Estimation of Basic Genetic Parameters

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1 Lecture 7 Estimatio of Basic Geetic Parameters Guilherme J. M. Rosa Uiversity of Wiscosi-Madiso Itroductio to Quatitative Geetics SISG, Seattle July 018 Estimatio of Basic Geetic Parameters 1

2 arrow vs. broad sese Heritability arrow sese: h = V A /V P Slope of midparet - offsprig regressio (sexual reproductio) Broad sese: H = V G /V P Slope of a paret - cloed offsprig regressio (asexual reproductio) Whe oe refers to heritability, the default is arrow-sese, h h is the measure of (easily) usable geetic variatio uder sexual reproductio

3 Why h istead of h? Blame Sewall Wright, who used h to deote the correlatio betwee pheotype ad breedig value. Hece, h is the total fractio of pheotypic variace due to breedig values r(a,p) = σ (A,P) = σ A = σ A = h σ A σ P σ A σ P σ P Heritabilities are fuctios of populatios Heritability values oly make sese i the cotet of the populatio for which it was measured Heritability measures the stadig geetic variatio of a populatio A zero heritability DOES OT imply that the trait is ot geetically determied Heritabilities are fuctios of the distributio of evirometal values (i.e., the uiverse of E values) Decreasig V P icreases h. Heritability values measured i oe eviromet (or distributio of eviromets) may ot be valid uder aother Measures of heritability for lab-reared idividuals may be very differet from heritability i ature 3

4 Heritability ad the Predictio of Breedig Values If P deotes a idividual s pheotype, the best liear predictor of their breedig value A is A = σ (P, A) σ P (P µ P ) + e = h (P µ P ) + e The residual variace is also a fuctio of h : σ e = (1 h )σ P The larger the heritability, the tighter the distributio of true breedig values aroud the value h (P - µ P ) predicted by a idividual s pheotype. Heritability ad Populatio Divergece Heritability is a completely ureliable predictor of log-term respose Measurig heritability values i two populatios that show a differece i their meas provides o iformatio o whether the uderlyig differece is geetic 4

5 Sample Heritabilities People Height 0.80 Serum IG 0.45 Pigs Back-fat 0.70 Weight gai 0.30 Litter size 0.05 Fruit Flies Abdomial Bristles 0.50 Body size 0.40 Ovary size 0.30 Egg productio 0.0 h Traits more closely associated with fitess ted to have lower heritabilities 5

6 Expected Value ad Variace Expected Value (Mea) otatio: E[X] = µ X Discrete radom variable, fiite case: k E[X] = x i p i p i = Pr[X = x i ], where (weighted average) If p 1 = p = = p k =1/ k the: E[X] = 1 k k x i (simple average) 6

7 Expected Value Discrete radom variable, coutable case: E[X] = x i p i ad E[g(X)] = g(x i )p i Cotiuous radom variable: E[X] = xf(x)dx ad E[g(X)] = g(x)f(x)dx where f(x) : probability desity fuctio Properties: Expected Value Costat c: E[c] = c E[cX] = ce[x] E[X + Y] = E[X]+ E[Y] E[X Y = y] = xpr(x = x Y = y) E[X] = E[E[X Y]] 7

8 Variace otatio: Var[X] = σ X Defiitio: expected value of the square deviatio from the mea, i.e. Var[X] = E[(X µ) ] Var[X] = E[(X E[X]) ] = E[X XE[X]+ (E[X]) ] = E[X ] E[X]E[X]+ (E[X]) = E[X ] (E[X]) = E[X ] µ Variace Discrete radom variable: Var[X] = (x i µ) p i = x i p i µ Cotiuous radom variable: Var[X] = (x µ) f(x)dx = x f(x)dx µ 8

9 Variace Properties: Costat c: Var[c] = 0 Var[c + X] = Var[X] Var[cX] = c Var[X] Var[X + Y] = Var[X]+ Var[Y]+ Cov[X, Y] Var[X Y] = Var[X]+ Var[Y] Cov[X, Y] Var[X] = E Y [Var[X Y]]+ Var Y [E[X Y]] Covariace otatio: Cov[X, Y] = σ X,Y Cov[X, Y] = E[(X µ X )(Y µ Y )] = E[XY] µ X µ Y Correlatio otatio: Corr[X, Y] = ρ X,Y ρ X,Y = Cov[X, Y] σ X σ Y 9

10 AOVA: Aalysis of Variace Partitioig of trait variace ito withi- ad amoggroup compoets Two key AOVA idetities Total variace = betwee-group variace + withigroup variace Var(T) = Var(B) + Var(W) Variace(betwee groups) = covariace (withi groups) Itraclass correlatio, t = Var(B)/Var(T) The more similar idividuals are withi a group (higher withi-group covariace), the larger their betweegroup differeces (variace i the group meas) Situatio 1 Situatio Var(B) =.5 Var(W) = 0. Var(T) =.7 Var(B) = 0 t =.5/.7 = 0.93 Var(W) =.7 t = 0 Var(T) =.7 10

11 Pheotypic Resemblace Betwee Relatives Relatives Covariace Regressio (b) or correlatio (t) Offsprig ad oe paret Offsprig ad mid-paret Half sibs Full sibs 1 V A 1 V A 1 4 V A 1 V A V D +V E c t = b = 1 V A V P b = V A V P t = 1 V A 4 V P 1 V + 1 A 4 V +V D E c V P Why cov(withi) = variace(amog)? Let z ij deote the jth member of group i. Here z ij = u + g i + e ij g i is the group effect e ij the residual error Covariace withi a group Cov(z ij,z ik ) = Cov(u + g i + e ij, u + g i + e ik ) = Cov(g i, g i ) as all other terms are ucorrelated Cov(g i, g i ) = Var(g) is the amog-group variace 11

12 Estimatio: Oe-way AOVA Simple (balaced) full-sib desig: full-sib families, each with offsprig: Oe-way AOVA model Trait value i sib j from family i Commo mea Deviatio of sib j from the family mea z ij = m + f i + w ij Effect for family i; deviatio of mea of i from the commo mea Covariace betwee members of the same group equals the variace amog (betwee) groups Cov(Full Sibs) = σ (z ij,z ik ) = σ [(µ + f i + w ij ),(µ + f i + w ik )] = σ ( f i, f i ) +σ ( f i,w ik ) +σ (w ij, f i ) +σ (w ij,w ik ) = σ f Hece, the variace amog family effects equals the covariace betwee full sibs σ f = σ A / +σ D / 4 +σ Ec 1

13 The withi-family variace σ w = σ P - σ f, σ w(fs) = σ P (σ A / +σ D / 4 +σ Ec ) = σ A +σ D +σ E (σ A / +σ D / 4 +σ Ec ) = (1 / )σ A + (3 / 4)σ D +σ E σ Ec Oe-way AOVA: families with sibs, T = Factor Degrees of freedom, df Sum of squares (SS) Mea squares (MS) E[MS] Amog family -1 SS f = (z i z) SS f /(-1) σ w + σ f Withi family T- SS SS w /(T-) σ w = (z ij z i ) w j=1 13

14 Appedix: Calculatig E(MS) Model: z ij = m + f i + w ij with m fixed E[m] = m, E[m ] = m, Var[m] = 0 f iid i ~ (0,σ f ) E[f i ] = 0, E[f i ] = Var[f i ] = σ f w iid ij ~ (0,σ w ) E[w ij ] = 0, E[w ij ] = Var[w ij ] = σ w Cov[f i, f i' ] = Cov[f i, w ij ] = Cov[w ij, w i' j' ] = 0 Sum of Squares: SS f = (z i z) = 1 z i 1 T z SS w = (z ij z i ) = z ij 1 j=1 j=1 z i z i = z = j=1 z ij j=1 z ij Key Expectatios: E z ij j=1, E 1 T z, ad E 1 z i E z ij j=1 = E z ij = E m + f i + w ij j=1 j=1 = E m + f i + w ij + mf i + mw ij + f i w ij j=1 ( ) = m + E[f i ]+ E[w ij ]+ me[f i ]+ me[w ij ]+ E[f i ]E[w ij ] j=1 ( ) = m + σ t + σ w j=1 = Tm + Tσ t + Tσ w 14

15 E 1 T z = 1 T E j=1 = 1 T E (m + f i + w ij ) j=1 z ij = 1 T E Tm + f i + j=1 w ij = 1 T E T m + f i + w ij + DPs j=1 = 1 T (T m + σ f + Tσ w + 0) = Tm + σ f + σ w E 1 z i = 1 = 1 = 1 = 1 E[z i ] = 1 E j=1 z ij E m + f i + z ij j=1 E m + f i + z ij + DPs j=1 ( m + σ f + σ w + 0) = Tm + Tσ f + σ w 15

16 Expected MS E[MS f ] = 1 1 E[SS ] = 1 f 1 E 1 E[MS w ] = z i 1 T z = 1 1 (Tm + Tσ f + σ w) (Tm + σ f + σ w ) = 1 1 ( 1)σ f + ( 1)σ w = σ f + σ w 1 T E[SS ] = 1 w T 1 E j=1 z ij 1 z i 1 = T E (Tm + Tσ t + Tσ w ) (Tm + Tσ f + σ w) 1 = T (T )σ w = σ w Estimatig the variace compoets: Var( f ) = MS f MS w Var(w) = MS w Var(z) = Var( f ) +Var(w) Sice σ f = σ A / +σ D / 4 +σ Ec Var(f) is a upper boud for the additive variace 16

17 Assigig stadard errors ( = square root of Var) Fu fact: Uder ormality, the (large-sample) variace for a mea-square is give by σ (MS x ) (MS x ) df x + Var[Var(w(FS))] = Var(MS w ) (MS w) Var[Var( f )] = Var MS f MS w T + (MS f ) +1 + (MS w) T + Estimatig heritability t FS = Var( f ) Var(z) = 1 h + σ D / 4 +σ Ec σ z Hece, h t FS A approximate large-sample stadard error for h is give by SE(h ) (1 t FS )[1+ ( 1)t FS ] / [ ( 1)] 17

18 18 Worked Example Factor df SS MS EMS Amog-families 9 SS f = σ w + 5 σ f Withi-families 40 SS w = σ w 10 full-sib families, each with 5 offsprig are measured Var( f ) = MS f MS w = = 5 Var(w) = MS w = 0 Var(z) = Var( f ) +Var(w) = 5 SE(h ) (1 0.4)[1+ (5 1)0.4] / [50(5 1)] = 0.31 V A < 10 h < (5/5) = 0.4 Full sib-half sib desig: ested AOVA 1 3 o o o. o 1 3 o o o. o 1 M 3 o o o. o o o o. o 1 3 o o o. o 1 M 3 o o o. o 1 Full-sibs Half-sibs

19 Estimatio: ested AOVA Balaced full-sib / half-sib desig: males (sires) are crossed to M dams each of which has offsprig: ested AOVA model Value of the kth offsprig from the jth dam for sire i Overall mea z ijk = m + s i + d ij + w ijk Effect of dam j of sire i; deviatio of mea of dam j from sire ad overall mea Effect of sire i; deviatio of mea of i s family from overall mea Withi-family deviatio of kth offsprig from the mea of the ij-th family ested AOVA Model z ijk = m + s i + d ij + w ijk σ s σ d σ w = betwee-sire variace = variace i sire family meas = variace amog dams withi sires = variace of dam meas for the same sire = withi-family variace σ T = σ s + σ d + σ w 19

20 ested AOVA: sires crossed to M dams, each with sibs, T = M Factor df SS MS E[MS] Sires -1 SS s SS s /(-1) σ w + σ d + Mσ s Dams(Sires) (M-1) SS d SS d /[(M-1)] σ w + σ d Sibs(Dams) T-M SS w SS w /(T-M) σ w where: SS s = M (z i z ) M SS d = (z ij z i ) ad SS w = (z ijk z ij ) j=1 j=1 k=1 M Estimatio of sire, dam, ad family variaces: Var(s) = MS s MS d M Var(d) = MS d MS w Var(e) = MS w Traslatig these ito the desired variace compoets: Var(Total) = Var(betwee FS families) + Var(withi FS) σ w = σ z Cov(FS) Var(Sires) = Cov(Pateral half-sibs) σ d = σ z σ s σ w = σ (FS) σ (PHS) 0

21 Summarizig: σ s = σ (PHS) σ d = σ z σ s σ w σ w = σ z σ (FS) = σ (FS) σ (PHS) Expressig these i terms of the geetic ad evirometal variaces: σ s σ A 4 σ d σ A 4 + σ D 4 + σ Ec σ w σ A + 3σ D 4 +σ Es Itraclass correlatios ad estimatig heritability t PHS = Cov(PHS) Var(z) = Var(s) Var(z) 4t PHS = h t FS = Cov(FS) Var(z) = Var(s) +Var(d) Var(z) h t FS ote that 4t PHS = t FS implies o domiace or shared family evirometal effects 1

Worked Example: = 10 sires, M = 3 dams, = 10 sibs/dam Factor df SS MS E[MS] Sires 9 4,30 470 σ w +10σ d + 30σ s Dams(Sires) 0 3,400 170 σ w +10σ d Withi Dams 70 5,400 0 σ w σ w = MS w = 0 σ d = MS d

22 Worked Example: = 10 sires, M = 3 dams, = 10 sibs/dam Factor df SS MS E[MS] Sires 9 4, σ w +10σ d + 30σ s Dams(Sires) 0 3, σ w +10σ d Withi Dams 70 5,400 0 σ w σ w = MS w = 0 σ d = MS d MS w σ s = MS s MS d = = 30 σ P = σ s +σ d +σ w = 45 =15 =10 σ d =15 = (1 / 4)σ A + (1 / 4)σ D +σ Ec =10 + (1 / 4)σ D +σ Ec σ A = 4σ s = 40 h = σ A σ = 40 P 45 = 0.89 σ D + 4σ Ec = 0 Beetle Example Messia ad Fry (003): 4 males each mated to 4 or 5 dams (differet for each sire), ad 5 female progey from each dam were measured for two traits, mass eclosio ad lifetime fecudity AOVA for fecudity Factor df SS MS Sires 3 33,983 1,477.5 Dams(Sires) 86 64, Sibs(Dams) , beetle example

23 Beetle Example Expected Mea Squares (EMS) Sires: σ R + σ D + qσ S Dams(Sires): σ R + σ D Sibs(Dams): σ R Approximately = 5 progey by matig, ad a average of q = 4.58 dams per sire, so: σ R = σ D = ( )/5 = σ S = (1, )/.9 = ote: AOVA method works oly with balaced or slightly ubalaced data sets; otherwise ML or REML should be preferred Beetle Example Estimatio of geetic (causal) parameters: σ S = V A /4 σ D = V A /4 + V D /4 + V Ec σ R = V A / + 3V D /4 + V Es For simplicity, assumig V D = 0, the followig estimates are obtaied for the causal compoets: V A = 4σ S = 17. V Ec = σ D - σ S = 8.56 V Es = σ R σ S = Heritability: h = V A /(σ R + σ D + σ S ) =

24 Paret-offsprig Regressio Sigle paret - offsprig regressio z oi = µ + b o p (z pi µ) + e i The expected slope of this regressio is: E(b o p ) = σ (z,z ) o p σ (z p ) (σ A / ) +σ (E o,e p ) = h σ z + σ (E,E ) o p σ z Residual error variace (spread aroud expected values) σ e = 1 h σ z The expected slope of this regressio is: E(b o p ) = σ (z,z ) o p σ (z p ) (σ A / ) +σ (E o,e p ) = h σ z + σ (E,E ) o p σ z Shared evirometal values To avoid this term, typically regressios are male-offsprig, as female-offsprig more likely to share evirometal values 4

25 Midparet-offsprig regressio: z z oi = µ + b mi + z fi o MP b o MP = Cov[z o,(z m + z f ) / ] Var[(z m + z f ) / ] = [Cov(z,z ) + Cov(z,z )] / o m o f [Var(z) +Var(z)] / 4 = Cov(z o,z p ) = b o p Var(z) µ + e i The expected slope of this regressio is h Residual error variace (spread aroud expected values) σ e = 1 h σ z Stadard Errors Sigle paret-offsprig regressio, parets, each with offsprig Square regressio slope Var(b o p ) (t b p p) + (1 t) Total umber of offsprig Sib correlatio t = t HS = h / 4 t FS = h / + σ D +σ Ec σ z for half-sibs for full-sibs Var(h ) = Var(b o p ) = 4Var(b o p ) 5

26 Midparet-offsprig regressio, sets of parets, each with offsprig Var(h ) = Var(b o MP ) [(t b FS o MP / ) + (1 t FS )] Midparet-offsprig variace half that of sigle paret-offsprig variace Var(h ) = Var(b o p ) = 4Var(b o p ) Estimatig Heritability i atural Populatios Ofte, sibs are reared i a laboratory eviromet, makig paret-offsprig regressios ad sib AOVA problematic for estimatig heritability Let b be the slope of the regressio of the values of lab-raised offsprig regressed i the trait values of their parets i the wild A lower boud ca be placed of heritability usig parets from ature ad their lab-reared offsprig, h mi = (b' o MP ) Var (z) Var l (A) Trait variace i ature Additive variace i lab 6

27 Why is this a lower boud? Covariace betwee breedig value i ature ad BV i lab (b' o MP ) Var (z) Var l (A) = Cov (A) l, Var (z) Var (z) Var l (A) = γ h where γ = Cov l, (A) Var (A)Var l (A) is the additive geetic covariace betwee eviromets ad hece ϒ 1 Defiig H for Plat Populatios Plat breeders ofte do ot measure idividual plats (especially with pure lies), but istead measure a plot or a block of idividuals. This ca result i icosistet measures of H eve for otherwise idetical populatios Geotype i Eviromet j Effect of plot k for geotype i i eviromet j z ijkl = G i + E j + GE ij + p ijk + e ijkl Iteractio betwee geotype i ad eviromet j Deviatios of idividual plats withi plots 7

28 z ijkl = G i + E j + GE ij + p ijk + e ijkl σ (z i ) = σ G +σ E + σ GE e + σ p er + σ e er e = umber of eviromets r = (replicates) umber of plots/eviromet = umber of idividuals per plot Hece, V P, ad hece H, depeds o our choice of e, r, ad 8

A random variable is a variable whose value is a numerical outcome of a random phenomenon.

A random variable is a variable whose value is a numerical outcome of a random phenomenon. The Practice of Statistics, d ed ates, Moore, ad Stares Itroductio We are ofte more iterested i the umber of times a give outcome ca occur tha i the possible outcomes themselves For example, if we toss