This document discusses various linear regression models for analyzing genotype-environment interactions: 1. Finlay and Wilkinson (1963) developed the first model where the mean yield of each genotype is regressed onto the mean yield of environments. 2. Eberhart and Russell (1966) improved on this by regressing the phenotypic value of each genotype onto an environmental index to estimate two stability parameters. 3. Perkins and Jinks (1968) extended the model to partition GE interaction into linear and nonlinear components to measure sensitivity to environmental changes. 4. The models are analyzed using ANOVA to test for significance of GE interaction and its linear and nonlinear components. Parameters like regression coefficients and deviation from regression are estimated to

1. Linear regression approach for
analysis of GE Interaction

2. 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.

3. • 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.
•

4. • 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.

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

6. • 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.

7. 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

8. • 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

9. • 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 interactions
I.
II.
III.
IV.
intra - population, micro-environment
Intra- population, macro- environment
Inter- population, micro- environment
Inter- population, macro- environment

10. • 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.

11. 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)

12. 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

13. • 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

14. • Regression coefficient
Bi
(
i
yij I j ) /
j
• Ij is the environmental index
Ij
yij
i
v
yij
ij
vb
I
2
j

15. Analysis of Variance (Finlay and
Wilkinson, 1963 Model)
Source of
variation
Degrees of
freedom
Sum of Squares
Environments pooled
over
1
2
CF
i Yi
b
Mean sum of
Square
Replication with
environments
b(n-1)
Genotypes(G)
(v-1)
Environments (E)
(b-1)
GE interaction (GE)
(v-1)(b-1)
Regressions
(v-1)
Deviation from
regression
(v-1)(b-2)
GE SS – Regression SS
MD
Error
b(v-1)(n-1)
Pooled
Me
Ij (
j
2
ij Yij
i
(
j
j
1
b
I j )2 / b
i
j
ME
MGE
Yi 2
Yij I j )2 /
MG
I2
j
MR

16. 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.

17. • 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.

18. Analysis of Variance
(Eberhart and Russell, 1966 Model)
Source of
variation
Degrees of
freedom
Genotypes(G)
(b-1)
GE interaction
(GE)
(v-1)(b-1)
Environment
(linear) (El)
1
GE (linear)
(GEl)
(v-1)
Pooled
Deviation (d)
v(b-2)
Deviation due
to genotype i
(v-2)
Pooled Error
b(v-1)(n-1)
1
2
CF
i Yi
b
1
2
CF
j yj
v
(v-1)
Environments
(E)
Sum of Squares
1
b
2
ij Yij
1
(
v
i
j
1
v
2
i Yi
(
y Ej ) /
i
j
y
ME
y. jE j )2 /
j ij
2
ij
MG
2
CF
j Yj
j
2
yi2
(
b
j
j
j
Mean sum
of Square
E
MR
E2
j
2
j
SSEnv(linear)
MGEI
Md
2
ij
E j )2 /
MGE
j
E2
j
Pooled over environments
Mi
Me

19. • The first stability parameter is a regression
coefficient, bi which can be estimated by
bi
j
yij E j /
E2
j
j
• The second stability parameter played a very
important role as the estimated variance,
2
di
S
[
j
ˆ2 /(b 2)] S 2 / n
ij
e

20. • Where
ˆ
ij
ˆ
[ yij
yi2
] (
b
j
yij E j )2 /
ˆ
E2
j
j
• δij is the deviation from regression of the ith
genotype in the jth environment

21. 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

22. • 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.

23. • Since di , Ej and gij are fixed effects over the
jth environment
• Therefore
i
di
0;
j
Ej
0.and
i
g ij
0
• If the ith genotype is regressed onto the jth
environment
g ij
Bi E j
ij

24. • 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

25. • 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 .

26. Source of variation
Degrees of
freedom
Genotypes(G)
(b-1)
GE interaction (GE)
d i2
i
MG
j
( E j )2
ME
(v-1)(b-1)
v
( yij ) 2
2
ij
Heterogeneity between
regression(H)
(v-1)
Reminder
(v-1) (b-2
Pooled Error
b(v-1)(n-1)
Mean
sum of
Square
b
(v-1)
Environments (E)
Sum of Squares
y ij
i
i
j
Bi2
j
ij
vb
2
ij
(E j ) 2
Pooled
MGE
SSG SSE
Mh
MD
Me

27. • 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.

28. • 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.

29. • 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.

30. • 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,
∑jδ2i / (b-2) a measure of the non-linear sensitivity

31. • 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
ˆ
y..
• Where y.. is the mean yield of all the genotypes in
all environments;

32. • di can be obtained as the deviation from the
mean yield for the ith genotype
ˆ
di
yi
y..
• Ej environmental index can be obtained as the
deviation from the mean yield for the jth
environment
Ej
y. j
y..

33. Refference
• Prem narain,Statistical genetics, wiley eastern
ltd
• http://shodhganga.inflibnet.ac.in/bitstream/1
0603/9794/10/10_chapter%205.pdf