ANOVA for estimating Nei’s diversity and related parameters in a fixed set of populations with an application in genetic resources conservation

Abstract

Genetic diversity parameters are used by plant breeders to develop efficient genetic resources sampling and conservation strategies. Extending previous developments on the use of ANOVA on allele frequencies in a fixed set of populations, we show that this approach allows unbiased estimation of diversity parameters, including Nei’s diversity parameters, H_S, the within-population diversity; D_ST, the between-population differentiation; H_T, the total gene diversity; and other related parameters well suited for guiding conservation decisions. We consider two cases: selfing plants and outcrossing plants. For outcrossing plants, this approach also allows the estimation of the average frequency of heterozygotes (H₀) and average departure of populations from a random mating equilibrium. These unbiased ANOVA estimators correspond to those derived by Nei and Chesser (Ann Hum Genet 47:253–259, 1983) by using properties related to the multinomial sampling of genotypes. With an equal number of individuals sampled per population, we first developed analyses of variation for each allele at one locus. Then, considering the whole set of alleles, we show the correspondence between the sum of the variances in allele frequencies over the alleles and Nei’s within- and between-population diversities. Considering large populations leads to Nei’s relationship, H_T = H_S + D_ST, which is a decomposition of the total variance in allele frequencies into within- and between-population variance components, variance meaning the sum of the variances of each allele over the whole set of alleles. Finally, we use theoretical results of the ANOVA approach to consider a genetic resources conservation design with only one individual per population, which allows Nei’s total gene diversity to be maintained.

Appendix: Derivation of the expected mean squares for the ANOVA on allele frequencies

Case of selfing plants

Using the indicator variable x_ij such that x_ij = 1 for an individual j of the population i if a given allele k is present and x_ij = 0 otherwise, according to the model (1), the sum of squares (SS_k) of the corresponding ANOVA for a given allele k are the following:

• - SS_k between populations (pop) = (SS_pop)_k = $n\sum\nolimits_{i} {x_{i.}^{2} } - ns\;x_{..}^{2}$ with (s−1) degrees of freedom (df).

• - SS_k within populations (ind/pop) = (SS_ind/pop)_k = $\sum\nolimits_{ij} {x_{ij}^{2} } - n\sum\nolimits_{i} {x_{i.}^{2} }$ with s(n − 1) degrees of freedom,

• - SS_k total = (SS_Tot)_k = $\sum\nolimits_{ij} {x_{ij}^{2} } - ns\;x_{..}^{2}$ with (sn − 1) degrees of freedom.

Corresponding expected mean squares can be derived for a fixed set of populations and for a given allele k, be E(MS)_k = E(SS_k/df), i.e. respectively, E(MS_pop)_k, E(MS_ind/pop)_k, and E(MS_Tot)_k, denoting by p_i the frequency of the considered allele in a population i and taking into account the following expressions: E_j(x_ij) = p_i, E_ij(x_ij) = $E_{ij} (x_{ij}^{2} )$ = $E_{i} (x_{i.} ) = E(x_{..} ) = \sum\nolimits_{i} {p_{i} } /s = \overline{p}$.

Expectation E(MS_ind/pop)_k = $\sigma_{w(k)}^{2}$,

For a given allele k, we can write

E(MS_ind/pop)_k = $E\left[\sum\nolimits_{ij} {x_{ij}^{2} } - \sum\nolimits_{i} {(\sum\nolimits_{j} {x_{ij} )^{2} } } /n\right]/s(n - 1)$

$$ E\left( {\sum\nolimits_{ij} {x_{ij}^{2} } } \right) = \sum\nolimits_{i} {n\;E(x_{ij}^{2} ) = \;} n\sum\nolimits_{i} {p_{i} } \,\,\,{\text{because}}\,\,\,E_{j} (x_{ij}^{2} ) = E_{j} (x_{ij} ) = p_{i} , $$

(A1)

$$ E\left[\sum\nolimits_{i} {(\sum\nolimits_{j} {x_{ij} } } )^{2} /n\right] = E\left[\sum\nolimits_{ij} {x_{ij}^{2} } \right]/n + E\left[\sum\nolimits_{i} {\sum\nolimits_{jj^{\prime}} {x_{ij} } } x_{ij^{\prime}} \right]/n = \sum\nolimits_{i} {p_{i} } + (n - 1)\sum\nolimits_{i} {p_{i}^{2} } , $$

(A2)

$$ \sigma_{w(k)}^{2} = [(n - 1)\sum\nolimits_{i} {p_{i} } - (n - 1)\sum\nolimits_{i} {p_{i}^{2} ]/s(n - 1)} = \sum\nolimits_{i} {p_{i} } (1 - p_{i} )/s, $$

or with the introduction of the subscript k for the considered allele

$$ \sigma_{w(k)}^{2} = [(n - 1)\sum\nolimits_{i} {p_{ik} } - (n - 1)\sum\nolimits_{i} {p_{ik}^{2} ]/s(n - 1)} = \sum\nolimits_{i} {p_{ik} } (1 - p_{ik} )/s = H_{S(k)} . $$

(A3)

Expectation E(MS_pop)_k = E[SSB_k/(s − 1)]

$$ E(MS_{pop} )_{k} = E[\sum\nolimits_{i} ( \sum\nolimits_{j} {x_{ij}^{{}} } )^{2} ]/n - (\sum\nolimits_{ij} {x_{ij} )^{2} } /ns]/(s - 1) $$

$$ E[\sum\nolimits_{i} {(\sum\nolimits_{j} {x_{ij} )^{2} } } /n = \sum\nolimits_{i} {p_{i} } + (n - 1)\sum\nolimits_{i} {p_{i}^{2} } \,\,\,{\text{from }}\left( {{\text{A2}}} \right), $$

(A4)

$E(\sum\nolimits_{ij} {x_{ij} } )^{2} /ns = \left[E(\sum\nolimits_{ij} {x_{ij}^{2} } ) + E(\sum\nolimits_{i} {\sum\nolimits_{jj^{\prime}} {x_{ij} } } x_{ij^{\prime}} ) + E(\sum\nolimits_{ii^{\prime}} {\sum\nolimits_{jj^{\prime}} {x_{ij} } } x_{i^{\prime}j^{\prime}} )\right]/ns$ with $i^{\prime} \ne i,$

$$ E(\sum\nolimits_{ij} {x_{ij} } )^{2} /ns = \sum\nolimits_{i} {p{}_{i}} /s + (n - 1)\sum\nolimits_{i} {p_{i}^{2} } /s + n\,\sum\nolimits_{i,i^{\prime} \ne i} {p_{i} } p_{i^{\prime}} /s, $$

(A5)

and then introducing the subscript k for the considered allele:

$$ E(MS_{pop} )_{k} = \sum\nolimits_{i} {p_{ik} } (1 - p_{ik} )/s + \left[n/s(s - 1)\right]\,\,\left[(s - 1)\sum\nolimits_{i} {p_{ik}^{2} } - \sum\nolimits_{i,i^{\prime} \ne i} {p_{ik} } p{}_{i^{\prime}k}\right] $$

(A6)

and as $D_{ST}^{\prime } = \,\left[(s - 1)\sum\nolimits_{i} {p_{ik}^{2} } - \sum\nolimits_{i,i^{\prime} \ne i} {p_{ik} } p{}_{{i^{\prime } k}}\right]/s(s - 1)$, $E(MS_{pop} )_{k} = \sigma_{w(k)}^{2} + n\;D_{ST(k)}^{\prime } = H_{S(k)} + n\;D_{ST(k)}^{\prime }$, which corresponds to the result given by Weir (1996) when the sample size of each population is constant (n_i = n).

Case of outcrossing plants

Using the indicator variable x_ijt, indexing with t the alleles at a locus within an individual j from a population i, such that x_ijt = 1 if a given allele is present and x_ijt = 0 otherwise, the table for the analysis of variation of this indicator variable, associated to model (9) is given Table 1.

Table 1 Analysis of variance layout for the variable x indicating a given allele in fixed populations, for s populations and n individuals per population

To derive the expected mean squares E(MS) for a fixed set of populations we derive the expectations of four terms: $E\left[\sum\nolimits_{ijt} {x_{ijt}^{2} } \right]$, $E\left[2\sum\nolimits_{ij} {x_{ij.}^{2} } \right]$, $E\left[2n\sum\nolimits_{i} {x_{i..}^{2} } \right]$ et $E\left[2ns\;x_{...}^{2} \right]$, noting that we have for the indicator variable x:$E_{ijt} (x_{ijt} ) = E_{ijt} (x_{ijt}^{2} ) = \overline{p} = \sum\nolimits_{i} {p_{i} } /s$, $E_{jt} (x_{ijt} ) = p_{i}.$

$${\mathbf{\cdot}} \,E\left[\sum\nolimits_{ijt} {x_{ijt}^{2} }\right] = 2ns\,\overline{p} = 2n\,\sum\nolimits_{i} {p_{i} } . $$

(A7)

$${\mathbf{\cdot}} \,E\left[2\sum\nolimits_{ij} {x_{ij.}^{2} } \right] $$

$$ 2\sum\nolimits_{{ij}} {x_{{ij.}}^{2} } = 2E_{{ij}} \left[ {\sum\nolimits_{t} {x_{{ijt}} } } \right]^{2} {\text{/}}4 = \left[ {E\sum\nolimits_{{ij}} {\left( {\sum\nolimits_{t} {x_{{ijt}}^{2} + \sum\nolimits_{{tt^{\prime } }} {x_{{ijt}} } x_{{ijt^{\prime } }} } } \right)} } \right]/2 = \left[ {n\sum\nolimits_{i} {p_{i} } + n\sum\nolimits_{i} {p_{i} } } \right]_{i}, $$

(A8)

with t $\ne$ t', because $E_{{tt^{\prime } }} (x_{{ijt}} x_{{ijt^{\prime } }} ) = P_{i}$, P_i being the frequency of homozygous genotypes for the considered allele in the population i;

$$ {\mathbf{\cdot}}\, 2n\,E\left[ {\sum\nolimits_{i} {x_{{i..}}^{2} } } \right] = E\left[ {\sum\nolimits_{i} {\left[ {\sum\nolimits_{{jt}} {x_{{ijt}} } } \right]} ^{2} } \right]/2n $$

$E[\sum\nolimits_{jt} {x_{ijt} } ]^{2} = E[\sum\nolimits_{jt} {x_{ijt}^{2} ]} + E[\sum\nolimits_{j} {\sum\nolimits_{tt^{\prime}} {x_{ijt} } } x_{ijt^{\prime}} ] + E[\sum\nolimits_{jj^{\prime}} {\sum\nolimits_{tt^{\prime}} {x_{ijt} } } x_{ij^{\prime}t^{\prime}} ]$, with j $\ne$ j', t $\ne$ t’,

$E\left[\sum\nolimits_{jt} {x_{ijt} } \right]^{2} = 2n\,p_{i}$,

$E\left[\sum\nolimits_{j} {\sum\nolimits_{kk^{\prime}} {x_{ijt} } } x_{ijt^{\prime}} \right] = 2n\,P_{i}$,

$E\left[\sum\nolimits_{jj^{\prime}} {\sum\nolimits_{tt^{\prime}} {x_{ijt} } } x_{ij^{\prime}t^{\prime}} \right] = 4n(n - 1)p_{i}^{2}$, and then it results:

$$ 2n\;E\left[\sum\nolimits_{i} {x_{i..}^{2} }\right ] = \sum\nolimits_{i} {p_{i} } + \sum\nolimits_{i} {P_{i} } + 2(n - 1)\sum\nolimits_{i} {p_{i}^{2} } . $$

(A9)

$$ \begin{aligned} {\mathbf{\cdot}} \,2ns\;E\left[ {x_{{...}}^{2} } \right] & = \left[ {\sum\nolimits_{{ijt}} {x_{{ijt}} } } \right]^{2} /2ns \\ E\left[ {\sum\nolimits_{{ijt}} {x_{{ijt}} ^{2} } } \right] & = E\left[ {\sum\nolimits_{{ijt}} {x_{{ijt}} ^{2} } } \right] + E\left[ {\sum\nolimits_{{ij}} {\sum\nolimits_{{tt^{\prime } }} {x_{{ijt}} } } x_{{ijt^{\prime } }} } \right] + E\left[ {\sum\nolimits_{i} {\sum\nolimits_{{jj^{\prime } tt^{\prime } }} {x_{{ijt}} } } x_{{ij^{\prime } t^{\prime } }} } \right] + E\left[ {\sum\nolimits_{{ii^{\prime } }} {\sum\nolimits_{{jj^{\prime } tt^{\prime } }} {x_{{ijt}} } } x_{{i^{\prime } j^{\prime } t^{\prime } }} } \right] \\ \end{aligned} $$

with i $\ne$ i', j $\ne$ j', t $\ne$ t';

$E\left[\sum\nolimits_{ijt} {x_{ijt}^{2} } \right] = 2n\,\sum\nolimits_{i} {p{}_{i}}$,

$E\left[\sum\nolimits_{ij} {\sum\nolimits_{tt^{\prime}} {x_{ijt} } } x_{ijt^{\prime}} \right] = 2n\,\sum\nolimits_{i} {P_{i} }$,

$E\left[\sum\nolimits_{i} {\sum\nolimits_{jj^{\prime}tt^{\prime}} {x_{ijt} } } x_{ij^{\prime}t^{\prime}} \right] = 4sn(n - 1)E_{i} \left[E_{jt} (x_{ijt} )\,E_{j^{\prime}t^{\prime}} (x_{ij^{\prime}t^{\prime}} )\right] = 4n(n - 1)\sum\nolimits_{i} {p_{i}^{2} } E\left[\sum\nolimits_{ii^{\prime}} {\sum\nolimits_{jj^{\prime}tt^{\prime}} {x_{ijt} } } x_{i^{\prime}j^{\prime}t^{\prime}} \right] = 4n^{2} \sum\nolimits_{ii^{\prime}} {E_{jt} (x_{ijt} )E_{j^{\prime}t^{\prime}} (x_{i^{\prime}j^{\prime}t^{\prime}} )} = 4n^{2} \sum\nolimits_{ii^{\prime}} {p_{i} } p_{i^{\prime}}$, then, it results:

$$ E\left[\sum\nolimits_{ijt} {x_{ijt}^{{}} }\right]^{2} /2ns = \sum\nolimits_{i} {p_{i} } /s + \sum\nolimits_{i} {P_{i} } /s + 2(n - 1)\;\sum\nolimits_{i} {p_{i}^{2} } /s + 2n\sum\nolimits_{ii^{\prime}} {p_{i} } p_{i^{\prime}} /s, $$

(A10)

Adding the subscript k to p_i and P_i for the frequency of allele k and the frequency of the homozygous genotype for this allele in population i, we get for the expected mean square E(MS) = E(SS)/df, in terms of p_ik and P_ik:

$$ E\left( {MS_{pop} } \right)_{k} = \sum\nolimits_{i} {p_{ik} } /s + \sum\nolimits_{i} {P_{ik} } /s + 2(n - 1)\sum\nolimits_{i} {p_{ik}^{2} /s} - \left[2n/s(s - 1)\right]\;\sum\nolimits_{ii^{\prime}} {p_{ik} } p_{i^{\prime}k} $$

(A11)

$$ E\left( {MS_{ind/pop} } \right)_{k} = \sum\nolimits_{i} {p_{ik} } /s + \sum\nolimits_{i} {P_{ik} } /s - 2\sum\nolimits_{i} {p_{ik}^{2} } /s, $$

(A12)

$$ E\left( {MS_{all/ind} } \right)_{k} = \sum\nolimits_{i} {p_{ik} } /s - \sum\nolimits_{i} {P_{ik} } /s, $$

(A13)

$$ E\left( {MS_{Tot} } \right)_{k} = \sum\nolimits_{i} {p_{ik} } /s - \left[\sum\nolimits_{i} {P_{ik} } /s + 2(n - 1)\,\sum\nolimits_{i} {p_{ik}^{2} /s} + 2n\;\sum\nolimits_{ii^{\prime}} {p_{ik} } p_{i^{\prime}k} /s\right]/\left( {{2}sn - {1}} \right). $$

(A14)

Summing over all alleles from (A13) we can write

$$ \sum\nolimits_{k} {E(MS_{all/ind} )_{k} = \,} \left[\sum\nolimits_{ik} {p_{ik} } (1 - p_{ik} ) + \sum\nolimits_{ik} {p_{ik}^{2} - \sum\nolimits_{ik} {P_{ik} } } \right]/s, $$

then with $\sum\nolimits_{ik} {p_{ik} } (1 - p_{ik} )/s = H_{S}$, $\sum\nolimits_{ik} {p_{ik}^{2} } /s = 1 - H_{S}$ and $\sum\nolimits_{ik} {P_{ik}^{{}} } /s = 1 - H_{0}$, $H_{0}$ being the average frequency of heterozygotes across populations, it results

$$ \sum\nolimits_{k} {E(MS_{all/ind} )_{k} = } \,H_{0} . $$

(A15)

Noting that $\sum\nolimits_{ik} {P_{ik} /s - \sum\nolimits_{ik} {p_{ik}^{2} /s} = } \,H{}_{S} - H_{0} = \overline{\Delta }$ the global departure from random mating equilibrium, from (A12) we can write

$$ \sum\limits_{k} {E(MS_{ind/pop} } )_{k} = H_{0} + 2\,\left(\sum\limits_{ik} {P_{ik} } - \sum\limits_{ik} {p_{ik}^{2} } \right) = H_{0} + 2\;(H{}_{S} - H_{0} ). $$

(A16)

Finally, by noting that $D_{ST}^{^{\prime}} = [(s - 1)\sum\nolimits_{ik} {p_{ik}^{2} } - \sum\nolimits_{ii^{\prime}k} {p_{ik} p_{i^{\prime}k} ]/\;} s(s - 1)$, from (A11) we can write in terms of Nei’s parameters the sum over all alleles of the between-population expected mean squares:

$$ \sum\nolimits_{k} {E(MS_{pop} )_{k} } = H_{0} + 2\,(H_{S} - H_{0} ) + 2n\;D_{ST}^{^{\prime}} . $$

(A17)