basis for comparison between two communities, so we then evaluate the average value of a unique species with respect to its deviation from an average value of a shared species. In contrast, Fox and Kerr effectively evaluate the average value of a unique species with respect to its deviation from the average value of any species in that community (averaging over both unique and shared species). In both decompositions, the “composition” components only exist if there is some difference in the average value of shared and unique species. We prefer our decomposition for this case because it works with that difference directly rather than indirectly via the difference between unique and all species (which is the average of unique and shared species). Moreover, our composition makes intuitive sense that the function of the “average” species is determined by the ones that are known to exist at both sites. A full comparison of the Fox and Kerr formulation and ours is provided in the Supplementary Materials.Statistical analysisA general function to conduct our new decomposition from a site-by-species biomass matrix, and a second function to perform the simulations, can be found here: https://gist.github.com/jslefche/76c076c1c7c5d200e5cb87113cdb9fb4.We first ordered all sites by decreasing total biomass. Beginning with the highest biomass site of all sites as the first reference site, we identified all other sites within a certain spatial radius (15-, 25-, 50-, or 100-km) to serve as the comparison sites. Setting the reference to be the site with the highest community biomass constrains the sum of the terms to be negative. This choice simplifies the language used to discuss the output13 and allows us to speak directly to the consequences of real-world activities like overharvesting (and their implications).We then computed the components for each set of comparisons. We standardized the output to the same scale (−1, 1) by first taking the sum of the absolute value of all components, and then dividing each component by this value. This relativization was done to account for the fact that raw biomass may differ substantially among sites and regions and to make our results comparable across the entire dataset. Once the scaled components were computed, the reference and comparison sites were removed from the ordered list from any further comparisons to prevent any bias that might arise from including the same site multiple times. We then moved onto the next most productive site in the list, identified the comparison sites within 100 km, computed the components, and so on, until all sites were analyzed. From these individual comparisons, we computed the means of all components while omitting any reference sites for which there were fewer than five comparison sites. We alternately averaged the components for all comparisons for each reference site and then took the grand mean of these averaged values, although this additional level of aggregation did not qualitatively change our results (Supplementary Fig. 6). We have chosen to present the raw values in the main text to demonstrate the full range of variability inherent in the individual comparisons, which might otherwise be condensed by showing only the means for each reference site. We repeated the analysis over multiple spatial radii to assess whether the spatial extent and therefore the size and composition of the species pool, might influence our results.We calculated the relative strength of the total diversity effect vs. the context-dependent effect for each comparison as the ratio of DIV/CDE, and of compositional vs. richness losses as:$${{{{{rm{Q}}}}}}=frac{(-{s}_{{uB}}{delta }_{B}{-s}_{{uB}}{bar{z}}_{{cB}})}{{-s}_{{uB}}{bar{z}}_{{cB}}}=frac{{bar{z}}_{{uB}}}{{bar{z}}_{{cB}}}$$ (4) In this case, Q = (COMP-L + RICH-L)/RICH-L, which reduces to the average value of unique species relative to the average value of shared species at the reference site. This quantity reflects the magnitude to which species unique to the reference site contribute to biomass relative to the “expected” contribution per species. To avoid biases associated with averaging ratios, we report the geometric mean of both quantities. Bootstrapped 95% confidence intervals were derived by randomly resampling DIV/CDE and Q for a total of 5000 times. For DIV/CDE, some values were negative, so we excluded them in both the original data and bootstrap samples. As an alternative approach that focused on the magnitude of effect, we examined the absolute value of |DIV | / | CDE | . In this case, the ratio was 6.9x with bootstrap 95% CIs of [6.2, 7.7].To explore the drivers of the components of our decomposition, we applied random forest analysis to account for potential collinearity and interactions among the suite of predictors previously selected in ref. 39. Depth was recorded on the surveys while the following predictors were obtained from the combination of remote sensed and in situ measurements compiled in the Bio-ORACLE database: mean, minimum, maximum, and range of sea surface temperature; mean, minimum and maximum for surface chlorophyll-a; mean salinity; mean PAR; mean dissolved oxygen; mean nitrate concentration; mean phosphate concentration40. Finally, an index of human population density was calculated by fitting a smoothly tapered surface to each settlement point on the year 2010 world-population density grid using a quadratic kernel function described previously41. Random forests were fit using the default settings in the randomForest package42 in R version 4.1.143. Variable importance was determined using the percent increase in the mean-square error after randomly permuting the predictor of interest for each tree in the random forest, averaging the error of the models, and then computing the difference relative to the accuracy of the original model.Null simulationsA key finding of our analysis is that compositional losses are considerably greater than losses due to other aspects of the reef fish community. We wanted to evaluate the possibility of whether such a result could be an artifact of applying our decomposition to a dataset in which we assign the site with the higher total biomass as the “reference” community and the site with lower total biomass as the “focal” community. To do so, we conducted simulations in which we created communities with species richness values matching the observed data, but for community compositions that were random. Following the same procedure we used with the real communities, we applied our decomposition to these simulated communities to generate null distributions for the average values of each of the five terms when community composition is random. Comparing our observed values to these null distributions tells us if the values of the compositional components (or indeed any component) we observed arose as an artifact of our procedure or, alternatively, because high-biomass sites actually contain more high-biomass species than expected under random community assembly.Our simulation procedure focused on the site-by-species biomass matrix from each set of comparisons used in the main 100-km analysis. We divided this matrix by the corresponding site-by-species abundance matrix to yield the observed per capita contribution of each species in each community. We then averaged the per capita contributions of each species across all communities where the species was present to yield a single vector representing mean per capita contributions for all S species within that set of comparisons.We initially constructed each simulated community by populating it with every species in the region (“maximum richness”). To determine the biomass of each species in each community we applied the following procedure. First, we identified the minimum and maximum observed abundance of each species across all communities where it is present. For a single community, we sampled an integer value between the minimum and maximum abundance for each species to yield a single vector of random abundance values of length S, and then multiplied this vector by the vector of average per capita contributions. This procedure yielded a new vector representing a new total contribution to biomass by every species. We repeated this for all n communities in the original site-by-species matrix and bound these vectors together in a new “maximum richness” version of the site-by-species matrix. For the ith row (community) in the original dataset, we calculated the richness, si. We then randomly subsampled si species at random from the simulated “maximum richness” site-by-species matrix and set the biomass of any remaining species to zero. We repeated this for each community to yield a simulated “observed richness” site-by-species matrix with the same dimensions as the original matrix. This procedure ensures that richness is held at the observed levels and that the biomass contribution of each species are within the observed range.These communities were intentionally constructed randomly with respect to composition as our goal was to test whether the observed compositional effects in the real data are significantly different than under this null hypothesis with respect to composition. Thus, using the simulated “observed richness” site-by-species matrix, we computed the (scaled) components as we had with the real data and took their means across all communities. We repeated the randomization procedure 1000 times to yield 1000 total average values of each component. We compared the observed mean to the distribution of expected means using a one-tailed t-test to determine whether the observed components were more or less extreme than would be expected by chance.Reporting SummaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More