Linear regression approach for analysis of GE Interaction

GE Interaction

• A dynamic approach to interpretation of varying environments was developed by Finlay and Wilkinson (1963)

• It leads to the discovery that the components of a genotype and environment interaction were linearly related to environmental effects.

• The regression technique was improved upon by Eberhart and Russell (1966) adding another stability parameter.

• The approaches given by Finlay and Wilkinson (1963) and Eberhart and Russell (1966) are purely statistical

• The components of this analysis have not been related to parameters in a biometrical genetical model.

•

• The second approach was developed by Bucio Alanis (1966), based on fitting of models which specify contributions of genetic, environmental and genotype-environment interactions to generation means, variances, etc.,

• Bucio Alanis and Hill (1966) extended the above model by including some parental effect for averaged dominance over all environments.

• Perkins and Jinks (1968a) extended the technique to include many inbred lines and considered analysis of GE interaction with different angle.

• Perkins and Jinks (1968a) extended the technique to include many inbred lines and considered analysis of GE interaction with different angle.

• The approach was known as ‘joint regression analysis’ as each genotype is regressed onto the environments, which is measured by the joint effect of all genotypes.

• This technique has been widely used to measure the contribution of genotypes to the GE interactions and predicting performance of genotypes over environments and generations.

GE Interaction

• Gene–environment interaction (or genotype–environment interaction or G×E) is the phenotypic effect of interactions between genes and the environment.

• There are two different conceptions of gene–environment interaction.

• biometric and developmental interaction,• uses the terms statistical and commonsense

interaction

• Biometric gene–environment interaction has particular use in population genetics and behavioral genetics

• Developmental gene–environment interaction is a concept more commonly used by developmental geneticists and developmental psychobiologists

• The GE interactions extensively used in many crop plants and animals to evaluate the stability of genotypes to varying environments.

• There are four types of GE interactionsI. intra - population, micro-environmentII. Intra- population, macro- environmentIII. Inter- population, micro- environmentIV. Inter- population, macro- environment

• The first and third types of interaction – very difficult to handle.

• The second and fourth types of interaction could be handled by performing well designed experiments.

Regression method

• Environments are defined clearly and distinguishable from one another.

• The changes in the expression of different genotypes can be related to changes from one environment to another by regression technique( Bucio Alanis 1966)

Finlay and Wilkinson (1963) Model

• In this technique, the mean yield of all genotypes for each location is considered for quantitative grading of the environments

• The linear regression of the mean values for each genotype onto the mean values for environments is estimated

• The model is defined as• yij = μi + BiIj + δij + eij

• where yij is the ith genotype mean in the jth environment, i=1,2,...,v; j=1,2,...,b

• μi is the overall mean of the ith genotype

• Bi is the regression coefficient that measures the response of the genotype of varying environments,

• δij is the deviation from regression of the ith

genotype at the jth environment;• eij is the error such that E(eij) = 0 and Var (eij) = V

• Regression coefficient

• Ij is the environmental index

2/)( jjjijii IIyB

vb

y

v

yI ij

ijij

ij

Analysis of Variance (Finlay and Wilkinson, 1963 Model)

Source of variation

Degrees of freedom

Sum of Squares Mean sum of Square

Replication with environments

b(n-1) Environments pooled over

Genotypes(G) (v-1) MG

Environments (E) (b-1) ME

GE interaction (GE) (v-1)(b-1) MGE

Regressions (v-1) MR

Deviation from regression

(v-1)(b-2) GE SS – Regression SS MD

Error b(v-1)(n-1) Pooled Me

CFYb ii 21

bII jjjj /)( 2

22 1iiijij Y

bY

22 /)( jjjijji IIY

Eberhart and Russell (1966) Model

• A linear relationship between phenotype and environment and measured its effect on the character

• Adopted another approach to obtain the phenotypic regression (bi) of the value yij on the environment Ej, as against to genotypic regression (Bi) of gij on Ej as formulated by Finlay and Wilkinson.

• The model by Eberhart and Russell may be written as

• yijk = μi + bi Ej + δij + eijk

• where yijk is the phenotypic value of the ith genotype at the jth

environment in the kth replicate (i = 1,2,...,v; j = 1,2,...,b; k = 1,2,...,n),

• μi is the mean of the ith genotype over all the environments,

• bi is the regression coefficient that measures the response of ith

genotype to the varying environments

• Ej is the environmental index obtained

• δij is the deviation from regression of the ith genotype in the jth

environment, and is the random component.

Analysis of Variance (Eberhart and Russell, 1966 Model)

Source of variation

Degrees of freedom

Sum of Squares Mean sum of Square

Genotypes(G) (v-1) MG

Environments (E)

(b-1) ME

GE interaction (GE)

(v-1)(b-1) MGE

Environment (linear) (El)

1 MR

GE (linear) (GEl)

(v-1) MGEI

Pooled Deviation (d)

v(b-2) Md

Deviation due to genotype i

(v-2) Mi

Pooled Error b(v-1)(n-1) Pooled over environments Me

CFYb ii 21

CFYv

Yb

Y jjiiijij 222 11

22 /).(1

jjjj EjEyv

CFyv jj 21

)(/)( 22 linearSSEnvEEy jjjijji

2ijji

222

2 /)( jjjji

ijj EEb

yy

• The first stability parameter is a regression coefficient, bi which can be estimated by

• The second stability parameter played a very important role as the estimated variance,

2/ jjjijji EEyb

nSbS eijjdi /)]2/(ˆ[ 222

• Where

• δij is the deviation from regression of the ith genotype in the jth environment

222

/)(]ˆ[ˆjEEy

b

yy jjijj

iijij

Perkins and Jinks (1968) Model

• Perkins and Jinks (1968) extended the technique of Bucio Alanis (1966) and Bucio Alanis and Hill (1966) by improving their models.

• They describe the performance of the ith

genotype in the jth environment as given by the model

• yij= μ + di + Ej + gij + eij

• where μ is the general mean over all genotypes and environments;

• di is the additive effect of the ith genotype ;

• Ej is the effect of the jth environment;

• gij is the GE interaction of the ith genotype with jth environment; and

• eij is the error terms.

• Since di , Ej and gij are fixed effects over the jth environment

• Therefore

• If the ith genotype is regressed onto the jth environment

0.0;0 ijijjii gandEd

ijjiij EBg

• where Bi is the linear regression coefficient for the ith genotype and δij is the deviation from regression for the ith genotype in the jth

environment. • Hence the model can be written as • yij= μ + di + (1 + Bi ) Ej + δij + eij

• By this approach two aspects of phenotypes are recognized: (i) linear sensitivity to change in environment as

measured by the regression coefficient, Bi and

(ii) non linear sensitivity to environmental changes which is expressed by δij .

Source of variation Degrees of freedom

Sum of Squares Mean sum of Square

Genotypes(G) (v-1) MG

Environments (E) (b-1) ME

GE interaction (GE) (v-1)(b-1) MGE

Heterogeneity between regression(H)

(v-1) Mh

Reminder (v-1) (b-2 MD

Pooled Error b(v-1)(n-1) Pooled Me

2ii db

SSESSGvb

yy ij

jiij ij

22 )(

2)( jj Ev

2)(2jjii EB

2ijij

• The analysis of variance consists of two parts: 1. A conventional analysis of variance so as to check

whether GE interaction is significant, and 2. To test whether GE interaction is a linear function of

the additive environmental component • For this purpose the GE interaction is partitioned

further into two parts: (i) heterogeneity between regression sum of squares,

and (ii) the reminder sum of squares.

• The two components of GE interaction can be tested for their significance against Me, which is the error mean square due to within genotype, and within environmental variations averaged over all environments.

• If either of the heterogeneity regression mean square, the remainder mean square or both are significant, one may conclude that GE interactions are significant.

• If heterogeneity mean square alone is significant one may derive the finding that GE interactions for each genotype may be treated as linear regression (for genotype) on the environmental values

• If the remainder mean square alone is significant, it may be assumed that there is no evidence of any relationship between the GE interactions and the environmental values and no prediction can be made with this approach.

• When both the components are significant the usefulness of any prediction will depend on the relative magnitude of these mean squares

• In this approach, two measures of sensitivity of the genotype to changes on environment are calculated:

(i) the linear regression coefficient, Bi, of the ith genotype to the environmental measure giving as a measure of the linear sensitivity, and

(ii) the deviation from regression mean square,∑jδ2

i / (b-2) a measure of the non-linear sensitivity

• If model of Eberhart and Russell (1966) and model of Jinks and Perkins (1968) are compared then the following relationships hold between the parameters:

• μi = (μ + di); bi = (1 + Bi) and δij.= δij • The estimates of the various parameters can be

obtained as

• Where is the mean yield of all the genotypes in all environments;

..ˆ y

..y

• di can be obtained as the deviation from the mean yield for the ith genotype

• Ej environmental index can be obtained as the deviation from the mean yield for the jth environment

..ˆ yyd ii

... yyE jj

Refference

• Prem narain,Statistical genetics, wiley eastern ltd

• http://shodhganga.inflibnet.ac.in/bitstream/10603/9794/10/10_chapter%205.pdf