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# QUANTITATIVE GENETICS BPSC 148

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Appendix A Statistics Review Variable Probability and Distribution Random variable A numerically valued function de ned over a sample space The value of a random variable will vary from trial to trial as the experiment repeats Examples outcome of tossing a coin body weight of a laboratory mouse Probability A probability on a sample space S is a specification of numbers PrAl which satisfy the following two conditions 1 0SPrAS1 k 2 Z PrAl l where k is the total number of possible events Distribution The association of probability to each value or outcome of a random variable It describes the relationship between the probability and a random variable There are two types of distributions 1 Discrete distribution A random variable X has a discrete distribution if X can take a finite number of values x1xk or at most an infinite sequence of different values x1x2 Probability lnction or pf is defined as f x PrX x Note that the letter in upper case denotes variable and that in lower case denotes an observation of the variable Properties of discrete distribution lfx is a probability ie fx 2 0 k 2 Z xl l ie the sum ofthe probabilities of all events is one 11 2 Continuous distribution IfA is an interval in the real line and Prx EA L fxdx thenXhas a continuous distribution and x is called the probability density function or pdf Properties of continuous distribution 1 f x Z 0 is not a probability The probability that x takes a particular value is zero 2 ffxdx 1 and Pra s x s b jab max 3 Multivariate distribution joint distribution In many experiments it is necessary to consider the properties of two or more random variables simultaneously The joint probability distribution of two random variables is called bivariate distribution When both variables are discrete their joint distribution is also discrete and defined as fxy PrX x amp Y y where Z xwyl 1 11 When both variables are continuous their joint distribution is also continuous Prxy eA1 Lfxydxdy where f x y is called the joint probability density function ie pdf where jfxydxdy l 4 Marginal distribution Marginal distribution is simply the distribution of a single variable but it is derived from the joint distribution For example the marginal distribution of X is obtained by summing up the joint distribution for all possible values of Y ie fx Zfxy y Similarly the marginal distribution of y is fy Zfxy The following table shows the relationship between the joint and marginal distributions X Y fx 0 l 0 f00 f01 f20 f00f01 fx 0 1 f10 f11 121 f10f11 fx 1 2 f20 f21 112 f20f21 fx 2 fy 0fy0 1fy1 Zfxyl 130 f00 f10 mo 2 mo and 31 f01 f11 f21 Z fx1 For continuous variables the marginal distributions are de ned as x fx j fxydy and y fy J fxydx Two variablesXand Y are said to be independent if fx y fxfy A typical example is the relationship between gene frequency and genotype frequency under random mating eg PrAa PrA Pra Ap aq A02 A140 Aaoq an aAqp aaq2 5 Conditional distribution Conditional distribution of X given Y is the distribution of variable X under the situation where the value of Y is xed to a particular value It is de ned by x mm f y f y Similarly the conditional distribution of Y givenX is de ned by x fylx f y f x The following table gives the joint and marginal distributions of X smoking habit and Y sex variables from which conditional distributions are culculated X Y Man Woman f x Smoker 825 45 1275 Nonsmoker 46 75 40 5 87 25 fy 55 45 100 The joint probability that a randomly sampled person is a man and he is also a smoker is f X s Y m 825 The marginal probability that a randomly sampled A3 Chapter 5 Genetic Variance Quantitative genetics centers around the study of its variation for it is in terms of variation that the primary questions are formulated The phenotypic value can be described by various forms of genetic model P A D 1 E G AD1E R Similarly the variance can be partitioned into various components VP VA VD VI VE VG VA VD VI VE VR where V phenotypic variance VG genetypic variance VA additive genetic variance VD dominance variance VI epistatic variance VE environmental variance V nonadditive variance Heritability Relative importance of each effect can only be answered if it is expressed in terms of the variance attributable to the different sources of variation Broad sense heritability Broad sense heritability is also called the degree of genetic determination It is defined by H2 2 Lo VP Narrow sense heritability Narrow sense heritability is simply called heritability It is defined by if P The narrow sense heritability is very important in animal and plant breeding because it re ects the proportion of genetic variance that can be transmitted to the next generation Total genetic variance The genetic variance can be obtained by using the following table Genotype Frequency Breeding value Dominance deviation Genotypic value A1141 p2 290 2qu a AIA2 2m q ma 2pqd d A2142 qz 270 2p2d a EG p261 qud q2a M EG2 117ch2 210qu q2a2 VG VarG EG2 EG2 2pqa dq Pl2 21W2 227903 21W2 EH my VA VD Recall that OL a dq p is the average effect of gene substitution Additive variance It is the variance of the breeding values E A 102 2w 2pqq ma 92 2100 0 EA2 061002 2pqq Mac2 qZ 2P0 2 quocz VA EA2 E2A 2pq0c2 Dominance variance It is the variance of the dominance deviations ED p2 Zqzd 2pq2pqd q22p2d 0 ED2 11722 126i2 2P92qu2 929211726172 2pqd2 VD ED2 E D 2PM Covariance between additive and dominance e ects E A X D p22qa2q2d 2pqq pa2pqd q22pa2p2d 4p2q3otd 4pzq2 p qad 4q2p3ad 4adp2q2 q q p p 0 CovAD EA gtltD EAED 0 0 X 0 0 Epistatic variance Epistasis can happen when two or more loci are considered Epistatic effects are defined as various interaction effects between loci in contrast to the dominance effect which is defined as the interaction between allelic effects within a locus The epistatic variance can be partitioned into different components Vi VAAKiDVDAVDDVAAAquot39 Note that Km is the variance of interaction between the additive effect of the first locus and the dominance effect of the second locus It is different from VDA which is defined as the variance of interaction between the dominance effect of the first locus and the additive effect of the second locus Genotype gtlt Environment interaction G X E A speci c difference of environment has different effect on different genotypes The genetic model is P G E G X E interaction The variance is partitioned similarly VP VG V V GXE 39 G X E interaction is important when a new cultivar is released from nursery to a farm It can be demonstrated using the following 2X2 table Environment Genotype E1 E2 G1 W11 fu W12 fu f G1 G2 W21 f21 W22 fzz f G2 fE1 fE2 G is the effect of the ith genotype E is the effect of the j th environment WU is the effect of the ith genotype raised in the j th environment it is the average value of all individuals with genotype 139 and raised in environment j j is the frequency of individuals with the ith genotype raised in the j th environment f G fl fl is the marginal frequency of genotype 139 f E j E fl is the marginal frequency of environment j The cundmunal effeds are de ned as fa 1 G WI 111 fur gmulypet 15 15 J Wu i Wi fur mmmnmm39al IKE 103 xExntemennnxspresenumeaeEdsn furallastuneufthetand cumbmauuns where d 15 Lhepupulauun man In uthervmrds Lhare 15 nu 5x12 mleramumeer rEpe malland Nutethat AW WV he57dE 7w Wrap h is de ned asthe G XEmlemcuun Nurlh suum gure 5 1 0x5 Imaman Different genulypes A and a pa39funn m aemly m diffa39em mmmnmmls Nunh and Snmh Genmype and Envimnmzmcnmhu39nn Cunelauun belwem G and E15 different mm G x Emlaacnun Currelauun can uncur fur example when a better gmulype is pmvlded fur a better envimnm nl In new I fur v cumbmauuns In uthervmrds Lha39exs nu G and E currelauumf 1 7 15 102 211 and When the G amp E correlation is present but without G X E interaction the model is P G E but the phenotypic variance is partitioned by VP VG VE 2CovGE where C0vG E is the covariance between G and E When both the G amp E correlation and the G X E interaction are present the model P G E G X E but the phenotypic variance is partitioned by VP VG VE 2CovGE VG XE39 This model has assumed that neither G nor E is correlated with the G X E effect Numerical example of G X E interaction E1 E2 G1 W11100 W1250 G1 75 M025 M025 f105 AW11 225 AW12 225 G2 W2170 W22l 10 G290 f21025 f22025 f205 AW21225 AWZZ 225 E185 E280 H 825 f105 f205 N 100 Calculate G X E interaction u gtlt100gtlt50 gtlt70 gtlt110825 fGlf1 fGzfz fE1f1fEzfz x70 gtlt110 NIH L L GI4gtlt1004gtlt5075GZ 90 L L 2 Z L 100L 70 L 50L 110 E14X 14X 85E2 X 1 80 AW11 w11 G1 E1u 225 Aw W12 G1 E2 H 225 Aw21 wz1 G2 E1u 225 Awn W22 G2 E2 p 225 Because AW 0 we conclude that there is G X E interaction Calculate G amp E correlation C G E GE L 1s the G amp E correlat10n To see whether the correlation 1s present or 6262 not we only need to look at the value of the covariance the numerator If CovG E 0 the correlation is of course equals zero EG fG1G1 fG2GZ x 75 x 90 825 EE fE1E1 fE2E2 x 85 x 80 825 EGE 75gtlt8575gtlt8085gtlt9080gtlt90 7590gtlt8580 825 x825 CovG E EGE EGEE 825 x825 825gtlt 825 0 The conclusion is that there is no G by E correlation Numerical example of G amp E correlation E1 E2 G1 W11100 W1250 G1 80 M030 M020 f105 AW11 190 AWl2 290 G2 w2170 sz110 G294 f21020 f22030 f205 AW21250 AWZZ 170 E188 13286 H 870 f105 f205 N 100 Chapter 18 Multivariate Selection The economical value of an animal or plant normally depends on several traits Selection must be considered on these traits simultaneously Common methods for multi trait selection 1 Tandem selection Select in turn for each character singly in successive generations 2 Independent culling levels Select for all the characters at the same time but independently Reject all individuals that fail to come up to a certain standard for each character regardless of their values for any other characters The advantage of independent culling level is that if traits are expressed in different stages this method will allow breeders to select in several stages referred to as multi stage selection Multistage selection is very practical in large animals and trees 3 Index selection Select an index which is a linear combination of the phenotypic values of all characters 4 Multistage index selection A combination of independent culling level with index selection Variance covariance matrix De ne X X1 X2 X KT as a vector of phenotypic values forK traits expressed in the same individuals A A1 A2 AKT a vector ofbreeding values ofthe K traits and E E1 E2 E KT a vector of environmental effects The multivariate model is XAE BecauseX are already as J viati from the 39 quot means EX 0 and l r r VarX VarA VarE T T T U PGE where VarX1 CovX1X2 CovX1XK P Varoo COVQQXZ Varng Cov2XK Cov1XK CovK2XK VarXK is the phenotypic variancecovariance matrix of the K traits VarA1 CovA1A2 CovA1AK G WIN1 Cov41A2 VarSAi C0v142AK COVz41AK COV141AK VartAK is the genetic variancecovariance matrix for the K characters and VarE1 CovE1E2 CovE1EK E WINE CovE1EZ VarE2 CovE2EK CovE1EK CovE2EK VarEK is the environmental variancecovariance matrix It should be that CovXA CovA E A CovAA CovEA VarA G 0 202 Index selection The selection criterion is I lel b2X2bKXK bTX These X s are the phenotypic values of the traits expressed in the same individual they are not different sources of information described Chapter 18 Selection Index The objective of the selection is a linear combination of all breeding values for all characters T H wlA1 w2A2wKAK w A where w w1 w2 WKT is called the vector of economic weights which are determined by the breeder H is the objective of selection called the aggregate breeding value The index weights b are found by maximizing the correlation between I and H rm After some derivations we found I VarX 1CovXH where VarX P and CovXH CovX wTA CovXAw VarAw Gw Therefore b PTIGW Individuals are ranked based on their index scores I bTX The index scores are treated as observed phenotypic value Only the best individuals are selected for breeding This selection method is called index selection 203 Response to index selection 1 Selection response of the aggregate breeding value The selection response of the aggregate breeding value is denoted by AH which can be predicted using the usual expression AHzIrIH6H where 62 VarH VarwTA wTVarAw wTGw and CovIH CovbTXwTA bTCovXAw bTGw bTPb 6i i H GHGI GHGI GHGI GHGI GHGI GHGI 6H Therefore 6 AHzI I H ZIGI H where 6 VarbTX bTPb 20 4 2 Selection responses of individual traits components of H The aggregate breeding value is decomposed as AH wlAA1w2AA2wKAAK where AA is the genetic change response of the ith character De ne AA AA1 AA1T which can be predicted by AA buds f bAIAIAI fc Gb II Notethat 1391 AlGI and C0vA I CovA bTX CovAXb Gb The detailed expression of AA is 1 21 CovA1Aj bj AAI VarA1 CovA1A2 CovA1AK b1 51 K AA 21 1 Cow11142 VarA2 CovA2AK b2 G IZj ColeZl bj 6 I AAK I CovA1AK CovA2AK VarAK bK l K 3 G IZj ColeKl bj I Chapter 21 Mapping quantitative Trait Loci 1 Regression Method Models for the variation of quantitative traits Infinitesimal model The genetic variation of a quantitative character is controlled by in nite number of loci each with an in nitely small effect Our previous discussion about quantitative genetics is actually based on this in nitesimal model Polygenic model The genetic variation of a quantitative character is controlled by many loci each with a small effect This model is similar to the in nitesimal model except that it is not stated as strong as in nitesimal model The key word of the polygenic model is many loci but not matter how many the number of loci is nite and countable Oligogenic model The genetic variation of a quantitative character is controlled by a few loci each with a relatively large effect and many modi er genes each with an unidentifiable effect Most genetic disease traits are considered to follow this oligogenic model M onogenic model The genetic variation of a character is controlled by the segregation of a single locus Quantitative genetics of nite loci There must be a nite number of loci to control a quantitative character These loci must be physically located on chromosomes If the number of loci is nite the effects of some gene must be estimable The nite number of identi able genes are called quantitative trait loci QTL Regression method of QTL mapping Single marker analysis Single marker analysis is similar to the candidate gene approach ie the marker is treated as a functional gene Here we use a backcross population as an example to demonstrate the procedure With a backcross population we cannot separate the dominance effect from the additive effect Thus we assume that the dominance effect is absent The backcross design is illustrated below Backcross design Pl A1141 X A2141 Pz l F1 141A2 gtlt A1141 l BC A1A1 1A2 Value at 0 The genetic model is y j u g j e j where y j is the phenotypic value of individual j u is the population mean g j is the genotypic value and e is the residual effect with a N 0 6 distribution If the marker is a quantitative trait locus the genotypic value would be de ned as 0 for 141A1 g a forAlA1 The statistical regression model is y b0 ij1 8 where b0 u is the intercept bl a is the regression coef cient the size of the QTL and 0 for 141A1 x j l for 141A1 The data will be xjyj for j l N Given the data set the estimated regression coefficient 51 and the variance of the estimate 6 can be obtained using the usual regression analysis To test whether the proposed candidate gene is actually a trait locus responsible for the variation of a quantitative trait a signi cance test is required We propose two hypotheses Null hypothesis H0b1 0 or a 0 Alternative hypothesis H A b1 i 0 or a 7i 0 An F test statistic is used to decide which hypothesis should be accepted The test statistic is Under the null hypothesis ie when H0 is true the test statistic follows an F distribution with 1 degree of freedom in the numerator and N 7 2 degrees of freedom in the denominator When the sample size is sufficiently large the F distribution approximately equals a chisquare distribution with 1 degree of freedom denoted by xim Therefore the test statistic can be compared to the critical value percentile of xim The critical value 1 0c percentile of the chisquare distribution is denoted by xg m where at is the Type I error rate For example when df 1 and at 005 x31 ve 95 3841 when df 2 and at 005 xfmliu x20 95 5991 We now compare Fto x1270 95 3841 ifF S 3841 we will accept H0 otherwise we will accept HA If HA is accepted we can declare a QTL The single marker analysis treats a marker as a gene the candidate gene approach If a marker is not a gene but linked to a gene the test statistic may still be significant depending on the size of the gene effect and the distance between the marker and the gene We now use BC as an example to show what we estimate and test if the marker is not a gene Let us assume that a single QTL d Morgan away from a polymorphic marker First we use the Haldane s mapping function to convert the genetic distance 6139 into recombination fraction r We then use x and m to denote the indicator variables of the genotype of QTL and the marker respectively ie 0 for 141112 0 for MIM2 and m 1 for 141A1 x 1 for MIM1 where M stands for the genotype of the marker In a backcross population there is an equal chance of0 and 1 for each ofthe x and m Thus Varx Varm 1 The covariance between x and m is C0vx m l 2r The QTL model is y be xb1 6 Because x is not observable and has been substituted by an estimate of x given marker information the actual model the marker model is y be Ebl e where 56 Exl m e 1m is the conditional expectation of x given m and 51 is the regression coefficient of x on m which is obtained by Covx m 1 2r Varm 51 l 2r Therefore the marker model becomes y b0 Acb1 e b0 o 1mb1 8 b0 30b1 1b1me b0 ob1 l 2rb1me f at 17 171 b mb e In single marker analysis what we estimate and test is bf 1 2rb1 1 2ra a confounding effect between the QTL effects a and linkage parameter r So a statistically significant b may be due to either a small QTL tightly linked to the marker or a large QTL loosely linked to the marker This is a drawback of the single marker analysis Flanking marker analysis interval mapping Consider two markers m1 and ml with a known recombination frequency r both linked to a hypothesized QTL with effect a on the trait Using the first marker we can estimate the confounding effect 1 2r1a similarly another confounding effect 1 2r2 a can be obtained by using the second marker where V1 and r are the recombination fractions of the QTL with the first and the second markers respectively The relationship between V V1 and r are 1 r1 r1 r2r1r2 r r11 r2r21r1 If both markers are used simultaneously we will be able to separate the position of the QTL from its effect because we now have two equations to solve for two unknowns This leads to the concept of interval mapping first proposed by Lander and Botstein 1989 and later extended by Haley and Knott 1992 The QTL model is still the same as before y b0 xb1 e We now use two markers to estimate x so that fc is used in place of x y be Ebl e where 9E Exlm1m2 e 1m1 2m2 Given the position of the putative QTL relative to the two markers these 539 s values can be solved using the following equations 51 Varm1 Covm1m2 1 Covm1x 52 Covmlm2 Varm2 Covm2x and e Ex 1Em1 zEmz In backcross design Varm1 Covm1m2 l Zr l l l 2r Covm1m2 Varm2 1 2r 4 l 2r l Covm1x 1 2r1 1 Ha C0vm2x l Zrz 4 l 2r2 E06 Eml E0712 and The explicit solutions of the 539 S values are B0 F10an 51 l ZmlZVDUZ l 52 44L 1 2r21 201 20 Since there are only four possible marker classes in the backcross design we can easily nd the explicit formulas of fc for each marker class as shown below Marker class 9 Exl m1 m2 m1lm2l m1lm20 m10m2l ml0m2 0 1 H1 V21 r 1 Hr2r na awr H zl r The derivation of fc at m1 l and m2 1 is illustrated below E e ml l mz z e 1 2 14 irz 1 1 W2 1 jnrz M14 M14 M14 newnnm 7K12n1ZGXIZVHz 7K1ZQlZHX12r K1 2n1 2a L 2ex1 2r 1 2nx1 2rn Kl zn1 2anl 1 2rn 2rK12n126H 0912 12V112V2l 11rrlr21rlr2 1V11V21V Note that r1 and r2 determine the position of the QTL within the interval bracketed by m1 and m2 Since fc is a function of the position of the QTL we can treat every point of the chromosome as a putative position of the QTL and evaluate the test statistic at every position We then have a test statistic pro le the plot of the test statistic against the position of the chromosome see Fig 261 The position that shows the maximum value of the test statistic is the estimated position of the QTL if the test statistic is greater than a predetermined threshold value Extension to F2 design We now extend the method into an F2 population The mating design is schematically illustrated below F2 design P1 A1141 gtlt P2 l Fl A1142 l 9 F2 MA MA MA Value at d a The genetic model is y j u g j e j where the genotypic value is de ned as a for g j d for 141A1 a for 239 The statistical regression model is y b0 xljb1 xzjb2 8 where b0 u is the intercept bl a is the additive effect bl d is the dominance effect of the QTL 1 for x1 0 for A1142 1 for 1411A1 and 0 for x2 1 for 1411A1 0 for The data will be x1x2yj for 1N Using standard multiple regression analysis we obtain the estimated regression coefficients and their variancecovariance matrix 13 and v Varb V WEEQ 3521 b2 quot C ovb1 b2 Varb2 Because there are two effects a and d we can propose three null hypotheses H1 b1 0test for additive effect H2 b2 0test for dominance effect H3 b1 b2 0test for overall QTL effect The tests for individual effects are conducted as usual with a test statistic of for k 1 2 Under the null hypothesis ie when Hk is true F k approximately follows a Xim distribution Therefore if Ii S 3841 Hk will be accepted To test the overall QTL effect we use F3 ffv 1 Under the null hypothesis ie when H3 is true F3 will approximately follow a xim distribution Because xfmiu x20 95 5991 for df 2 and at 005 H3 will be accepted if E S 5991 2310 When the QTL genotype is not known and it is inferred from anking markers the independent variables in the statistical regression model are replaced by their conditional expectations given marker information ie y b0JE1jb1JE2jb2 6 where 1 Ex1jlm1jm2j and 2 Ex2jlm1j mzj The estimation and statistical test of QTL effects can be accomplished using conventional multiple regression analysis by treating 1 and 2 as observed values The only additional work left for us is to infer x1 Exljlmlj mzj and x2 Ex2jlm1jm2j from markers Although we can use a linear regression approach to estimate x1 and x as we did in backcross populations it 2 9 is more convenient to take a different approach here Recall that l for 0 for x1 0 for 141A1 and x2 l for 141A1 1 for 141A1 0 for 141A1 We will first infer the conditional probability of QTL genotype given genotype of anking markers The conditional probability denoted by Prx1jlmljmzj pjx1 is a function of r1 and r2 and thus a function of the position of the QTL Given p 1 x1 the conditional expectation of x1 and x2 are obtained by 561 Exljlm1 m2 1Pj10Pj01Pj1 Pj1 Pj1 and 562 EOCZJlmpmz 0117 11Pj00P1 PO In words 9E1 takes the difference between the probabilities of the two homozygotes All1 A2A2 and 2 takes the probability of the heterozygote A1142 2311 We now describe the calculation of Prx1jl mljmzj pjx1 There are three possible genotypes for each marker leading to nine possible marker classes for anking markers Under each marker class there are three possible genotypes of QTL So the total number of conditional probabilities to be derived is 33 27 The derivations of these formulas are achieved using the Bayesian theorem 2 PFOCIJWFWU ijlxlj Prx1Prmljlx1Prmzjlx11 z PrxPrmm2x z Prx1PrmUlx1Prijlx1 PrXIl m1 M2 where Prx1j stands for the prior unconditional probability of QTL genotype and PrmU l x11 denotes the transition probability from a genotype of the QTL to a genotype of the ith marker for 139 12 The prior probabilities are Prx1j l Prx1j ll4 and Prx1j 0 12 These transition probabilities are arranged in an array of matrix PrmU llx1j 1 PrmU lelj 1 PrmU llx1j 1 PrMlX PrmU llx1j 0 PrmU 0 x1 0 PrmU ll x1 0 PrmU llx1j l PrmU 0 x1 l PrmU llx1j 1 002 2410 n2 r1r lr2r2 r1r r2 ladr 002 This can be illustrated by using the following example Prx l Pr m lx l Pr m 0x l W1 l mljz ym2120 1 1 l1 2 l1 ZkPrOcl kPrm1 llx1 kPrm2 le1 k for kl 0 1 2312 First we need to calculate Prx1j lPrmlj llx1j lPrm2j 0lxlj ll r12r2l r2 PFOCU 0Prm1 1le 0Prmzj lelj 0V11 V1V22 1 V22 Prx1j lPrmlj llx1j lPrm2j lelj lr12r2l r2 Therefore the denominator is 1r12r21r2r11r1rf1n2nza1n 7r21r2rf1n2n1n1r22 r21rzrfV21G1H2n1n62n1n1G2 WZMGVJMUH1n1nn1nn10 r11r2r21r1r1r2 1r11r2 r 14 rl V which leads to i l 2 l l 2 l Wl 2W1 21ij 0 2 r1 r2 r2 r1 r2 r2 rl r rl r Formulas for all the 27 probabilities are listed in the following table 2313 Table l Conditional probability of QTL genotype given anking marker genotypes pjx1 Prx1lm1am2 QTL genotype Marker genotype p 1 p 0 p 1 m1 lmzj l glrl lrzgz Zrlrz lrl lrz r1er2 13902 13902 13902 m1 1 m2 0 r2139r12139r2 r1139f1l139f22fzzl fizfz l39fz rlr rlr rlr m1 lmzj l rzzglrlgz Zrlrz lrl lrz rlzglrzgz 12 12 r2 m1 Oamzj 1 r1139r1139r22 r2139r2l139f12fizl fifzzi l39fil rlr r lr rlr m1 0m2j 0 Zrlrzglrlgglrzg 1r121r22r12r22 2r1r2 lr1 lr2 2 2 2 r 1r r221r12r121r22 12139r 111r2 r2 lrz lr1 2r12 rlglrl lrzgz rlr m1 0m2j l rlrzzglrlg rlr rlr r221 1r1 2 2 m1 lm2j 1 r12 lrz 2 Zrlrz lrl lrz 11 11 r m1 lm2j 0 rlzrzglrzg rlglrlz g1r222r22 rzglrlgzg lrzl rlr rlr rlr m1 1 m2 1 r1er2 Zrlrz lrl lrz glrl g2 lrzg2 102 102 102 2314 Chapter 15 Methods of Arti cial Selection Objective of selection The variable to be improved ie the breeding value of a candidate Criteria of selection Breeding value is not observed and thus cannot be directly selected We must select some variables which can be observed and must be correlated with the objective Methods of selection 1 Individual selection The selection criterion is solely the phenotypic value of the candidate Individual selection is also called mass selection This selection method is preferred when the heritability of the trait is high 2 Family selection The selection criterion is the mean of the family in which the candidate belongs Note that all candidates in the same family have the same criterion Therefore the whole family will be selected or rejected as a unit Families can be fullsib or halfsib families This method can be efficient when the heritability of the trait is low and there is little common environmental variance 3 Sib selection The selection criterion is the mean of fullsibs or halfsibs of the candidate Sib selection differs from family selection in that the candidate does not contribute to the estimate of the mean of sibs When the family size is large this method is very close to the method of family selection It can be efficient when the heritability of the trait is low and there is little common environmental variance 4 Progeny testing The selection criterion is the mean of progenies of the candidate This method is also similar to family selection It can be used when the heritability of the trait is low 5 Within family selection The selection criterion is the deviation of the candidate from the mean of the family in which the candidate belongs Similar to family selection this method can be used when the heritability of the trait is low However it is only used when family members share common environmental effects ie the common environmental variance is high 6 Pedigree selection The selection criterion is determined by the phenotypic values of parents grandparents and so on This method of selection can be used in early stage of development ie an individual can be selected before it is fully grown or even before it is born 7 Combined selection The selection criterion is determined based on all available information including candidate s own phenotype family information pedigrees and so on All information is used but information from each source is properly weighted The method is also called index selection Index selection will be discussed later This is the most efficient method of selection Evaluation of selection methods In individual selection the objective of selection is A the breeding value of a candidate and the selection criterion is P the phenotypic value of the candidate As we know the selection equation is RihaA 6 where h quot wh1ch can be shown to be the correlatlon betweenA and P 6 Ah 61613 61613 61613 61613 6P CovAP CovAAR CovAA i 6 AP Therefore the selection equation is R irAPO39A In general the selection equation can be expressed as RX 139 XVAXG A where 139 X is the selection intensity for a particular selection method A is the breeding value of the candidate the objective of the selection X is the criterion of selection it can be anything that is related to A 039 A is standard deviation of the breeding value the selection objective Derivation of R ixrAxaA ZS ZbAX Sy EH EH R S CovAX COVAX6A 6A M VAX VarX GAGX 6X 6X 7 7 MGA VAXZ39XGA TS Z bAXQ S Y ruampXS Y as R 3 6X Example Family selection The objective of selection is the breeding value of the candidate The selection criterion is the mean of 71 members of the family in which the candidate belongs Note that the 71 members include the candidate itself The response to selection is given in the text as l n lr 1nl n lt where r for fullsib family and r for halfsib family 71 is the number of sibs in the family including the candidate I is the intraclass correlation phenotypic correlation between sibs When common environmental effect is absent t rhz The derivation is shown below Rz39039Ph2 Recall that R 139 XVAXG A where X is the mean phenotypic value of 71 family members ie 1 n X B quot14 where B is the phenotypic value of the ith sib CovAX AX GAGX where 6X2 VarX Var ZPl 11 nizVarZPl VarBZZ CovPlPj 11 11 1 i2n6 2 CoWSIB n 1 n 2n6 nn 1t6 1n 1t6 7 because t and thus CovSIB t6 C0vSIB 6 CovAX CovAZPI 11 1 1 CovAZB ZCovAB 7 11 7 11 1 quot1 ltzm 6 7 11 1 2 n 1r 16A n Therefore 1n Dr 62 i6A 1quot 1t 16th 101 11quot GAJ n 6 M n r n 176 Example Within family selection The objective of selection is the breeding value of the candidate The selection criterion is the deviation of the phenotypic value of the candidate individual from the mean of 71 members of the family to which the candidate belongs Note that the 71 members include the candidate itself The response to selection is given in the text as n l n1 t R i039Ph21 r where r for fullsib family and r for halfsib family 71 is the number of sibs in the family including the candidate I is the intraclass correlation phenotypic correlation between sibs which is I rh2 c where c 039 039 and 039 is the common environmental variance When c 0 t rh2 The derivation is actually based on the assumption of c 0 as shown below Recall that R 139 XVAXG A where X is the deviation of the phenotypic value of the candidate individual 139 from the mean phenotypic value of 71 family members ie XP liP 7111 where B is the phenotypic value of the candidate and P is the phenotypic value of the jth family member CovAX GAGX a VarX VarB 1213 71 F1 VarBVar 2CovBlZPj n H n F1 where VarB 039 VarlZPj1n 1to n F1 n Cov13iZPiz CovBPjCov1313 7 11 7 m r0 VarB 1n 1r0391 0123 n M n n 1rh2 1o 178

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