Analysis of Genetic Designs

Biometrics and Statistics Unit

Francisco Manuel Rodríguez Huerta ([email protected])

Centro Internacional de Mejoramiento de Maíz y Trigo, Internacional

1 Basic model 2 North Carolina NCI NCII 3 Line by Tester 4 Gring Diallel Methods Models

Basic genetic model.

yik = µ + sk+ gi+ eik

Interaction model.

yik = µ + sk + gi + sk∗ gi + eik

yijk is the observed value

µ is the mean

sk is the non random eect (k = 1, 2, ..., s)

gi is the genotype eect (i = 1, 2, ..., g)

North Carolina I

In this type of design due to Comstock and Robinson (1952), a number m of males are selected from an F2 gen-eration or an advanced gengen-eration by random mating, and each of the se-lected males are crossed with two or more f dierent females. This gives mf full sib (FS) families.

Balanced: the number of crosses is m ∗ f crosses, in the example f = 4 Unbalanced: the number of crosses is lower than m ∗ f , example: no in-formation for cross 1x2

RCBD can be modeled using:

yijk= µ + REPk + gij+ eij

= µ + REPk + mi + fj (i )+ eijk

yijk is the observed value

µ is the mean

REPk is the replicate eect (k = 1, 2, ..., r)

mi is the male eect (i = 1, 2, ..., m)

fj (i ) is the eect of female j in male i (j = 1, 2, ..., f )

North Carolina I

Alpha Lattice can be modeled using:

yijk = µ + REPk + gij + BLK (REPk) + eij

= µ + REPk + mi + fj (i )+ BLK (REPk) + eijk

yijk is the observed value

µ is the mean

REPk the replicate eect (k = 1, 2, ..., r)

mi is the male eect (i = 1, 2, ..., m)

fj (i ) is the eect of female j in male i (j = 1, 2, ..., f )

BLK (REPk) random eect of block nested in replicate k

VC Henderson (balanced) REML (Balanced or unbalanced) σ_m2 (MSm− MSf (m))/rm σˆ2m σ_{f (m)}2 (MSf (m)− MSe)/r σˆ_{f (m)}2 σ_A2 4σ2_m σ2_D 4(σ_{f (m)}2 − σ_m2) σ2_E σe2 r h_b2 σ2A+σD2 σ2_A+σ_D2+σ2_E hn2 σ_A2 σ2_A+σ_D2+σ2_E

North Carolina I

Multi-environment RCBD (interaction model) can be modeled using:

yijkd = µ + Ed+ REPk(Ed) + gij + Ed∗ gij + eijkd

= µ + Ed+ REPk(Ed) + mi+ fj (i )+ Ed∗ mi+ Ed∗ fj (i )+ eijkd

yijkd is the observed value

µ is the mean

Ed environmental eect (d = 1, 2, ..., s)

REPk(Ed)the eect of replicate k nested in environment d

(k = 1, 2, ..., r)

mi is the male eect (i = 1, 2, ..., m)

fj (i ) is the eect of female j in male i (j = 1, 2, ..., f )

Multi-environment Alpha Lattice can be modeled using:

yijkd = µ + Ed+ REPk(Ed) + gij + Ed∗ gij + BLK (REPkEd) + eijkd

= µ + Ed+ REPk(Ed) + mi+ fj (i )+ Ed∗ mi+ Ed∗ fj (i )

+ BLK (REPkEd) + eijkd

yijkd is the observed value

µ is the mean

Ed environmental eect (d = 1, 2, ..., s)

REPk(Ed)the eect of replicate k nested in environment d

(k = 1, 2, ..., r)

mi is the male eect (i = 1, 2, ..., m)

f_{j (i )} is the eect of female j in male i (j = 1, 2, ..., f ) BLK (REPkEn) random eect of block nested in replicate k

nested in environment d eijkd residual.

North Carolina I

VC Henderson (balanced) REML (Balanced or unbalanced)

σ_m2 (MSm− MSf (m))/srm σˆ2m σ2_{f (m)} (MSf (m)− MSe)/sr σˆ2_{f (m)} σ2_A 4σ2_m σ_D2 4(σ_{f (m)}2 − σ_m2) σ2_E σ2gxe s + σ2_e sr h2_b σ2A+σD2 σ2_A+σ_D2+σ2_E h2n σ_A2 σ2_A+σ_D2+σ2_E

In this design (Comstock and Robinson (1952)), a set of males m and a set of females f are selected, say from an F2 population. Each of the males is crossed with every female in the set, and thus mf FS families are generated.

Balanced: the number of crosses is m ∗ f crosses

Unbalanced: the number of crosses is lower than m ∗ f , example: no information for cross 1x2

Male\Female 1 2 . . . f

1 x₁₁ x₁₂ . . . x_1f

2 x₂₁ x₂₂ . . . x_2f

... ... ... ... ...

North Carolina II

RCBD can be modeled using:

yijk = µ + REPk+ gij + eij

= µ + REPk+ mi+ fj + mi ∗ fj + eijk

yijk is the observed value

µ is the mean

REPk the replicate eect (k = 1, 2, ..., r)

mi is the male eect (i = 1, 2, ..., m)

fj is the female eect (j = 1, 2, ..., f )

Alpha Lattica can be modeled using: yijk = µ + REPk+ gij + BLK (REPk)eij

= µ + REPk+ mi + fj + mi∗ fj + BLK (REPk) + eijk

yijk is the observed value

µ is the mean

REPk the replicate eect (k = 1, 2, ..., r)

mi is the male eect (i = 1, 2, ..., m)

fj is the female eect (j = 1, 2, ..., f )

BLK (REPk) random eect of block nested in rep

North Carolina II

VC Henderson (balanced) REML (Balanced or unbalanced)

σ_m2 (MSm− MSmf)/rm σˆ2m σ2_f (MSf − MSmf)/rf σˆ2f σ2_mf (MSmf − MSe)/r σˆ2mf σ2_g (m−1)MSm+(f −1)MSf−(m+f −2)MSmf r (2mf −m−f ) σˆ2g σ_A2 4σ_g2 σ_D2 4σ_mf2 σ2_E σ2_e/r h_b2 σA2+σ2D σ_A2+σ2_D+σ_E2 h_n2 σA2 σ_A2+σ2_D+σ_E2

Multi-environment RCBD can be modeled using:

yijkd = µ + Ed+ REPk(Ed) + gij + Ed∗ gij + eijkd

= µ + Ed+ REPk(Ed) + mi+ fj + mi ∗ fj

+ Ed∗ mi+ Ed∗ fj + Ed ∗ mi ∗ fj + eijkd

yijkd is the observed value

µ is the mean

Ed environmental eect (d = 1, 2, ..., s)

REPk(Ed)the eect of replicate k nested in environment d

(k = 1, 2, ..., r)

mi is the male eect (i = 1, 2, ..., m)

fj is the female eect (j = 1, 2, ..., f )

North Carolina II

Multi-environment Alpha Lattica can be modeled using:

yijkd = µ + Ed+ REPk(Ed) + gij + S ∗ gij + BLK (REPkEd) + eijkd

= µ + Ed+ REPk(Ed) + mi+ fj+ mi∗ fj

+ Ed∗ mi+ Ed ∗ fj + Ed∗ mi ∗ fj + BLK (REPkEd) + eijkd

yijkd is the observed value

µ is the mean

Ed environmental eect (d = 1, 2, ..., s)

REPk(Ed)the eect of replicate k nested in environment d

(k = 1, 2, ..., r)

mi is the male eect (i = 1, 2, ..., m)

fj is the female eect (j = 1, 2, ..., f )

BLK (REPkEd) random eect of block nested in replicate k

nested in environment d eijkd residual.

VC Henderson (balanced) REML (Balanced or unbalanced) σ_m2 (MSm− MSmf)/srm σˆ2m σ2_f (MSf − MSmf)/srf σˆ2f σ2_mf (MSmf − MSe)/sr σˆ2mf σ2_g (m−1)MSm+(f −1)MSf−(m+f −2)MSmf sr (2mf −m−f ) σˆ2g σ_A2 4σ_g2 σ_D2 4σ_mf2 σ2_E σ2gxe s + σ2_e sr h_b2 σA2+σ2D σ_A2+σ2_D+σ_E2 h_n2 σA2 σ_A2+σ2_D+σ_E2

Line by Tester

Kempthorne (1957) proposed a line Ö tester experiment, with similar mating pattern as the North Carolina design II.

Balanced: the number of crosses is l ∗ t crosses, in the example

f =4

Unbalanced: the number of crosses is lower than l ∗ t , example: no information for cross 1x2

Line\Tester 1 2 . . . t

1 x₁₁ x₁₂ . . . x_1t

2 x₂₁ x₂₂ . . . x_2t

... ... ... ... ...

RCBD can be modeled using:

yijk = µ + REPk+ li+ tj+ li∗ tj + eijk

yijk is the observed value

µ is the mean

REPk the replicate eect (k = 1, 2, ..., r)

li is the male eect (i = 1, 2, ..., l)

tj is the female eect (j = 1, 2, ..., t)

Line by Tester

Alpha Latice can be modeled using:

yijk = µ + REPk+ li + tj + li ∗ tj + BLK (REPk) + eijk

yijk is the observed value

µ is the mean

REPk is the replicate eect (k = 1, 2, ..., r)

li is the male eect (i = 1, 2, ..., l)

tj is the female eect (j = 1, 2, ..., t)

BLK (REPk) random eect of block nested in replicate k

VC Henderson (balanced) REML (Balanced or unbalanced) σ_l2 (MSl− MSlt)/rt σˆ2l σ_t2 (MSt− MSlt)/rl σˆ2t σ2_lt (MSlt− MSe)/r σˆlt2 σ_g2 (l −1)MSl+(t−1)MSt−(l+t−2)MSlt r (2lt−l−t) σˆ2g σ2_A 4σ_g2 σ_D2 4σ_lt2 σ2_E σ2e r h2_b σA2+σ2D σ_A2+σ2_D+σ2_E h2_n σ2A σ_A2+σ2_D+σ2_E

Line by Tester

Multi-environment RCBD can be modeled using:

yijkd = µ + Ed+ REPk(Ed) + gij + Ed∗ gij + eijkd

= µ + Ed+ REPk(Ed) + li+ tj+ li∗ tj

+ Ed∗ li+ Ed∗ tj + Ed∗ li ∗ tj + eijkd

yijkd is the observed value

µ is the mean

Ed environmental eect (d = 1, 2, ..., s)

REPk(Ed)the eect of replicate k nested in environment d

(k = 1, 2, ..., r)

li is the male eect (i = 1, 2, ..., l)

tj is the female eect (j = 1, 2, ..., t)

Multi-Environment Alpha Latice can be modeled using:

yijkd = µ + Ed+ REPk(Ed) + gij + Ed∗ gij + BLK (REPkEd) + eijkd

= µ + Ed+ REPk(Ed) + li+ tj + li∗ tj

+ Ed∗ li+ Ed∗ tj + Ed∗ li ∗ tj + BLK (REPkEd) + eijkd,

yijkd is the observed value

µ is the mean

Ed environmental eect (d = 1, 2, ..., s)

REPk(Ed)the eect of replicate k nested in environment d

(k = 1, 2, ..., r)

li is the male eect (i = 1, 2, ..., l)

tj is the female eect (j = 1, 2, ...t)

BLK (REPkEd) random eect of block nested in replicate k

nested in environment d eijkd residual.

Line by Tester

VC Henderson (balanced) REML (Balanced or unbalanced)

σ_l2 (MSl− MSlt)/srt σˆ2l σ_t2 (MSt− MSlt)/srl σˆ2t σ2_lt (MSlt− MSe)/sr σˆlt2 σ_g2 (l −1)MSl+(t−1)MSt−(l+t−2)MSlt sr (2lt−l−t) σˆ2g σ2_A 4σ_g2 σ_D2 4σ_lt2 σ2_E σ2gxe s + σ_e2 sr h2_b σA2+σ2D σ_A2+σ2_D+σ2_E h2_n σ2A σ_A2+σ2_D+σ2_E

Gring (1956) considered four methods for generating the crosses among a set of p inbred lines and their analysis for comparative and exploratory experiments. Gring presented four mating designs, called methods, and four models by taking the genetic eects as xed or random.

Methods

Diallel crossing techniques may vary depending upon whether or not the parental inbreeds or the reciprocal F0

1s are included or both.

With this as a basis for classication there are four diallel crossing systems or experimental methods:

a) Method 1: Parents, one set of F0

1s and reciprocal F10s are

included (all p2 _{combinations)} b) Method 2: Parents and one set of F0

1s are included but

reciprocal F0

1s are not (p(p + 1)/2 combinations) c) Method 3: One set of F0

1s and reciprocal F10s are included but

not the parents (p(p − 1) combinations)

d) Method 4: One set of F0

1s but neither parents nor reciprocal

a) Parents 1 2 3 1 x₁₁ x₁₂ x₁₃ 2 x₂₁ x₂₂ x₂₃ 3 x₃₁ x₃₂ x₃₃ b) Parents 1 2 3 1 x₁₁ x₁₂ x₁₃ 2 x₂₂ x₂₃ 3 x₃₃ c) Parents 1 2 3 1 x₁₂ x₁₃ 2 x₂₁ x₂₃ 3 x₃₁ x₃₂ d) Parents 1 2 3 1 x₁₂ x₁₃ 2 x₂₃ 3

Models

Model 1: the genotype and block eects are constants Model 2: the genotype and block eects are both random variables

Model A (mixed A): the genotype eects are random variables and the block eects are constants

Model B (mixed B): the genotype eects are constants and block eects are random variables

RCBD or Alpha Lattice can be modeled respectively using: yijk = µ + REPk+ gij + eijk

= µ + REPk+ gcai + scaij + mi+ rij + eijk

or

yijk= µ + REPk + gcai+ scaij + mi + rij + BLK (REPk) + eijk

yijk is the observed value

µ is the mean

REPk is the replicate eect (k = 1, 2, ..., r)

gcai is the general combining ability eect (i = 1, 2, ..., p)

scaij is the specic combining ability eect (i = 1, 2, ..., p and

j =1, 2, ..., p)

mi is the maternal eect (i = 1, 2, ..., p)

rij is the reciprocal eect (i = 1, 2, ..., p and j = 1, 2, ..., p)

BLK (REPk) random eect of block nested in replicate k

Variance components Method 1

VC model 1 and mixed B model 2 and mixed A σ_gca2 _2rp1 (MSgca− MSe) _2rp1  MSgca− MSe−p(p−1)MSsca (p(p−1)+1)  σ2_sca (MSsca− MSe)/r p 2 2r(p(p−1)+1)(MSsca− MSe) σ_m2 (MSm− MSr)/2pr (MSm− MSr)/2pr σ2_r (MSr− MSe)/2r (MSr− MSe)/2r σ2_ρ 2σ_gca2 + σ2_sca+2σ_m2 + σ2_r +σ2e r σ_A2 2σ2_gca σ2_D σ2_sca h_b2 σA2+σ2D σ_ρ2 h2_n σA2 σ_ρ2

VC model 1 and mixed B model 2 and mixed A σ_gca2 _2r(p−2)1 (MSgca− MSe) _2r(p−2)1 (MSgca− MSsca)

σ2_sca (MSsca− MSe)/2r (MSsca− MSe)/2r

σ2m (MSm− MSr)/2pr (MSm− MSr)/2pr σ2r (MSr− MSe)/2r (MSr− MSe)/2r σ2_ρ 2σ_gca2 + σ2_sca+2σ_m2 + σ2_r +σ2e r σ_A2 2σ2_gca σ2_D σ2_sca h2_b σA2+σ2D σ_ρ2 h2n σ_A2 σ_ρ2

Models for Method 1 and Method 3 Multi-environment

RCBD or Alpha Lattice can be modeled respectively using: yijkd = µ + Ed+ REPk(Ed) + gij + Ed∗ gij + eijkd

= µ + Ed+ REPk(Ed) + gcai + scaij + mi+ rij

+ Ed∗ gcai + Ed∗ scaij + Ed∗ mi + Ed∗ rij + eijkd

or

yijkd = µ + Ed+ REPk(Ed) + gcai+ scaij + mi + rij

+ Ed∗ gcai+ Ed ∗ scaij + Ed∗ mi+ Ed∗ rij

+ BLK (REPkEd) + eijkd

Ed environmental eect (d = 1, 2, ..., s)

REPk(Ed)the eect of replicate k nested in environment d

(k = 1, 2, ..., r)

BLK (REPkEd) random eect of block nested in replicate k

VC model 1 and mixed B model 2 and mixed A σ2_{gca 2srp}1 (MSgca− MSe) _2srp1  MSgca− MSe−p(p−1)MSsca (p(p−1)+1)  σ_sca2 _sr1(MSsca− MSe) p 2 2sr(p(p−1)+1)(MSsca− MSe) σ_m2 _2srp1 (MSm− MSr) _2srp1 (MSm− MSr) σ2_{r 2sr}1 (MSr− MSe) _2sr1 (MSr− MSe)

σ2_ρ 2σ2_gca+ σ_sca2 +2σ2_m+ σ_r2+σgxe2

s + σ2_e sr σ_A2 2σ2_gca σ_D2 σ2_sca h_b2 σA2+σ2D σ_ρ2 h_n2 σA2 σ_ρ2

Variance components Method 3 Multi-environment

VC model 1 and mixed B model 2 and mixed A σ2_{gca 2sr(p−2)}1 (MSgca− MSe) _2sr(p−2)1 (MSgca− MSsca)

σ_sca2 _2sr1 (MSsca− MSe) _2sr1 (MSsca− MSe)

σm2 _2srp1 (MSm− MSr) _2srp1 (MSm− MSr) σ2r _2sr1 (MSr− MSe) _2sr1 (MSr − MSe) σ2_ρ 2σ2_gca+ σ_sca2 +2σ2_m+ σ_r2+σ 2 gxe s + σ2_e sr σ_A2 2σ2_gca σ_D2 σ2_sca h_b2 σA2+σ2D σ_ρ2 hn2 σ_A2 σ_ρ2

RCBD or Alpha Lattice can be modeled respectively using: yijk = µ + REPk+ gij + eijk

= µ + REPk+ gcai+ scaij + eijk

or

yijk = µ + REPk + gcai+ scaij + BLK (REPk) + eijk

yijk is the observed value

µ is the mean

REPk is the replicate eect (k = 1, 2, ..., r)

gcai is the general combining ability eect (i = 1, 2, ..., p)

scaij is the specic combining ability eect (i = 1, 2, ..., p and

j =1, 2, ..., p)

BLK (REPk) random eect of block nested in replicate k

Variance components Method 2

VC model 1 and mixed B model 2 and mixed A σgca2 (p+12)r(MSgca− MSe) r (p+12)(MSgca− MSsca)

σ2_sca 1_r(MSsca− MSe) 1_r(MSsca− MSe)

σ2_ρ 2σ_gca2 + σ2_sca+σ2e r σ_A2 2σ2_gca σ2_D σ2_sca h2_b σA2+σ2D σ_ρ2 h2n σ_A2 σ_ρ2

VC model 1 and mixed B model 2 and mixed A σ2gca (p−12)r(MSgca− MSe) r (p−12)(MSgca− MSsca)

σ_sca2 1_r(MSsca− MSe) 1_r(MSsca− MSe)

σ2_ρ 2σ_gca2 + σ2_sca+σ2e r σ_A2 2σ2_gca σ_D2 σ2_sca h_b2 σA2+σ2D σ_ρ2 hn2 σ_A2 σ_ρ2

Models for Method 2 and Method 4 Multi-environment

RCBD or Alpha Lattice can be modeled respectively using: yijkd = µ + Ed+ REPk(Ed) + gij + Ed∗ gij + eijkd

= µ + REPk+ gcai + scaij

+ Ed∗ gcai + Ed∗ scaij + eijkd

or

yijkd = µ + Ed+ REPk(Ed) + gcai+ scaij

+ Ed∗ gcai+ Ed∗ scaij + BLK (REPkEd) + eijkd

Ed environmental eect (d = 1, 2, ..., s)

REPk(Ed)the eect of replicate k nested in environment d

(k = 1, 2, ..., r)

BLK (REPkEd) random eect of block nested in replicate k

VC model 1 and mixed B model 2 and mixed A σgca2 sr (p+1 2)(MSgca− MSe) sr (p+1 2)(MSgca− MSsca)

σ2_{sca sr}1(MSsca− MSe) _sr1(MSsca− MSe)

σ_ρ2 2σ2_gca+ σ_sca2 +2σ2_m+ σ_r2+σgxe2

s + σ2_e sr σ2_A 2σ2_gca σ2_D σ2_sca h2_b σA2+σ2D σ_ρ2 h2n σ_A2 σ_ρ2

Variance components Method 4 Multi-environment

VC model 1 and mixed B model 2 and mixed A σ2gca sr (p−1 2)(MSgca− MSe) sr (p−1 2)(MSgca− MSsca)

σ_sca2 _sr1(MSsca− MSe) _sr1(MSsca− MSe)

σ_ρ2 2σ2_gca+ σ_sca2 +2σ2_m+ σ_r2+σgxe2

s + σ2_e sr σ2_A 2σ2_gca σ_D2 σ2_sca h2_b σA2+σ2D σ_ρ2 h2n σ_A2 σ_ρ2

H. Hinkelmann.

Design and Analysis Handbook of Experiments Volume 3 Special Designs and applications.

WILEY, 2012.

B. Ggring. 1956b

Concept of general and specic combining ability in relation to diallel crossing systems

Martínez, G. A.

Diseño y análisis de los experimentos de cruzas dialélicas.