A cross such as Mendel's, where true-breeding (largely homozygous) opposite parents are crossed in a controlled way to produce an F1, is a special case of hybrid structure. The F1 is often regarded as "entirely heterozygous" for the gene under consideration. However, this is an over-simplification and does not apply generally—for example when individual parents are not homozygous, or when  populations  inter-hybridise to form  hybrid swarms . [ 24 ]  The general properties of intra-species hybrids (F1) and F2 (both "autogamous" and "allogamous") are considered in a later section.


Having noticed that the pea is naturally self-pollinated, we cannot continue to use it as an example for illustrating random fertilization properties. Self-fertilization ("selfing") is a major alternative to random fertilization, especially within Plants. Most of the Earth's cereals are naturally self-pollinated (rice, wheat, barley, for example), as well as the pulses. Considering the millions of individuals of each of these on Earth at any time, it is obvious that self-fertilization is at least as significant as random fertilization. Self-fertilization is the most intensive form of  inbreeding , which arises whenever there is restricted independence in the genetical origins of gametes. Such reduction in independence arises if parents are already related, and/or from genetic drift or other spatial restrictions on gamete dispersal. Path analysis demonstrates that these are tantamount to the same thing. [ 26 ] [ 27 ]  Arising from this background, the  inbreeding coefficient  (often symbolized as  F  or  f ) quantifies the effect of inbreeding from whatever cause. There are several formal definitions of  f , and some of these are considered in later sections. For the present, note that for a long-term self-fertilized species  f  =  1 .
Natural self-fertilized populations are not single "  pure lines  ", however, but mixtures of such lines. This becomes particularly obvious when considering more than one gene at a time. Therefore, allele frequencies ( p  and  q ) other than  1  or  0  are still relevant in these cases (refer back to the Mendel Cross section). The genotype frequencies take a different form, however.


In general, the genotype frequencies become

 [

 p

 2

 (
 1
 −
 f
 )
 +
 p
 f
 ]

 {\textstyle [p^{2}(1-f)+pf]}

  for  AA  and

 2
 p
 q
 (
 1
 −
 f
 )

 {\textstyle 2pq(1-f)}

  for  Aa  and

 [

 q

 2

 (
 1
 −
 f
 )
 +
 q
 f
 ]

 {\textstyle [q^{2}(1-f)+qf]}

  for  aa . [ 13 ] : 65


Notice that the frequency of the heterozygote declines in proportion to  f . When  f  = 1 , these three frequencies become respectively  p ,  0  and  q  Conversely, when  f = 0 , they reduce to the random-fertilization quadratic expansion shown previously.


The population mean shifts the central reference point from the homozygote midpoint ( mp ) to the mean of a sexually reproduced population. This is important not only to relocate the focus into the natural world, but also to use a measure of  central tendency  used by Statistics/Biometrics. In particular, the square of this mean is the Correction Factor, which is used to obtain the genotypic variances later. [ 9 ]


For each genotype in turn, its allele effect is multiplied by its genotype frequency; and the products are accumulated across all genotypes in the model. Some algebraic simplification usually follows to reach a succinct result.


The contribution of  AA  is

 p

 2

 (
 +
 )
 a

 {\textstyle p^{2}(+)a}

 , that of  Aa  is

 2
 p
 q
 d

 {\textstyle 2pqd}

 , and that of  aa  is

 q

 2

 (
 −
 )
 a

 {\textstyle q^{2}(-)a}

 . Gathering together the two  a  terms and accumulating over all, the result is:

 a
 (

 p

 2

 −

 q

 2

 )
 +
 2
 p
 q
 d

 {\textstyle a(p^{2}-q^{2})+2pqd}

 . Simplification is achieved by noting that

 (

 p

 2

 −

 q

 2

 )
 =
 (
 p
 −
 q
 )
 (
 p
 +
 q
 )

 {\textstyle (p^{2}-q^{2})=(p-q)(p+q)}

 , and by recalling that

 (
 p
 +
 q
 )
 =
 1

 {\textstyle (p+q)=1}

 , thereby reducing the right-hand term to

 (
 p
 −
 q
 )

 {\textstyle (p-q)}

 .


The succinct result is therefore

 G
 =
 a
 (
 p
 −
 q
 )
 +
 2
 p
 q
 d

 {\textstyle G=a(p-q)+2pqd}

 . [ 14 ]   : 110


This defines the population mean as an "offset" from the homozygote midpoint (recall  a  and  d  are defined as  deviations  from that midpoint). The Figure depicts  G  across all values of  p  for several values of  d , including one case of slight over-dominance. Notice that  G  is often negative, thereby emphasizing that it is itself a  deviation  (from  mp ).


Finally, to obtain the  actual  Population Mean in "phenotypic space", the midpoint value is added to this offset:

 P
 =
 G
 +
 m
 p

 {\textstyle P=G+mp}

 .


An example arises from data on ear length in maize. [ 28 ] : 103   Assuming for now that one gene only is represented,  a  = 5.45 cm,  d  = 0.12 cm [virtually "0", really],  mp  = 12.05 cm. Further assuming that  p  = 0.6 and  q  = 0.4 in this example population, then:


G  = 5.45 (0.6 − 0.4) + (0.48)0.12 =  1.15 cm  (rounded); and


P  = 1.15 + 12.05 =  13.20 cm  (rounded).


The contribution of  AA  is

 p
 (
 +
 a
 )

 {\textstyle p(+a)}

 , while that of  aa  is

 q
 (
 −
 a
 )

 {\textstyle q(-a)}

 . [See above for the frequencies.] Gathering these two  a  terms together leads to an immediately very simple final result:


G

 (
 f
 =
 1
 )

 =
 a
 (
 p
 −
 q
 )

 {\textstyle G_{(f=1)}=a(p-q)}

 . As before,

 P
 =
 G
 +
 m
 p

 {\textstyle P=G+mp}

 .


Often, "G (f=1) " is abbreviated to "G 1 ".


Mendel's peas can provide us with the allele effects and midpoint (see previously); and a mixed self-pollinated population with  p  = 0.6 and  q  = 0.4 provides example frequencies. Thus:


G (f=1)  = 82 (0.6 − .04) = 59.6 cm (rounded); and


P (f=1)  = 59.6 + 116 = 175.6 cm (rounded).


A general formula incorporates the inbreeding coefficient  f , and can then accommodate any situation. The procedure is exactly the same as before, using the weighted genotype frequencies given earlier. After translation into our symbols, and further rearrangement: [ 13 ]   : 77–78


G

 f

 =
 a
 (
 q
 −
 p
 )
 +
 [
 2
 p
 q
 d
 −
 f
 (
 2
 p
 q
 d
 )
 ]

 =
 a
 (
 p
 −
 q
 )
 +
 (
 1
 −
 f
 )
 2
 p
 q
 d

 =

 G

 0

 −
 f

 2
 p
 q
 d

 {\displaystyle {\begin{aligned}G_{f}&=a(q-p)+[2pqd-f(2pqd)]\\&=a(p-q)+(1-f)2pqd\\&=G_{0}-f\ 2pqd\end{aligned}}}

Here,  G 0  is  G , which was given earlier. (Often, when dealing with inbreeding, "G 0 " is preferred to "G".)


Supposing that the maize example [given earlier] had been constrained on a holme (a narrow riparian meadow), and had partial inbreeding to the extent of  f  =  0.25 , then, using the third version (above) of  G f :


G 0.25  = 1.15 − 0.25 (0.48) 0.12 = 1.136 cm (rounded), with  P 0.25  = 13.194 cm (rounded).


There is hardly any effect from inbreeding in this example, which arises because there was virtually no dominance in this attribute ( d  → 0). Examination of all three versions of  G f  reveals that this would lead to trivial change in the Population mean. Where dominance was notable, however, there would be considerable change.


Genetic drift was introduced when discussing the likelihood of panmixia being widely extant as a natural fertilization pattern. [See section on Allele and Genotype frequencies.] Here the sampling of gametes from the  potential  gamodeme is discussed in more detail. The sampling involves random fertilization between pairs of random gametes, each of which may contain either an  A  or an  a  allele. The sampling is therefore binomial sampling. [ 13 ] : 382–395  [ 14 ] : 49–63  [ 29 ] : 35  [ 30 ] : 55   Each sampling "packet" involves  2N  alleles, and produces  N  zygotes (a "progeny" or a "line") as a result. During the course of the reproductive period, this sampling is repeated over and over, so that the final result is a mixture of sample progenies. The result is  dispersed random fertilization

 (
 ⨀
 )

 {\displaystyle \left(\bigodot \right)}

  These events, and the overall end-result, are examined here with an illustrative example.


The "base" allele frequencies of the example are those of the  potential gamodeme : the frequency of  A  is  p g  = 0.75 , while the frequency of  a  is  q g  = 0.25 . [ White label  " 1 " in the diagram.] Five example actual gamodemes are binomially sampled out of this base ( s  = the number of samples = 5), and each sample is designated with an "index"  k : with  k = 1 .... s  sequentially. (These are the sampling "packets" referred to in the previous paragraph.) The number of gametes involved in fertilization varies from sample to sample, and is given as  2N k  [at  white label  " 2 " in the diagram]. The total (Σ) number of gametes sampled overall is 52 [ white label  " 3 " in the diagram]. Because each sample has its own size,  weights  are needed to obtain averages (and other statistics) when obtaining the overall results. These are

 ω

 k

 =
 2

 N

 k

 /

 (

 ∑

 k

 s

 2

 N

 k

 )

 {\textstyle \omega _{k}=2N_{k}/(\sum _{k}^{s}2N_{k})}

 , and are given at  white label  " 4 " in the diagram.


Following completion of these five binomial sampling events, the resultant actual gamodemes each contained different allele frequencies—( p k  and  q k ). [These are given at  white label  " 5 " in the diagram.] This outcome is actually the genetic drift itself. Notice that two samples (k = 1 and 5) happen to have the same frequencies as the  base  ( potential ) gamodeme. Another (k = 3) happens to have the  p  and  q  "reversed". Sample (k = 2) happens to be an "extreme" case, with  p k  = 0.9  and  q k  = 0.1 ; while the remaining sample (k = 4) is "middle of the range" in its allele frequencies. All of these results have arisen only by "chance", through binomial sampling. Having occurred, however, they set in place all the downstream properties of the progenies.


Because sampling involves chance, the  probabilities  (  ∫ k  ) of obtaining each of these samples become of interest. These binomial probabilities depend on the starting frequencies ( p g  and  q g ) and the sample size ( 2N k ). They are tedious to obtain, [ 13 ] : 382–395  [ 30 ] : 55   but are of considerable interest. [See  white label  " 6 " in the diagram.] The two samples (k = 1, 5), with the allele frequencies the same as in the  potential gamodeme , had higher "chances" of occurring than the other samples. Their binomial probabilities did differ, however, because of their different sample sizes (2N k ). The "reversal" sample (k = 3) had a very low Probability of occurring, confirming perhaps what might be expected. The "extreme" allele frequency gamodeme (k = 2) was not "rare", however; and the "middle of the range" sample (k=4)  was  rare. These same Probabilities apply also to the progeny of these fertilizations.


Here, some  summarizing  can begin. The  overall allele frequencies  in the progenies bulk are supplied by weighted averages of the appropriate frequencies of the individual samples. That is:

 p

 ⋅

 =

 ∑

 k

 s

 ω

 k

 p

 k

 {\textstyle p_{\centerdot }=\sum _{k}^{s}\omega _{k}\ p_{k}}

  and

 q

 ⋅

 =

 ∑

 k

 s

 ω

 k

 q

 k

 {\textstyle q_{\centerdot }=\sum _{k}^{s}\omega _{k}\ q_{k}}

 . (Notice that  k  is replaced by  •  for the overall result—a common practice.) [ 9 ]  The results for the example are  p •  = 0.631 and  q •  = 0.369 [ black label  " 5 " in the diagram]. These values are quite different to the starting ones ( p g  and  q g ) [ white label  " 1 "]. The sample allele frequencies also have variance as well as an average. This has been obtained using the  sum of squares (SS)  method  [ 31 ]  [See to the right of  black label  " 5 " in the diagram]. [Further discussion on this variance occurs in the section below on Extensive genetic drift.]


The  genotype frequencies  of the five sample progenies are obtained from the usual quadratic expansion of their respective allele frequencies ( random fertilization ). The results are given at the diagram's  white label  " 7 " for the homozygotes, and at  white label  " 8 " for the heterozygotes. Re-arrangement in this manner prepares the way for monitoring inbreeding levels. This can be done either by examining the level of  total  homozygosis [( p 2 k  + q 2 k ) = ( 1 − 2p k q k )], or by examining the level of heterozygosis  ( 2p k q k ), as they are complementary. [ 32 ]     Notice that samples  k= 1, 3, 5  all had the same level of heterozygosis, despite one being the "mirror image" of the others with respect to allele frequencies. The "extreme" allele-frequency case (k=  2 ) had the most homozygosis (least heterozygosis) of any sample. The "middle of the range" case (k=  4 ) had the least homozygosity (most heterozygosity): they were each equal at 0.50, in fact.


The  overall summary  can continue by obtaining the  weighted average  of the respective genotype frequencies for the progeny bulk. Thus, for  AA , it is

 p

 ⋅

 2

 =

 ∑

 k

 s

 ω

 k

 p

 k

 2

 {\textstyle p_{\centerdot }^{2}=\sum _{k}^{s}\omega _{k}\ p_{k}^{2}}

 , for  Aa , it is

 2

 p

 ⋅

 q

 ⋅

 =

 ∑

 k

 s

 ω

 k

 2

 p

 k

 q

 k

 {\textstyle 2p_{\centerdot }q_{\centerdot }=\sum _{k}^{s}\omega _{k}\ 2p_{k}q_{k}}

  and for  aa , it is

 q

 ⋅

 2

 =

 ∑

 k

 s

 ω

 k

 q

 k

 2

 {\textstyle q_{\centerdot }^{2}=\sum _{k}^{s}\omega _{k}\ q_{k}^{2}}

 . The example results are given at  black label  " 7 " for the homozygotes, and at  black label  " 8 " for the heterozygote. Note that the heterozygosity mean is  0.3588 , which the next section uses to examine inbreeding resulting from this genetic drift.


The next focus of interest is the dispersion itself, which refers to the "spreading apart" of the progenies'  population means . These are obtained as

 G

 k

 =
 a
 (

 p

 k

 −

 q

 k

 )
 +
 2

 p

 k

 q

 k

 d

 {\textstyle G_{k}=a(p_{k}-q_{k})+2p_{k}q_{k}d}

  [see section on the Population mean], for each sample progeny in turn, using the example gene effects given at  white label  " 9 " in the diagram. Then, each

 P

 k

 =

 G

 k

 +
 m
 p

 {\textstyle P_{k}=G_{k}+mp}

  is obtained also [at  white label  " 10 " in the diagram]. Notice that the "best" line (k = 2) had the  highest  allele frequency for the "more" allele ( A ) (it also had the highest level of homozygosity). The  worst  progeny (k = 3) had the highest frequency for the "less" allele ( a ), which accounted for its poor performance. This "poor" line was less homozygous than the "best" line; and it shared the same level of homozygosity, in fact, as the two  second-best  lines (k = 1, 5). The progeny line with both the "more" and the "less" alleles present in equal frequency (k = 4) had a mean below the  overall average  (see next paragraph), and had the lowest level of homozygosity. These results reveal the fact that the alleles most prevalent in the "gene-pool" (also called the "germplasm") determine performance, not the level of homozygosity per se. Binomial sampling alone effects this dispersion.


The  overall summary  can now be concluded by obtaining

 G

 ⋅

 =

 ∑

 k

 s

 ω

 k

 G

 k

 {\textstyle G_{\centerdot }=\sum _{k}^{s}\omega _{k}\ G_{k}}

  and

 P

 ⋅

 =

 ∑

 k

 s

 ω

 k

 P

 k

 {\textstyle P_{\centerdot }=\sum _{k}^{s}\omega _{k}\ P_{k}}

 . The example result for  P •  is 36.94 ( black label  " 10 " in the diagram). This later is used to quantify  inbreeding depression  overall, from the gamete sampling. [See the next section.] However, recall that some "non-depressed" progeny means have been identified already (k = 1, 2, 5). This is an enigma of inbreeding—while there may be "depression" overall, there are usually superior lines among the gamodeme samplings.


Included in the  overall summary  were the average allele frequencies in the mixture of progeny lines ( p •  and  q • ). These can now be used to construct a hypothetical panmictic equivalent. [ 13 ] : 382–395  [ 14 ] : 49–63  [ 29 ] : 35   This can be regarded as a "reference" to assess the changes wrought by the gamete sampling. The example appends such a panmictic to the right of the Diagram. The frequency of  AA  is therefore  (p • ) 2  = 0.3979. This is less than that found in the dispersed bulk (0.4513 at  black label  " 7 "). Similarly, for  aa ,  (q • ) 2  = 0.1303—again less than the equivalent in the progenies bulk (0.1898). Clearly,  genetic drift  has increased the overall level of homozygosis by the amount (0.6411 − 0.5342) = 0.1069. In a complementary approach, the heterozygosity could be used instead. The panmictic equivalent for  Aa  is  2 p •  q •  = 0.4658, which is  higher  than that in the sampled bulk (0.3588) [ black label  " 8 "]. The sampling has caused the heterozygosity to decrease by 0.1070, which differs trivially from the earlier estimate because of rounding errors.


The  inbreeding coefficient  ( f ) was introduced in the early section on Self Fertilization. Here, a formal definition of it is considered:  f  is the probability that two "same" alleles (that is  A  and  A , or  a  and  a ), which fertilize together are of common ancestral origin—or (more formally)  f  is the probability that two homologous alleles are autozygous. [ 14 ] [ 27 ]  Consider any random gamete in the  potential  gamodeme that has its syngamy partner restricted by binomial sampling. The probability that that second gamete is homologous autozygous to the first is  1/(2N) , the reciprocal of the gamodeme size. For the five example progenies, these quantities are 0.1, 0.0833, 0.1, 0.0833 and 0.125 respectively, and their weighted average is  0.0961 . This is the  inbreeding coefficient  of the example progenies bulk, provided it is  unbiased  with respect to the full binomial distribution. An example based upon  s = 5  is likely to be biased, however, when compared to an appropriate entire binomial distribution based upon the sample number ( s ) approaching infinity ( s → ∞ ). Another derived definition of  f  for the full Distribution is that  f  also equals the rise in homozygosity, which equals the fall in heterozygosity. [ 33 ]  For the example, these frequency changes are  0.1069  and  0.1070 , respectively. This result is different to the above, indicating that bias with respect to the full underlying distribution is present in the example. For the example  itself , these latter values are the better ones to use, namely  f •  =  0.10695 .


The  population mean  of the equivalent panmictic is found as  [a (p • -q • ) + 2 p • q •  d] + mp . Using the example  gene effects  ( white label  " 9 " in the diagram), this mean is

 P

 ⋅

 =

 {\textstyle P_{\centerdot }=}

  37.87. The equivalent mean in the dispersed bulk is 36.94 ( black label  " 10 "), which is depressed by the amount  0.93 . This is the  inbreeding depression  from this Genetic Drift. However, as noted previously, three progenies were  not  depressed (k = 1, 2, 5), and had means even greater than that of the panmictic equivalent. These are the lines a plant breeder looks for in a line selection programme. [ 34 ]


If the number of binomial samples is large ( s → ∞  ),  then  p •  → p g  and  q •  → q g . It might be queried whether panmixia would effectively re-appear under these circumstances. However, the  sampling of allele frequencies  has  still occurred , with the result that  σ 2 p, q  ≠  0 . [ 35 ]  In fact, as  s → ∞ , the

 σ

 p
 ,

 q

 2

 →

 p

 g

 q

 g

 2
 N

 {\textstyle \sigma _{p,\ q}^{2}\to {\tfrac {p_{g}q_{g}}{2N}}}

 , which is the  variance  of the  whole binomial distribution . [ 13 ] : 382–395  [ 14 ] : 49–63   Furthermore, the "Wahlund equations" show that the progeny-bulk  homozygote  frequencies can be obtained as the sums of their respective average values ( p 2 •  or  q 2 • )  plus   σ 2 p, q . [ 13 ] : 382–395   Likewise, the bulk  heterozygote  frequency is  (2 p •  q • )   minus   twice  the  σ 2 p, q . The variance arising from the binomial sampling is conspicuously present. Thus, even when  s → ∞ , the progeny-bulk  genotype  frequencies still reveal  increased homozygosis ,  and  decreased heterozygosis , there is still  dispersion of progeny means , and still  inbreeding  and  inbreeding depression . That is, panmixia is  not  re-attained once lost because of genetic drift (binomial sampling). However, a new  potential  panmixia can be initiated via an allogamous F2 following hybridization. [ 36 ]


Previous discussion on genetic drift examined just one cycle (generation) of the process. When the sampling continues over successive generations, conspicuous changes occur in  σ 2 p ,  q  and  f . Furthermore, another "index" is needed to keep track of "time":  t  =  1 .... y  where  y  = the number of "years" (generations) considered. The methodology often is to add the current binomial increment ( Δ  = " de novo ") to what has occurred previously. [ 13 ]  The entire Binomial Distribution is examined here. [There is no further benefit to be had from an abbreviated example.]


Earlier this variance (σ  2 p,q [ 35 ] ) was seen to be:-

 σ

 p
 ,
 q

 2

 =

 p

 g

 q

 g

 /

 2
 N

 =

 p

 g

 q

 g

 (

 1

 2
 N

 )

 =

 p

 g

 q

 g

 f

 =

 p

 g

 q

 g

 Δ
 f

 when used in recursive equations

 {\displaystyle {\begin{aligned}\sigma _{p,q}^{2}&=p_{g}q_{g}\ /\ 2N\\&=p_{g}q_{g}\left({\frac {1}{2N}}\right)\\&=p_{g}q_{g}\ f\\&=p_{g}q_{g}\ \Delta f\ \scriptstyle {\text{when used in recursive equations}}\end{aligned}}}


With the extension over time, this is also the result of the  first  cycle, and so is

 σ

 1

 2

 {\textstyle \sigma _{1}^{2}}

  (for brevity). At cycle 2, this variance is generated yet again—this time becoming the  de novo  variance (

 Δ

 σ

 2

 {\textstyle \Delta \sigma ^{2}}

 )—and accumulates to what was present already—the "carry-over" variance. The  second  cycle variance (

 σ

 2

 2

 {\textstyle \sigma _{2}^{2}}

 ) is the weighted sum of these two components, the weights being

 1

 {\textstyle 1}

  for the  de novo  and

 (

 1
 −

 1

 2
 N

 )

 {\textstyle \left(1-{\tfrac {1}{2N}}\right)}

  =

 (

 1
 −
 Δ
 f

 )

 {\textstyle \left(1-\Delta f\right)}

  for the"carry-over".


Thus,


The extension to generalize to any time  t , after considerable simplification, becomes: [ 13 ] : 328  -


Because it was this variation in allele frequencies that caused the "spreading apart" of the progenies' means ( dispersion ), the change in  σ 2 t  over the generations indicates the change in the level of the  dispersion .


The method for examining the inbreeding coefficient is similar to that used for  σ  2 p,q . The same weights as before are used respectively for  de novo f  (  Δ f  ) [recall this is  1/(2N)  ]  and  carry-over f . Therefore,

 f

 2

 =

 (
 1
 )

 Δ
 f
 +

 (

 1
 −
 Δ
 f

 )

 f

 1

 {\textstyle f_{2}=\left(1\right)\Delta f+\left(1-\Delta f\right)f_{1}}

  , which is similar to  Equation (1)  in the previous sub-section.


In general, after rearrangement, [ 13 ]

 f

 t

 =
 Δ
 f
 +

 (

 1
 −
 Δ
 f

 )

 f

 t
 −
 1

 =
 Δ
 f

 (

 1
 −

 f

 t
 −
 1

 )

 +

 f

 t
 −
 1

 {\displaystyle {\begin{aligned}f_{t}&=\Delta f+\left(1-\Delta f\right)f_{t-1}\\&=\Delta f\left(1-f_{t-1}\right)+f_{t-1}\end{aligned}}}

  The graphs to the left show levels of inbreeding over twenty generations arising from genetic drift for various  actual gamodeme  sizes (2N).


Still further rearrangements of this general equation reveal some interesting relationships.


(A)  After some simplification, [ 13 ]

 (

 f

 t

 −

 f

 t
 −
 1

 )

 =
 Δ
 f

 (

 1
 −

 f

 t
 −
 1

 )

 =
 δ

 f

 t

 {\textstyle \left(f_{t}-f_{t-1}\right)=\Delta f\left(1-f_{t-1}\right)=\delta f_{t}}

 . The left-hand side is the difference between the current and previous levels of inbreeding: the  change in inbreeding  ( δf t ). Notice, that this  change in inbreeding  ( δf t ) is equal to the  de novo inbreeding  ( Δf ) only for the first cycle—when f t-1  is  zero .


(B)  An item of note is the  (1-f t-1 ) , which is an "index of  non-inbreeding ". It is known as the  panmictic index . [ 13 ] [ 14 ]

 P

 t
 −
 1

 =

 (

 1
 −

 f

 t
 −
 1

 )

 {\textstyle P_{t-1}=\left(1-f_{t-1}\right)}

 .


(C)  Further useful relationships emerge involving the  panmictic index . [ 13 ] [ 14 ]

 Δ
 f

 =

 δ

 f

 t

 P

 t
 −
 1

 =
 1
 −

 P

 t

 P

 t
 −
 1

 {\displaystyle {\begin{aligned}\Delta f&={\frac {\delta f_{t}}{P_{t-1}}}\\&=1-{\frac {P_{t}}{P_{t-1}}}\end{aligned}}}

 .
 (D)  A key link emerges between  σ  2 p,q  and  f . Firstly... [ 13 ]

 f

 t

 =
 1
 −

 (

 1
 −
 1
 Δ
 f

 )

 t

 (

 1
 −

 f

 0

 )

 {\displaystyle {\begin{aligned}f_{t}&=1-\left(1-1\Delta f\right)^{t}\left(1-f_{0}\right)\end{aligned}}}

Secondly, presuming that  f 0  =  0 , the right-hand side of this equation reduces to the section within the brackets of  Equation (2)  at the end of the last sub-section. That is, if initially there is no inbreeding,

 σ

 t

 2

 =

 p

 g

 q

 g

 f

 t

 {\textstyle \sigma _{t}^{2}=p_{g}q_{g}f_{t}}

   !  Furthermore, if this then is rearranged,

 f

 t

 =

 σ

 t

 2

 p

 g

 q

 g

 {\textstyle f_{t}={\tfrac {\sigma _{t}^{2}}{p_{g}q_{g}}}}

 . That is, when initial inbreeding is zero, the two principal viewpoints of  binomial gamete sampling  (genetic drift) are directly inter-convertible.


It is easy to overlook that  random fertilization  includes self-fertilization. Sewall Wright showed that a proportion  1/N  of  random fertilizations  is actually  self fertilization

 (
 ⨂
 )

 {\displaystyle \left(\bigotimes \right)}

 , with the remainder  (N-1)/N  being  cross fertilization

 (

 X

 )

 {\displaystyle \left({\mathsf {X}}\right)}

 .  Following path analysis and simplification, the new view   random fertilization inbreeding  was found to be:

 f

 t

 =
 Δ
 f

 (

 1
 +

 f

 t
 −
 1

 )

 +

 N
 −
 1

 N

 f

 t
 −
 1

 {\textstyle f_{t}=\Delta f\left(1+f_{t-1}\right)+{\tfrac {N-1}{N}}f_{t-1}}

 . [ 27 ] [ 37 ]  Upon further rearrangement, the earlier results from the binomial sampling were confirmed, along with some new arrangements. Two of these were potentially very useful, namely:  (A)

 f

 t

 =
 Δ
 f

 [

 1
 +

 f

 t
 −
 1

 (

 2
 N
 −
 1

 )

 ]

 {\textstyle f_{t}=\Delta f\left[1+f_{t-1}\left(2N-1\right)\right]}

 ; and  (B)

 f

 t

 =
 Δ
 f

 (

 1
 −

 f

 t
 −
 1

 )

 +

 f

 t
 −
 1

 {\textstyle f_{t}=\Delta f\left(1-f_{t-1}\right)+f_{t-1}}

 .


The recognition that selfing may   intrinsically be a part of  random fertilization leads to some issues about the use of the previous  random fertilization  'inbreeding coefficient'. Clearly, then, it is inappropriate for any species incapable of  self fertilization , which includes plants with self-incompatibility mechanisms, dioecious plants, and  bisexual animals . The equation of Wright was modified later to provide a version of random fertilization that involved only  cross fertilization  with no  self fertilization . The proportion  1/N  formerly due to  selfing  now defined the  carry-over  gene-drift inbreeding arising from the previous cycle. The new version is: [ 13 ] : 166

 f

 X

 t

 =

 f

 t
 −
 1

 +
 Δ
 f

 (

 1
 +

 f

 t
 −
 2

 −
 2

 f

 t
 −
 1

 )

 {\displaystyle f_{{\mathsf {X}}_{t}}=f_{t-1}+\Delta f\left(1+f_{t-2}-2f_{t-1}\right)}

 .


The graphs to the right depict the differences between standard  random fertilization   RF , and random fertilization adjusted for "cross fertilization alone"  CF . As can be seen, the issue is non-trivial for small gamodeme sample sizes.


It now is necessary to note that not only is "panmixia"  not  a synonym for "random fertilization", but also that "random fertilization" is  not  a synonym for "cross fertilization".


In the sub-section on "The sample gamodemes – Genetic drift", a series of gamete samplings was followed, an outcome of which was an increase in homozygosity at the expense of heterozygosity. From this viewpoint, the rise in homozygosity was due to the gamete samplings. Levels of homozygosity can be viewed also according to whether homozygotes arose allozygously or autozygously. Recall that autozygous alleles have the same allelic origin, the likelihood (frequency) of which  is  the  inbreeding coefficient ( f )  by definition. The proportion arising  allozygously  is therefore  (1-f) . For the  A -bearing gametes, which are present with a general frequency of  p , the overall frequency of those that are autozygous is therefore ( f   p ). Similarly, for  a -bearing gametes, the autozygous frequency is ( f   q ). [ 38 ]  These two viewpoints regarding genotype frequencies must be connected to establish consistency.


Following firstly the  auto/allo  viewpoint, consider the  allozygous  component. This occurs with the frequency of  (1-f) , and the alleles unite according to the  random fertilization  quadratic expansion. Thus:

 (

 1
 −
 f

 )

 [

 p

 0

 +

 q

 0

 ]

 2

 =

 (

 1
 −
 f

 )

 [

 p

 0

 2

 +

 q

 0

 2

 ]

 +

 (

 1
 −
 f

 )

 [

 2

 p

 0

 q

 0

 ]

 {\displaystyle \left(1-f\right)\left[p_{0}+q_{0}\right]^{2}=\left(1-f\right)\left[p_{0}^{2}+q_{0}^{2}\right]+\left(1-f\right)\left[2p_{0}q_{0}\right]}

  Consider next the  autozygous  component. As these alleles  are   autozygous , they are effectively  selfings , and produce either  AA  or  aa  genotypes, but no heterozygotes. They therefore produce

 f

 p

 0

 {\textstyle fp_{0}}

   "AA"  homozygotes plus

 f

 q

 0

 {\textstyle fq_{0}}

   "aa"  homozygotes. Adding these two components together results in:

 [

 (

 1
 −
 f

 )

 p

 0

 2

 +
 f

 p

 0

 ]

 {\textstyle \left[\left(1-f\right)p_{0}^{2}+fp_{0}\right]}

  for the  AA  homozygote;

 [

 (

 1
 −
 f

 )

 q

 0

 2

 +
 f

 q

 0

 ]

 {\textstyle \left[\left(1-f\right)q_{0}^{2}+fq_{0}\right]}

  for the  aa  homozygote; and

 (

 1
 −
 f

 )

 2

 p

 0

 q

 0

 {\textstyle \left(1-f\right)2p_{0}q_{0}}

  for the  Aa  heterozygote. [ 13 ] : 65  [ 14 ]  This is the same equation as that presented earlier in the section on "Self fertilization – an alternative". The reason for the decline in heterozygosity is made clear here. Heterozygotes can arise  only  from the allozygous component, and its frequency in the sample bulk is just  (1-f) : hence this must also be the factor controlling the frequency of the heterozygotes.


Secondly, the  sampling  viewpoint is re-examined. Previously, it was noted that the decline in heterozygotes was

 f

 (

 2

 p

 0

 q

 0

 )

 {\textstyle f\left(2p_{0}q_{0}\right)}

 . This decline is distributed equally towards each homozygote; and is added to their basic  random fertilization  expectations. Therefore, the genotype frequencies are:

 (

 p

 0

 2

 +
 f

 p

 0

 q

 0

 )

 {\textstyle \left(p_{0}^{2}+fp_{0}q_{0}\right)}

  for the  "AA"  homozygote;

 (

 q

 0

 2

 +
 f

 p

 0

 q

 0

 )

 {\textstyle \left(q_{0}^{2}+fp_{0}q_{0}\right)}

  for the  "aa"  homozygote;  and

 2

 p

 0

 q

 0

 −
 f

 (

 2

 p

 0

 q

 0

 )

 {\textstyle 2p_{0}q_{0}-f\left(2p_{0}q_{0}\right)}

  for the heterozygote.


Thirdly, the  consistency  between the two previous viewpoints needs establishing. It is apparent at once [from the corresponding equations above] that the heterozygote frequency is the same in both viewpoints.  However, such a straightforward result is not immediately apparent for the homozygotes. Begin by considering the  AA  homozygote's final equation in the  auto/allo  paragraph above:-

 [

 (

 1
 −
 f

 )

 p

 0

 2

 +
 f

 p

 0

 ]

 {\textstyle \left[\left(1-f\right)p_{0}^{2}+fp_{0}\right]}

 . Expand the brackets, and follow by re-gathering [within the resultant] the two new terms with the common-factor  f  in them. The result is:

 p

 0

 2

 −
 f

 (

 p

 0

 2

 −

 p

 0

 )

 {\textstyle p_{0}^{2}-f\left(p_{0}^{2}-p_{0}\right)}

 . Next, for the parenthesized "  p 2 0  ", a  (1-q)  is substituted for a  p , the result becoming

 p

 0

 2

 −
 f

 [

 p

 0

 (

 1
 −

 q

 0

 )

 −

 p

 0

 ]

 {\textstyle p_{0}^{2}-f\left[p_{0}\left(1-q_{0}\right)-p_{0}\right]}

 . Following that substitution, it is a straightforward matter of multiplying-out, simplifying and watching signs. The end result is

 p

 0

 2

 +
 f

 p

 0

 q

 0

 {\textstyle p_{0}^{2}+fp_{0}q_{0}}

 , which is exactly the result for  AA  in the  sampling  paragraph. The two viewpoints are therefore  consistent  for the  AA  homozygote. In a like manner, the consistency of the  aa  viewpoints can also be shown. The two viewpoints are consistent for all classes of genotypes.


In previous sections, dispersive random fertilization ( genetic drift ) has been considered comprehensively, and self-fertilization and hybridizing have been examined to varying degrees. The diagram to the left depicts the first two of these, along with another "spatially based" pattern:  islands . This is a pattern of  random fertilization  featuring  dispersed gamodemes , with the addition of "overlaps" in which  non-dispersive  random fertilization occurs. With the  islands  pattern, individual gamodeme sizes ( 2N ) are observable, and overlaps ( m ) are minimal. This is one of Sewall Wright's array of possibilities. [ 37 ]  In addition to "spatially" based patterns of fertilization, there are others based on either "phenotypic" or "relationship" criteria. The  phenotypic  bases include  assortative  fertilization (between similar phenotypes) and  disassortative  fertilization (between opposite phenotypes). The  relationship  patterns include  sib crossing ,  cousin crossing  and  backcrossing —and are considered in a separate section.  Self fertilization  may be considered both from a spatial or relationship point of view.


The breeding population consists of  s  small  dispersed random fertilization  gamodemes of sample size

 2

 N

 k

 {\textstyle 2N_{k}}

  (  k  = 1 ...  s  ) with "  overlaps  " of proportion

 m

 k

 {\textstyle m_{k}}

  in which  non-dispersive random fertilization   occurs. The   dispersive proportion   is thus

 (

 1
 −

 m

 k

 )

 {\textstyle \left(1-m_{k}\right)}

 . The bulk population consists of  weighted averages  of sample sizes, allele and genotype frequencies and progeny means, as was done for genetic drift in an earlier section. However, each  gamete sample size  is reduced to allow for the  overlaps , thus finding a

 2

 N

 k

 {\textstyle 2N_{k}}

  effective for

 (

 1
 −

 m

 k

 )

 {\textstyle \left(1-m_{k}\right)}

 .


For brevity, the argument is followed further with the subscripts omitted. Recall that

 1

 2
 N

 {\textstyle {\tfrac {1}{2N}}}

  is

 Δ
 f

 {\textstyle \Delta f}

  in general. [Here, and following, the  2N  refers to the  previously defined  sample size, not to any "islands adjusted" version.]


After simplification, [ 37 ]

 i
 s
 l
 a
 n
 d
 s

 Δ
 f
 =

 (

 1
 −
 m

 )

 2

 2
 N
 −

 m

 2

 (

 2
 N
 −
 1

 )

 {\displaystyle ^{\mathsf {islands}}\Delta f={\frac {\left(1-m\right)^{2}}{2N-m^{2}\left(2N-1\right)}}}

Notice that when  m = 0  this reduces to the previous  Δ f . The reciprocal of this furnishes an estimate of the "

 2

 N

 k

 {\textstyle 2N_{k}}

   effective for

 (

 1
 −

 m

 k

 )

 {\textstyle \left(1-m_{k}\right)}

  ", mentioned above.


This Δf is also substituted into the previous  inbreeding coefficient  to obtain  [ 37 ]

 i
 s
 l
 a
 n
 d
 s

 f

 t

 =

 i
 s
 l
 a
 n
 d
 s

 Δ

 f

 t

 +

 (

 1
 −

 i
 s
 l
 a
 n
 d
 s

 Δ

 f

 t

 )

 i
 s
 l
 a
 n
 d
 s

 f

 t
 −
 1

 {\displaystyle {^{\mathsf {islands}}f_{t}}=\ {^{\mathsf {islands}}\Delta f_{t}}+\left(1-\ {^{\mathsf {islands}}\Delta f_{t}}\right)\ {^{\mathsf {islands}}f_{t-1}}}

  where  t  is the index over generations, as before.


The effective  overlap proportion  can be obtained also, [ 37 ]  as

 m

 t

 =
 1
 −

 [

 2
 N

 i
 s
 l
 a
 n
 d
 s

 Δ

 f

 t

 (

 2
 N
 −
 1

 )

 i
 s
 l
 a
 n
 d
 s

 Δ

 f

 t

 +
 1

 ]

 1
 2

 {\displaystyle m_{t}=1-\left[{\frac {2N\ {^{\mathsf {islands}}\Delta f_{t}}}{\left(2N-1\right)\ {^{\mathsf {islands}}\Delta f_{t}+1}}}\right]^{\tfrac {1}{2}}}


The graphs to the right show the  inbreeding  for a gamodeme size of  2N = 50  for  ordinary dispersed random fertilization    (RF)   (m=0) , and for  four overlap levels ( m = 0.0625, 0.125, 0.25, 0.5 )  of  islands   random fertilization . There has indeed been reduction in the inbreeding resulting from the  non-dispersed random fertilization  in the overlaps. It is particularly notable as  m → 0.50 . Sewall Wright suggested that this value should be the limit for the use of this approach. [ 37 ]


The  gene-model  examines the heredity pathway from the point of view of "inputs" (alleles/gametes) and "outputs" (genotypes/zygotes), with fertilization being the "process" converting one to the other. An alternative viewpoint concentrates on the "process" itself, and considers the zygote genotypes as arising from allele shuffling. In particular, it regards the results as if one allele had "substituted" for the other during the shuffle, together with a residual that deviates from this view. This formed an integral part of Fisher's method, [ 8 ]  in addition to his use of frequencies and effects to generate his genetical statistics. [ 14 ]  A discursive derivation of the  allele substitution  alternative follows. [ 14 ] : 113


Suppose that the usual random fertilization of gametes in a "base" gamodeme—consisting of  p  gametes ( A ) and  q  gametes ( a )—is replaced by fertilization with a "flood" of gametes all containing a single allele ( A  or  a , but not both). The zygotic results can be interpreted in terms of the "flood" allele having "substituted for" the alternative allele in the underlying "base" gamodeme. The diagram assists in following this viewpoint: the upper part pictures an  A  substitution, while the lower part shows an  a  substitution. (The diagram's "RF allele" is the allele in the "base" gamodeme.)


Consider the upper part firstly. Because  base   A  is present with a frequency of  p , the  substitute   A  fertilizes it with a frequency of  p  resulting in a zygote  AA  with an allele effect of  a . Its contribution to the outcome, therefore, is the product

 (

 p

 a

 )

 {\textstyle \left(p\ a\right)}

 . Similarly, when the  substitute  fertilizes  base   a  (resulting in  Aa  with a frequency of  q  and heterozygote effect of  d ), the contribution is

 (

 q

 d

 )

 {\textstyle \left(q\ d\right)}

 . The overall result of substitution by  A  is, therefore,

 (

 p

 a
 +
 q

 d

 )

 {\textstyle \left(p\ a+q\ d\right)}

 . This is now oriented towards the population mean [see earlier section] by expressing it as a deviate from that mean :

 (

 p

 a
 +
 q

 d

 )

 −
 G

 {\textstyle \left(p\ a+q\ d\right)-G}


After some algebraic simplification, this becomes

 β

 A

 =
 q

 [

 a
 +

 (

 q
 −
 p

 )

 d

 ]

 {\displaystyle \beta _{A}=q\ \left[a+\left(q-p\right)d\right]}

- the  substitution effect  of  A .


A parallel reasoning can be applied to the lower part of the diagram, taking care with the differences in frequencies and gene effects. The result is the   substitution effect  of  a , which is

 β

 a

 =
 −

 p

 [

 a
 +

 (

 q
 −
 p

 )

 d

 ]

 {\displaystyle \beta _{a}=-\ p\left[a+\left(q-p\right)d\right]}

The common factor inside the brackets is the  average allele substitution effect , [ 14 ] : 113   and is

 β
 =
 a
 +

 (

 q
 −
 p

 )

 d

 {\displaystyle \beta =a+\left(q-p\right)d}

It can also be derived in a more direct way, but the result is the same. [ 39 ]


In subsequent sections, these substitution effects help define the gene-model genotypes as consisting of a partition predicted by these new effects ( substitution  expectations ), and a residual ( substitution deviations ) between these expectations and the previous gene-model effects. The  expectations  are also called the  breeding values  and the deviations are also called  dominance deviations .


Ultimately, the variance arising from the  substitution expectations  becomes the so-called  Additive genetic variance (σ 2 A ) [ 14 ]  (also the  Genic variance   [ 40 ] )— while that arising from the  substitution deviations  becomes the so-called  Dominance variance (σ 2 D ) . It is noticeable that neither of these terms reflects the true meanings of these variances. The   "genic variance"   is less dubious than the   additive genetic variance , and more in line with Fisher's own name for this partition. [ 8 ] [ 29 ] : 33   A less-misleading name for the  dominance deviations variance  is the   "quasi-dominance variance"   [see following sections for further discussion]. These  latter terms are preferred herein.


The gene-model effects ( a ,  d  and  -a ) are important soon in the derivation of the  deviations from substitution , which were first discussed in the previous  Allele Substitution  section. However, they need to be redefined themselves before they become useful in that exercise. They firstly need to be re-centralized around the population mean ( G ), and secondly they need to be re-arranged as functions of  β , the  average allele substitution effect .


Consider firstly the re-centralization. The re-centralized effect for  AA  is  a• = a - G  which, after simplification, becomes  a• = 2 q (a- p d) . The similar effect for  Aa  is  d• = d - G = a( q - p ) + d(1-2 pq ) , after simplification. Finally, the re-centralized effect for  aa  is  (-a)• = -2 p (a+ q d) . [ 14 ] : 116–119


Secondly, consider the re-arrangement of these re-centralized effects as functions of  β . Recalling from the "Allele Substitution" section that β = [a +(q-p)d], rearrangement gives  a = [β -(q-p)d] . After substituting this for  a  in  a•  and simplifying, the final version becomes  a•• = 2q(β-qd) . Similarly,  d•  becomes  d•• = β(q-p) + 2pqd ; and   (-a)•  becomes  (-a)•• = -2p(β+pd) . [ 14 ] : 118


The zygote genotypes are the target of all this preparation. The homozygous genotype  AA  is a union of two  substitution effects of A , one from each sex. Its  substitution expectation  is therefore  β AA  = 2β A  = 2 q β  (see previous sections). Similarly, the  substitution expectation  of  Aa  is  β Aa  = β A  + β a  = ( q - p )β ; and for  aa ,  β aa  = 2β a  = -2 p β . These  substitution expectations  of the genotypes are also called  breeding values . [ 14 ] : 114–116


Substitution deviations  are the differences between these  expectations  and the  gene effects  after their two-stage redefinition in the previous section. Therefore,  d AA  = a•• - β AA  = -2 q 2 d  after simplification. Similarly,  d Aa  = d•• - β Aa  = 2 pq d  after simplification. Finally,  d aa  = (-a)•• - β aa  = -2 p 2 d  after simplification. [ 14 ] : 116–119   Notice that all of these  substitution deviations  ultimately are functions of the gene-effect  d —which accounts for the use of ["d" plus subscript] as their symbols. However, it is a serious  non sequitur  in logic to regard them as accounting for the dominance (heterozygosis) in the entire gene model : they are simply  functions  of "d" and not an  audit  of the "d" in the system. They  are  as derived:  deviations from the substitution expectations !


The "substitution expectations" ultimately give rise to the  σ 2 A  (the so-called "Additive" genetic variance); and the "substitution deviations" give rise to the  σ 2 D  (the so-called "Dominance" genetic variance). Be aware, however, that the average substitution effect (β) also contains "d" [see previous sections], indicating that dominance is also embedded within the "Additive" variance [see following sections on the Genotypic Variance for their derivations]. Remember also [see previous paragraph] that the "substitution deviations" do not account for the dominance in the system (being nothing more than deviations from the  substitution expectations ), but which happen to consist algebraically of functions of "d". More appropriate names for these respective variances might be  σ 2 B  (the  "Breeding expectations" variance) and  σ 2 δ  (the "Breeding deviations" variance). However, as noted previously, "Genic" (σ  2 A ) and "Quasi-Dominance" (σ  2 D ), respectively, will be preferred herein.


There are two major approaches to defining and partitioning  genotypic variance . One is based on the  gene-model effects , [ 40 ]  while the other is based on the  genotype substitution effects [ 14 ]  They are algebraically inter-convertible with each other. [ 36 ]  In this section, the basic  random fertilization  derivation is considered, with the effects of inbreeding and dispersion set aside. This is dealt with later to arrive at a more general solution. Until this  mono-genic  treatment is replaced by a  multi-genic  one, and until  epistasis  is resolved in the light of the findings of  epigenetics , the Genotypic variance has only the components considered here.


It is convenient to follow the biometrical approach, which is based on correcting the  unadjusted sum of squares (USS)  by subtracting the  correction factor (CF) . Because all effects have been examined through frequencies, the USS can be obtained as the sum of the products of each genotype's frequency' and the square of its  gene-effect . The CF in this case is the mean squared. The result is the SS, which, again because of the use of frequencies, is also immediately the  variance . [ 9 ]


The

 U
 S
 S

 =

 p

 2

 a

 2

 +
 2
 p
 q

 d

 2

 +

 q

 2

 (
 −
 a

 )

 2

 {\textstyle {\mathsf {USS}}=p^{2}a^{2}+2pqd^{2}+q^{2}(-a)^{2}}

 ,  and the

 C
 F

 =

 G

 2

 {\textstyle {\mathsf {CF}}={\mathsf {G}}^{2}}

 .  The

 S
 S

 =

 U
 S
 S

 −

 C
 F

 {\textstyle {\mathsf {SS}}={\mathsf {USS}}-{\mathsf {CF}}}


After partial simplification,

 σ

 G

 2

 =
 2
 p
 q

 a

 2

 +
 (
 q
 −
 p
 )
 4
 p
 q
 a
 d
 +
 2
 p
 q

 d

 2

 +
 (
 2
 p
 q

 )

 2

 d

 2

 =

 σ

 a

 2

 +
 (

 weighted-covariance

 )

 a
 d

 +

 σ

 d

 2

 +

 σ

 D

 2

 =

 1
 2

 D

 +

 1
 2

 F

 ′

 +

 1
 2

 H

 1

 +

 1
 4

 H

 2

 {\displaystyle {\begin{aligned}\sigma _{G}^{2}&=2pqa^{2}+(q-p)4pqad+2pqd^{2}+(2pq)^{2}d^{2}\\&=\sigma _{a}^{2}+({\text{weighted-covariance}})_{ad}+\sigma _{d}^{2}+\sigma _{D}^{2}\\&={\tfrac {1}{2}}{\mathsf {D}}+{\tfrac {1}{2}}{\mathsf {F}}^{\prime }+{\tfrac {1}{2}}{\mathsf {H}}_{1}+{\tfrac {1}{4}}{\mathsf {H}}_{2}\end{aligned}}}

The last line is in Mather's terminology. [ 40 ] : 212  [ 41 ] [ 42 ]


Here,  σ 2 a  is the  homozygote  or  allelic  variance, and  σ 2 d  is the  heterozygote  or  dominance  variance. The  substitution deviations  variance ( σ 2 D ) is also present. The  (weighted_covariance) ad [ 43 ]  is abbreviated hereafter to "  cov ad  ".


These components are plotted across all values of  p  in the accompanying figure. Notice that  cov ad  is  negative  for  p > 0.5 .


Most of these components are affected by the change of central focus from  homozygote mid-point  ( mp ) to  population mean  ( G ), the latter being the basis of the  Correction Factor . The  cov ad  and  substitution deviation  variances are simply artifacts of this shift. The  allelic  and  dominance  variances are genuine genetical partitions of the original gene-model, and are the only eu-genetical components. Even then, the algebraic formula for the  allelic  variance is effected by the presence of  G : it is only the  dominance  variance (i.e. σ 2 d  ) which is unaffected by the shift from  mp  to  G . [ 36 ]  These insights are commonly not appreciated.


Further gathering of terms [in Mather format] leads to

 1
 2

 D

 +

 1
 2

 F

 ′

 +

 1
 2

 H

 3

 +

 1
 4

 H

 2

 {\textstyle {\tfrac {1}{2}}{\mathsf {D}}+{\tfrac {1}{2}}{\mathsf {F}}^{\prime }+{\tfrac {1}{2}}{\mathsf {H}}_{3}+{\tfrac {1}{4}}{\mathsf {H}}_{2}}

 , where

 1
 2

 H

 3

 =
 (
 q
 −
 p

 )

 2

 1
 2

 H

 1

 =
 (
 q
 −
 p

 )

 2

 2
 p
 q

 d

 2

 {\textstyle {\tfrac {1}{2}}{\mathsf {H}}_{3}=(q-p)^{2}{\tfrac {1}{2}}{\mathsf {H}}_{1}=(q-p)^{2}2pqd^{2}}

 . It is useful later in Diallel analysis, which is an experimental design for estimating these genetical statistics. [ 44 ]


If, following the last-given rearrangements, the first three terms are amalgamated together, rearranged further and simplified, the result is the variance of the Fisherian  substitution expectation .


That is:

 σ

 A

 2

 =

 σ

 a

 2

 +

 c
 o
 v

 a
 d

 +

 σ

 d

 2

 {\displaystyle \sigma _{A}^{2}=\sigma _{a}^{2}+{\mathsf {cov}}_{ad}+\sigma _{d}^{2}}


Notice particularly that  σ 2 A  is not   σ 2 a . The first is the   substitution expectations  variance, while the second is the  allelic  variance. [ 45 ]   Notice also that  σ 2 D  (the  substitution-deviations  variance) is  not   σ 2 d  (the  dominance  variance), and recall that it is an artifact arising from the use of  G  for the Correction Factor. [See the "blue paragraph" above.] It now will be referred to as the "quasi-dominance" variance.


Also note that   σ 2 D  <  σ 2 d  ("2pq" being always a fraction); and note that (1)  σ 2 D  =  2pq σ 2 d , and that (2)  σ 2 d  =  σ 2 D  / (2pq) . That is: it is confirmed that σ 2 D  does not quantify the dominance variance in the model. It is σ 2 d  which does that. However, the dominance variance (σ 2 d ) can be estimated readily from the σ 2 D  if  2pq  is available.


From the Figure, these results can be visualized as accumulating  σ 2 a ,  σ 2 d  and  cov ad   to obtain  σ 2 A , while leaving the  σ 2 D  still separated. It is clear also in the Figure that  σ 2 D  <  σ 2 d , as expected from the equations.


The overall result (in Fisher's format) is

 σ

 G

 2

 =
 2
 p
 q

 [

 a
 +
 (
 q
 −
 p
 )
 d

 ]

 2

 +

 (

 2
 p
 q

 )

 2

 d

 2

 =

 σ

 A

 2

 +

 σ

 D

 2

 =

 [

 (

 σ

 a

 2

 +

 c
 o
 v

 a
 d

 +

 σ

 d

 2

 )

 ]

 +

 [

 2
 p
 q

 σ

 d

 2

 ]

 {\displaystyle {\begin{aligned}\sigma _{G}^{2}&=2pq\left[a+(q-p)d\right]^{2}+\left(2pq\right)^{2}d^{2}\\&=\sigma _{A}^{2}+\sigma _{D}^{2}\\&=\left[\left(\sigma _{a}^{2}+{\mathsf {cov}}_{ad}+\sigma _{d}^{2}\right)\right]+\left[2pq\ \sigma _{d}^{2}\right]\end{aligned}}}

The Fisherian components have just been derived, but their derivation via the  substitution effects  themselves is given also, in the next section.


Reference to the several earlier sections on allele substitution reveals that the two ultimate effects are  genotype substitution   expectations and  genotype substitution deviations . Notice that these are each already defined as deviations from the  random fertilization  population mean ( G ). For each genotype in turn therefore, the product of the frequency and the square of the relevant effect is obtained, and these are accumulated to obtain directly a  SS  and  σ 2 . [ 46 ]  Details follow.


σ 2 A  =  p 2  β AA 2  +  2 pq  β Aa 2  +  q 2  β aa 2 , which simplifies to  σ 2 A  = 2 pq β 2 —the Genic variance.


σ 2 D  =  p 2  d AA 2  +  2 pq  d Aa 2  +  q  d aa 2 , which simplifies to  σ 2 D  = (2 pq ) 2  d 2 —the quasi-Dominance variance.


Upon accumulating these results,  σ 2 G  =  σ 2 A  + σ 2 D   . These components are visualized in the graphs to the right. The  average allele substitution  effect is graphed also, but the symbol is "α" (as is common in the citations) rather than "β" (as is used herein).


Once again, however, refer to the earlier discussions about the true meanings and identities of these components. Fisher himself did not use these modern terms for his components. The  substitution expectations  variance he named the   "genetic"   variance; and the  substitution deviations  variance he regarded simply as the unnamed  residual  between the "genotypic" variance (his name for it) and his "genetic" variance. [ 8 ] [ 29 ] : 33  [ 47 ] [ 48 ]  [The terminology and derivation used in this article are completely in accord with Fisher's own.] Mather's term for the  expectations  variance— "genic" [ 40 ] —is obviously derived from Fisher's term, and avoids using "genetic" (which has become too generalized in usage to be of value in the present context). The origin is obscure of the modern misleading terms "additive" and "dominance" variances.


Note that this allele-substitution approach defined the components separately, and then totaled them to obtain the final Genotypic variance. Conversely, the gene-model approach derived the whole situation (components and total) as one exercise. Bonuses arising from this were (a) the revelations about the real structure of  σ 2 A , and (b) the real meanings and relative sizes of  σ 2 d  and  σ 2 D  (see previous sub-section). It is also apparent that a "Mather" analysis is more informative, and that a "Fisher" analysis can always be constructed from it. The opposite conversion is not possible, however, because information about  cov ad  would be missing.


In the section on genetic drift, and in other sections that discuss inbreeding, a major outcome from allele frequency sampling has been the   dispersion  of progeny means. This collection of means has its own average, and also has a variance: the  amongst-line variance . (This is a variance of the attribute itself, not of  allele frequencies .) As dispersion develops further over succeeding generations, this amongst-line variance would be expected to increase. Conversely, as homozygosity rises, the within-lines variance would be expected to decrease. The question arises therefore as to whether the total variance is changing—and, if so, in what direction. To date, these issues have been presented in terms of the  genic (σ  2 A  )  and  quasi-dominance (σ  2 D  )  variances rather than the gene-model components. This will be done herein as well.


The crucial  overview equation  comes from Sewall Wright, [ 13 ]   : 99, 130  [ 37 ]  and is the outline of the  inbred genotypic variance  based on a  weighted average of its extremes , the weights being quadratic with respect to the  inbreeding coefficient

 f

 {\textstyle f}

 . This equation is:


σ

 G

 f

 2

 =

 (

 1
 −
 f

 )

 σ

 G

 0

 2

 +
 f

 σ

 G

 1

 2

 +
 f

 (

 1
 −
 f

 )

 [

 G

 0

 −

 G

 1

 ]

 2

 {\displaystyle \sigma _{G_{f}}^{2}=\left(1-f\right)\sigma _{G_{0}}^{2}+f\ \sigma _{G_{1}}^{2}+f\left(1-f\right)\left[G_{0}-G_{1}\right]^{2}}


where

 f

 {\textstyle f}

  is the inbreeding coefficient,

 σ

 G

 0

 2

 {\textstyle \sigma _{G_{0}}^{2}}

  is the genotypic variance at
 f=0 ,

 σ

 G

 1

 2

 {\textstyle \sigma _{G_{1}}^{2}}

  is the genotypic variance at  f=1 ,

 G

 0

 {\textstyle G_{0}}

  is the population mean at  f=0 , and

 G

 1

 {\textstyle G_{1}}

  is the population mean at  f=1 .


The

 (

 1
 −
 f

 )

 {\textstyle \left(1-f\right)}

  component [in the equation above] outlines the reduction of variance within progeny lines. The

 f

 {\textstyle f}

   component addresses the increase in variance amongst progeny lines. Lastly, the

 f

 (

 1
 −
 f

 )

 {\textstyle f\left(1-f\right)}

  component is seen (in the next line) to address the  quasi-dominance  variance. [ 13 ]   : 99 & 130   These components can be expanded further thereby revealing additional insight. Thus:-


σ

 G

 f

 2

 =

 (

 1
 −
 f

 )

 [

 σ

 A

 0

 2

 +

 σ

 D

 0

 2

 ]

 +
 f

 (

 4
 p
 q

 a

 2

 )

 +
 f

 (

 1
 −
 f

 )

 [

 2
 p
 q

 d

 ]

 2

 {\displaystyle \sigma _{G_{f}}^{2}=\left(1-f\right)\left[\sigma _{A_{0}}^{2}+\sigma _{D_{0}}^{2}\right]+f\ \left(4pq\ a^{2}\right)+f\ \left(1-f\right)\left[2pq\ d\right]^{2}}


Firstly,  σ 2 G(0)  [in the equation above] has been expanded to show its two sub-components [see section on "Genotypic variance"]. Next, the  σ 2 G(1)  has been converted to  4pqa 2   , and is derived in a section following. The third component's substitution is the difference between the two "inbreeding extremes" of the population mean [see section on the "Population Mean"]. [ 36 ]


Summarising: the  within-line  components are

 (

 1
 −
 f

 )

 σ

 A

 0

 2

 {\textstyle \left(1-f\right)\sigma _{A_{0}}^{2}}

  and

 (

 1
 −
 f

 )

 σ

 D

 0

 2

 {\textstyle \left(1-f\right)\sigma _{D_{0}}^{2}}

 ; and the  amongst-line  components are

 2
 f

 σ

 a

 0

 2

 {\textstyle 2f\ \sigma _{a_{0}}^{2}}

  and

 (

 f
 −

 f

 2

 )

 σ

 D

 0

 2

 {\textstyle \left(f-f^{2}\right)\sigma _{D_{0}}^{2}}

 . [ 36 ]


Rearranging gives the following:

 σ

 A

 i
 n
 b
 r
 e
 d

 2

 =

 (

 1
 −
 f

 )

 σ

 A

 0

 2

 +
 2
 f

 σ

 a

 0

 2

 =

 (

 1
 +
 f

 )

 σ

 A

 f

 2

 {\displaystyle {\begin{aligned}\sigma _{A_{inbred}}^{2}&=\left(1-f\right)\sigma _{A_{0}}^{2}+2f\sigma _{a_{0}}^{2}\\&=\left(1+f\right)\sigma _{A_{f}}^{2}\end{aligned}}}

The version in the last line is discussed further in a subsequent section.


Similarly,

 σ

 D

 i
 n
 b
 r
 e
 d

 2

 =

 (

 1
 −
 f

 )

 σ

 D

 0

 2

 +

 (

 f
 −

 f

 2

 )

 σ

 D

 0

 2

 =

 (

 1
 −

 f

 2

 )

 σ

 D

 0

 2

 {\displaystyle {\begin{aligned}\sigma _{D_{inbred}}^{2}&=\left(1-f\right)\sigma _{D_{0}}^{2}+\left(f-f^{2}\right)\sigma _{D_{0}}^{2}\\&=\left(1-f^{2}\right)\sigma _{D_{0}}^{2}\end{aligned}}}


Graphs to the left show these three genic variances, together with the three quasi-dominance variances,  across all values of  f , for  p = 0.5  (at which the quasi-dominance variance is at a maximum). Graphs to the right show the  Genotypic  variance partitions (being the sums of the respective  genic  and  quasi-dominance  partitions) changing over ten generations with an example  f = 0.10 .


Answering, firstly, the questions posed at the beginning about the  total variances  [the  Σ  in the graphs] : the  genic variance  rises linearly with the  inbreeding coefficient , maximizing at twice its starting level. The  quasi-dominance variance  declines at the rate of  (1 − f 2  )  until it finishes at zero. At low levels of  f , the decline is very gradual, but it accelerates with higher levels of  f .


Secondly, notice the other trends. It is probably intuitive that the  within line  variances decline to zero with continued inbreeding, and this is seen to be the case (both at the same linear rate  (1-f)  ). The  amongst line  variances both increase with inbreeding up to  f = 0.5 , the  genic variance  at the rate of  2f , and the  quasi-dominance variance  at the rate of  (f − f 2 ) . At  f > 0.5 , however, the trends change. The  amongst line   genic variance  continues its linear increase until it equals the  total   genic variance . But, the  amongst line   quasi-dominance variance  now declines towards  zero ,  because  (f − f 2 )  also declines with  f > 0.5 . [ 36 ]


Recall that when  f=1 , heterozygosity is zero, within-line variance is zero, and all genotypic variance is thus  amongst-line  variance and deplete of dominance variance. In other words,  σ 2 G(1)  is the variance amongst fully inbred line means. Recall further [from "The mean after self-fertilization" section] that such means (G 1 's, in fact) are  G = a(p-q) . Substituting  (1-q)  for the  p , gives  G 1  = a (1 − 2q)  =  a − 2aq . [ 14 ] : 265   Therefore, the  σ 2 G(1)  is the  σ 2 (a-2aq)  actually. Now, in general, the  variance of a difference (x-y)  is   [ σ 2 x  + σ 2 y  − 2 cov xy  ] . [ 49 ] : 100  [ 50 ]   : 232   Therefore,  σ 2 G(1)  = [ σ 2 a  + σ 2 2aq  − 2 cov (a, 2aq)  ]  . But  a  (an allele  effect ) and  q  (an allele  frequency ) are  independent —so this covariance is zero. Furthermore,  a  is a constant from one line to the next, so  σ 2 a  is also zero. Further,  2a  is another constant (k), so the  σ 2 2aq  is of the type  σ 2 k X . In general, the variance  σ 2 k X  is equal to  k 2  σ 2 X   . [ 50 ] : 232   Putting all this together reveals that   σ 2 (a-2aq)  = (2a) 2  σ 2 q   . Recall [from the section on "Continued genetic drift"] that  σ 2 q  = pq f  . With  f=1  here within this present derivation, this becomes  pq 1  (that is  pq ), and this is substituted into the previous.


The final result is:  σ 2 G(1)  = σ 2 (a-2aq)  = 4a 2  pq = 2(2pq a 2 ) = 2 σ 2 a   .


It follows immediately that  f  σ 2 G(1)  =  f  2 σ 2 a   . [This last  f  comes from the  initial Sewall Wright equation  : it is  not  the  f   just set to "1" in the derivation concluded two lines above.]


Previous sections found that the  within line   genic variance  is based upon the  substitution-derived  genic variance  ( σ 2 A  ) —but the  amongst line   genic variance  is based upon the  gene model  allelic variance  ( σ 2 a  ) . These two cannot simply be added to get  total genic variance . One approach in avoiding this problem was to re-visit the derivation of the  average allele substitution effect , and to construct a version,  ( β   f    ) , that incorporates the effects of the dispersion. Crow and Kimura achieved this [ 13 ]   : 130–131   using the re-centered allele effects ( a•, d•, (-a)•  ) discussed previously ["Gene effects re-defined"]. However, this was found subsequently to under-estimate slightly the  total Genic variance , and a new variance-based derivation led to a refined version. [ 36 ]


The  refined  version is:   β   f    = { a 2  + [(1− f  ) / (1 +  f  )] 2(q − p ) ad + [(1- f  ) / (1 +  f  )] (q − p ) 2  d 2  }  (1/2)


Consequently,  σ 2 A(f)  = (1 +  f  ) 2pq β f   2    does now agree with  [ (1-f) σ 2 A(0)  +  2f σ 2 a(0)  ]  exactly.


The  total genic variance  is of intrinsic interest in its own right. But, prior to the refinements by Gordon, [ 36 ]  it had had another important use as well. There had been no extant estimators for the "dispersed" quasi-dominance. This had been estimated as the difference between Sewall Wright's  inbred genotypic variance   [ 37 ]  and the total "dispersed" genic variance [see the previous sub-section]. An anomaly appeared, however, because the  total quasi-dominance variance  appeared to increase early in inbreeding despite the decline in heterozygosity. [ 14 ]   : 128    : 266


The refinements in the previous sub-section corrected this anomaly. [ 36 ]  At the same time, a direct solution for the  total quasi-dominance variance  was obtained, thus avoiding the need for the "subtraction" method of previous times. Furthermore, direct solutions for the  amongst-line  and  within-line  partitions of the  quasi-dominance variance  were obtained also, for the first time. [These have been presented in the section "Dispersion and the genotypic variance".]


The environmental variance is phenotypic variability, which cannot be ascribed to genetics. This sounds simple, but the experimental design needed to separate the two needs very careful planning. Even the "external" environment can be divided into spatial and temporal components ("Sites" and "Years"); or into partitions such as "litter" or "family", and "culture" or "history". These components are very dependent upon the actual experimental model used to do the research. Such issues are very important when doing the research itself, but in this article on quantitative genetics this overview may suffice.


It is an appropriate place, however, for a summary:


Phenotypic variance = genotypic variances + environmental variances +  genotype-environment interaction  + experimental "error" variance


i.e.,  σ 2 P  = σ 2 G  + σ 2 E  + σ 2 GE  + σ 2


or
σ 2 P  = σ 2 A  + σ 2 D  + σ 2 I  + σ 2 E  + σ 2 GE  +  σ 2


after partitioning the genotypic variance (G) into component variances "genic" (A), "quasi-dominance" (D), and "epistatic" (I). [ 51 ]


The environmental variance will appear in other sections, such as "Heritability" and "Correlated attributes".


The  heritability  of a trait is the proportion of the total (phenotypic) variance (σ 2   P ) that is attributable to genetic variance, whether it be the full genotypic variance, or some component of it. It quantifies the degree to which phenotypic variability is due to genetics: but the precise meaning depends upon which genetical variance partition is used in the numerator of the proportion. [ 52 ]  Research estimates of heritability  have standard errors, just as have all estimated statistics. [ 53 ]


Where the numerator variance is the whole Genotypic variance (  σ 2 G   ), the heritability is known as the "broadsense" heritability ( H 2 ). It quantifies the degree to which variability in an attribute is determined by genetics as a whole.

 H

 2

 =

 σ

 G

 2

 σ

 P

 2

 =

 σ

 A

 2

 +

 σ

 D

 2

 σ

 P

 2

 =

 [

 σ

 a

 2

 +

 σ

 d

 2

 +
 c
 o

 v

 a
 d

 ]

 +

 σ

 D

 2

 σ

 P

 2

 {\displaystyle {\begin{aligned}H^{2}&={\frac {\sigma _{G}^{2}}{\sigma _{P}^{2}}}\\&={\frac {\sigma _{A}^{2}+\sigma _{D}^{2}}{\sigma _{P}^{2}}}\\&={\frac {\left[\sigma _{a}^{2}+\sigma _{d}^{2}+cov_{ad}\right]+\sigma _{D}^{2}}{\sigma _{P}^{2}}}\end{aligned}}}

  [See section on the Genotypic variance.]


If only genic variance ( σ 2 A ) is used in the numerator, the heritability may be called "narrow sense" (h 2 ). It quantifies the extent to which phenotypic variance is determined by Fisher's  substitution expectations  variance.

 h

 2

 =

 σ

 A

 2

 σ

 P

 2

 =

 σ

 a

 2

 +

 σ

 d

 2

 +
 c
 o

 v

 a
 d

 σ

 P

 2

 {\displaystyle {\begin{aligned}h^{2}&={\frac {\sigma _{A}^{2}}{\sigma _{P}^{2}}}\\&={\frac {\sigma _{a}^{2}+\sigma _{d}^{2}+cov_{ad}}{\sigma _{P}^{2}}}\end{aligned}}}

 Fisher proposed that this narrow-sense heritability might be appropriate in considering the results of natural selection, focusing as it does on change-ability,  that is  upon "adaptation". [ 29 ]  He proposed it with regard to quantifying Darwinian evolution.


Recalling that the  allelic  variance ( σ  2 a ) and the  dominance  variance ( σ  2 d ) are eu-genetic components of the gene-model [see section on the Genotypic variance], and that  σ  2 D  (the  substitution deviations  or   "quasi-dominance"   variance) and  cov ad  are due to changing from the homozygote midpoint ( mp ) to the population mean ( G ), it can be seen that the real meanings of these heritabilities are obscure. The heritabilities

 H

 e
 u

 2

 =

 σ

 a

 2

 +

 σ

 d

 2

 σ

 P

 2

 {\textstyle H_{eu}^{2}={\tfrac {\sigma _{a}^{2}+\sigma _{d}^{2}}{\sigma _{P}^{2}}}}

   and

 h

 e
 u

 2

 =

 σ

 a

 2

 σ

 P

 2

 {\textstyle h_{eu}^{2}={\tfrac {\sigma _{a}^{2}}{\sigma _{P}^{2}}}}

  have unambiguous meaning.


Narrow-sense heritability has been used also for predicting generally the results of  artificial selection . In the latter case, however, the broadsense heritability may be more appropriate, as the whole attribute is being altered: not just adaptive capacity. Generally, advance from selection is more rapid the higher the heritability. [See section on "Selection".] In animals, heritability of reproductive traits is typically low, while heritability of disease resistance and production are moderately low to moderate, and heritability of body conformation is high.


Repeatability (r 2 ) is the proportion of phenotypic variance attributable to differences in repeated measures of the same subject, arising from later records. It is used particularly for long-lived species. This value can only be determined for traits that manifest multiple times in the organism's lifetime, such as adult body mass, metabolic rate or litter size. Individual birth mass, for example, would not have a repeatability value: but it would have a heritability value. Generally, but not always, repeatability indicates the upper level of the heritability. [ 54 ]


r 2  = (s 2 G  + s 2 PE )/s 2 P


where s 2 PE  = phenotype-environment interaction = repeatability.


The above concept of repeatability is, however, problematic for traits that necessarily change greatly between measurements. For example, body mass increases greatly in many organisms between birth and adult-hood. Nonetheless, within a given age range (or life-cycle stage), repeated measures could be done, and repeatability would be meaningful within that stage.


From the heredity perspective, relations are individuals that inherited genes from one or more common ancestors. Therefore, their "relationship" can be  quantified  on the basis of the probability that they each have inherited a copy of an allele from the common ancestor. In earlier sections, the  Inbreeding coefficient  has been defined as, "the probability that two  same  alleles (  A  and  A , or  a  and  a  ) have a common origin"—or, more formally, "The probability that two homologous alleles are autozygous." Previously, the emphasis was on an individual's likelihood of having two such alleles, and the coefficient was framed accordingly. It is obvious, however, that this probability of autozygosity for an individual must also be the probability that each of its  two parents  had this autozygous allele. In this re-focused form, the probability is called the  co-ancestry coefficient  for the two individuals  i  and  j  (  f   ij  ). In this form, it can be used to quantify the relationship between two individuals, and may also be known as the  coefficient of kinship  or the  consanguinity coefficient . [ 13 ] : 132–143
 [ 14 ] : 82–92


Pedigrees  are diagrams of familial connections between individuals and their ancestors, and possibly between other members of the group that share genetical inheritance with them. They are relationship maps. A pedigree can be analyzed, therefore, to reveal coefficients of inbreeding and co-ancestry. Such pedigrees actually are informal depictions of  path diagrams  as used in  path analysis , which was invented by Sewall Wright when he formulated his studies on inbreeding. [ 55 ] : 266–298   Using the adjacent diagram, the probability that individuals "B" and "C" have received autozygous alleles from ancestor "A" is  1/2  (one out of the two diploid alleles). This is the "de novo" inbreeding ( Δf Ped ) at this step. However, the other allele may have had "carry-over" autozygosity from previous generations, so the probability of this occurring is ( de novo complement  multiplied by the  inbreeding of ancestor A  ), that is   (1 − Δf Ped  ) f A    =   (1/2) f A   . Therefore, the total probability of autozygosity in B and C, following the bi-furcation of the pedigree, is the sum of these two components, namely   (1/2) + (1/2)f A    =   (1/2) (1+f  A  )  . This can be viewed as the probability that two random gametes from ancestor A carry autozygous alleles, and in that context is called the  coefficient of parentage  (  f AA   ). [ 13 ] : 132–143  [ 14 ] : 82–92   It appears often in the following paragraphs.


Following the "B" path, the probability that any autozygous allele is "passed on" to each successive parent is again  (1/2)  at each step (including the last one to the "target"  X  ). The overall probability of transfer down the "B path" is therefore   (1/2) 3   . The power that (1/2) is raised to can be viewed as "the number of intermediates in the path between  A  and  X  ",  n B  = 3  . Similarly, for the "C path",   n C  = 2  , and the "transfer probability" is   (1/2) 2   . The combined probability of autozygous transfer from  A  to  X  is therefore  [ f AA  (1/2) (n B )  (1/2) (n C )  ]  . Recalling that   f AA  = (1/2) (1+f  A  )  ,   f X  = f PQ    =   (1/2) (n B  + n C  + 1)  (1 + f A  )  . In this example, assuming that f A  = 0,   f X  = 0.0156   (rounded) =  f PQ   , one measure of the "relatedness" between  P  and  Q .


In this section, powers of ( 1/2 ) were used to represent the "probability of autozygosity". Later, this same method will be used to represent the proportions of ancestral gene-pools which are inherited down a pedigree [the section on "Relatedness between relatives"].


In the following sections on sib-crossing and similar topics, a number of "averaging rules" are useful. These derive from  path analysis . [ 55 ]   The rules show that any co-ancestry coefficient can be obtained as the average of  cross-over co-ancestries  between appropriate grand-parental and parental combinations. Thus, referring to the adjacent diagram,  Cross-multiplier 1  is that  f PQ  = average of (  f AC ,  f AD ,  f BC ,  f BD  ) =   (1/4) [f AC  + f AD  + f BC  + f BD  ]   =  f Y   .  In a similar fashion,  cross-multiplier 2  states that  f PC  = (1/2) [ f AC  + f BC  ] —while  cross-multiplier 3  states that  f PD  = (1/2) [ f AD  + f BD  ]  . Returning to the first multiplier, it can now be seen also to be  f PQ  = (1/2) [ f PC  + f PD  ] , which, after substituting multipliers 2 and 3, resumes its original form.


In much of the following, the grand-parental generation is referred to as  (t-2) , the parent generation as  (t-1) , and the "target" generation as  t .


The diagram to the right shows that  full sib crossing  is a direct application of  cross-Multiplier 1 , with the slight modification that  parents A and B  repeat (in lieu of  C and D ) to indicate that individuals  P1  and  P2  have both of  their  parents in common—that is they are  full siblings . Individual  Y   is the result of the crossing of two full siblings. Therefore,  f Y  = f P1,P2  = (1/4) [ f AA  + 2 f AB  + f BB  ]  . Recall that  f AA  and  f BB  were defined earlier (in Pedigree analysis) as  coefficients of parentage , equal to   (1/2)[1+f A  ]   and   (1/2)[1+f B  ]   respectively, in the present context. Recognize that, in this guise, the grandparents  A  and  B  represent  generation (t-2)  . Thus, assuming that in any one generation all levels of inbreeding are the same, these two  coefficients of parentage   each represent  (1/2) [1 + f (t-2)  ]  .


Now, examine  f AB   . Recall that this also is  f P1    or  f P2   , and so represents  their   generation -  f (t-1)   . Putting it all together,  f t  = (1/4) [ 2 f AA  + 2 f AB  ]   =   (1/4) [ 1 + f (t-2)  + 2 f (t-1)  ]  . That is the  inbreeding coefficient   for  Full-Sib crossing  . [ 13 ] : 132–143  [ 14 ] : 82–92   The graph to the left shows the rate of this inbreeding over twenty repetitive generations. The "repetition" means that the progeny after cycle  t  become the crossing parents that generate cycle ( t+1  ), and so on successively. The graphs also show the inbreeding for  random fertilization 2N=20  for comparison. Recall that this inbreeding coefficient for progeny  Y  is also the  co-ancestry coefficient  for its parents, and so is a measure of the  relatedness of the two Fill siblings .


Derivation of the  half sib crossing  takes a slightly different path to that for Full sibs. In the adjacent diagram, the two half-sibs at generation (t-1) have only one parent in common—parent "A" at generation (t-2). The  cross-multiplier 1  is used again, giving  f Y  = f (P1,P2)  = (1/4) [ f AA  + f AC  + f BA  + f BC  ]  . There is just one  coefficient of parentage  this time, but three  co-ancestry coefficients  at the (t-2) level (one of them—f BC —being a "dummy" and not representing an actual individual in the (t-1) generation). As before, the  coefficient of parentage  is  (1/2)[1+f A  ]  , and the three  co-ancestries  each represent  f (t-1)   . Recalling that   f A    represents   f (t-2)   , the final gathering and simplifying of terms gives   f Y  = f t  = (1/8) [ 1 + f (t-2)  + 6 f (t-1)  ]  . [ 13 ] : 132–143  [ 14 ] : 82–92   The graphs at left include this  half-sib (HS) inbreeding  over twenty successive generations.


As before, this also quantifies the  relatedness  of the two half-sibs at generation (t-1) in its alternative form of  f (P1, P2)   .


A pedigree diagram for selfing is on the right. It is so straightforward it does not require any cross-multiplication rules. It employs just the basic juxtaposition of the  inbreeding coefficient  and its alternative the  co-ancestry coefficient ; followed by recognizing that, in this case, the latter is also a  coefficient of parentage . Thus,   f Y  = f (P1, P1)  = f t  = (1/2) [ 1 + f (t-1)  ]  . [ 13 ] : 132–143  [ 14 ] : 82–92   This is the fastest rate of inbreeding of all types, as can be seen in the graphs above. The selfing curve is, in fact, a graph of the  coefficient of parentage .


These are derived with methods similar to those for siblings. [ 13 ] : 132–143  [ 14 ] : 82–92   As before, the  co-ancestry  viewpoint of the  inbreeding coefficient  provides a measure of "relatedness" between the parents  P1  and  P2  in these cousin expressions.


The pedigree for  First Cousins (FC)  is given to the right. The prime equation is  f Y  = f t  = f P1,P2  = (1/4) [ f 1D  + f 12  + f CD  + f C2  ] . After substitution with corresponding inbreeding coefficients, gathering of terms and simplifying, this becomes   f t  = (1/4) [ 3 f (t-1)  + (1/4) [2 f (t-2)  + f (t-3)  + 1 ]]  , which is a version for iteration—useful for observing the general pattern, and for computer programming. A "final" version is   f t  = (1/16) [ 12 f (t-1)  + 2 f (t-2)  + f (t-3)  + 1 ]  .


The  Second Cousins (SC)  pedigree is on the left. Parents in the pedigree not related to the  common Ancestor  are indicated by numerals instead of letters. Here, the prime equation is   f Y  = f t  = f P1,P2  = (1/4) [ f 3F  + f 34  + f EF  + f E4  ] . After working through the appropriate algebra, this becomes   f t  = (1/4) [ 3 f (t-1)  + (1/4) [3 f (t-2)  + (1/4) [2 f (t-3)  + f (t-4)  + 1 ]]]  , which is the iteration version. A "final" version is   f t  = (1/64) [ 48 f (t-1)  + 12 f (t-2)  + 2 f (t-3)  + f (t-4)  + 1 ]  .


To visualize the  pattern in full cousin  equations, start the series with the  full sib  equation re-written in iteration form:   f t  = (1/4)[2 f (t-1)  + f (t-2)  + 1 ] . Notice that this is the "essential plan" of the last term in each of the cousin iterative forms: with the small difference that the generation indices increment by "1" at each cousin "level". Now, define the  cousin level  as  k = 1  (for First cousins),  = 2  (for Second cousins),  = 3  (for Third cousins), etc., etc.; and  = 0  (for Full Sibs, which are "zero level cousins"). The  last term  can be written now as:   (1/4) [ 2 f (t-(1+k))  + f (t-(2+k))  + 1]  . Stacked in front of this  last term  are one or more  iteration increments  in the form  (1/4) [ 3 f (t-j)  + ...  , where  j  is the  iteration index  and takes values from  1 ... k  over the successive iterations as needed. Putting all this together provides a general formula for all levels of  full cousin  possible, including  Full Sibs . For  k th  level  full cousins,  f{k} t  =  Ιter j = 1 k  { (1/4) [ 3 f (t-j)  + } j  + (1/4) [ 2 f (t-(1+k))  + f (t-(2+k))  + 1]  . At the commencement of iteration, all f (t- x )  are set at "0", and each has its value substituted as it is calculated through the generations. The graphs to the right show the successive inbreeding for several levels of Full Cousins.


For  first half-cousins (FHC) , the pedigree is to the left.  Notice there is just one common ancestor (individual  A ). Also, as for  second cousins , parents not related to the common ancestor are indicated by numerals. Here, the prime equation is   f Y  = f t  = f P1,P2  = (1/4) [ f 3D  + f 34  + f CD  + f C4  ] . After working through the appropriate algebra, this becomes   f t  = (1/4) [ 3 f (t-1)  + (1/8) [6 f (t-2)  + f (t-3)  + 1 ]]  , which is the iteration version. A "final" version is   f t  = (1/32) [ 24 f (t-1)  + 6 f (t-2)  + f (t-3)  + 1 ]  . The iteration algorithm is similar to that for  full cousins , except that the last term is  (1/8) [ 6 f (t-(1+k))  + f (t-(2+k))  + 1 ]  . Notice that this last term is basically similar to the half sib equation, in parallel to the pattern for full cousins and full sibs. In other words, half sibs are "zero level" half cousins.


There is a tendency to regard cousin crossing with a human-oriented point of view, possibly because of a wide interest in Genealogy. The use of pedigrees to derive the inbreeding perhaps reinforces this "Family History" view. However, such kinds of inter-crossing occur also in natural populations—especially those that are sedentary, or have a "breeding area" that they re-visit from season to season. The progeny-group of a harem with a dominant male, for example, may contain elements of sib-crossing, cousin crossing, and backcrossing, as well as genetic drift, especially of the "island" type. In addition to that, the occasional "outcross" adds an element of hybridization to the mix. It is  not  panmixia.


Following the hybridizing between  A  and  R , the  F1  (individual  B ) is crossed back ( BC1 ) to an original parent ( R ) to produce the  BC1  generation (individual  C ). [It is usual to use the same label for the act of  making  the back-cross  and  for the generation produced by it. The act of back-crossing is here in  italics . ] Parent  R  is the  recurrent  parent. Two successive backcrosses are depicted, with individual  D  being the  BC2  generation. These generations have been given  t  indices also, as indicated. As before,  f D  = f t  = f CR  = (1/2) [ f RB  + f RR  ]  , using  cross-multiplier 2  previously given.  The  f RB  just defined is the one that involves generation  (t-1)  with  (t-2) . However, there is another such  f RB  contained wholly  within  generation  (t-2)  as well, and it is  this  one that is used now: as the  co-ancestry  of the  parents  of individual  C  in generation  (t-1) . As such, it is also the  inbreeding coefficient  of  C , and hence is  f (t-1) . The remaining  f RR  is the  coefficient of parentage  of the  recurrent parent , and so is  (1/2) [1 + f R  ]  . Putting all this together :  f t  = (1/2) [ (1/2) [ 1 + f R  ] + f (t-1)  ]   =   (1/4) [ 1 + f R  + 2 f (t-1)  ]  . The graphs at right illustrate Backcross inbreeding over twenty backcrosses for three different levels of (fixed) inbreeding in the Recurrent parent.


This routine is commonly used in Animal and Plant Breeding programmes. Often after making the hybrid (especially if individuals are short-lived), the recurrent parent needs separate "line breeding" for its maintenance as a future recurrent parent in the backcrossing. This maintenance may be through selfing, or through full-sib or half-sib crossing, or through restricted randomly fertilized populations, depending on the species' reproductive possibilities. Of course, this incremental rise in  f R  carries-over into the  f t  of the backcrossing. The result is a more gradual curve rising to the asymptotes than shown in the present graphs, because the  f R  is not at a fixed level from the outset.


In the section on "Pedigree analysis",

 (

 1
 2

 )

 n

 {\textstyle \left({\tfrac {1}{2}}\right)^{n}}

  was used to represent probabilities of autozygous allele descent over  n  generations down branches of the pedigree. This formula arose because of the rules imposed by sexual reproduction:  (i)  two parents contributing virtually equal shares of autosomal genes, and  (ii)  successive dilution for each generation between the zygote and the "focus" level of parentage. These same rules apply also to any other viewpoint of descent in a two-sex reproductive system. One such is the proportion of any ancestral gene-pool (also known as 'germplasm') which is contained within any zygote's genotype.


Therefore, the proportion of an  ancestral genepool  in a genotype is:

 γ

 n

 =

 (

 1
 2

 )

 n

 {\displaystyle \gamma _{n}=\left({\frac {1}{2}}\right)^{n}}

where  n  = number of sexual generations between the zygote and the focus ancestor.


For example, each parent defines a genepool contributing

 (

 1
 2

 )

 1

 {\textstyle \left({\tfrac {1}{2}}\right)^{1}}

  to its offspring; while each great-grandparent contributes

 (

 1
 2

 )

 3

 {\textstyle \left({\tfrac {1}{2}}\right)^{3}}

  to its great-grand-offspring.


The zygote's total genepool ( Γ ) is, of course, the sum of the sexual contributions to its descent.


Γ

 =

 ∑

 n
 =
 1

 2

 n

 γ

 n

 =

 ∑

 n
 =
 1

 2

 n

 (

 1
 2

 )

 n

 {\displaystyle {\begin{aligned}\Gamma &=\sum _{n=1}^{2^{n}}{\gamma _{n}}\\&=\sum _{n=1}^{2^{n}}{\left({\frac {1}{2}}\right)^{n}}\end{aligned}}}


Individuals descended from a common ancestral genepool obviously are related. This is not to say they are identical in their genes (alleles), because, at each level of ancestor, segregation and assortment will have occurred in producing gametes. But they will have originated from the same pool of alleles available for these meioses and subsequent fertilizations. [This idea was encountered firstly in the sections on pedigree analysis and relationships.]  The genepool contributions [see section above] of their  nearest common ancestral genepool (an  ancestral node ) can therefore be used to define their relationship. This leads to an intuitive definition of relationship which conforms well with familiar notions of "relatedness" found in family-history; and permits comparisons of the "degree of relatedness" for complex patterns of relations arising from such  genealogy.


The only modifications necessary (for each individual in turn) are in Γ and are due to the shift to "shared  common  ancestry" rather than "individual  total  ancestry". For this, define  Ρ  (in lieu of  Γ );   m = number of ancestors-in-common  at the node (i.e. m = 1 or 2 only); and an "individual index"  k . Thus:


P

 k

 =

 ∑

 m
 =
 1

 1
 ,
 2

 γ

 n

 =

 ∑

 m
 =
 1

 1
 ,
 2

 (

 1
 2

 )

 n

 {\displaystyle {\begin{aligned}\mathrm {P} _{k}&=\sum _{m=1}^{1,2}{\gamma _{n}}\\&=\sum _{m=1}^{1,2}{\left({\frac {1}{2}}\right)^{n}}\end{aligned}}}


where, as before,   n = number of sexual generations   between the individual and the ancestral node.


An example is provided by two first full-cousins. Their nearest common ancestral node is their grandparents which gave rise to their two sibling parents, and they have both of these grandparents in common. [See earlier pedigree.] For this case,  m=2  and  n=2 , so for each of them


P

 k

 =

 ∑

 m
 =
 1

 2

 γ

 2

 =

 ∑

 m
 =
 1

 2

 (

 1
 2

 )

 2

 =

 1
 2

 {\displaystyle {\begin{aligned}\mathrm {P} _{k}&=\sum _{m=1}^{2}{\gamma _{2}}\\&=\sum _{m=1}^{2}{\left({\frac {1}{2}}\right)^{2}}\\&={\frac {1}{2}}\end{aligned}}}


In this simple case, each cousin has numerically the same Ρ .


A second example might be between two full cousins, but one ( k=1 ) has three generations back to the ancestral node (n=3), and the other ( k=2 ) only two (n=2) [i.e. a second and first cousin relationship].  For both, m=2 (they are full cousins).


P

 1

 =

 ∑

 m
 =
 1

 2

 γ

 3

 =

 ∑

 m
 =
 1

 2

 (

 1
 2

 )

 3

 =

 1
 4

 {\displaystyle {\begin{aligned}\mathrm {P} _{1}&=\sum _{m=1}^{2}{\gamma _{3}}\\&=\sum _{m=1}^{2}{\left({\frac {1}{2}}\right)^{3}}\\&={\frac {1}{4}}\end{aligned}}}


and


P

 2

 =

 ∑

 m
 =
 1

 2

 γ

 2

 =

 ∑

 m
 =
 1

 2

 (

 1
 2

 )

 2

 =

 1
 2

 {\displaystyle {\begin{aligned}\mathrm {P} _{2}&=\sum _{m=1}^{2}{\gamma _{2}}\\&=\sum _{m=1}^{2}{\left({\frac {1}{2}}\right)^{2}}\\&={\frac {1}{2}}\end{aligned}}}


Notice each cousin has a different Ρ  k .


In any pairwise relationship estimation, there is one  Ρ k  for each individual: it remains to average them in order to combine them into a single "Relationship coefficient". Because each  Ρ   is a  fraction of a total genepool , the appropriate average for them is the  geometric mean   [ 56 ] [ 57 ] : 34–55   This average is their  Genepool Relationship Coefficient —the "GRC".


For the first example (two full first-cousins), their GRC = 0.5; for the second case (a full first and second cousin), their GRC = 0.3536.


All of these relationships (GRC) are applications of path-analysis. [ 55 ] : 214–298   A summary of some levels of relationship (GRC) follow.


These, in like manner to the Genotypic variances, can be derived through either the gene-model ("Mather") approach or the allele-substitution ("Fisher") approach.  Here, each method is demonstrated for alternate cases.


These can be viewed either as the covariance between any offspring and  any one  of its parents ( PO ), or as the covariance between any offspring and the   "mid-parent"   value of both its parents ( MPO ).


This can be derived as the  sum of cross-products   between parent gene-effects and  one-half  of the progeny expectations using the allele-substitution approach. The  one-half  of the progeny expectation accounts for the fact that  only one of the two parents  is being considered. The appropriate parental gene-effects are therefore the second-stage redefined gene effects used to define the genotypic variances earlier, that is:  a″ = 2q(a − qd)  and  d″ = (q-p)a + 2pqd  and also  (-a)″ = -2p(a + pd)  [see section "Gene effects redefined"]. Similarly, the appropriate progeny effects,  for allele-substitution expectations  are one-half of the earlier  breeding values , the latter being:  a AA  = 2qa , and  a Aa  = (q-p)a  and also  a aa  = -2pa  [see section on "Genotype substitution – Expectations and Deviations"].


Because all of these effects are defined already as deviates from the genotypic mean, the cross-product sum using { genotype-frequency * parental gene-effect * half-breeding-value } immediately provides the  allele-substitution-expectation covariance  between any one parent and its offspring. After careful gathering of terms and simplification, this becomes  cov(PO) A  = pqa 2  =   1 / 2   s 2 A   . [ 13 ]   : 132–141  [ 14 ]   : 134–147


Unfortunately, the  allele-substitution-deviations  are usually overlooked, but they have not "ceased to exist" nonetheless! Recall that these deviations are:  d AA  = -2q 2  d , and  d Aa  = 2pq d  and also  d aa  = -2p 2  d  [see section on "Genotype substitution – Expectations and Deviations"].  Consequently, the cross-product sum using { genotype-frequency * parental gene-effect * half-substitution-deviations } also immediately provides the allele-substitution-deviations covariance between any one parent and its offspring. Once more, after careful gathering of terms and simplification, this becomes  cov(PO) D  = 2p 2 q 2 d 2  =   1 / 2   s 2 D   .


It follows therefore that:  cov(PO) = cov(PO) A  + cov(PO) D  =   1 / 2   s 2 A  +   1 / 2   s 2 D   , when dominance is  not  overlooked !


Because there are many combinations of parental genotypes, there are many different mid-parents and offspring means to consider, together with the varying frequencies of obtaining each parental pairing. The gene-model approach is the most expedient in this case. Therefore, an  unadjusted sum of cross-products (USCP) —using all products {  parent-pair-frequency * mid-parent-gene-effect * offspring-genotype-mean  }—is adjusted by subtracting the  {overall genotypic mean} 2    as  correction factor  (CF) . After multiplying out all the various combinations, carefully gathering terms, simplifying, factoring and cancelling-out where applicable, this becomes:


cov(MPO) = pq [a + (q-p)d ] 2  = pq a 2  =   1 / 2   s 2 A   , with no dominance having been overlooked in this case, as it had been used-up in defining the  a . [ 13 ]   : 132–141  [ 14 ]   : 134–147


The most obvious application is an experiment that contains all parents and their offspring, with or without reciprocal crosses, preferably replicated without bias, enabling estimation of all appropriate means, variances and covariances,  together with their standard errors. These estimated statistics can then be used to estimate the genetic variances. Twice  the difference between the estimates of the two forms of (corrected) parent-offspring covariance  provides an estimate of  s 2 D ; and twice the  cov(MPO)  estimates  s 2 A . With appropriate experimental design and analysis, [ 9 ] [ 49 ] [ 50 ]  standard errors can be obtained for these genetical statistics as well. This is the basic core of an experiment known as  Diallel analysis , the Mather, Jinks and Hayman version of which is discussed in another section.


A second application involves using  regression analysis , which estimates from statistics the ordinate (Y-estimate), derivative (regression coefficient) and constant (Y-intercept) of calculus. [ 9 ] [ 49 ] [ 58 ] [ 59 ]  The  regression coefficient  estimates the  rate of change  of the function predicting  Y  from  X , based on minimizing the residuals between the fitted curve and the observed data (MINRES). No alternative method of estimating such a function satisfies this basic requirement of MINRES.  In general, the regression coefficient is estimated as  the ratio of the covariance(XY) to the variance of the determinator (X) . In practice, the sample size is usually the same for both X and Y, so this can be written as  SCP(XY) / SS(X) , where all terms have been defined previously. [ 9 ] [ 58 ] [ 59 ]  In the present context, the parents are viewed as the "determinative variable" (X), and the offspring as the "determined variable" (Y), and the regression coefficient as the "functional relationship" (ß PO ) between the two. Taking  cov(MPO) =   1 / 2   s 2 A    as  cov(XY) , and   s 2 P  / 2   (the variance of the mean of two parents—the mid-parent) as  s 2 X , it can be seen that  ß MPO  = [  1 / 2   s 2 A ] / [  1 / 2   s 2 P ] = h 2   . [ 60 ]  Next, utilizing  cov(PO) = [   1 / 2   s 2 A  +   1 / 2   s 2 D  ]  as  cov(XY) , and   s 2 P  as  s 2 X , it is seen that   2 ß PO  = [ 2 (  1 / 2   s 2 A  +   1 / 2   s 2 D  )] / s 2 P  = H 2   .


Analysis of  epistasis  has previously been attempted via an  interaction variance  approach of the type   s 2 AA   , and   s 2 AD  and also   s 2 DD . This has been integrated with these present covariances in an effort to provide estimators for the epistasis variances. However, the findings of epigenetics suggest that this may not be an appropriate way to define epistasis.


Covariance between half-sibs ( HS ) is defined easily using allele-substitution methods; but, once again, the dominance contribution has historically been omitted. However, as with the mid-parent/offspring covariance, the covariance between full-sibs ( FS ) requires a "parent-combination" approach, thereby necessitating the use of the gene-model corrected-cross-product method; and the dominance contribution has not historically been overlooked. The superiority of the gene-model derivations is as evident here as it was for the Genotypic variances.


The sum of the cross-products  { common-parent frequency * half-breeding-value of one half-sib * half-breeding-value of any other half-sib in that same common-parent-group }  immediately provides one of the required covariances, because the effects used [ breeding values —representing the allele-substitution expectations] are already defined as deviates from the genotypic mean [see section on "Allele substitution – Expectations and deviations"]. After simplification. this becomes:   cov(HS) A  =   1 / 2   pq a 2  =   1 / 4   s 2 A   . [ 13 ]   : 132–141  [ 14 ]   : 134–147   However, the  substitution deviations  also exist, defining the sum of the cross-products  { common-parent frequency * half-substitution-deviation of one half-sib * half-substitution-deviation of any other half-sib in that same common-parent-group } , which ultimately leads to:   cov(HS) D  = p 2  q 2  d 2  =   1 / 4   s 2 D   . Adding the two components gives:


cov(HS) = cov(HS) A  + cov(HS) D  =   1 / 4   s 2 A  +   1 / 4   s 2 D   .


As explained in the introduction, a method similar to that used for mid-parent/progeny covariance is used. Therefore, an  unadjusted sum of cross-products  (USCP) using all products—{  parent-pair-frequency * the square of the offspring-genotype-mean  }—is adjusted by subtracting the  {overall genotypic mean} 2    as  correction factor (CF) . In this case, multiplying out all combinations, carefully gathering terms, simplifying, factoring, and cancelling-out is very protracted. It eventually becomes:


cov(FS) = pq a 2  + p 2  q 2  d 2  =   1 / 2   s 2 A   +   1 / 4   s 2 D   , with no dominance having been overlooked. [ 13 ]   : 132–141  [ 14 ]   : 134–147


The most useful application here for genetical statistics is the  correlation between half-sibs . Recall that the correlation coefficient ( r ) is the ratio of the covariance to the variance [see section on "Associated attributes" for example]. Therefore,   r HS  = cov(HS) / s 2 all HS together    =   [  1 / 4   s 2 A   +   1 / 4   s 2 D  ] / s 2 P  =   1 / 4   H 2   . [ 61 ]  The correlation between full-sibs is of little utility, being   r FS  = cov(FS) / s 2 all FS together    =   [  1 / 2   s 2 A   +   1 / 4   s 2 D  ] / s 2 P   . The suggestion that it "approximates" (  1 / 2   h 2 ) is poor advice.


Of course, the correlations between siblings are of intrinsic interest in their own right, quite apart from any utility they may have for estimating heritabilities or genotypic variances.


It may be worth noting that  [ cov(FS) − cov(HS)] =   1 / 4   s 2 A   . Experiments consisting of FS and HS families could utilize this by using intra-class correlation to equate experiment variance components to these covariances [see section on "Coefficient of relationship as an intra-class correlation" for the rationale behind this].


The earlier comments regarding epistasis apply again here [see section on "Applications (Parent-offspring"].


Selection operates on the attribute (phenotype), such that individuals that equal or exceed a selection threshold  (z P )  become effective parents for the next generation. The  proportion  they represent of the base population is the  selection pressure . The  smaller  the proportion, the  stronger  the pressure. The  mean of the selected group   (P s )  is superior to the  base-population mean   (P 0 )  by the difference called the  selection differential (S) . All these quantities are phenotypic. To "link" to the underlying genes, a  heritability   (h 2 )  is used, fulfilling the role of a  coefficient of determination  in the biometrical sense. The  expected genetical change —still expressed in  phenotypic units of measurement —is called the  genetic advance (ΔG) , and is obtained by the product of the  selection differential (S)  and its  coefficient of determination   (h 2 ) . The expected  mean of the progeny   (P 1 )  is found by adding the  genetic advance (ΔG)  to the  base mean (P 0 ) . The graphs to the right show how the (initial) genetic advance is greater with stronger selection pressure (smaller  probability ). They also show how progress from successive cycles of selection (even at the same selection pressure) steadily declines, because the Phenotypic variance and the Heritability are being diminished by the selection itself. This is discussed further shortly.


Thus

 Δ
 G
 =
 S

 h

 2

 {\displaystyle \Delta G=Sh^{2}}

 . [ 14 ]   : 1710–181
and

 P

 1

 =

 P

 0

 +
 Δ
 G

 {\displaystyle P_{1}=P_{0}+\Delta G}

 . [ 14 ]   : 1710–181


The  narrow-sense heritability (h 2 )  is usually used, thereby linking to the  genic variance (σ 2 A )  . However, if appropriate, use of the  broad-sense heritability (H 2 )  would connect to the  genotypic variance (σ 2 G ) ; and even possibly an  allelic heritability [ h 2 eu  = (σ 2 a ) / (σ 2 P ) ]  might be contemplated, connecting to ( σ 2 a  ). [See section on Heritability.]


To apply these concepts  before  selection actually takes place, and so predict the outcome of alternatives (such as choice of  selection threshold , for example), these phenotypic statistics are re-considered against the properties of the Normal Distribution, especially those concerning truncation of the  superior tail  of the Distribution. In such consideration, the  standardized  selection differential (i)″ and the  standardized  selection threshold (z)″ are used instead of the previous "phenotypic" versions. The  phenotypic standard deviate (σ P(0) )  is also needed. This is described in a subsequent section.