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<?xml version="1.0"?>
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<title>Notes on Population Genetics</title>
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<h1 class="ltx_title ltx_title_document">Notes on Population Genetics</h1>
<div class="ltx_date ltx_role_creation"/>
<div class="ltx_creator ltx_role_author">
<div class="ltx_personname">Graham Coop<math xmlns="http://www.w3.org/1998/Math/MathML" id="m1" class="ltx_Math" display="inline" alttext="{}^{1}"><semantics><msup><mi/><mn>1</mn></msup><annotation encoding="application/x-tex">{}^{1}</annotation></semantics></math>
<br class="ltx_break"/><math xmlns="http://www.w3.org/1998/Math/MathML" id="m2" class="ltx_Math" display="inline" alttext="{}^{1}"><semantics><msup><mi/><mn>1</mn></msup><annotation encoding="application/x-tex">{}^{1}</annotation></semantics></math><span class="ltx_text ltx_font_small"> Department of Evolution and Ecology & Center for Population Biology,
<br class="ltx_break"/>University of California, Davis.
<br class="ltx_break"/>To whom correspondence should be addressed: <span class="ltx_text ltx_font_typewriter">gmcoop@ucdavis.edu
<br class="ltx_break"/></span>This work is licensed under a Creative Commons Attribution 3.0 Unported License.
<br class="ltx_break"/>http://creativecommons.org/licenses/by/3.0/
<br class="ltx_break"/>i.e. you are free to reuse and remix this work, but please include an attribution to the original.
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<h6>Contents:</h6>
<ul class="ltx_toclist">
<li class="ltx_tocentry"><a href="#S1" title="1 Allele and Genotype frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Allele and Genotype frequencies</span></a>
<ul class="ltx_toclist ltx_toc_section">
<li class="ltx_tocentry"><a href="#S1.SS1" title="1.1 Allele frequencies ‣ 1 Allele and Genotype frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Allele frequencies</span></a></li>
<li class="ltx_tocentry"><a href="#S1.SS2" title="1.2 Hardy-Weinberg ‣ 1 Allele and Genotype frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.2 </span>Hardy-Weinberg</span></a></li>
<li class="ltx_tocentry"><a href="#S1.SS3" title="1.3 Relatedness coefficients ‣ 1 Allele and Genotype frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.3 </span>Relatedness coefficients</span></a></li>
<li class="ltx_tocentry"><a href="#S1.SS4" title="1.4 Inbreeding ‣ 1 Allele and Genotype frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.4 </span>Inbreeding</span></a></li>
<li class="ltx_tocentry"><a href="#S1.SS5" title="1.5 Calculating inbreeding coefficients from data ‣ 1 Allele and Genotype frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.5 </span>Calculating inbreeding coefficients from data</span></a></li>
<li class="ltx_tocentry"><a href="#S1.SS6" title="1.6 Summarizing Population structure ‣ 1 Allele and Genotype frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.6 </span>Summarizing Population structure</span></a></li>
</ul></li>
<li class="ltx_tocentry"><a href="#S2" title="2 The phenotypic resemblance between relatives ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>The phenotypic resemblance between relatives</span></a>
<ul class="ltx_toclist ltx_toc_section">
<li class="ltx_tocentry"><a href="#S2.SS6.SSS1" title="2.0.1 Additive genetic variance and heritability ‣ 2 The phenotypic resemblance between relatives ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.0.1 </span>Additive genetic variance and heritability</span></a></li>
<li class="ltx_tocentry"><a href="#S2.SS6.SSS2" title="2.0.2 The covariance between relatives ‣ 2 The phenotypic resemblance between relatives ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.0.2 </span>The covariance between relatives</span></a></li>
<li class="ltx_tocentry"><a href="#S2.SS6.SSS3" title="2.0.3 The response to selection ‣ 2 The phenotypic resemblance between relatives ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.0.3 </span>The response to selection</span></a></li>
<li class="ltx_tocentry"><a href="#S2.SS6.SSS4" title="2.0.4 Hamiliton’s Rule and the evolution of altruistic and selfish behaviours ‣ 2 The phenotypic resemblance between relatives ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.0.4 </span>Hamiliton’s Rule and the evolution of altruistic and
selfish behaviours</span></a></li>
</ul></li>
<li class="ltx_tocentry"><a href="#S3" title="3 Correlations between loci, linkage disequilibrium, and recombination. ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Correlations between loci, linkage disequilibrium, and recombination.</span></a></li>
<li class="ltx_tocentry"><a href="#S4" title="4 One locus models of selection ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>One locus models of selection</span></a>
<ul class="ltx_toclist ltx_toc_section">
<li class="ltx_tocentry"><a href="#S4.SS1" title="4.1 fitness ‣ 4 One locus models of selection ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>fitness</span></a></li>
<li class="ltx_tocentry"><a href="#S4.SS2" title="4.2 Haploid selection model ‣ 4 One locus models of selection ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Haploid selection model</span></a></li>
<li class="ltx_tocentry"><a href="#S4.SS3" title="4.3 Diploid model ‣ 4 One locus models of selection ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Diploid model</span></a>
<ul class="ltx_toclist ltx_toc_subsection">
<li class="ltx_tocentry"><a href="#S4.SS3.SSS1" title="4.3.1 Diploid directional selection ‣ 4.3 Diploid model ‣ 4 One locus models of selection ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3.1 </span>Diploid directional selection</span></a></li>
<li class="ltx_tocentry"><a href="#S4.SS3.SSS2" title="4.3.2 heterozygote advantage ‣ 4.3 Diploid model ‣ 4 One locus models of selection ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3.2 </span>heterozygote advantage</span></a></li>
</ul></li>
<li class="ltx_tocentry"><a href="#S4.SS4" title="4.4 Mutation Selection Balance ‣ 4 One locus models of selection ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.4 </span>Mutation Selection Balance</span></a>
<ul class="ltx_toclist ltx_toc_subsection">
<li class="ltx_tocentry"><a href="#S4.SS4.SSS1" title="4.4.1 Inbreeding depression ‣ 4.4 Mutation Selection Balance ‣ 4 One locus models of selection ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.4.1 </span>Inbreeding depression</span></a></li>
</ul></li>
<li class="ltx_tocentry"><a href="#S4.SS5" title="4.5 Migration-Selection Balance ‣ 4 One locus models of selection ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.5 </span>Migration-Selection Balance</span></a>
<ul class="ltx_toclist ltx_toc_subsection">
<li class="ltx_tocentry"><a href="#S4.SS5.SSS1" title="4.5.1 Some theory of the spatial distribution of allele frequencies under deterministic models of selection ‣ 4.5 Migration-Selection Balance ‣ 4 One locus models of selection ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.5.1 </span>Some theory of the spatial distribution of allele
frequencies under deterministic models of selection</span></a></li>
</ul></li>
</ul></li>
<li class="ltx_tocentry"><a href="#S5" title="5 Stochasticity and Genetic Drift in allele frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Stochasticity and Genetic Drift in allele frequencies</span></a>
<ul class="ltx_toclist ltx_toc_section">
<li class="ltx_tocentry"><a href="#S5.SS1" title="5.1 Stochastic loss of strongly selected alleles ‣ 5 Stochasticity and Genetic Drift in allele frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1 </span>Stochastic loss of strongly selected alleles</span></a></li>
<li class="ltx_tocentry"><a href="#S5.SS2" title="5.2 The interaction between genetic drift and weak selection. ‣ 5 Stochasticity and Genetic Drift in allele frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2 </span>The interaction between genetic drift and weak selection.</span></a>
<ul class="ltx_toclist ltx_toc_subsection">
<li class="ltx_tocentry"><a href="#S5.SS2.SSS1" title="5.2.1 The fixation of slightly deleterious alleles. ‣ 5.2 The interaction between genetic drift and weak selection. ‣ 5 Stochasticity and Genetic Drift in allele frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2.1 </span>The fixation of slightly deleterious alleles.</span></a></li>
<li class="ltx_tocentry"><a href="#S5.SS2.SSS2" title="5.2.2 A Sketch Proof of the probability of fixation of weakly selected alleles ‣ 5.2 The interaction between genetic drift and weak selection. ‣ 5 Stochasticity and Genetic Drift in allele frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2.2 </span>A Sketch Proof of the probability of fixation of
weakly selected alleles</span></a></li>
</ul></li>
</ul></li>
<li class="ltx_tocentry"><a href="#S6" title="6 Genetic drift and Neutral alleles ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Genetic drift and Neutral alleles</span></a>
<ul class="ltx_toclist ltx_toc_section">
<li class="ltx_tocentry"><a href="#S6.SS1" title="6.1 The fixation of neutral alleles ‣ 6 Genetic drift and Neutral alleles ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.1 </span>The fixation of neutral alleles</span></a></li>
<li class="ltx_tocentry"><a href="#S6.SS2" title="6.2 Loss of heterozygosity due to to drift. ‣ 6 Genetic drift and Neutral alleles ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.2 </span>Loss of heterozygosity due to to drift.</span></a></li>
<li class="ltx_tocentry"><a href="#S6.SS3" title="6.3 Levels of diversity maintained by a balance between mutation and drift. ‣ 6 Genetic drift and Neutral alleles ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.3 </span>Levels of diversity maintained by a balance between
mutation and drift.</span></a></li>
<li class="ltx_tocentry"><a href="#S6.SS4" title="6.4 The effective population size. ‣ 6 Genetic drift and Neutral alleles ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.4 </span>The effective population size.</span></a></li>
<li class="ltx_tocentry"><a href="#S6.SS5" title="6.5 The coalescent process of a sample of alleles. ‣ 6 Genetic drift and Neutral alleles ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.5 </span>The coalescent process of a sample of alleles.</span></a></li>
<li class="ltx_tocentry"><a href="#S6.SS6" title="6.6 Deviations from the constant population model. ‣ 6 Genetic drift and Neutral alleles ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.6 </span>Deviations from the constant population model.</span></a></li>
<li class="ltx_tocentry"><a href="#S6.SS7" title="6.7 The coalescent and population structure ‣ 6 Genetic drift and Neutral alleles ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.7 </span>The coalescent and population structure</span></a></li>
</ul></li>
<li class="ltx_tocentry"><a href="#S7" title="7 The effect of linked selection on patterns of neutral diversity ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7 </span>The effect of linked selection on patterns of neutral diversity</span></a>
<ul class="ltx_toclist ltx_toc_section">
<li class="ltx_tocentry"><a href="#S7.SS1" title="7.1 A simple recurrent model of selective sweeps ‣ 7 The effect of linked selection on patterns of neutral diversity ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7.1 </span>A simple recurrent model of selective sweeps</span></a></li>
</ul></li>
</ul>
<div id="S1" class="ltx_section">
<h2 class="ltx_title ltx_title_section"><span class="ltx_tag ltx_tag_section">1 </span>Allele and Genotype frequencies</h2>
<div id="S1.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection"><span class="ltx_tag ltx_tag_subsection">1.1 </span>Allele frequencies</h3>
<div id="S1.SS1.p1" class="ltx_para">
<p class="ltx_p">Consider a diploid autosomal locus segregating two alleles (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS1.p1.m1" class="ltx_Math" display="inline" alttext="1"><semantics><mn>1</mn><annotation encoding="application/x-tex">1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS1.p1.m2" class="ltx_Math" display="inline" alttext="2"><semantics><mn>2</mn><annotation encoding="application/x-tex">2</annotation></semantics></math>). Let’s say that <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS1.p1.m3" class="ltx_Math" alttext="f_{11}" display="inline"><semantics><msub><mi>f</mi><mn>11</mn></msub><annotation encoding="application/x-tex">f_{11}</annotation></semantics></math> is the frequency of <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS1.p1.m4" class="ltx_Math" alttext="11" display="inline"><semantics><mn>11</mn><annotation encoding="application/x-tex">11</annotation></semantics></math> homozygotes and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS1.p1.m5" class="ltx_Math" alttext="f_{12}" display="inline"><semantics><msub><mi>f</mi><mn>12</mn></msub><annotation encoding="application/x-tex">f_{12}</annotation></semantics></math> is the frequency of <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS1.p1.m6" class="ltx_Math" display="inline" alttext="12"><semantics><mn>12</mn><annotation encoding="application/x-tex">12</annotation></semantics></math> heterozygotes. The frequency of allele <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS1.p1.m7" class="ltx_Math" display="inline" alttext="1"><semantics><mn>1</mn><annotation encoding="application/x-tex">1</annotation></semantics></math> in the population is</p>
<table id="S1.E1" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.E1.m1" class="ltx_Math" display="block" alttext="p=f_{11}+f_{12}/2"><semantics><mrow><mi>p</mi><mo>=</mo><mrow><msub><mi>f</mi><mn>11</mn></msub><mo>+</mo><mrow><msub><mi>f</mi><mn>12</mn></msub><mo>/</mo><mn>2</mn></mrow></mrow></mrow><annotation encoding="application/x-tex">p=f_{11}+f_{12}/2</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(1)</span></td></tr>
</table>
<p class="ltx_p">note that this makes no assumption of Hardy-Weinberg [see below]. The frequency of the alternate allele (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS1.p1.m8" class="ltx_Math" alttext="2" display="inline"><semantics><mn>2</mn><annotation encoding="application/x-tex">2</annotation></semantics></math>) is then just <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS1.p1.m9" class="ltx_Math" alttext="q=1-p" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mrow><mn>1</mn><mo>-</mo><mi>p</mi></mrow></mrow><annotation encoding="application/x-tex">q=1-p</annotation></semantics></math>.</p>
</div>
</div>
<div id="S1.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection"><span class="ltx_tag ltx_tag_subsection">1.2 </span>Hardy-Weinberg</h3>
<div id="S1.SS2.p1" class="ltx_para">
<p class="ltx_p">Imagine a population mating at random with respect to our genotypes, i.e.
no inbreeding, no population structure, no sex differences in allele frequencies.</p>
</div>
<div id="S1.SS2.p2" class="ltx_para">
<p class="ltx_p">The frequency of allele <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m1" class="ltx_Math" alttext="1" display="inline"><semantics><mn>1</mn><annotation encoding="application/x-tex">1</annotation></semantics></math> in the population at the time of reproduction is <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m2" class="ltx_Math" display="inline" alttext="p"><semantics><mi>p</mi><annotation encoding="application/x-tex">p</annotation></semantics></math>.
A <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m3" class="ltx_Math" display="inline" alttext="11"><semantics><mn>11</mn><annotation encoding="application/x-tex">11</annotation></semantics></math> genotype is made by reaching out into our population and selecting two <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m4" class="ltx_Math" alttext="1" display="inline"><semantics><mn>1</mn><annotation encoding="application/x-tex">1</annotation></semantics></math> alelle gametes to form a zygote. Therefore, the probability that
our individual is a <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m5" class="ltx_Math" alttext="11" display="inline"><semantics><mn>11</mn><annotation encoding="application/x-tex">11</annotation></semantics></math> homozygote is <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m6" class="ltx_Math" display="inline" alttext="p^{2}"><semantics><msup><mi>p</mi><mn>2</mn></msup><annotation encoding="application/x-tex">p^{2}</annotation></semantics></math>. This probability is also
the expected frequency of the <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m7" class="ltx_Math" display="inline" alttext="11"><semantics><mn>11</mn><annotation encoding="application/x-tex">11</annotation></semantics></math> homozygote in the population.The
expected frequency of our three genotypes is
</p>
<table class="ltx_tabular ltx_centering">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m8" class="ltx_Math" alttext="f_{11}" display="inline"><semantics><msub><mi>f</mi><mn>11</mn></msub><annotation encoding="application/x-tex">f_{11}</annotation></semantics></math></td>
<td class="ltx_td ltx_align_center ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m9" class="ltx_Math" display="inline" alttext="f_{12}"><semantics><msub><mi>f</mi><mn>12</mn></msub><annotation encoding="application/x-tex">f_{12}</annotation></semantics></math></td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m10" class="ltx_Math" alttext="f_{22}" display="inline"><semantics><msub><mi>f</mi><mn>22</mn></msub><annotation encoding="application/x-tex">f_{22}</annotation></semantics></math></td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_l ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m11" class="ltx_Math" alttext="p^{2}" display="inline"><semantics><msup><mi>p</mi><mn>2</mn></msup><annotation encoding="application/x-tex">p^{2}</annotation></semantics></math></td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m12" class="ltx_Math" alttext="2pq" display="inline"><semantics><mrow><mn>2</mn><mo>⁢</mo><mi>p</mi><mo>⁢</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">2pq</annotation></semantics></math></td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m13" class="ltx_Math" display="inline" alttext="q^{2}"><semantics><msup><mi>q</mi><mn>2</mn></msup><annotation encoding="application/x-tex">q^{2}</annotation></semantics></math></td></tr>
</tbody>
</table>
<p class="ltx_p">Note that we only need to assume random mating with
respect to our allele in order for these expected frequencies to hold,
as long at <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m14" class="ltx_Math" display="inline" alttext="p"><semantics><mi>p</mi><annotation encoding="application/x-tex">p</annotation></semantics></math> is the frequency of the <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS2.p2.m15" class="ltx_Math" display="inline" alttext="1"><semantics><mn>1</mn><annotation encoding="application/x-tex">1</annotation></semantics></math> allele in the population at
the time when gametes fuse.</p>
</div>
<div id="S1.SS2.fig1" class="ltx_figure"><object data="x1.png" id="S1.SS2.g1" class="ltx_graphics ltx_centering" width="538" height="327" alt=""/>
</div>
<div id="S1.SS2.fig2" class="ltx_figure"><object data="x2.png" id="S1.SS2.g2" class="ltx_graphics ltx_centering" width="338" height="280" alt=""/>
</div>
</div>
<div id="S1.SS3" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection"><span class="ltx_tag ltx_tag_subsection">1.3 </span>Relatedness coefficients</h3>
<div id="S1.SS3.p1" class="ltx_para">
<p class="ltx_p">We will define two alleles to be identical by descent if they are
identical due to a common ancestor in the past few generations (for the moment ignoring the
possibility of mutation). For example parent and child share exactly
one allele identical by descent at a locus (assuming that the two
parents of the child are randomly mated individuals from the population).
<br class="ltx_break"/></p>
</div>
<div id="S1.SS3.p2" class="ltx_para">
<p class="ltx_p">A key quantity is the probability that our pair of individuals share
0, 1, or 2 alleles identical by descent. We denote these probabilities
by <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS3.p2.m1" class="ltx_Math" display="inline" alttext="r_{0}"><semantics><msub><mi>r</mi><mn>0</mn></msub><annotation encoding="application/x-tex">r_{0}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS3.p2.m2" class="ltx_Math" alttext="r_{1}" display="inline"><semantics><msub><mi>r</mi><mn>1</mn></msub><annotation encoding="application/x-tex">r_{1}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS3.p2.m3" class="ltx_Math" alttext="r_{2}" display="inline"><semantics><msub><mi>r</mi><mn>2</mn></msub><annotation encoding="application/x-tex">r_{2}</annotation></semantics></math> respectively. See Table <a href="#S1.T1" title="Table 1 ‣ 1.3 Relatedness coefficients ‣ 1 Allele and Genotype frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a> for
some examples.
<br class="ltx_break"/></p>
</div>
<div id="S1.SS3.p3" class="ltx_para">
<p class="ltx_p">One summary of relatedness, which will be of help to us, is the probability that one allele picked at
random from each of our two individuals is identical by descent.
We call this quantity the coefficient of kinship of individual <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS3.p3.m1" class="ltx_Math" display="inline" alttext="i"><semantics><mi>i</mi><annotation encoding="application/x-tex">i</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS3.p3.m2" class="ltx_Math" alttext="j" display="inline"><semantics><mi>j</mi><annotation encoding="application/x-tex">j</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS3.p3.m3" class="ltx_Math" display="inline" alttext="F_{ij}"><semantics><msub><mi>F</mi><mrow><mi>i</mi><mo>⁢</mo><mi>j</mi></mrow></msub><annotation encoding="application/x-tex">F_{ij}</annotation></semantics></math>, and can
we calculate it as</p>
<table id="S1.E2" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.E2.m1" class="ltx_Math" display="block" alttext="F_{ij}=0\times r_{0}+\frac{1}{4}r_{1}+\frac{1}{2}r_{2}"><semantics><mrow><msub><mi>F</mi><mrow><mi>i</mi><mo>⁢</mo><mi>j</mi></mrow></msub><mo>=</mo><mrow><mrow><mn>0</mn><mo>×</mo><msub><mi>r</mi><mn>0</mn></msub></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>⁢</mo><msub><mi>r</mi><mn>1</mn></msub></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><msub><mi>r</mi><mn>2</mn></msub></mrow></mrow></mrow><annotation encoding="application/x-tex">F_{ij}=0\times r_{0}+\frac{1}{4}r_{1}+\frac{1}{2}r_{2}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(2)</span></td></tr>
</table>
</div>
<div id="S1.SS3.p4" class="ltx_para">
<p class="ltx_p">This quantity will appear multiple times, in both our discussion of
inbreeding and our discussion of the phenotypic resemblance between relatives.
<br class="ltx_break"/></p>
</div>
<div id="S1.T1" class="ltx_table">
<table class="ltx_tabular ltx_centering">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left ltx_border_l ltx_border_r ltx_border_t">Relationship (i,j)<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.T1.m1" class="ltx_Math" alttext="{}^{*}" display="inline"><semantics><msup><mi/><mo>*</mo></msup><annotation encoding="application/x-tex">{}^{*}</annotation></semantics></math></th>
<th class="ltx_td ltx_align_center ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.T1.m2" class="ltx_Math" display="inline" alttext="r_{0}"><semantics><msub><mi>r</mi><mn>0</mn></msub><annotation encoding="application/x-tex">r_{0}</annotation></semantics></math></th>
<th class="ltx_td ltx_align_center ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.T1.m3" class="ltx_Math" display="inline" alttext="r_{1}"><semantics><msub><mi>r</mi><mn>1</mn></msub><annotation encoding="application/x-tex">r_{1}</annotation></semantics></math></th>
<th class="ltx_td ltx_align_center ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.T1.m4" class="ltx_Math" alttext="r_{2}" display="inline"><semantics><msub><mi>r</mi><mn>2</mn></msub><annotation encoding="application/x-tex">r_{2}</annotation></semantics></math></th>
<th class="ltx_td ltx_align_center ltx_border_r ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.T1.m5" class="ltx_Math" alttext="F_{ij}" display="inline"><semantics><msub><mi>F</mi><mrow><mi>i</mi><mo>⁢</mo><mi>j</mi></mrow></msub><annotation encoding="application/x-tex">F_{ij}</annotation></semantics></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_r ltx_border_t">parent-child</td>
<td class="ltx_td ltx_align_center ltx_border_t">0</td>
<td class="ltx_td ltx_align_center ltx_border_t">1</td>
<td class="ltx_td ltx_align_center ltx_border_t">0</td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t">1/4</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_r">full siblings</td>
<td class="ltx_td ltx_align_center">1/4</td>
<td class="ltx_td ltx_align_center">1/2</td>
<td class="ltx_td ltx_align_center">1/4</td>
<td class="ltx_td ltx_align_center ltx_border_r">1/4</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_l ltx_border_r">identical (monzygotic) twins</td>
<td class="ltx_td ltx_align_center">0</td>
<td class="ltx_td ltx_align_center">0</td>
<td class="ltx_td ltx_align_center">1</td>
<td class="ltx_td ltx_align_center ltx_border_r">1/2</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left ltx_border_b ltx_border_l ltx_border_r"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.T1.m6" class="ltx_Math" display="inline" alttext="1^{st}"><semantics><msup><mn>1</mn><mrow><mi>s</mi><mo>⁢</mo><mi>t</mi></mrow></msup><annotation encoding="application/x-tex">1^{st}</annotation></semantics></math> cousins</td>
<td class="ltx_td ltx_align_center ltx_border_b">3/4</td>
<td class="ltx_td ltx_align_center ltx_border_b">1/4</td>
<td class="ltx_td ltx_align_center ltx_border_b">0</td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r">1/16</td></tr>
</tbody>
</table>
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 1: </span>Probability that two individuals of a given relationship share 0, 1, or 2 alleles
identical by descent. <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.T1.m8" class="ltx_Math" display="inline" alttext="{}^{*}"><semantics><msup><mi/><mo>*</mo></msup><annotation encoding="application/x-tex">{}^{*}</annotation></semantics></math> assuming this is the only relationship
the pair of individuals share (above that expected from randomly
sampling individuals from the population). </div>
</div>
</div>
<div id="S1.SS4" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection"><span class="ltx_tag ltx_tag_subsection">1.4 </span>Inbreeding</h3>
<div id="S1.SS4.p1" class="ltx_para">
<p class="ltx_p">We can define an inbred individual as an individual whose parents are
more closely related to each other than two random individuals drawn
from some reference population.
<br class="ltx_break"/></p>
</div>
<div id="S1.SS4.p2" class="ltx_para">
<p class="ltx_p">When two related individuals produce an offspring, that individual can
receive two alleles that are identical by descent, i.e. they
can be homozygous by descent (sometimes termed autozygous), due to the
fact that they have two copies of an allele through different paths
through the pedigree. This increased likelihood of being homozygous
relative to an outbred individual is the most obvious effect of
inbreeding, and the one that will be of most interest to us as it
underlies a lot of our ideas about inbreeding depression and
population structure.
<br class="ltx_break"/></p>
</div>
<div id="S1.SS4.p3" class="ltx_para">
<p class="ltx_p">As the offspring receives a random allele from each parent (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS4.p3.m1" class="ltx_Math" alttext="i" display="inline"><semantics><mi>i</mi><annotation encoding="application/x-tex">i</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS4.p3.m2" class="ltx_Math" alttext="j" display="inline"><semantics><mi>j</mi><annotation encoding="application/x-tex">j</annotation></semantics></math>) the
probability that those two alleles are identical by descent is our
kinship coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS4.p3.m3" class="ltx_Math" alttext="F_{ij}" display="inline"><semantics><msub><mi>F</mi><mrow><mi>i</mi><mo>⁢</mo><mi>j</mi></mrow></msub><annotation encoding="application/x-tex">F_{ij}</annotation></semantics></math> of the two parents (i.e. the quantity we
defined in eqn. <a href="#S1.E2" title="(2) ‣ 1.3 Relatedness coefficients ‣ 1 Allele and Genotype frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>). This follows from the fact that
our child’s genotype is made by sampling an allele at random from each
of our parents.
<br class="ltx_break"/></p>
</div>
<div id="S1.SS4.p4" class="ltx_para">
<p class="ltx_p">The only way the offspring can be heterozygous (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS4.p4.m1" class="ltx_Math" display="inline" alttext="A_{1}A_{2}"><semantics><mrow><msub><mi>A</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>A</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_{1}A_{2}</annotation></semantics></math>) is if their two alleles at a
locus, are not IBD (otherwise they’d necessarily be homozygous). Therefore, the probability that they are
heterozygous is</p>
<table id="S1.E3" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.E3.m1" class="ltx_Math" alttext="(1-F)2pq." display="block"><semantics><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>F</mi></mrow><mo>)</mo></mrow><mo>⁢</mo><mn>2</mn><mo>⁢</mo><mi>p</mi><mo>⁢</mo><mi>q</mi></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex">(1-F)2pq.</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(3)</span></td></tr>
</table>
<p class="ltx_p">Our offspring can homozygous for the <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS4.p4.m2" class="ltx_Math" display="inline" alttext="A_{1}"><semantics><msub><mi>A</mi><mn>1</mn></msub><annotation encoding="application/x-tex">A_{1}</annotation></semantics></math> allele two different ways,
they can have two non-IBD alleles which happen to be the <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS4.p4.m3" class="ltx_Math" alttext="A_{1}" display="inline"><semantics><msub><mi>A</mi><mn>1</mn></msub><annotation encoding="application/x-tex">A_{1}</annotation></semantics></math> allele ,
or their two alleles can be IBD, such that they inherited the <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS4.p4.m4" class="ltx_Math" display="inline" alttext="A_{1}"><semantics><msub><mi>A</mi><mn>1</mn></msub><annotation encoding="application/x-tex">A_{1}</annotation></semantics></math> by
two different routes from the same ancestor. Thus the probability they
are homozygous is</p>
<table id="S1.E4" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.E4.m1" class="ltx_Math" display="block" alttext="(1-F)p^{2}+Fp"><semantics><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>F</mi></mrow><mo>)</mo></mrow><mo>⁢</mo><msup><mi>p</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mi>F</mi><mo>⁢</mo><mi>p</mi></mrow></mrow><annotation encoding="application/x-tex">(1-F)p^{2}+Fp</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(4)</span></td></tr>
</table>
<p class="ltx_p">Therefore, our three genotype probabilities can be written as given in
Table <a href="#S1.T2" title="Table 2 ‣ 1.4 Inbreeding ‣ 1 Allele and Genotype frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>, generalizing our Hardy Weinberg
proportions.
<br class="ltx_break"/></p>
</div>
<div id="S1.T2" class="ltx_table">
<table class="ltx_tabular ltx_centering">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_l ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.T2.m1" class="ltx_Math" display="inline" alttext="f_{11}"><semantics><msub><mi>f</mi><mn>11</mn></msub><annotation encoding="application/x-tex">f_{11}</annotation></semantics></math></td>
<td class="ltx_td ltx_align_center ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.T2.m2" class="ltx_Math" display="inline" alttext="f_{12}"><semantics><msub><mi>f</mi><mn>12</mn></msub><annotation encoding="application/x-tex">f_{12}</annotation></semantics></math></td>
<td class="ltx_td ltx_align_center ltx_border_r ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.T2.m3" class="ltx_Math" alttext="f_{22}" display="inline"><semantics><msub><mi>f</mi><mn>22</mn></msub><annotation encoding="application/x-tex">f_{22}</annotation></semantics></math></td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_l ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.T2.m4" class="ltx_Math" alttext="(1-F)p^{2}+Fp" display="inline"><semantics><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>F</mi></mrow><mo>)</mo></mrow><mo>⁢</mo><msup><mi>p</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mi>F</mi><mo>⁢</mo><mi>p</mi></mrow></mrow><annotation encoding="application/x-tex">(1-F)p^{2}+Fp</annotation></semantics></math></td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.T2.m5" class="ltx_Math" alttext="(1-F)2pq" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>F</mi></mrow><mo>)</mo></mrow><mo>⁢</mo><mn>2</mn><mo>⁢</mo><mi>p</mi><mo>⁢</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">(1-F)2pq</annotation></semantics></math></td>
<td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.T2.m6" class="ltx_Math" alttext="(1-F)q^{2}+Fq" display="inline"><semantics><mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mi>F</mi></mrow><mo>)</mo></mrow><mo>⁢</mo><msup><mi>q</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mi>F</mi><mo>⁢</mo><mi>q</mi></mrow></mrow><annotation encoding="application/x-tex">(1-F)q^{2}+Fq</annotation></semantics></math></td></tr>
</tbody>
</table>
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 2: </span><span class="ltx_text ltx_font_bold">Generalized Hardy Weinberg</span></div>
</div>
<div id="S1.SS4.p5" class="ltx_para">
<p class="ltx_p">Note that the generalized Hardy Weinberg proportions completely
specify our genotype probabilities, as there are two parameters (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS4.p5.m1" class="ltx_Math" display="inline" alttext="p"><semantics><mi>p</mi><annotation encoding="application/x-tex">p</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS4.p5.m2" class="ltx_Math" display="inline" alttext="F"><semantics><mi>F</mi><annotation encoding="application/x-tex">F</annotation></semantics></math>)
and two degrees of freedom (as our frequencies have to sum to one).
Therefore, any combination of genotype frequencies at a biallelic site
can be specified by a combination of <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS4.p5.m3" class="ltx_Math" alttext="p" display="inline"><semantics><mi>p</mi><annotation encoding="application/x-tex">p</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS4.p5.m4" class="ltx_Math" display="inline" alttext="F"><semantics><mi>F</mi><annotation encoding="application/x-tex">F</annotation></semantics></math>.
<br class="ltx_break"/></p>
</div>
</div>
<div id="S1.SS5" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection"><span class="ltx_tag ltx_tag_subsection">1.5 </span>Calculating inbreeding coefficients from data</h3>
<div id="S1.SS5.p1" class="ltx_para">
<p class="ltx_p">If the observed heterozygosity is <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS5.p1.m1" class="ltx_Math" alttext="f_{12}" display="inline"><semantics><msub><mi>f</mi><mn>12</mn></msub><annotation encoding="application/x-tex">f_{12}</annotation></semantics></math> then an estimate of
our inbreeding coefficient is</p>
<table id="S1.E5" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.E5.m1" class="ltx_Math" display="block" alttext="\hat{F}=1-\frac{f_{12}}{2pq}=\frac{2pq-f_{12}}{2pq}"><semantics><mrow><mover accent="true"><mi>F</mi><mo>^</mo></mover><mo>=</mo><mrow><mn>1</mn><mo>-</mo><mfrac><msub><mi>f</mi><mn>12</mn></msub><mrow><mn>2</mn><mo>⁢</mo><mi>p</mi><mo>⁢</mo><mi>q</mi></mrow></mfrac></mrow><mo>=</mo><mfrac><mrow><mrow><mn>2</mn><mo>⁢</mo><mi>p</mi><mo>⁢</mo><mi>q</mi></mrow><mo>-</mo><msub><mi>f</mi><mn>12</mn></msub></mrow><mrow><mn>2</mn><mo>⁢</mo><mi>p</mi><mo>⁢</mo><mi>q</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\hat{F}=1-\frac{f_{12}}{2pq}=\frac{2pq-f_{12}}{2pq}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(5)</span></td></tr>
</table>
<p class="ltx_p">where <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS5.p1.m2" class="ltx_Math" alttext="p" display="inline"><semantics><mi>p</mi><annotation encoding="application/x-tex">p</annotation></semantics></math> is the frequency of the allele in our reference
population. This can be rewritten in terms of the observed heterozygosity (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS5.p1.m3" class="ltx_Math" display="inline" alttext="H_{O}=f_{12}"><semantics><mrow><msub><mi>H</mi><mi>O</mi></msub><mo>=</mo><msub><mi>f</mi><mn>12</mn></msub></mrow><annotation encoding="application/x-tex">H_{O}=f_{12}</annotation></semantics></math>)
and expected heterozygosity (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS5.p1.m4" class="ltx_Math" display="inline" alttext="H_{E}=2pq"><semantics><mrow><msub><mi>H</mi><mi>E</mi></msub><mo>=</mo><mrow><mn>2</mn><mo>⁢</mo><mi>p</mi><mo>⁢</mo><mi>q</mi></mrow></mrow><annotation encoding="application/x-tex">H_{E}=2pq</annotation></semantics></math>)</p>
<table id="S1.E6" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.E6.m1" class="ltx_Math" display="block" alttext="\hat{F}=\frac{H_{E}-H_{O}}{H_{E}}."><semantics><mrow><mrow><mover accent="true"><mi>F</mi><mo>^</mo></mover><mo>=</mo><mfrac><mrow><msub><mi>H</mi><mi>E</mi></msub><mo>-</mo><msub><mi>H</mi><mi>O</mi></msub></mrow><msub><mi>H</mi><mi>E</mi></msub></mfrac></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex">\hat{F}=\frac{H_{E}-H_{O}}{H_{E}}.</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(6)</span></td></tr>
</table>
<p class="ltx_p">If we have multiple loci we can replace <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS5.p1.m5" class="ltx_Math" alttext="H_{O}" display="inline"><semantics><msub><mi>H</mi><mi>O</mi></msub><annotation encoding="application/x-tex">H_{O}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS5.p1.m6" class="ltx_Math" display="inline" alttext="H_{E}"><semantics><msub><mi>H</mi><mi>E</mi></msub><annotation encoding="application/x-tex">H_{E}</annotation></semantics></math> by their means
over loci <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS5.p1.m7" class="ltx_Math" display="inline" alttext="\bar{H}_{O}"><semantics><msub><mover accent="true"><mi>H</mi><mo>¯</mo></mover><mi>O</mi></msub><annotation encoding="application/x-tex">\bar{H}_{O}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS5.p1.m8" class="ltx_Math" alttext="\bar{H}_{E}" display="inline"><semantics><msub><mover accent="true"><mi>H</mi><mo>¯</mo></mover><mi>E</mi></msub><annotation encoding="application/x-tex">\bar{H}_{E}</annotation></semantics></math>.
<br class="ltx_break"/></p>
</div>
</div>
<div id="S1.SS6" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection"><span class="ltx_tag ltx_tag_subsection">1.6 </span>Summarizing Population structure</h3>
<div id="S1.SS6.p1" class="ltx_para">
<p class="ltx_p">Our estimated inbreeding coefficient gives us a nice way to take a
first look at population structure.
<br class="ltx_break"/></p>
</div>
<div id="S1.SS6.p2" class="ltx_para">
<p class="ltx_p">We defined inbreeding as having parents that are
more closely related to each other than two random individuals drawn
from some reference population. So the question naturally arises,
which reference population should we use? While I might not look inbred in
comparison to allele frequencies in the UK (where I’m from), my
parents certainly aren’t too random individuals drawn from across the
world-wide population. If we calculated <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p2.m1" class="ltx_Math" alttext="F" display="inline"><semantics><mi>F</mi><annotation encoding="application/x-tex">F</annotation></semantics></math> using allele frequencies
within the UK, the inbreeding coefficient for me <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p2.m2" class="ltx_Math" alttext="F" display="inline"><semantics><mi>F</mi><annotation encoding="application/x-tex">F</annotation></semantics></math> would (hopefully)
be close to zero, but would likely be larger if we used world-wide
frequencies. That’s because there’s a somewhat lower level of
heterozygosity within the UK than in the human population across the world as a whole.
<br class="ltx_break"/></p>
</div>
<div id="S1.SS6.p3" class="ltx_para">
<p class="ltx_p">Wright (1943, 1951) developed a set of ‘F-statistics’ (fixation statistics) that formalized these ideas
about inbreeding. Wright defined <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m1" class="ltx_Math" display="inline" alttext="F_{XY}"><semantics><msub><mi>F</mi><mrow><mi>X</mi><mo>⁢</mo><mi>Y</mi></mrow></msub><annotation encoding="application/x-tex">F_{XY}</annotation></semantics></math> as:
the correlation between random gametes, drawn from the same <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m2" class="ltx_Math" display="inline" alttext="X"><semantics><mi>X</mi><annotation encoding="application/x-tex">X</annotation></semantics></math>,
relative to <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m3" class="ltx_Math" alttext="Y" display="inline"><semantics><mi>Y</mi><annotation encoding="application/x-tex">Y</annotation></semantics></math>. We’ll return to why <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m4" class="ltx_Math" display="inline" alttext="F"><semantics><mi>F</mi><annotation encoding="application/x-tex">F</annotation></semantics></math> statistics are statements
about correlations between alleles in just a moment. One commonly use <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m5" class="ltx_Math" display="inline" alttext="F_{IS}"><semantics><msub><mi>F</mi><mrow><mi>I</mi><mo>⁢</mo><mi>S</mi></mrow></msub><annotation encoding="application/x-tex">F_{IS}</annotation></semantics></math> for the inbreeding
coefficient between an individual (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m6" class="ltx_Math" display="inline" alttext="I"><semantics><mi>I</mi><annotation encoding="application/x-tex">I</annotation></semantics></math>) and the subpopulation
(<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m7" class="ltx_Math" alttext="S" display="inline"><semantics><mi>S</mi><annotation encoding="application/x-tex">S</annotation></semantics></math>). Consider a single locus, where in a sub-population (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m8" class="ltx_Math" display="inline" alttext="S"><semantics><mi>S</mi><annotation encoding="application/x-tex">S</annotation></semantics></math>) a fraction <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m9" class="ltx_Math" alttext="H_{I}=f_{12}" display="inline"><semantics><mrow><msub><mi>H</mi><mi>I</mi></msub><mo>=</mo><msub><mi>f</mi><mn>12</mn></msub></mrow><annotation encoding="application/x-tex">H_{I}=f_{12}</annotation></semantics></math> of our individuals
are heterozygotes. In this sub-population (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m10" class="ltx_Math" alttext="S" display="inline"><semantics><mi>S</mi><annotation encoding="application/x-tex">S</annotation></semantics></math>) the frequency of
allele <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m11" class="ltx_Math" alttext="1" display="inline"><semantics><mn>1</mn><annotation encoding="application/x-tex">1</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m12" class="ltx_Math" alttext="p_{S}" display="inline"><semantics><msub><mi>p</mi><mi>S</mi></msub><annotation encoding="application/x-tex">p_{S}</annotation></semantics></math> and the expected heterozygosity is <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m13" class="ltx_Math" alttext="H_{S}" display="inline"><semantics><msub><mi>H</mi><mi>S</mi></msub><annotation encoding="application/x-tex">H_{S}</annotation></semantics></math>. We will write <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m14" class="ltx_Math" alttext="F_{IS}" display="inline"><semantics><msub><mi>F</mi><mrow><mi>I</mi><mo>⁢</mo><mi>S</mi></mrow></msub><annotation encoding="application/x-tex">F_{IS}</annotation></semantics></math> as</p>
<table id="S1.E7" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.E7.m1" class="ltx_Math" display="block" alttext="F_{IS}=1-\frac{H_{I}}{H_{S}}=1-\frac{f_{12}}{2p_{S}q_{S}}"><semantics><mrow><msub><mi>F</mi><mrow><mi>I</mi><mo>⁢</mo><mi>S</mi></mrow></msub><mo>=</mo><mrow><mn>1</mn><mo>-</mo><mfrac><msub><mi>H</mi><mi>I</mi></msub><msub><mi>H</mi><mi>S</mi></msub></mfrac></mrow><mo>=</mo><mrow><mn>1</mn><mo>-</mo><mfrac><msub><mi>f</mi><mn>12</mn></msub><mrow><mn>2</mn><mo>⁢</mo><msub><mi>p</mi><mi>S</mi></msub><mo>⁢</mo><msub><mi>q</mi><mi>S</mi></msub></mrow></mfrac></mrow></mrow><annotation encoding="application/x-tex">F_{IS}=1-\frac{H_{I}}{H_{S}}=1-\frac{f_{12}}{2p_{S}q_{S}}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(7)</span></td></tr>
</table>
<p class="ltx_p">the direct analog of eqn. <a href="#S1.E5" title="(5) ‣ 1.5 Calculating inbreeding coefficients from data ‣ 1 Allele and Genotype frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_tag">5</span></a>, which compares the observed
heterozygosity to that expected under random mating within the sub-population. We could also compare our
heterozygosity in individuals (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m15" class="ltx_Math" alttext="H_{I}" display="inline"><semantics><msub><mi>H</mi><mi>I</mi></msub><annotation encoding="application/x-tex">H_{I}</annotation></semantics></math>) to that expected in the total
population (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m16" class="ltx_Math" display="inline" alttext="H_{T}"><semantics><msub><mi>H</mi><mi>T</mi></msub><annotation encoding="application/x-tex">H_{T}</annotation></semantics></math>). If the frequency of our allele in our total
population is <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m17" class="ltx_Math" display="inline" alttext="p_{T}"><semantics><msub><mi>p</mi><mi>T</mi></msub><annotation encoding="application/x-tex">p_{T}</annotation></semantics></math>, then we can write <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m18" class="ltx_Math" display="inline" alttext="F_{IT}"><semantics><msub><mi>F</mi><mrow><mi>I</mi><mo>⁢</mo><mi>T</mi></mrow></msub><annotation encoding="application/x-tex">F_{IT}</annotation></semantics></math> as</p>
<table id="S1.E8" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.E8.m1" class="ltx_Math" display="block" alttext="F_{IT}=1-\frac{H_{I}}{H_{T}}=1-\frac{f_{12}}{2p_{T}q_{T}}"><semantics><mrow><msub><mi>F</mi><mrow><mi>I</mi><mo>⁢</mo><mi>T</mi></mrow></msub><mo>=</mo><mrow><mn>1</mn><mo>-</mo><mfrac><msub><mi>H</mi><mi>I</mi></msub><msub><mi>H</mi><mi>T</mi></msub></mfrac></mrow><mo>=</mo><mrow><mn>1</mn><mo>-</mo><mfrac><msub><mi>f</mi><mn>12</mn></msub><mrow><mn>2</mn><mo>⁢</mo><msub><mi>p</mi><mi>T</mi></msub><mo>⁢</mo><msub><mi>q</mi><mi>T</mi></msub></mrow></mfrac></mrow></mrow><annotation encoding="application/x-tex">F_{IT}=1-\frac{H_{I}}{H_{T}}=1-\frac{f_{12}}{2p_{T}q_{T}}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(8)</span></td></tr>
</table>
<p class="ltx_p">which compares heterozygosity in individuals to that expected in the
total population. As a simple extension of this we could imagine
comparing the expected heterozygosity in the subpopulation (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m19" class="ltx_Math" display="inline" alttext="H_{S}"><semantics><msub><mi>H</mi><mi>S</mi></msub><annotation encoding="application/x-tex">H_{S}</annotation></semantics></math>) to
that expected our total population (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m20" class="ltx_Math" alttext="H_{T}" display="inline"><semantics><msub><mi>H</mi><mi>T</mi></msub><annotation encoding="application/x-tex">H_{T}</annotation></semantics></math>), via <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m21" class="ltx_Math" alttext="F_{ST}" display="inline"><semantics><msub><mi>F</mi><mrow><mi>S</mi><mo>⁢</mo><mi>T</mi></mrow></msub><annotation encoding="application/x-tex">F_{ST}</annotation></semantics></math></p>
<table id="S1.E9" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.E9.m1" class="ltx_Math" display="block" alttext="F_{ST}=1-\frac{H_{S}}{H_{T}}=1-\frac{2p_{S}q_{S}}{2p_{T}q_{T}}"><semantics><mrow><msub><mi>F</mi><mrow><mi>S</mi><mo>⁢</mo><mi>T</mi></mrow></msub><mo>=</mo><mrow><mn>1</mn><mo>-</mo><mfrac><msub><mi>H</mi><mi>S</mi></msub><msub><mi>H</mi><mi>T</mi></msub></mfrac></mrow><mo>=</mo><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mo>⁢</mo><msub><mi>p</mi><mi>S</mi></msub><mo>⁢</mo><msub><mi>q</mi><mi>S</mi></msub></mrow><mrow><mn>2</mn><mo>⁢</mo><msub><mi>p</mi><mi>T</mi></msub><mo>⁢</mo><msub><mi>q</mi><mi>T</mi></msub></mrow></mfrac></mrow></mrow><annotation encoding="application/x-tex">F_{ST}=1-\frac{H_{S}}{H_{T}}=1-\frac{2p_{S}q_{S}}{2p_{T}q_{T}}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(9)</span></td></tr>
</table>
<p class="ltx_p">If our total population contains our sub-population then, as we’ll see
below due to the Wahlund effect (to be added)
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m22" class="ltx_Math" alttext="2p_{S}q_{S}\leq 2p_{T}q_{T}" display="inline"><semantics><mrow><mrow><mn>2</mn><mo>⁢</mo><msub><mi>p</mi><mi>S</mi></msub><mo>⁢</mo><msub><mi>q</mi><mi>S</mi></msub></mrow><mo>≤</mo><mrow><mn>2</mn><mo>⁢</mo><msub><mi>p</mi><mi>T</mi></msub><mo>⁢</mo><msub><mi>q</mi><mi>T</mi></msub></mrow></mrow><annotation encoding="application/x-tex">2p_{S}q_{S}\leq 2p_{T}q_{T}</annotation></semantics></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m23" class="ltx_Math" alttext="F_{IS}\leq F_{IT}" display="inline"><semantics><mrow><msub><mi>F</mi><mrow><mi>I</mi><mo>⁢</mo><mi>S</mi></mrow></msub><mo>≤</mo><msub><mi>F</mi><mrow><mi>I</mi><mo>⁢</mo><mi>T</mi></mrow></msub></mrow><annotation encoding="application/x-tex">F_{IS}\leq F_{IT}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m24" class="ltx_Math" display="inline" alttext="F_{ST}\geq 0"><semantics><mrow><msub><mi>F</mi><mrow><mi>S</mi><mo>⁢</mo><mi>T</mi></mrow></msub><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">F_{ST}\geq 0</annotation></semantics></math>. We can
relate our three <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p3.m25" class="ltx_Math" alttext="F" display="inline"><semantics><mi>F</mi><annotation encoding="application/x-tex">F</annotation></semantics></math> statistics together as</p>
<table id="S1.E10" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.E10.m1" class="ltx_Math" display="block" alttext="(1-\hat{F_{IT}})=\frac{H_{I}}{H_{S}}\frac{H_{S}}{H_{T}}=(1-F_{IS})(1-F_{ST})"><semantics><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><mover accent="true"><msub><mi>F</mi><mrow><mi>I</mi><mo>⁢</mo><mi>T</mi></mrow></msub><mo>^</mo></mover></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mfrac><msub><mi>H</mi><mi>I</mi></msub><msub><mi>H</mi><mi>S</mi></msub></mfrac><mo>⁢</mo><mfrac><msub><mi>H</mi><mi>S</mi></msub><msub><mi>H</mi><mi>T</mi></msub></mfrac></mrow><mo>=</mo><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msub><mi>F</mi><mrow><mi>I</mi><mo>⁢</mo><mi>S</mi></mrow></msub></mrow><mo>)</mo></mrow><mo>⁢</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msub><mi>F</mi><mrow><mi>S</mi><mo>⁢</mo><mi>T</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex">(1-\hat{F_{IT}})=\frac{H_{I}}{H_{S}}\frac{H_{S}}{H_{T}}=(1-F_{IS})(1-F_{ST})</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(10)</span></td></tr>
</table>
<p class="ltx_p">i.e. the reduction in heterozygosity in our individuals compared to that expected
in the total population can be decomposed to the reduction in
heterozygosity of individuals compared to the sub-population, and the reduction in
heterozygosity from the total population to that in the sub-population.
<br class="ltx_break"/></p>
</div>
<div id="S1.SS6.p4" class="ltx_para">
<p class="ltx_p">If we want a summary of
population structure across multiple sub-populations we can average <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p4.m1" class="ltx_Math" alttext="H_{I}" display="inline"><semantics><msub><mi>H</mi><mi>I</mi></msub><annotation encoding="application/x-tex">H_{I}</annotation></semantics></math>
and/or <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p4.m2" class="ltx_Math" alttext="H_{S}" display="inline"><semantics><msub><mi>H</mi><mi>S</mi></msub><annotation encoding="application/x-tex">H_{S}</annotation></semantics></math> across populations, and use a <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p4.m3" class="ltx_Math" display="inline" alttext="p_{T}"><semantics><msub><mi>p</mi><mi>T</mi></msub><annotation encoding="application/x-tex">p_{T}</annotation></semantics></math> calculated by
averaging <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p4.m4" class="ltx_Math" display="inline" alttext="p_{S}"><semantics><msub><mi>p</mi><mi>S</mi></msub><annotation encoding="application/x-tex">p_{S}</annotation></semantics></math> across sub-populations. Furthermore, if we have multiple sites we can replace <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p4.m5" class="ltx_Math" alttext="H_{I}" display="inline"><semantics><msub><mi>H</mi><mi>I</mi></msub><annotation encoding="application/x-tex">H_{I}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p4.m6" class="ltx_Math" display="inline" alttext="H_{S}"><semantics><msub><mi>H</mi><mi>S</mi></msub><annotation encoding="application/x-tex">H_{S}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p4.m7" class="ltx_Math" alttext="H_{T}" display="inline"><semantics><msub><mi>H</mi><mi>T</mi></msub><annotation encoding="application/x-tex">H_{T}</annotation></semantics></math> with
their averages across loci (as above).
<br class="ltx_break"/></p>
</div>
<div id="S1.SS6.p5" class="ltx_para">
<p class="ltx_p">Lets now return to Wright’s definition of the <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p5.m1" class="ltx_Math" alttext="F_{XY}" display="inline"><semantics><msub><mi>F</mi><mrow><mi>X</mi><mo>⁢</mo><mi>Y</mi></mrow></msub><annotation encoding="application/x-tex">F_{XY}</annotation></semantics></math> statistic as correlation between random gametes, drawn from the same <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p5.m2" class="ltx_Math" alttext="X" display="inline"><semantics><mi>X</mi><annotation encoding="application/x-tex">X</annotation></semantics></math>,
relative to <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p5.m3" class="ltx_Math" display="inline" alttext="Y"><semantics><mi>Y</mi><annotation encoding="application/x-tex">Y</annotation></semantics></math>. With out loss of generality lets think about <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p5.m4" class="ltx_Math" alttext="X" display="inline"><semantics><mi>X</mi><annotation encoding="application/x-tex">X</annotation></semantics></math> as
individuals and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p5.m5" class="ltx_Math" display="inline" alttext="S"><semantics><mi>S</mi><annotation encoding="application/x-tex">S</annotation></semantics></math> as the sub-population.
Rewriting <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p5.m6" class="ltx_Math" display="inline" alttext="F_{ST}"><semantics><msub><mi>F</mi><mrow><mi>S</mi><mo>⁢</mo><mi>T</mi></mrow></msub><annotation encoding="application/x-tex">F_{ST}</annotation></semantics></math> in terms of our homozygote frequencies observed in
individuals (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p5.m7" class="ltx_Math" display="inline" alttext="f_{11}"><semantics><msub><mi>f</mi><mn>11</mn></msub><annotation encoding="application/x-tex">f_{11}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p5.m8" class="ltx_Math" display="inline" alttext="f_{22}"><semantics><msub><mi>f</mi><mn>22</mn></msub><annotation encoding="application/x-tex">f_{22}</annotation></semantics></math>) we find</p>
<table id="S1.E11" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.E11.m1" class="ltx_Math" display="block" alttext="F_{IS}=\frac{2p_{S}q_{S}-f_{12}}{2p_{S}q_{S}}=\frac{f_{11}+f_{22}-p_{S}^{2}-q_% {S}^{2}}{2p_{S}q_{S}}"><semantics><mrow><msub><mi>F</mi><mrow><mi>I</mi><mo>⁢</mo><mi>S</mi></mrow></msub><mo>=</mo><mfrac><mrow><mrow><mn>2</mn><mo>⁢</mo><msub><mi>p</mi><mi>S</mi></msub><mo>⁢</mo><msub><mi>q</mi><mi>S</mi></msub></mrow><mo>-</mo><msub><mi>f</mi><mn>12</mn></msub></mrow><mrow><mn>2</mn><mo>⁢</mo><msub><mi>p</mi><mi>S</mi></msub><mo>⁢</mo><msub><mi>q</mi><mi>S</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>f</mi><mn>11</mn></msub><mo>+</mo><msub><mi>f</mi><mn>22</mn></msub><mo>-</mo><msubsup><mi>p</mi><mi>S</mi><mn>2</mn></msubsup><mo>-</mo><msubsup><mi>q</mi><mi>S</mi><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><mo>⁢</mo><msub><mi>p</mi><mi>S</mi></msub><mo>⁢</mo><msub><mi>q</mi><mi>S</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">F_{IS}=\frac{2p_{S}q_{S}-f_{12}}{2p_{S}q_{S}}=\frac{f_{11}+f_{22}-p_{S}^{2}-q_%
{S}^{2}}{2p_{S}q_{S}}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(11)</span></td></tr>
</table>
<p class="ltx_p">using the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p5.m9" class="ltx_Math" display="inline" alttext="p^{2}+2pq+q^{2}=1"><semantics><mrow><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><mrow><mn>2</mn><mo>⁢</mo><mi>p</mi><mo>⁢</mo><mi>q</mi></mrow><mo>+</mo><msup><mi>q</mi><mn>2</mn></msup></mrow><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p^{2}+2pq+q^{2}=1</annotation></semantics></math>. The form of this
(eqn. <a href="#S1.E11" title="(11) ‣ 1.6 Summarizing Population structure ‣ 1 Allele and Genotype frequencies ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_tag">11</span></a>) is the covariance between pairs of alleles
found in an individual, divided by the
expected variance under binomial sampling. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p5.m10" class="ltx_Math" display="inline" alttext="F"><semantics><mi>F</mi><annotation encoding="application/x-tex">F</annotation></semantics></math> statistics can be
understood as the correlation between alleles drawn from a population
(or an individual) above that expected by chance (i.e. drawing alleles
sampled at random from some broader population).
<br class="ltx_break"/></p>
</div>
<div id="S1.SS6.p6" class="ltx_para">
<p class="ltx_p">We can also see <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p6.m1" class="ltx_Math" display="inline" alttext="F"><semantics><mi>F</mi><annotation encoding="application/x-tex">F</annotation></semantics></math> statistics as proportions of variance explained by
substructure. To see this lets think about <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p6.m2" class="ltx_Math" alttext="F_{ST}" display="inline"><semantics><msub><mi>F</mi><mrow><mi>S</mi><mo>⁢</mo><mi>T</mi></mrow></msub><annotation encoding="application/x-tex">F_{ST}</annotation></semantics></math> averaged over <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p6.m3" class="ltx_Math" display="inline" alttext="K"><semantics><mi>K</mi><annotation encoding="application/x-tex">K</annotation></semantics></math>
subpopulations, whose frequencies are <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p6.m4" class="ltx_Math" display="inline" alttext="p_{1},~{}\cdots,~{}p_{K}"><semantics><mrow><msub><mi>p</mi><mn>1</mn></msub><mo separator="true">, </mo><mi mathvariant="normal">⋯</mi><mo separator="true">, </mo><msub><mi>p</mi><mi>K</mi></msub></mrow><annotation encoding="application/x-tex">p_{1},~{}\cdots,~{}p_{K}</annotation></semantics></math>. The
frequency in the total population is <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p6.m5" class="ltx_Math" alttext="p_{T}=\bar{p}=\frac{1}{K}\sum_{i=1}^{K}p_{i}" display="inline"><semantics><mrow><msub><mi>p</mi><mi>T</mi></msub><mo>=</mo><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mo>=</mo><mrow><mfrac><mn>1</mn><mi>K</mi></mfrac><mo>⁢</mo><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></msubsup><msub><mi>p</mi><mi>i</mi></msub></mrow></mrow></mrow><annotation encoding="application/x-tex">p_{T}=\bar{p}=\frac{1}{K}\sum_{i=1}^{K}p_{i}</annotation></semantics></math>.
Then we can
write</p>
<table id="S1.E12" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.E12.m1" class="ltx_Math" alttext="F_{ST}=\frac{2\bar{p}\bar{q}-\frac{1}{K}\sum_{i=1}^{K}2p_{i}q_{i}}{2\bar{p}% \bar{q}}=\frac{\left(\frac{1}{K}\sum_{i=1}^{K}p_{i}^{2}+\frac{1}{K}\sum_{i=1}^% {K}q_{i}^{2}\right)-\bar{p}^{2}-\bar{q}^{2}}{2\bar{p}\bar{q}}=\frac{Var(p_{i})% }{Var(\bar{p})}" display="block"><semantics><mrow><msub><mi>F</mi><mrow><mi>S</mi><mo>⁢</mo><mi>T</mi></mrow></msub><mo>=</mo><mfrac><mrow><mrow><mn>2</mn><mo>⁢</mo><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mo>⁢</mo><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow><mo>-</mo><mrow><mfrac><mn>1</mn><mi>K</mi></mfrac><mo>⁢</mo><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></msubsup><mrow><mn>2</mn><mo>⁢</mo><msub><mi>p</mi><mi>i</mi></msub><mo>⁢</mo><msub><mi>q</mi><mi>i</mi></msub></mrow></mrow></mrow></mrow><mrow><mn>2</mn><mo>⁢</mo><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mo>⁢</mo><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow></mfrac><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><mrow><mrow><mfrac><mn>1</mn><mi>K</mi></mfrac><mo>⁢</mo><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></msubsup><msubsup><mi>p</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mi>K</mi></mfrac><mo>⁢</mo><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>K</mi></msubsup><msubsup><mi>q</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mrow></mrow><mo>)</mo></mrow><mo>-</mo><msup><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mn>2</mn></msup><mo>-</mo><msup><mover accent="true"><mi>q</mi><mo>¯</mo></mover><mn>2</mn></msup></mrow><mrow><mn>2</mn><mo>⁢</mo><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mo>⁢</mo><mover accent="true"><mi>q</mi><mo>¯</mo></mover></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>p</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow><mrow><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><mover accent="true"><mi>p</mi><mo>¯</mo></mover><mo>)</mo></mrow></mrow></mfrac></mrow><annotation encoding="application/x-tex">F_{ST}=\frac{2\bar{p}\bar{q}-\frac{1}{K}\sum_{i=1}^{K}2p_{i}q_{i}}{2\bar{p}%
\bar{q}}=\frac{\left(\frac{1}{K}\sum_{i=1}^{K}p_{i}^{2}+\frac{1}{K}\sum_{i=1}^%
{K}q_{i}^{2}\right)-\bar{p}^{2}-\bar{q}^{2}}{2\bar{p}\bar{q}}=\frac{Var(p_{i})%
}{Var(\bar{p})}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(12)</span></td></tr>
</table>
<p class="ltx_p">i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.SS6.p6.m6" class="ltx_Math" display="inline" alttext="F_{ST}"><semantics><msub><mi>F</mi><mrow><mi>S</mi><mo>⁢</mo><mi>T</mi></mrow></msub><annotation encoding="application/x-tex">F_{ST}</annotation></semantics></math> is the proportion of the variance explained by the
subpopulation labels.</p>
</div>
<div class="ltx_pagination ltx_role_newpage"/>
</div>
</div>
<div id="S2" class="ltx_section">
<h2 class="ltx_title ltx_title_section"><span class="ltx_tag ltx_tag_section">2 </span>The phenotypic resemblance between relatives</h2>
<div id="S2.p1" class="ltx_para">
<p class="ltx_p">We can use our understanding the sharing of alleles between relatives
to understand the phenotypic resemblance between relatives in
quantitative phenotypes. We can then use this to understand the
evolutionary change in quantitative phenotypes in response to selection.
<br class="ltx_break"/></p>
</div>
<div id="S2.p2" class="ltx_para">
<p class="ltx_p">Let’s imagine that the genetic component of the variation in our trait
is controlled by <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p2.m1" class="ltx_Math" alttext="L" display="inline"><semantics><mi>L</mi><annotation encoding="application/x-tex">L</annotation></semantics></math> autosomal loci that act in an additive manner. The frequency of allele <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p2.m2" class="ltx_Math" alttext="1" display="inline"><semantics><mn>1</mn><annotation encoding="application/x-tex">1</annotation></semantics></math> at locus <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p2.m3" class="ltx_Math" display="inline" alttext="l"><semantics><mi>l</mi><annotation encoding="application/x-tex">l</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p2.m4" class="ltx_Math" display="inline" alttext="p_{l}"><semantics><msub><mi>p</mi><mi>l</mi></msub><annotation encoding="application/x-tex">p_{l}</annotation></semantics></math>, with each copy of
allele <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p2.m5" class="ltx_Math" alttext="1" display="inline"><semantics><mn>1</mn><annotation encoding="application/x-tex">1</annotation></semantics></math> at this locus increasing your trait value by <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p2.m6" class="ltx_Math" alttext="a_{l}" display="inline"><semantics><msub><mi>a</mi><mi>l</mi></msub><annotation encoding="application/x-tex">a_{l}</annotation></semantics></math>.
The phenotype of an individual, let’s call her <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p2.m7" class="ltx_Math" display="inline" alttext="i"><semantics><mi>i</mi><annotation encoding="application/x-tex">i</annotation></semantics></math>, is <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p2.m8" class="ltx_Math" display="inline" alttext="Y_{i}"><semantics><msub><mi>Y</mi><mi>i</mi></msub><annotation encoding="application/x-tex">Y_{i}</annotation></semantics></math>.
Her genotype at SNP <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p2.m9" class="ltx_Math" alttext="l" display="inline"><semantics><mi>l</mi><annotation encoding="application/x-tex">l</annotation></semantics></math>, is
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p2.m10" class="ltx_Math" display="inline" alttext="G_{i,l}"><semantics><msub><mi>G</mi><mrow><mi>i</mi><mo>,</mo><mi>l</mi></mrow></msub><annotation encoding="application/x-tex">G_{i,l}</annotation></semantics></math>. Here <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p2.m11" class="ltx_Math" alttext="G_{i,l}=0,~{}1," display="inline"><semantics><mrow><mrow><msub><mi>G</mi><mrow><mi>i</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>=</mo><mrow><mn>0</mn><mo separator="true">, </mo><mn>1</mn></mrow></mrow><mo>,</mo></mrow><annotation encoding="application/x-tex">G_{i,l}=0,~{}1,</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p2.m12" class="ltx_Math" alttext="2" display="inline"><semantics><mn>2</mn><annotation encoding="application/x-tex">2</annotation></semantics></math> represents the number of copies of allele <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p2.m13" class="ltx_Math" display="inline" alttext="1"><semantics><mn>1</mn><annotation encoding="application/x-tex">1</annotation></semantics></math> she
has at this SNP. Her expected phenotype, given her genotype, is then</p>
<table id="S2.E13" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E13.m1" class="ltx_Math" alttext="X_{A,i}=\mathbb{E}(X_{i}|G_{i,1},\cdots,G_{i,L})=\sum_{l=1}^{L}G_{i,l}a_{l}" display="block"><semantics><mrow><msub><mi>X</mi><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><mi>𝔼</mi><mrow><mo>(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>|</mo><msub><mi>G</mi><mrow><mi>i</mi><mo>,</mo><mn>1</mn></mrow></msub><mo>,</mo><mi mathvariant="normal">⋯</mi><mo>,</mo><msub><mi>G</mi><mrow><mi>i</mi><mo>,</mo><mi>L</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><munderover><mo movablelimits="false">∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>L</mi></munderover><msub><mi>G</mi><mrow><mi>i</mi><mo>,</mo><mi>l</mi></mrow></msub><msub><mi>a</mi><mi>l</mi></msub></mrow><annotation encoding="application/x-tex">X_{A,i}=\mathbb{E}(X_{i}|G_{i,1},\cdots,G_{i,L})=\sum_{l=1}^{L}G_{i,l}a_{l}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(13)</span></td></tr>
</table>
<p class="ltx_p">Now in reality the genetic phenotype is a function of the
expression of those alleles in a particular environment. Therefore, we
can think of this expected phenotype as being an average across a set
of environments that occur in the population.
<br class="ltx_break"/></p>
</div>
<div id="S2.p3" class="ltx_para">
<p class="ltx_p">When we measure our individual’s phenotype we see</p>
<table id="S2.E14" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E14.m1" class="ltx_Math" display="block" alttext="X_{i}=X_{A,i}+X_{E,i}"><semantics><mrow><msub><mi>X</mi><mi>i</mi></msub><mo>=</mo><mrow><msub><mi>X</mi><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mi>E</mi><mo>,</mo><mi>i</mi></mrow></msub></mrow></mrow><annotation encoding="application/x-tex">X_{i}=X_{A,i}+X_{E,i}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(14)</span></td></tr>
</table>
<p class="ltx_p">where <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p3.m1" class="ltx_Math" display="inline" alttext="X_{E}"><semantics><msub><mi>X</mi><mi>E</mi></msub><annotation encoding="application/x-tex">X_{E}</annotation></semantics></math> is the deviation from the mean phenotype due to the
environment. This <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p3.m2" class="ltx_Math" alttext="X_{E}" display="inline"><semantics><msub><mi>X</mi><mi>E</mi></msub><annotation encoding="application/x-tex">X_{E}</annotation></semantics></math> included the systematic effects of the environment
our individual finds herself in and all of the noise during
development, growth, and the various random insults that life throws
at our individual. If a reasonable number of loci contribute to
variation in our trait then we can approximate the distribution of
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p3.m3" class="ltx_Math" display="inline" alttext="X_{A,i}"><semantics><msub><mi>X</mi><mrow><mi>A</mi><mo>,</mo><mi>i</mi></mrow></msub><annotation encoding="application/x-tex">X_{A,i}</annotation></semantics></math> by a normal distribution due to the central limit theorem (see R exercise). Thus if we can
approximate the distribution of the effect of environmental variation
on our trait (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p3.m4" class="ltx_Math" alttext="X_{E,i}" display="inline"><semantics><msub><mi>X</mi><mrow><mi>E</mi><mo>,</mo><mi>i</mi></mrow></msub><annotation encoding="application/x-tex">X_{E,i}</annotation></semantics></math>) also by a normal distribution, which is
reasonable as there are many small environmental effects, then the
distribution of phenotypes within the population (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p3.m5" class="ltx_Math" display="inline" alttext="X_{i}"><semantics><msub><mi>X</mi><mi>i</mi></msub><annotation encoding="application/x-tex">X_{i}</annotation></semantics></math>) will be
normally distributed (see Figure <a href="#S2.F1" title="Figure 1 ‣ 2 The phenotypic resemblance between relatives ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>).
<br class="ltx_break"/></p>
</div>
<div id="S2.F1" class="ltx_figure"><object data="x3.png" id="S2.F1.g1" class="ltx_graphics ltx_centering" width="539" height="210" alt=""/>
<div class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>The convergence of the phenotypic distribution to a normal distribution.</div>
</div>
<div id="S2.p4" class="ltx_para">
<p class="ltx_p">Note that as this is an additive model we can decompose eqn. <a href="#S2.E14" title="(14) ‣ 2 The phenotypic resemblance between relatives ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_tag">14</span></a> into the
effects of the two alleles at each locus, in particular we can rewrite
it as</p>
<table id="S2.E15" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E15.m1" class="ltx_Math" display="block" alttext="X_{i}=X_{iM}+X_{iP}+X_{iE}"><semantics><mrow><msub><mi>X</mi><mi>i</mi></msub><mo>=</mo><mrow><msub><mi>X</mi><mrow><mi>i</mi><mo>⁢</mo><mi>M</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mi>i</mi><mo>⁢</mo><mi>P</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mi>i</mi><mo>⁢</mo><mi>E</mi></mrow></msub></mrow></mrow><annotation encoding="application/x-tex">X_{i}=X_{iM}+X_{iP}+X_{iE}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(15)</span></td></tr>
</table>
<p class="ltx_p">where <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p4.m1" class="ltx_Math" alttext="X_{iM}" display="inline"><semantics><msub><mi>X</mi><mrow><mi>i</mi><mo>⁢</mo><mi>M</mi></mrow></msub><annotation encoding="application/x-tex">X_{iM}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.p4.m2" class="ltx_Math" display="inline" alttext="X_{iP}"><semantics><msub><mi>X</mi><mrow><mi>i</mi><mo>⁢</mo><mi>P</mi></mrow></msub><annotation encoding="application/x-tex">X_{iP}</annotation></semantics></math> are the contribution to the phenotype of
the allele that our individual received from her mother (maternal
alleles) and father (paternal alleles) respectively. This will come in
handy in just a moment when we start thinking about the phenotype covariance of relatives.
<br class="ltx_break"/></p>
</div>
<div id="S2.p5" class="ltx_para">
<p class="ltx_p">Now obviously this model seems silly at first sight as alleles don’t
only act in an additive manner, as they interact with alleles at the
same loci (dominance) and at different loci (epistasis). Later we’ll
relax this assumption,
however, we’ll find that for our interests in evolutionary change actually it is only the “additive
component” of genetic variation that will concern us.
We will define this more formally later on, but for the moment
we can offer the intuition that parents only get to pass on a single
allele at each locus on to the next generation. As such, it is the
effect of these transmitted alleles, averaged over possible matings,
that is an individual’s average contribution to the next generation
(i.e. the additive effect of the alleles that their genotype consists of).</p>
</div>
<div id="S2.SS6.SSS1" class="ltx_subsubsection">
<h4 class="ltx_title ltx_title_subsubsection"><span class="ltx_tag ltx_tag_subsubsection">2.0.1 </span>Additive genetic variance and heritability</h4>
<div id="S2.SS6.SSS1.p1" class="ltx_para">
<p class="ltx_p">As we are talking about an additive genetic model we’ll talk about the
additive genetic variance (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS1.p1.m1" class="ltx_Math" display="inline" alttext="V_{A}"><semantics><msub><mi>V</mi><mi>A</mi></msub><annotation encoding="application/x-tex">V_{A}</annotation></semantics></math>), the variance due to the additive
effects of segregating genetic variation. This is a subset of the total genetic
variance if we allow for non-additive effects.
<br class="ltx_break"/></p>
</div>
<div id="S2.SS6.SSS1.p2" class="ltx_para">
<p class="ltx_p">The variance of our phenotype across individuals (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS1.p2.m1" class="ltx_Math" display="inline" alttext="V"><semantics><mi>V</mi><annotation encoding="application/x-tex">V</annotation></semantics></math>) can write this as</p>
<table id="S2.E16" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E16.m1" class="ltx_Math" alttext="V=Var(X_{A})+Var(X_{E})=V_{A}+V_{E}" display="block"><semantics><mrow><mi>V</mi><mo>=</mo><mrow><mrow><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>X</mi><mi>A</mi></msub><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>X</mi><mi>E</mi></msub><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><msub><mi>V</mi><mi>A</mi></msub><mo>+</mo><msub><mi>V</mi><mi>E</mi></msub></mrow></mrow><annotation encoding="application/x-tex">V=Var(X_{A})+Var(X_{E})=V_{A}+V_{E}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(16)</span></td></tr>
</table>
<p class="ltx_p">in doing writing this we are assuming that there is no covariance between <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS1.p2.m2" class="ltx_Math" alttext="X_{G,i}" display="inline"><semantics><msub><mi>X</mi><mrow><mi>G</mi><mo>,</mo><mi>i</mi></mrow></msub><annotation encoding="application/x-tex">X_{G,i}</annotation></semantics></math>
and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS1.p2.m3" class="ltx_Math" alttext="X_{E,i}" display="inline"><semantics><msub><mi>X</mi><mrow><mi>E</mi><mo>,</mo><mi>i</mi></mrow></msub><annotation encoding="application/x-tex">X_{E,i}</annotation></semantics></math> i.e. there is no covariance between genotype and
environment.
<br class="ltx_break"/></p>
</div>
<div id="S2.SS6.SSS1.p3" class="ltx_para">
<p class="ltx_p">Our additive genetic variance can be written as</p>
<table id="S2.E17" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E17.m1" class="ltx_Math" alttext="V_{A}=\sum_{l=1}^{L}Var(G_{i,l}a_{l})" display="block"><semantics><mrow><msub><mi>V</mi><mi>A</mi></msub><mo>=</mo><mrow><munderover><mo movablelimits="false">∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>L</mi></munderover><mrow><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>G</mi><mrow><mi>i</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>⁢</mo><msub><mi>a</mi><mi>l</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></mrow><annotation encoding="application/x-tex">V_{A}=\sum_{l=1}^{L}Var(G_{i,l}a_{l})</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(17)</span></td></tr>
</table>
<p class="ltx_p">where <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS1.p3.m1" class="ltx_Math" alttext="Var(G_{i,l}a_{l})" display="inline"><semantics><mrow><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>G</mi><mrow><mi>i</mi><mo>,</mo><mi>l</mi></mrow></msub><mo>⁢</mo><msub><mi>a</mi><mi>l</mi></msub></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">Var(G_{i,l}a_{l})</annotation></semantics></math> is the contribution to the additive
variance among individuals of the <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS1.p3.m2" class="ltx_Math" alttext="l" display="inline"><semantics><mi>l</mi><annotation encoding="application/x-tex">l</annotation></semantics></math> locus. Assuming random mating we
can write our additive genetic variance as</p>
<table id="S2.E18" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E18.m1" class="ltx_Math" alttext="V_{A}=\sum_{l=1}^{L}a_{l}^{2}2p_{l}(1-p_{l})" display="block"><semantics><mrow><msub><mi>V</mi><mi>A</mi></msub><mo>=</mo><mrow><munderover><mo movablelimits="false">∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>L</mi></munderover><mrow><msubsup><mi>a</mi><mi>l</mi><mn>2</mn></msubsup><mo>⁢</mo><mn>2</mn><mo>⁢</mo><msub><mi>p</mi><mi>l</mi></msub><mo>⁢</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msub><mi>p</mi><mi>l</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></mrow><annotation encoding="application/x-tex">V_{A}=\sum_{l=1}^{L}a_{l}^{2}2p_{l}(1-p_{l})</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(18)</span></td></tr>
</table>
<p class="ltx_p">where the <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS1.p3.m3" class="ltx_Math" alttext="2p_{l}(1-p_{l})" display="inline"><semantics><mrow><mn>2</mn><mo>⁢</mo><msub><mi>p</mi><mi>l</mi></msub><mo>⁢</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msub><mi>p</mi><mi>l</mi></msub></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">2p_{l}(1-p_{l})</annotation></semantics></math> term follows the binomial sampling of two
alleles per individual at each locus.
<br class="ltx_break"/></p>
</div>
<div id="S2.SS6.SSS1.P1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">The narrow sense heritability</h5>
<div id="S2.SS6.SSS1.P1.p1" class="ltx_para">
<p class="ltx_p">We would like a way to think about what proportion of the variation
in our phenotype across individuals is due to genetic differences as
opposed to environmental differences. Such a quantity will be key in
helping us think about the evolution of phenotypes. For example, if
variation in our phenotype had no genetic basis then no matter how
much selection changes the mean phenotype within a generation
the trait will not change over generations.
<br class="ltx_break"/></p>
</div>
<div id="S2.SS6.SSS1.P1.p2" class="ltx_para">
<p class="ltx_p">We’ll call the proportion of the variance that is genetic the
heritability, and denote it by <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS1.P1.p2.m1" class="ltx_Math" alttext="h^{2}" display="inline"><semantics><msup><mi>h</mi><mn>2</mn></msup><annotation encoding="application/x-tex">h^{2}</annotation></semantics></math>. We can then write this as</p>
<table id="S2.E19" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E19.m1" class="ltx_Math" alttext="h^{2}=\frac{Var(X_{A})}{V}=\frac{V_{A}}{V}" display="block"><semantics><mrow><msup><mi>h</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>X</mi><mi>A</mi></msub><mo>)</mo></mrow></mrow><mi>V</mi></mfrac><mo>=</mo><mfrac><msub><mi>V</mi><mi>A</mi></msub><mi>V</mi></mfrac></mrow><annotation encoding="application/x-tex">h^{2}=\frac{Var(X_{A})}{V}=\frac{V_{A}}{V}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(19)</span></td></tr>
</table>
<p class="ltx_p">remember that we thinking about a trait where all of the alleles act
in a perfectly additive manner. In this case our heritability <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS1.P1.p2.m2" class="ltx_Math" alttext="h^{2}" display="inline"><semantics><msup><mi>h</mi><mn>2</mn></msup><annotation encoding="application/x-tex">h^{2}</annotation></semantics></math> is
referred to as the narrow sense heritability, the proportion of the
variance explained by the additive effect of our loci.
When we allow dominance
and epistasis into our model we’ll also have to define the broad sense
heritability (the total proportion of the phenotypic variance
attributable to genetic variation).
<br class="ltx_break"/></p>
</div>
<div id="S2.SS6.SSS1.P1.p3" class="ltx_para">
<p class="ltx_p">The narrow sense heritability of a trait is a useful quantity, indeed
we’ll see shortly that it is exactly what we need to understand the
evolutionary response to selection on a quantitative phenotype. We can
calculate the narrow sense heritability by using the resemblance between
relatives. For example, if our phenotype was totally environmental we
should not expect relatives to resemble each other any more than random
individuals drawn from the population. Now the obvious caveat here is
that relatives also share an environment, so may resemble each other
due to shared environmental effects.
<br class="ltx_break"/></p>
</div>
</div>
</div>
<div id="S2.SS6.SSS2" class="ltx_subsubsection">
<h4 class="ltx_title ltx_title_subsubsection"><span class="ltx_tag ltx_tag_subsubsection">2.0.2 </span>The covariance between relatives</h4>
<div id="S2.SS6.SSS2.p1" class="ltx_para">
<p class="ltx_p">So we’ll go ahead and calculate the covariance in phenotype between two individuals
(<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.p1.m1" class="ltx_Math" display="inline" alttext="1"><semantics><mn>1</mn><annotation encoding="application/x-tex">1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.p1.m2" class="ltx_Math" display="inline" alttext="2"><semantics><mn>2</mn><annotation encoding="application/x-tex">2</annotation></semantics></math>) who have a phenotype <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.p1.m3" class="ltx_Math" alttext="X_{1}" display="inline"><semantics><msub><mi>X</mi><mn>1</mn></msub><annotation encoding="application/x-tex">X_{1}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.p1.m4" class="ltx_Math" alttext="X_{2}" display="inline"><semantics><msub><mi>X</mi><mn>2</mn></msub><annotation encoding="application/x-tex">X_{2}</annotation></semantics></math> respectively.</p>
<table id="S2.E20" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E20.m1" class="ltx_Math" display="block" alttext="Cov(X_{1},X_{2})=Cov\left((X_{1M}+X_{1P}+X_{1E}),((X_{2M}+X_{2P}+X_{2E})\right)"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mrow><mo>(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>=</mo><mi>C</mi><mi>o</mi><mi>v</mi><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>P</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>E</mi></mrow></msub><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>P</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>E</mi></mrow></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex">Cov(X_{1},X_{2})=Cov\left((X_{1M}+X_{1P}+X_{1E}),((X_{2M}+X_{2P}+X_{2E})\right)</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(20)</span></td></tr>
</table>
<p class="ltx_p">We can expand this out in terms of the covariance between the various
components in these sums.
<br class="ltx_break"/></p>
</div>
<div id="S2.SS6.SSS2.p2" class="ltx_para">
<p class="ltx_p">To make our task easier we (and most analyses) will assume two things</p>
<ol id="I1" class="ltx_enumerate">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;"><span class="ltx_tag ltx_tag_enumerate">1.</span>
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p">that we can ignore the covariance of the environments
between individuals (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m1" class="ltx_Math" alttext="Cov(X_{1E},X_{2E})=0" display="inline"><semantics><mrow><mrow><mi>C</mi><mo>⁢</mo><mi>o</mi><mo>⁢</mo><mi>v</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>E</mi></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>E</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Cov(X_{1E},X_{2E})=0</annotation></semantics></math>)</p>
</div></li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;"><span class="ltx_tag ltx_tag_enumerate">2.</span>
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p">that we can ignore the covariance
between the environment variation experience by an individual and the
genetic variation in another individual (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i2.p1.m1" class="ltx_Math" display="inline" alttext="Cov(X_{1E},(X_{2M}+X_{2P}))=0"><semantics><mrow><mrow><mi>C</mi><mo>⁢</mo><mi>o</mi><mo>⁢</mo><mi>v</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>E</mi></mrow></msub><mo>,</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>P</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Cov(X_{1E},(X_{2M}+X_{2P}))=0</annotation></semantics></math>).</p>
</div></li>
</ol>
</div>
<div id="S2.SS6.SSS2.p3" class="ltx_para">
<p class="ltx_p">The failure of these assumptions
to hold can severely undermine our estimates of heritability, but we’ll
return to that later. Moving forward with these assumptions, we can
write our phenotypic covariance between our pair of individuals as</p>
<table id="S2.E21" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E21.m1" class="ltx_Math" display="block" alttext="Cov(X_{1},X_{2})=Cov((X_{1M},X_{2M})+Cov(X_{1M},X_{2P})+Cov(X_{1P},X_{2M})+Cov% (X_{1P},X_{2P})"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mrow><mo>(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>=</mo><mi>C</mi><mi>o</mi><mi>v</mi><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>)</mo></mrow><mo>+</mo><mi>C</mi><mi>o</mi><mi>v</mi><mrow><mo>(</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>P</mi></mrow></msub><mo>)</mo></mrow><mo>+</mo><mi>C</mi><mi>o</mi><mi>v</mi><mrow><mo>(</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>P</mi></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>)</mo></mrow><mo>+</mo><mi>C</mi><mi>o</mi><mi>v</mi><mrow><mo>(</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>P</mi></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>P</mi></mrow></msub><mo>)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex">Cov(X_{1},X_{2})=Cov((X_{1M},X_{2M})+Cov(X_{1M},X_{2P})+Cov(X_{1P},X_{2M})+Cov%
(X_{1P},X_{2P})</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(21)</span></td></tr>
</table>
<p class="ltx_p">This is saying that under our simple additive model we can see the
covariance in phenotypes between individuals as the covariance between
the allelic effects in our individuals. We can use our results about
the sharing of alleles between relatives to obtain these terms.
But before we write down the general case lets quickly work through some
examples.
<br class="ltx_break"/></p>
</div>
<div id="S2.SS6.SSS2.P1" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">The covariance between Identical Twins</h5>
<div id="S2.SS6.SSS2.P1.p1" class="ltx_para">
<p class="ltx_p">Lets first consider the case of a pair of identical twins from two
unrelated parents. Our pair of twins share their maternal and paternal
allele identical by descent (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P1.p1.m1" class="ltx_Math" display="inline" alttext="X_{1M}=X_{2M}"><semantics><mrow><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>=</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>M</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{1M}=X_{2M}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P1.p1.m2" class="ltx_Math" alttext="X_{1P}=X_{2P}" display="inline"><semantics><mrow><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>P</mi></mrow></msub><mo>=</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>P</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{1P}=X_{2P}</annotation></semantics></math>). As their maternal and
paternal alleles are not correlated draws from the population,
i.e. have no probability of being <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P1.p1.m3" class="ltx_Math" alttext="IBD" display="inline"><semantics><mrow><mi>I</mi><mo>⁢</mo><mi>B</mi><mo>⁢</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">IBD</annotation></semantics></math> as we’ve said the parents are unrelated, the
covariance between their effects on the phenotype is zero
(i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P1.p1.m4" class="ltx_Math" alttext="Cov(X_{1P},X_{2M})=Cov(X_{1M},X_{2P})=0" display="inline"><semantics><mrow><mrow><mi>C</mi><mo>⁢</mo><mi>o</mi><mo>⁢</mo><mi>v</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>P</mi></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>M</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi>C</mi><mo>⁢</mo><mi>o</mi><mo>⁢</mo><mi>v</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>P</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Cov(X_{1P},X_{2M})=Cov(X_{1M},X_{2P})=0</annotation></semantics></math>). In that case
eqn. <a href="#S2.E21" title="(21) ‣ 2.0.2 The covariance between relatives ‣ 2 The phenotypic resemblance between relatives ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_tag">21</span></a> is</p>
<table id="S2.E22" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E22.m1" class="ltx_Math" display="block" alttext="Cov(X_{1},X_{2})=Cov((X_{1M},X_{2M})+Cov(X_{1P},X_{2P})=2Var(X_{1M})=V_{A}"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mrow><mo>(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>=</mo><mi>C</mi><mi>o</mi><mi>v</mi><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>)</mo></mrow><mo>+</mo><mi>C</mi><mi>o</mi><mi>v</mi><mrow><mo>(</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>P</mi></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>P</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>2</mn><mi>V</mi><mi>a</mi><mi>r</mi><mrow><mo>(</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><msub><mi>V</mi><mi>A</mi></msub></mrow></mrow><annotation encoding="application/x-tex">Cov(X_{1},X_{2})=Cov((X_{1M},X_{2M})+Cov(X_{1P},X_{2P})=2Var(X_{1M})=V_{A}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(22)</span></td></tr>
</table>
<p class="ltx_p">Now in general identical twins are not going to be super helpful for
us in estimating <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P1.p1.m5" class="ltx_Math" display="inline" alttext="h^{2}"><semantics><msup><mi>h</mi><mn>2</mn></msup><annotation encoding="application/x-tex">h^{2}</annotation></semantics></math> as under models with non-additive effects
identical twins have higher covariance than we’d expect as they
resemble each other also because of the dominance effects as they
don’t just share alleles they share their entire genotype.
<br class="ltx_break"/></p>
</div>
</div>
<div id="S2.SS6.SSS2.P2" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">The covariance in phenotype between mother and child</h5>
<div id="S2.SS6.SSS2.P2.p1" class="ltx_para">
<p class="ltx_p">.
If the mother and father are unrelated individuals (i.e. are two
random draws from the population) then the mother and a child share
one allele IBD at each locus (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P2.p1.m1" class="ltx_Math" alttext="r_{1}=1" display="inline"><semantics><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">r_{1}=1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P2.p1.m2" class="ltx_Math" display="inline" alttext="r_{0}=r_{2}=0"><semantics><mrow><msub><mi>r</mi><mn>0</mn></msub><mo>=</mo><msub><mi>r</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">r_{0}=r_{2}=0</annotation></semantics></math>). Half the
time our mother transmits her paternal allele to the child, in which
case <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P2.p1.m3" class="ltx_Math" display="inline" alttext="X_{P1}=X_{M1}"><semantics><mrow><msub><mi>X</mi><mrow><mi>P</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>X</mi><mrow><mi>M</mi><mo>⁢</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{P1}=X_{M1}</annotation></semantics></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P2.p1.m4" class="ltx_Math" display="inline" alttext="Cov(X_{P1},X_{M2})=Var(X_{P1})"><semantics><mrow><mrow><mi>C</mi><mo>⁢</mo><mi>o</mi><mo>⁢</mo><mi>v</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mi>P</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mi>M</mi><mo>⁢</mo><mn>2</mn></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mi>P</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex">Cov(X_{P1},X_{M2})=Var(X_{P1})</annotation></semantics></math> and all
the other covariances in eqn. <a href="#S2.E21" title="(21) ‣ 2.0.2 The covariance between relatives ‣ 2 The phenotypic resemblance between relatives ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_tag">21</span></a> zero, and half
the time she transmits her maternal allele to the child
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P2.p1.m5" class="ltx_Math" alttext="Cov(X_{M1},X_{M2})=Var(X_{P1})" display="inline"><semantics><mrow><mrow><mi>C</mi><mo>⁢</mo><mi>o</mi><mo>⁢</mo><mi>v</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mi>M</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mi>M</mi><mo>⁢</mo><mn>2</mn></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mi>P</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex">Cov(X_{M1},X_{M2})=Var(X_{P1})</annotation></semantics></math> and all the other terms zero. By this
argument <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P2.p1.m6" class="ltx_Math" alttext="Cov(X_{1},X_{2})=\tfrac{1}{2}Var(X_{M1})+\tfrac{1}{2}Var(X_{P1})=\tfrac{1}{2}V% _{A}" display="inline"><semantics><mrow><mrow><mi>C</mi><mo>⁢</mo><mi>o</mi><mo>⁢</mo><mi>v</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mi>M</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mi>P</mi><mo>⁢</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><msub><mi>V</mi><mi>A</mi></msub></mrow></mrow><annotation encoding="application/x-tex">Cov(X_{1},X_{2})=\tfrac{1}{2}Var(X_{M1})+\tfrac{1}{2}Var(X_{P1})=\tfrac{1}{2}V%
_{A}</annotation></semantics></math>.
<br class="ltx_break"/></p>
</div>
</div>
<div id="S2.SS6.SSS2.P3" class="ltx_paragraph">
<h5 class="ltx_title ltx_title_paragraph">The covariance between general pairs of relatives under an
additive model</h5>
<div id="S2.SS6.SSS2.P3.p1" class="ltx_para">
<p class="ltx_p">The two examples make clear that to understand the covariance between
phenotypes of relatives we simply need to think about the alleles they
share IBD. Consider a pair of relatives (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p1.m1" class="ltx_Math" alttext="x" display="inline"><semantics><mi>x</mi><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p1.m2" class="ltx_Math" alttext="y" display="inline"><semantics><mi>y</mi><annotation encoding="application/x-tex">y</annotation></semantics></math>) with a probability <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p1.m3" class="ltx_Math" alttext="r_{0}" display="inline"><semantics><msub><mi>r</mi><mn>0</mn></msub><annotation encoding="application/x-tex">r_{0}</annotation></semantics></math>,
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p1.m4" class="ltx_Math" display="inline" alttext="r_{1}"><semantics><msub><mi>r</mi><mn>1</mn></msub><annotation encoding="application/x-tex">r_{1}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p1.m5" class="ltx_Math" alttext="r_{2}" display="inline"><semantics><msub><mi>r</mi><mn>2</mn></msub><annotation encoding="application/x-tex">r_{2}</annotation></semantics></math> of sharing zero, one, or two alleles IBD
respectively. When they share zero alleles
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p1.m6" class="ltx_Math" alttext="Cov((X_{1M}+X_{1P}),(X_{2M}+X_{2P}))=0" display="inline"><semantics><mrow><mrow><mi>C</mi><mo>⁢</mo><mi>o</mi><mo>⁢</mo><mi>v</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>P</mi></mrow></msub></mrow><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>P</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Cov((X_{1M}+X_{1P}),(X_{2M}+X_{2P}))=0</annotation></semantics></math>, when they share one allele
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p1.m7" class="ltx_Math" display="inline" alttext="Cov((X_{1M}+X_{1P}),(X_{2M}+X_{2P}))=\tfrac{1}{2}Var(X_{1M})=\frac{1}{4}V_{A}"><semantics><mrow><mrow><mi>C</mi><mo>⁢</mo><mi>o</mi><mo>⁢</mo><mi>v</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>P</mi></mrow></msub></mrow><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>P</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>⁢</mo><msub><mi>V</mi><mi>A</mi></msub></mrow></mrow><annotation encoding="application/x-tex">Cov((X_{1M}+X_{1P}),(X_{2M}+X_{2P}))=\tfrac{1}{2}Var(X_{1M})=\frac{1}{4}V_{A}</annotation></semantics></math>, and when they share two alleles <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p1.m8" class="ltx_Math" alttext="Cov((X_{1M}+X_{1P}),(X_{2M}+X_{2P}))=Var(X_{1M})=\tfrac{1}{2}V_{A}" display="inline"><semantics><mrow><mrow><mi>C</mi><mo>⁢</mo><mi>o</mi><mo>⁢</mo><mi>v</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>P</mi></mrow></msub></mrow><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>+</mo><msub><mi>X</mi><mrow><mn>2</mn><mo>⁢</mo><mi>P</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mn>1</mn><mo>⁢</mo><mi>M</mi></mrow></msub><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><msub><mi>V</mi><mi>A</mi></msub></mrow></mrow><annotation encoding="application/x-tex">Cov((X_{1M}+X_{1P}),(X_{2M}+X_{2P}))=Var(X_{1M})=\tfrac{1}{2}V_{A}</annotation></semantics></math>. Therefore, the general covariance between two
relatives is</p>
<table id="S2.E23" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E23.m1" class="ltx_Math" alttext="Cov(X_{1},X_{2})=r_{0}\times 0+r_{1}\frac{1}{4}V_{A}+r_{2}\tfrac{1}{2}V_{A}=F_% {x,y}V_{A}" display="block"><semantics><mrow><mrow><mi>C</mi><mo>⁢</mo><mi>o</mi><mo>⁢</mo><mi>v</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mn>1</mn></msub><mo>,</mo><msub><mi>X</mi><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi>r</mi><mn>0</mn></msub><mo>×</mo><mn>0</mn></mrow><mo>+</mo><mrow><msub><mi>r</mi><mn>1</mn></msub><mo>⁢</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>⁢</mo><msub><mi>V</mi><mi>A</mi></msub></mrow><mo>+</mo><mrow><msub><mi>r</mi><mn>2</mn></msub><mo>⁢</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>⁢</mo><msub><mi>V</mi><mi>A</mi></msub></mrow></mrow><mo>=</mo><mrow><msub><mi>F</mi><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>⁢</mo><msub><mi>V</mi><mi>A</mi></msub></mrow></mrow><annotation encoding="application/x-tex">Cov(X_{1},X_{2})=r_{0}\times 0+r_{1}\frac{1}{4}V_{A}+r_{2}\tfrac{1}{2}V_{A}=F_%
{x,y}V_{A}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(23)</span></td></tr>
</table>
<br class="ltx_break"/>
</div>
<div id="S2.SS6.SSS2.P3.p2" class="ltx_para">
<p class="ltx_p">So under a simple additive model of the genetic basis of a phenotype
to measure the narrow sense heritability we need to measure the
covariance between a set of pairs of relatives (assuming that we can remove the effect of
shared environmental noise). From the covariance between relatives we
can calculate <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p2.m1" class="ltx_Math" alttext="V_{A}" display="inline"><semantics><msub><mi>V</mi><mi>A</mi></msub><annotation encoding="application/x-tex">V_{A}</annotation></semantics></math>, we can then divide this by the total phenotypic
variance to get <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p2.m2" class="ltx_Math" display="inline" alttext="h^{2}"><semantics><msup><mi>h</mi><mn>2</mn></msup><annotation encoding="application/x-tex">h^{2}</annotation></semantics></math>.
<br class="ltx_break"/></p>
</div>
<div id="S2.SS6.SSS2.P3.p3" class="ltx_para">
<p class="ltx_p">Another way that we can estimate the narrow sense heritability is
through the regression of child’s phenotype on the parental mid-point
phenotype. The parental mid-point phenotype is simple the average of
the mum and dad’s phenotype. Denoting the child’s phenotype by <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p3.m1" class="ltx_Math" alttext="X_{kid}" display="inline"><semantics><msub><mi>X</mi><mrow><mi>k</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><annotation encoding="application/x-tex">X_{kid}</annotation></semantics></math> and mid-point
phenotype by <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p3.m2" class="ltx_Math" display="inline" alttext="X_{mid}"><semantics><msub><mi>X</mi><mrow><mi>m</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><annotation encoding="application/x-tex">X_{mid}</annotation></semantics></math> so that if we take the regression <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p3.m3" class="ltx_Math" display="inline" alttext="X_{kid}\sim X_{mid}"><semantics><mrow><msub><mi>X</mi><mrow><mi>k</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>∼</mo><msub><mi>X</mi><mrow><mi>m</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{kid}\sim X_{mid}</annotation></semantics></math> this
regression has slope <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p3.m4" class="ltx_Math" alttext="\beta=Cov(X_{kid},X_{mid})/Var(X_{mid})" display="inline"><semantics><mrow><mi>β</mi><mo>=</mo><mrow><mrow><mrow><mi>C</mi><mo>⁢</mo><mi>o</mi><mo>⁢</mo><mi>v</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mi>k</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mi>m</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>/</mo><mi>V</mi></mrow><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mi>m</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></mrow><annotation encoding="application/x-tex">\beta=Cov(X_{kid},X_{mid})/Var(X_{mid})</annotation></semantics></math>.
The covariance of <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p3.m5" class="ltx_Math" display="inline" alttext="Cov(X_{kid},X_{mid})=\tfrac{1}{2}V_{A}"><semantics><mrow><mrow><mi>C</mi><mo>⁢</mo><mi>o</mi><mo>⁢</mo><mi>v</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mi>k</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mi>m</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><msub><mi>V</mi><mi>A</mi></msub></mrow></mrow><annotation encoding="application/x-tex">Cov(X_{kid},X_{mid})=\tfrac{1}{2}V_{A}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS2.P3.p3.m6" class="ltx_Math" alttext="Var(X_{mid})=\tfrac{1}{2}V" display="inline"><semantics><mrow><mrow><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mi>m</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>⁢</mo><mi>V</mi></mrow></mrow><annotation encoding="application/x-tex">Var(X_{mid})=\tfrac{1}{2}V</annotation></semantics></math> as by taking the average of the
parents we have halved the variance, such that the slope of the
regression is</p>
<table id="S2.E24" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E24.m1" class="ltx_Math" alttext="\beta=\frac{Cov(X_{kid},X_{mid})}{Var(X_{mid})}=\frac{V_{A}}{V}=h^{2}" display="block"><semantics><mrow><mi>β</mi><mo>=</mo><mfrac><mrow><mi>C</mi><mo>⁢</mo><mi>o</mi><mo>⁢</mo><mi>v</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>X</mi><mrow><mi>k</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mi>m</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mrow><mi>V</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>r</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mi>m</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><msub><mi>V</mi><mi>A</mi></msub><mi>V</mi></mfrac><mo>=</mo><msup><mi>h</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\beta=\frac{Cov(X_{kid},X_{mid})}{Var(X_{mid})}=\frac{V_{A}}{V}=h^{2}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(24)</span></td></tr>
</table>
<p class="ltx_p">i.e. the regression of the child’s phenotype on the parental midpoint
phenotype is an estimate of the narrow sense heritability. This is a
common way to estimate heritability, although it doesn’t bypass the
need to control for environmental correlations between relatives.
<br class="ltx_break"/></p>
</div>
<div id="S2.SS6.SSS2.P3.p4" class="ltx_para">
<p class="ltx_p">Our regression allows us to attempt to predict the phenotype of the
child given the child; how well we can do this depends on the
slope. If the slope close to zero then the parental phenotypes hold no
information about the phenotype of the child, while if the slope is
close to one then the parental mid-point is a good guess at the child’s
phenotype.
<br class="ltx_break"/></p>
</div>
<div id="S2.F2" class="ltx_figure"><object data="x4.png" id="S2.F2.g1" class="ltx_graphics ltx_centering" width="538" height="320" alt=""/>
<div class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Regression of parental mid-point phenotype on child’s
phenotype.</div>
</div>
<div id="S2.SS6.SSS2.P3.p5" class="ltx_para">
<p class="ltx_p">More formally the expected phenotype of the child given the parental
phenotypes is</p>
<table id="S2.E25" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E25.m1" class="ltx_Math" alttext="\mathbb{E}(X_{kid}|X_{mum},X_{dad})=\mu+\beta(X_{mid}-\mu)=\mu+h^{2}(X_{mid}-\mu)" display="block"><semantics><mrow><mi>𝔼</mi><mrow><mo>(</mo><msub><mi>X</mi><mrow><mi>k</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>|</mo><msub><mi>X</mi><mrow><mi>m</mi><mo>⁢</mo><mi>u</mi><mo>⁢</mo><mi>m</mi></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mi>d</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>μ</mi><mo>+</mo><mi>β</mi><mrow><mo>(</mo><msub><mi>X</mi><mrow><mi>m</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>-</mo><mi>μ</mi><mo>)</mo></mrow><mo>=</mo><mi>μ</mi><mo>+</mo><msup><mi>h</mi><mn>2</mn></msup><mrow><mo>(</mo><msub><mi>X</mi><mrow><mi>m</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>-</mo><mi>μ</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathbb{E}(X_{kid}|X_{mum},X_{dad})=\mu+\beta(X_{mid}-\mu)=\mu+h^{2}(X_{mid}-\mu)</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(25)</span></td></tr>
</table>
<p class="ltx_p">this follows from the definition of linear regression. So to find the
child’s predicted phenotype we simply take the mean phenotype and add
on the difference between our parental mid-point multiplied by our
narrow sense heritability.
<br class="ltx_break"/></p>
</div>
</div>
</div>
<div id="S2.SS6.SSS3" class="ltx_subsubsection">
<h4 class="ltx_title ltx_title_subsubsection"><span class="ltx_tag ltx_tag_subsubsection">2.0.3 </span>The response to selection</h4>
<div id="S2.SS6.SSS3.p1" class="ltx_para">
<p class="ltx_p">Evolution by natural selection requires:</p>
<ol id="I2" class="ltx_enumerate">
<li id="I2.i1" class="ltx_item" style="list-style-type:none;"><span class="ltx_tag ltx_tag_enumerate">1.</span>
<div id="I2.i1.p1" class="ltx_para">
<p class="ltx_p">Variation in a phenotype</p>
</div></li>
<li id="I2.i2" class="ltx_item" style="list-style-type:none;"><span class="ltx_tag ltx_tag_enumerate">2.</span>
<div id="I2.i2.p1" class="ltx_para">
<p class="ltx_p">That survival is non-random with respect to this phenotypic
variation.</p>
</div></li>
<li id="I2.i3" class="ltx_item" style="list-style-type:none;"><span class="ltx_tag ltx_tag_enumerate">3.</span>
<div id="I2.i3.p1" class="ltx_para">
<p class="ltx_p">That this variation is heritable.</p>
</div></li>
</ol>
<p class="ltx_p">Points 1 and 2 encapsulate our idea of Natural Selection, but evolution by natural
selection will only occur if the 3rd condition is met. It is the
heritable nature of variation that couples change within a generation
due to natural selection, to change across generations (evolutionary
change).
<br class="ltx_break"/></p>
</div>
<div id="S2.SS6.SSS3.p2" class="ltx_para">
<p class="ltx_p">Lets start by thinking about the change within a generation due
to directional selection, where selection acts to change the mean
phenotype within a generation. For example, a decrease in mean height within a
generation, due to taller organisms having a lower chance of surviving
to reproduction than shorter organisms. Specifically, we’ll denote our mean phenotype at
reproduction by <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p2.m1" class="ltx_Math" display="inline" alttext="\mu_{S}"><semantics><msub><mi>μ</mi><mi>S</mi></msub><annotation encoding="application/x-tex">\mu_{S}</annotation></semantics></math>, i.e. after selection has acted, and our mean
phenotype before selection acts by <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p2.m2" class="ltx_Math" display="inline" alttext="\mu_{BS}"><semantics><msub><mi>μ</mi><mrow><mi>B</mi><mo>⁢</mo><mi>S</mi></mrow></msub><annotation encoding="application/x-tex">\mu_{BS}</annotation></semantics></math>. This second quantity may be hard to
measure, as obviously selection acts throughout the life-cycle, so it
might be easier to think of this as the mean phenotype if selection
hadn’t acted. So the mean phenotype changes within a generation is <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p2.m3" class="ltx_Math" display="inline" alttext="\mu_{S}-\mu_{BS}=S"><semantics><mrow><mrow><msub><mi>μ</mi><mi>S</mi></msub><mo>-</mo><msub><mi>μ</mi><mrow><mi>B</mi><mo>⁢</mo><mi>S</mi></mrow></msub></mrow><mo>=</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">\mu_{S}-\mu_{BS}=S</annotation></semantics></math>.
<br class="ltx_break"/></p>
</div>
<div id="S2.SS6.SSS3.p3" class="ltx_para">
<p class="ltx_p">We are interested in predicting the distribution of phenotypes in next
generation, in particular we are interested in the mean phenotype in
the next generation to understand how directional selection has
contributed to evolutionary change. We’ll denote the mean phenotype in
offspring, i.e. the mean phenotype in the next generation before selection acts,
as <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p3.m1" class="ltx_Math" alttext="\mu_{NG}" display="inline"><semantics><msub><mi>μ</mi><mrow><mi>N</mi><mo>⁢</mo><mi>G</mi></mrow></msub><annotation encoding="application/x-tex">\mu_{NG}</annotation></semantics></math>. The change across generations we’ll call the response
to selection <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p3.m2" class="ltx_Math" display="inline" alttext="R"><semantics><mi>R</mi><annotation encoding="application/x-tex">R</annotation></semantics></math> and put this equal to <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p3.m3" class="ltx_Math" alttext="\mu_{NG}-\mu_{BS}" display="inline"><semantics><mrow><msub><mi>μ</mi><mrow><mi>N</mi><mo>⁢</mo><mi>G</mi></mrow></msub><mo>-</mo><msub><mi>μ</mi><mrow><mi>B</mi><mo>⁢</mo><mi>S</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mu_{NG}-\mu_{BS}</annotation></semantics></math>.
<br class="ltx_break"/></p>
</div>
<div id="S2.SS6.SSS3.p4" class="ltx_para">
<p class="ltx_p">The mean phenotype in the next generation is</p>
<table id="S2.E26" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E26.m1" class="ltx_Math" display="block" alttext="\mu_{NG}=\mathbb{E}\left(\mathbb{E}(X_{kid}|X_{mum},X_{dad})\right)"><semantics><mrow><msub><mi>μ</mi><mrow><mi>N</mi><mo>⁢</mo><mi>G</mi></mrow></msub><mo>=</mo><mi>𝔼</mi><mrow><mo>(</mo><mi>𝔼</mi><mrow><mo>(</mo><msub><mi>X</mi><mrow><mi>k</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>|</mo><msub><mi>X</mi><mrow><mi>m</mi><mo>⁢</mo><mi>u</mi><mo>⁢</mo><mi>m</mi></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><mi>d</mi><mo>⁢</mo><mi>a</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mu_{NG}=\mathbb{E}\left(\mathbb{E}(X_{kid}|X_{mum},X_{dad})\right)</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(26)</span></td></tr>
</table>
<p class="ltx_p">where the outer expectation is over the randomly mating of individuals
who survive to reproduce. We can use eqn. <a href="#S2.E25" title="(25) ‣ The covariance between general pairs of relatives under an additive model ‣ 2.0.2 The covariance between relatives ‣ 2 The phenotypic resemblance between relatives ‣ Notes on Population Genetics" class="ltx_ref"><span class="ltx_text ltx_ref_tag">25</span></a> to obtain
an expression for this</p>
<table id="S2.E27" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E27.m1" class="ltx_Math" display="block" alttext="\mu_{NG}=\mu_{BS}+\beta(\mathbb{E}(X_{mid})-\mu_{BS})"><semantics><mrow><msub><mi>μ</mi><mrow><mi>N</mi><mo>⁢</mo><mi>G</mi></mrow></msub><mo>=</mo><mrow><msub><mi>μ</mi><mrow><mi>B</mi><mo>⁢</mo><mi>S</mi></mrow></msub><mo>+</mo><mrow><mi>β</mi><mo>⁢</mo><mrow><mo>(</mo><mrow><mrow><mi>𝔼</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mi>m</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow><mo>-</mo><msub><mi>μ</mi><mrow><mi>B</mi><mo>⁢</mo><mi>S</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow></mrow></mrow><annotation encoding="application/x-tex">\mu_{NG}=\mu_{BS}+\beta(\mathbb{E}(X_{mid})-\mu_{BS})</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(27)</span></td></tr>
</table>
<p class="ltx_p">so to obtain <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p4.m1" class="ltx_Math" display="inline" alttext="\mu_{NG}"><semantics><msub><mi>μ</mi><mrow><mi>N</mi><mo>⁢</mo><mi>G</mi></mrow></msub><annotation encoding="application/x-tex">\mu_{NG}</annotation></semantics></math> we need to compute <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p4.m2" class="ltx_Math" alttext="\mathbb{E}(X_{mid})" display="inline"><semantics><mrow><mi>𝔼</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>X</mi><mrow><mi>m</mi><mo>⁢</mo><mi>i</mi><mo>⁢</mo><mi>d</mi></mrow></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\mathbb{E}(X_{mid})</annotation></semantics></math> the expected
mid-point phenotype of pairs of individuals who survive to
reproduce. Well this is just the expected phenotype in the individuals
who survived to reproduce (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p4.m3" class="ltx_Math" alttext="\mu_{S}" display="inline"><semantics><msub><mi>μ</mi><mi>S</mi></msub><annotation encoding="application/x-tex">\mu_{S}</annotation></semantics></math>), so</p>
<table id="S2.E28" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E28.m1" class="ltx_Math" alttext="\mu_{NG}=\mu_{BS}+h^{2}(\mu_{S}-\mu_{BS})" display="block"><semantics><mrow><msub><mi>μ</mi><mrow><mi>N</mi><mo>⁢</mo><mi>G</mi></mrow></msub><mo>=</mo><mrow><msub><mi>μ</mi><mrow><mi>B</mi><mo>⁢</mo><mi>S</mi></mrow></msub><mo>+</mo><mrow><msup><mi>h</mi><mn>2</mn></msup><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>μ</mi><mi>S</mi></msub><mo>-</mo><msub><mi>μ</mi><mrow><mi>B</mi><mo>⁢</mo><mi>S</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow></mrow></mrow><annotation encoding="application/x-tex">\mu_{NG}=\mu_{BS}+h^{2}(\mu_{S}-\mu_{BS})</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(28)</span></td></tr>
</table>
<p class="ltx_p">So we can write our response to selection as</p>
<table id="S2.E29" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E29.m1" class="ltx_Math" alttext="R=\mu_{NG}-\mu_{BS}=h^{2}(\mu_{S}-\mu_{BS})=h^{2}S" display="block"><semantics><mrow><mi>R</mi><mo>=</mo><mrow><msub><mi>μ</mi><mrow><mi>N</mi><mo>⁢</mo><mi>G</mi></mrow></msub><mo>-</mo><msub><mi>μ</mi><mrow><mi>B</mi><mo>⁢</mo><mi>S</mi></mrow></msub></mrow><mo>=</mo><mrow><msup><mi>h</mi><mn>2</mn></msup><mo>⁢</mo><mrow><mo>(</mo><mrow><msub><mi>μ</mi><mi>S</mi></msub><mo>-</mo><msub><mi>μ</mi><mrow><mi>B</mi><mo>⁢</mo><mi>S</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi>h</mi><mn>2</mn></msup><mo>⁢</mo><mi>S</mi></mrow></mrow><annotation encoding="application/x-tex">R=\mu_{NG}-\mu_{BS}=h^{2}(\mu_{S}-\mu_{BS})=h^{2}S</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(29)</span></td></tr>
</table>
<p class="ltx_p">So our response to selection is proportional to our selection
differential, and the constant of proportionality is the narrow sense
heritability. This equation is sometimes termed the Breeders
equation. It is a statement that the evolutionary change across
generations (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p4.m4" class="ltx_Math" alttext="R" display="inline"><semantics><mi>R</mi><annotation encoding="application/x-tex">R</annotation></semantics></math>) is proportional to the change caused by directional selection
within a generation, and the strength of this relationship is
determined by the narrow sense heritability.
<br class="ltx_break"/></p>
</div>
<div id="S2.SS6.SSS3.p5" class="ltx_para">
<p class="ltx_p">Using the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p5.m1" class="ltx_Math" display="inline" alttext="h^{2}=V_{A}/V"><semantics><mrow><msup><mi>h</mi><mn>2</mn></msup><mo>=</mo><mrow><msub><mi>V</mi><mi>A</mi></msub><mo>/</mo><mi>V</mi></mrow></mrow><annotation encoding="application/x-tex">h^{2}=V_{A}/V</annotation></semantics></math> we can rewrite this in a different form as</p>
<table id="S2.E30" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E30.m1" class="ltx_Math" alttext="R=V_{A}\frac{S}{V}" display="block"><semantics><mrow><mi>R</mi><mo>=</mo><mrow><msub><mi>V</mi><mi>A</mi></msub><mo>⁢</mo><mfrac><mi>S</mi><mi>V</mi></mfrac></mrow></mrow><annotation encoding="application/x-tex">R=V_{A}\frac{S}{V}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(30)</span></td></tr>
</table>
<p class="ltx_p">i.e. our response to selection is the additive genetic variance of our
trait (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p5.m2" class="ltx_Math" display="inline" alttext="V_{A}"><semantics><msub><mi>V</mi><mi>A</mi></msub><annotation encoding="application/x-tex">V_{A}</annotation></semantics></math>) multiplied by the change within a generation as a
fraction of the total phenotypic variance (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p5.m3" class="ltx_Math" alttext="S/V" display="inline"><semantics><mrow><mi>S</mi><mo>/</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">S/V</annotation></semantics></math>).
<br class="ltx_break"/></p>
</div>
<div id="S2.SS6.SSS3.fig1" class="ltx_figure"><object data="x5.png" id="S2.SS6.SSS3.g1" class="ltx_graphics ltx_centering" width="538" height="320" alt=""/>
</div>
<div id="S2.SS6.SSS3.p6" class="ltx_para">
<p class="ltx_p">A change in mean phenotype within a generation occurs because of the
differential fitness of our organisms. To think more carefully about this change within a
generation lets think about a simple fitness model where our phenotype affects the
viability of our organisms (i.e. the probability they survive to
reproduce). The probability that an individual has a phenotype <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p6.m1" class="ltx_Math" display="inline" alttext="X"><semantics><mi>X</mi><annotation encoding="application/x-tex">X</annotation></semantics></math>
before selection is <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p6.m2" class="ltx_Math" display="inline" alttext="p(X)"><semantics><mrow><mi>p</mi><mo>⁢</mo><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">p(X)</annotation></semantics></math>, so that the mean phenotype before
selection is</p>
<table id="S2.E31" class="ltx_equation">
<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E31.m1" class="ltx_Math" display="block" alttext="\mu_{BS}=\mathbb{E}[X]=\int_{-\infty}^{\infty}xp(x)dx"><semantics><mrow><msub><mi>μ</mi><mrow><mi>B</mi><mo>⁢</mo><mi>S</mi></mrow></msub><mo>=</mo><mrow><mi>𝔼</mi><mo>⁢</mo><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow></mrow><mo>=</mo><mrow><msubsup><mo>∫</mo><mrow><mo>-</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></msubsup><mrow><mi>x</mi><mo>⁢</mo><mi>p</mi><mo>⁢</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>⁢</mo><mi>d</mi><mo>⁢</mo><mi>x</mi></mrow></mrow></mrow><annotation encoding="application/x-tex">\mu_{BS}=\mathbb{E}[X]=\int_{-\infty}^{\infty}xp(x)dx</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/>
<td rowspan="1" class="ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation">(31)</span></td></tr>
</table>
<p class="ltx_p">The probability that an organism with a phenotype <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p6.m3" class="ltx_Math" display="inline" alttext="X"><semantics><mi>X</mi><annotation encoding="application/x-tex">X</annotation></semantics></math> survives to
reproduce is <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS6.SSS3.p6.m4" class="ltx_Math" display="inline" alttext="w(X)"><semantics><mrow><mi>w</mi><mo>⁢</mo><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">w(X)</annotation></semantics></math>, and we’ll think about this as the fitness of