Lecture 6 Heritability and Field...

Lecture 6 Heritability and Field Design

Lucia Gutierrez lecture notesTucson Winter Institute

1

Selection Response

0µSµ

Y)(selection toresponse R

X)( ldiferentiaselection S

pointn truncatio c

sindividual selected ofprogeny a ofmean

population R.M. from sindividual selected a ofmean

population Mating Random initial theofmean

1

0

∆=∆=

====

µµµ

S

1µ

S

R

2,

,

)(

hS

Rb

SbR

XbY

yx

yx

==

=∆=∆

Heritability: “Expected proportion of selection differential to be achieved as a gain from selection” (Hanson, 1963).

Holland et al. 20102

c

Heritability

REALIZED HERITABILITYThe relation between the observed response to selection and the selection differential.

NARROW SENSE HERITABILITYThe fraction of all trait variation due to variation in breeding values (additive variance).

BROAD SENSE HERITABILITYThe fraction of all trait variation due to variation in genotypic values (genetic variance).

P

A

V

Vh =2

P

G

V

VH =2

S

Rh

ˆ

ˆ2 =

3

Holland et al. 2010

Heritability

FUNCTION OF POPULATION

o They are functions of both genetic and environmental variance, therefore a property of the population.

o Suppose one inbred line with a Mendelian inherited trait. How much is the genetic variance? Will the trait segregate? How much is the h2? Does it mean that the trait is not genetically determined? Since there is no genetic variation within this population, h2=H2=0. However, it does not mean that the trait is not genetically determined.

o Additionally, it is unreliable to predict future response to selection because while conducting selection the population changes.

4

Holland et al. 2010

Heritability

ENVIRONMENTAL VARIANCE

Since h2 is also a function of environmental variance, and decreasing environmental variance increases h2, controlled conditions would be optimal for identifying superior genotypes (predicting breeding values). However, caution should be exercised because GxE is important for many traits and therefore selecting in a non-targeted environment could be detrimental.

5

Holland et al. 2010

Parent-Offspring Regression

( ) ipipooi ezbz +−+= µµ |

deviationse

offspringth i ofparent of phenotypez

slope offspring-parent regression b

mean populationµ

offspringth -i theof phenotypez

i

pi

po|

oi

=

−=

==

=

LW 17

Offspring

Parent

-20

-15

-10

-5

0

5

10

15

20

-20 -15 -10 -5 0 5 10 15 20

6

Parent-Offspring Regression

( ) ( )( )

( )po

z

AAApo bhh

EpEobE |

222

24

122

1

p2

po| 2 ,

2

1,

z

z,zˆ ≅≅++≅=σ

σσσσ

σ

Regression one parent on offspring – no environment correlation among parent and offspring.

( ) ( )( )

( )( ) po

z

AAApo bhhbE |

222

21

24

122

1

P221

P1212

P221

P121

o

p2

po| ,

zz

zz,z

z

z,zˆ ≅=+=++==

σσσ

σσ

σσ

Regression one parent – offspring (one offspring or the mean of multiple offspring).

( ) ( )( ) po

z

AAApo bhhbE |

222

24

122

1

p2

po| 2 ,

2

1

z

z,zˆ ≅≅+≅=σ

σσσ

σ

Regression mid parent on offspring – no environment correlation among parent and offspring.

( ) ( )( ) 0|1:0

222

2212

122

12

S02

1:S0S00|1:0 ,

z

z,zˆSS

z

AADDASS bhhbE ≅≅+++==

σσσσσ

σσ

Regression parent -offspring inbreeding – no environment correlation.

LW 177

Mating Designs

FULL-SIB DESIGN: N full-sib families with n offspring each.

ijiij wfz ++= µ

iancefamily var-ithin w

on)contributi talenvironmen dominance, on,(segregatierror residualw

familyth -i theofeffect f

mean populationµ

familyth -i theof offspringth -j theof phenotypez

ij

i

ij

===

=

SoV df SS MS EMS

Among-families N-1 SSf/df(f)

Within-families n(N-1) SSw/df(w)

( )∑ −=i if zznSS 2

...

∑ −=ji iijw zzSS

,

2. )( 2

)(FSwσ

22)( fFSw nσσ +

LW 188

Mating Designs


ijiij wfz ++= µ

( )( )( ) ( ) ( ) ( )2

, ,, ,

,

,)(

f

ijijiijijiii

ijiiji

ijij

wwfwwfff

wfwf

zzFSCov

σ

σσσσµµσ

σ

=

+++=

++++=

=

224

122

12EcDAf σσσσ ++=

( )( )

2224

322

1

224

122

1222

224

122

122)(

EcEDA

EcDAEDA

EcDAPFSw

σσσσσσσσσσ

σσσσσ

+++=

++−++=

++−=

2)(

22FSwfP σσσ +=

LW 189

Mating Designs


( )( )( ) ( ) ( )wfz

w

f wf

VarVarVar

MSVarn

MSMSVar

w

+==

−=

( )( )

( )[ ] [ ])1(211)1(2)(

2

2

1

Var

Var

2

2

2

224

12

2

224

122

1

−−+−≅

≅

++=++==

nNntnthSE

th

hz

ft

FSFS

FS

z

EcD

z

EcDAFS σ

σσσ

σσσ

ijiij wfz ++= µ

SoV df SS MS EMS




...


,

2. )( 2

)(FSwσ

22)( fFSw nσσ +

LW 1810

Mating Designs

HALF-SIB DESIGN: N half-sib families with n offspring each.

ijiij wfz ++= µ




mean populationµ


ij

i

ij

===

=

SoV df SS MS EMS




...


,

2. )( 2

)(FSwσ

22)( fFSw nσσ +

LW 1811

Mating Designs

HALF-SIB DESIGN: N half-sib families with n offspring each.

ijiij wfz ++= µ




mean populationµ


ij

i

ij

===

=

SoV df SS MS EMS




...


,

2. )( 2

)(FSwσ

22)( fFSw nσσ +

( )( )( ) ( ) ( )wfz

w

f wf

VarVarVar

MSVarn

MSMSVar

w

+==

−= ( )

( )HS

z

AHS

th

hz

ft

4

4

1

Var

Var

2

22

24

1

≅

===σσ

LW 1812

Mating Designs

NORTH CAROLINA DESIGN I: Each male (N sire) is mated to several unrelated females (M dams) to produce n offspring per dam.

ijkijiijk wdsz +++= )(µ

)deviations iancefamily var-(withinerror residualw

sireth -i the tomated damth -j theofeffect d

sireth -i theofeffect s

mean populationµ

damth -j and sireth -i theoffamily thefrom offspringth -k theof phenotypez

ijk

ij

i

ijk

=

===

=

LW 1813

Mating Designs

NORTH CAROLINA DESIGN I: Each male (N sire) is mated to several unrelated females (M dams) to produce n offspring per dam.

ijkijiijk wdsz +++= )(µ

SoV df SS MS EMS

Sires N-1 MSs/df(s)

Dams(Sire) N(M-1) MSd/df(d)

Sibs(dams) T-NM MSw/df(w)

( )∑ −=ji is zzMnSS

,

2...

∑ −=ji iijd zzSS

,

2. )(

222sdw Mnn σσσ ++

( )

( )( )( ) ( ) ( ) ( )wdsz

w

d

s

Wd

dS

VarVarVarVar

MSVarn

MSMSVar

Mn

MSMSVar

w

++==

−=

−=( )( )

( ) ( )( )

PHS

z

EcD

z

EcDAFS

z

APHS

th

hz

dst

hz

st

4

2

1

Var

VarVar

4

1

Var

Var

2

2

224

12

2

224

122

1

22

24

1

≅

++=++=+=

===

σσσ

σσσσ

σσ

∑ −=kji ijijkw zzSS

,,

2. )(

22dw nσσ +

2wσ

LW 1814

Mating Designs

NORTH CAROLINA DESIGN II: A group of sires (NS sires) are mated to an independent group of dams (ND dams) to produce n offspring

.

ijkijjiijk wIdsz ++++= µ


damth -j theand sireth -i ebetween thn interactio theofeffect Iij

damth -j theofeffect d

sireth -i theofeffect s

mean populationµ

damth -j and sireth -i theoffamily thefrom offspringth -k theof phenotypez

ijk

i

i

ijk

=====

=

LW 2015

Mating Designs

NORTH CAROLINA DESIGN II: A group of sires (NS sires) are mated to an independent group of dams (ND dams) to produce n offspring

.ijkijjiijk wIdsz ++++= µ

SoV df SS EMS

Sires Ns-1

Dams

Interaction

Nd-1

(Ns-1)(Nd-1)

Sibs NsNd(n-1)

∑ −−−=ji jiijI zzzzSS

,

2........ )(

222sdIw nNn σσσ ++

2

24

1

z

APHSt

σσ=

∑ −=kji ijijkw zzSS

,,

2. )(

22Iw nσσ +

2wσ

( )∑ −=j jsd zznNSS 2

.....

( )∑ −=i ids zznNSS 2

.....

222dsIw nNn σσσ ++

2

2224

1

z

EcGmAMHSt

σσσσ ++=

2

24

1

z

DIt σ

σ=

LW 2016

Full Diallele (all selfed and reciprocal crosses are made)Incomplete Diallele – no selfed crosses Incomplete Diallele – no selfed, no reciprocal crosses

Mating Designs

DIALLELS: A group of individuals (N) are mated to the same set of individuals (N) to produce n offspring

ijkijjiijk wsggz ++++= µ


th-j andth -i parents ofability combining specific s

th-jparent ofability combining general g

th-iparent ofability combining general g

mean populationµ

damth -j and parentsth -j andth -i thefrom offspringth -k theof phenotypez

ijk

ij

j

i

ijk

=

=

===

=

LW 2017

Mating Designs

DIALLELS: A group of individuals (N) are mated to the same set of individuals (N) to produce n offspring. Analysis for incomplete diallelewithout selfed or reciprocal crosses.

ijkijjiijk wsggz ++++= µ

SoV df SS EMS

GCA N-1

SCA N(N-3)/2

Sibs (n-1)[N(N-1)/2-1]

( ) 222 2 GCASGAw Nnn σσσ −++

2

24

1

z

AGCAt

σσ=

∑ <−=

kji ijijkw zzSS,

2. )(

2wσ

( ) GCAji ijSCA SSzznSS −−= ∑ <2

....

( ) ( )∑ −−−=

i iGCA zzN

NnSS 2

.....

2

2

1

22SCAw nσσ +

2

24

1

z

DSCAt

σσ=

LW 2018

Genotypic Means

GENOTYPIC MEANS:

The environment includes non-genetic factors that affect the phenotype, and usually has a large influence on quantitative traits. o Micro-environment. Environment of a single plant. Need to be

controlled with experimental design.o Maco-environment. Environment associated to a location and

time. GxE is the norm and not the exception in plants. Therefore defining the target environments is a crucial part in plant breeding, both for variance component estimation and identifying superior genotypes.

ijklijklijjiijkl pGEEGz ε++++=

B819

Micro -environmental variation

HERITABILITY: For related individuals, heritability can be calculated as previously shown. The previous calculation assumes individual plants are measured, and heritability on an individual plant basis is estimated. However, because quantitative traits measured on individual plants have large non-genetic effects, heritability on a mean basis is higher.

22222weFEFP σσσσσ +++=

Individual plant basis, n=1, r=1, e=1

2222

2

2

2

2

ˆˆˆˆ

ˆ1

4

ˆ

ˆ1

4

ˆweFEF

FP

p

FP

F

FFh

σσσσ

σ

σ

σ

+++

+=

+=

Plot basis, n=n, r=1, e=1

n

hw

eFEF

F

p

FF 2

222

2

2

22

ˆˆˆˆ

ˆ

ˆ

ˆ

σσσσ

σσσ

+++==

nw

eFEFP

22222 σσσσσ +++=

Plot basis, multiple env, n=n, r=r, e=e

ernere

hweFE

F

F

p

FF 222

2

2

2

22

ˆˆˆˆ

ˆ

ˆ

ˆ

σσσσ

σσσ

+++==

ernereweFE

FP

22222 σσσσσ +++=

Holland et al., 2010, B620

Experimental Design and Analysis

So we need good experimental designs to test genotypes!

FISHER’S PRINCIPLES. Statistical theory for efficient estimation (i.e. unbiased and of minimum variance) of treatment mean differences are based on 3 principles:

– Randomization , random assignment of treatments to experimental units provides valid estimation of experimental error, unbiased comparisons of treatments, and justifies statistical inference.

– Replication allows estimation of experimental error variance, and decreases mean variances (i.e. variance of a mean = variance of the observation divided by the number of replications).

– Local control is the grouping of homogenous experimental units. Well chosen blocks will decrease error variance.

21

Experimental Design and AnalysisCLASSIFICATION OF EXPERIMENTAL DESIGNS: 1. Experimental Units.

1. Homogenous –Complete Randomized Design (CRD)2. Heterogenous in one way – Randomized Complete Block Design (RCBD)3. Heterogenous in more than one way – Latin squares or latinized designs.

2. Large number of treatments.1. Incomplete Block Designs (IBD or Alpha)2. Unreplicated experiments (or Federer)

3. Modeling (post-blocking, spatial analysis)

CRD RCBD IBD

13

12

243

4 14

42

213

3 14

42

213

3

ijiijy εαµ ++= ijjiijy εβαµ +++= 22ijkjkjiijky εγβαµ ++++= )(

Complete Randomized Design (CRD)

TREATMENT ASSIGNMENT: Each treatment is assigned atrandom to the experimental units.� Experimental unit: one plot.

RICE EXAMPLE. YIELD COMPARISON OF 4 RICE CULTIVARSTreatments: 4 cultivars

Replications: 4 homogenous experimental plots per treatment

Experimental design: CRD

Dependent variable : Y = Grain yield (Kg ha-1)

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

33

33

2 22

2

11

11

444

4

EXPERIMENTAL PRINCIPLES1. Randomization2. Replication3. Local control

23


ijiijY εαµ ++=

(residual)error alexperimentε

ntth treatme-i theofeffect α

mean populationµ

repth -j on thent th treatme-i theof responseY

ij

i

ij

==

=

=

ASSUMPTIONS:

1. TO THE MODEL:• Is correct (in relation to the experimental units)• Is additive

2. EXPERIMENTAL ERRORS:• Are random variables• εij ~ N• E(εij) = 0 for every i, j• V(εij) = σ2 for every i, j• Are independent

3. “BY DEFINITION” ααααi = µi - µ

24


T1 T2 T3 T4

47 50 57 54

52 54 53 65

62 67 69 74

51 57 57 59

53 57 59 63

SoV df SS MS F p

Geno (t-1) SST/gl(T) MST/MsE

Error t(r-1) SSE/gl(E)

Total tr-1 SSTOT/gl(TOT)

2... )(∑ −=

iiT yyrSS

∑ ∑= =

−=t

i

r

jiijE yySS

1 1

2. )(

2.. )(∑ −=

ijijTOT yySS

SoV df SS MS F p

Geno 3 208 69.3 1.29 0.323

Error 3 646 59.8

Total 15 854

58

25

Randomized Complete Block Design (RCBD)

TREATMENT ASSIGNMENT: Within a block, each treatment isassigned randomly to the experiemntal units within a block.Randomization in each block is independent.� Experimental unit: one plot.

WHEAT EXAMPLE. PLANT HEIGHT OF 5 ADVANCED LINES OF WHEAT IN 5 BLOCKS

Treatment: 5 advanced wheat linesBlock: 5 diferent blocksExperimental design: RCBDDependent variable: Y = cm

3 3

3

3

2 22

2 11

11

4

44

4

3

2

14

55

5

5

5B 1 B 2 B 3 B 4 B 5

EXPERIMENTAL PRINCIPLES1. Randomization2. Replication3. Local control

26


ijjiijY εβαµ +++=

ASSUMPTIONS:

1. TO THE MODEL:• Is correct (in relation to the experimental units)• Is additive• There is NO treatment by block interaction

2. EXPERIMENTAL ERRORS:• Are random variables• εij ~ N• E(εij) = 0 for every i, j• V(εij) = σ2 for every i, j• Are independent

3. “BY DEFINITION” ααααi = µi - µ


blockth -j theofeffect


mean populationµ

blockth -j on thent th treatme-i theof responseY

ij

j

i

ij

=

==

=

=

β

27


T1 T2 T3 T4 T5

B1 8 10 8 9 10 9

B2 7 9 8 8 9 8.2

B3 6 8 9 8 8 7.8

B4 6 7 9 8 7 7.4

B5 7 9 10 7 9 8.4

6.8 8.6 8.8 8.0 8.6 8.16

SoV df SS MS F p

Block (r-1) SSB/gl(B)

Geno (t-1) SST/gl(T) MST/MSE

Error (t-1)(r-1) SSE/gl(E)

Total tr-1

2... )(∑ −=

iiT yyrSS

∑ ∑= =

−−−=t

i

r

jjiijE yyyySS

1 1

2.... )(

2.. )(∑ −=

ijijTOT yySS

SoV df SS MS F p

Block 4 7.36 1.84 2.77 0.0637

Geno 4 13.36 3.34 5.02 0.0081

Error 16 10.64 0.665

Total 24 31.36

2... )(∑ −=

jjB yytSS

28

Number of treatments

HOW TO DEAL WITH HIGH NUMBER OF TREATMENTS?

1. STRATIFICATION: Group genotypes with similar characteristics(maturity, color, family), compare within groups. NO BETWEEN GROUP COMPARISONS.

2. PRODUCE HOMOGENOUS EXPERIMENTAL UNITS: Make everyeffort to homogenize experimental area (look for soil similarity, fieldconditions to reduce variation, choose seeds of similar vigor).

3. USE REPEATED CHECKS: You may use checks in a systematicway to control or model soil heterogeneity.

4. EXPERIMENTAL DESIGN WITH SPATIAL CONSIDERATIONS. Use experimental designs that include a large number of treatmentswhile controling variability (i.e. alpha designs, unrep, etc.).

29

Incomplete Block Designs (IBD)

IBD CLASIFICATION.1. Based on their balance.

A. Balanced: each treatment is compared with another one the samenumber of times in an incomplete block (usually once). A large numberof reps are needed.

B. Partially unbalanced: Not all pair of treatments are compared inan incomplete block the same number of times.Precision for mean comparisons is differentdepending on the pair. Statistical analysis morecomplex.

BI 1 BI 2 BI 3 BI 4 BI 5 BI 6 BI 7 BI 8 BI 9 BI 10 BI 11 BI 12

1 4 7 1 2 3 1 2 3 1 2 3

2 5 8 4 5 6 5 6 4 6 4 5

3 6 9 7 8 9 9 7 8 8 9 7

BI 1 BI 2 BI 3 BI 4 BI 5 BI 6

1 4 7 1 2 3

2 5 8 4 5 6

3 6 9 7 8 930


2. Based on their structure.A. Resolubles: incomplete blocks could be grouped in complete

replications.

B. Non resolubles: incomplete blocks cannot be grouped in completereplications.

BI 1 BI 2 BI 3 BI 4 BI 5 BI 6 BI 7 BI 8 BI 9 BI 10 BI 11 BI 12

1 4 7 1 2 3 1 2 3 1 2 3

2 5 8 4 5 6 5 6 4 6 4 5

3 6 9 7 8 9 9 7 8 8 9 7

BI 1 BI 2 BI 4 BI 5 BI 7 BI 8 BI 10 BI 11

1 4 1 2 1 2 1 2

2 5 4 5 5 6 6 4

3 6 7 8 9 7 8 9 31

Incomplete Block Designs (IBD)SUNFLOWER EXAMPLE. OIL YIELD COMPARISON OF 20 ADVANCED INBRED LINES.

Treatment: 20 sunflower ILExperimental design: IBDResoluble with r=3 and s=5Dependent variable : Y = L ha-1

Incomplete block1 2 3 4 51 2 3 4 5

R1 6 7 8 9 10 11 12 13 14 1516 17 18 19 20

1 2 3 4 5R2 7 8 9 10 6

13 14 15 11 1219 20 16 17 18

1 2 3 4 5R3 8 9 10 6 7

15 11 12 13 1417 18 19 20 16

TREATMENT ASSIGNMENT:Each treatment is assigned randomly to the experimental units in the first rep. In the following reps, restrictions in the randomization are conducted such that each pair of treatment is compared the same number of times within an incomplete block.

32


ijkjkjiijkY εγβαµ ++++= )(


repth -j in theblock with incompleteth -k theofeffect

repth -j theofeffect


mean populationµ

block incompleteth -k theand repth -j on thent th treatme-i theof responseY

ijk

k(j)

j

i

ijk

=

=

==

=

=

γβ

33

Field Designs and Heritability

ijkijkijkjijk PEPEy εβµ +++++= )(

SoV df EMS

Environment (e-1)

Block(Env) e(r-1)

Progeny (n-1)

ProgenyXEnv (n-1)(e-1)

Pooled error (n-1)(r-1)e

ProgenyPEεProgeny reVrVVMS ++=

FACTORIAL DESIGNS (HS, FS, RIL, DH, Clones)

PEεPE rVVMS +=

εError VMS =

( )D

2

AProgeny V4

1V

2

1V:FS

FF +++=

AProgeny V4

1V:HS

F+=

AProgeny V2V:RIL =ernere

hweFE

F

F

p

FF 222

2

2

2

22

ˆˆˆˆ

ˆ

ˆ

ˆ

σσσσ

σσσ

+++==

B634

Other Considerations (Design )

1. PLOT SIZE:a. Big enough to have plants growing normally (i.e. one-plant does

not work in crops but works fine for trees).b. Big enough to represent trait variation (i.e. you might need larger

plots for yield than for maturity).c. Not too big as to have considerable within-plot variation (i.e.

increasing plot size increases within-plot variation).d. Balance between cost of increasing plot size and increasing

number of plots.

EXAMPLE. Mohammad et al. (2001) compared plot size and number of replications to detect differences in wheat. Bigger plots require less reps.

r=2

r=3

r=4

r=5

Difference to be detected

Plo

t siz

e

35


2. PLOT SHAPE: a. Balance between plot border and plants within plot (i.e.

rectangular plots have more border than squared ones).b. Take into account gradients (i.e. if unidirectional gradient like

fertility long and narrow plots might be better).

EXAMPLE. Haapanen(1992) compared plot size and shape for height in pines

36


3. ROW vs. HILL PLOTS:

EXAMPLE. Tragoonrung et al., 1990 compared row and hill-plots for different traits in barley. Hill-plots are ok for most traits but not for yield.

37


4. BORDER ROW. It is a good idea to include a row of plants as a border row to avoid having plots in the margins of the experiment performing differently due to lack of competition.

5. OPERATORS NOISE. If measurements on the experiment will be performed by different people it is a good idea to consider operators noise by having different people measuring for example in different blocks.

6. EARLY TESTING. Small amount of seeds are available in early generations. Therefore replicating is a challenge. Taking this into consideration when deciding which traits will be measured early is important.

7. OTHER DESIGNS AND SPATIAL CORRELATIONS. More efficient designs exist for testing large numbers of genotypes in fields that might not be completely heterogeneous. Additionally, spatial corrections might improve estimations of genotypic means. 38

Other Considerations (Estimation )

1. MIXED MODELS. Careful interpretation of Expected Means Squares should be exercised especially when using Mixed Models. EMS is different if factors (i.e. environments, genotypes, etc. ) are defined as random or fixed effects:

a. E, G, GxE random:

within env:

a. G, GxE random, E fixed:

across env.:

222)( GIeG nNnEMS σσσ ++=

22)( GeG nNEMS σσ +=

( )222

222

2 IGe

IG

Fh

σσσσσ

+++=

( )2222

22

2 EIGe

G

Fh

σσσσσ

+++=

LW 2239

Other Considerations (Estimation )

2. NON-BALANCED DESIGNS . When designs are not balanced due to their design, missing plots or missing data, MS calculations are not straight-forward. There are four ways to estimate MS.

MIXED MODELS - INCOMPLETE BLOCKS – UNBALANCED

With unbalanced data, mixed models, or correlation among genotypes, variance component estimation of heritability is not related to the gain from selection. Using the concept of “effective error variance” (Cochran and Cox, 1957), Cullis et al (2006) proposed to use a generalized method for heritability:

22

21

G

BLUPhσ

υ−=

Piepho and Mohring 2007

BLUP twoof difference a of ncemean varia:BLUPυ

40

Lecture 6 Heritability and Field...

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