Stability Variance and its Bayesian Estimation for Genotype × Environment Interaction in m × m Gracro-latin Square Layout

In the plant breeding experiments, researchers always keen to develop new as well as modified varieties or genotypes which are favorable to the diverse environmental under consideration. The study of genotype × environment (G × E)  Interaction plays a vital role along with prior information on the exiting genotype to estimate its stability in particular environment. Cotes et al., (2006) used the Bayesian methodology for computing Shukla’s phenotypic stability variance Shukla (1972) and incorporate prior information on the parameter for better estimation. Edwards and Jannink (2006) tested a Bayesian approach to estimate heterogeneous error and (G × E) interaction variance. Their results show that error variances were highly heterogeneous among environments and the (G×E) interaction variances were heterogeneous among environment and genotypes. Elizalde et al., (2011) studied usefulness of general Bayesian approach for breeding trials and detection of genotypes groups and environments that showed significance in (G × E) interaction. They also advocated that they may be extended to other linear-bilinear models by fixing certain conditions. Birla and Ramgiry (2015) used AMMI analysis to comprehend genotype by environment (G × E) interaction in rain fed grown soybean [Glycine max (L.) Merrill]. Bhushan and Samnotra (2017) studies the stability for yield and quality trails in brinjal (Solanum melongena L). In that particular experiment, the performance of genotypes in terms of yield as well as quality across seasons and years under wide range of environments through phenotypic stability studies using Eberhart and Russell regression model.

In general, in many plant breeding programmes, new genotypes are developed in controlled conditions and as well as in open environmental conditions where it is very important to study the behavior and stability of genotypes under consideration. The new genotype under various situation and conditions received different treatments, Chetan et al., (2018) studies the stability variance obtained through (G × E) interaction in m × m Latin square layout with one treatment effect. Mamata et al., (2019) proposed new index for evaluation of G × E interaction in pearl millet using AMMI and GGE bilot analysis. Chetan et al., (2019) studied the general Bayesian estimation for (G × E)  interaction in m × m Latin square layout with one treatment effect. Still there are many situations where the experimenters need to study more than one treatment. Therefore, in the present study, an attempt is made to study the situation where two treatments are act simultaneously with m × m Graceo Latin Square layout with two treatment effects. For estimating the new explanatory power about the unknown parameters, the maximal data information prior and asymptotically locally invariant prior are also studied for the obtained stability variance in such layout.

The present study was carried out at Department of Statistics, M.D. University, Rohtak and Department of Statistics, Sri Venkateswara College, University of Delhi during 2020-21. Let y_jkr (i, j, k, r = 1, 2……..m) denote the observation on the experimental unit of the ith genotype in the jth environment receiving two different treatments i.e. kth and rth. In this situation, general least square layout is most suitable one with m² experimental units to be used in the experiment instead of m⁵ possible experimental units needed in a complete layout. Therefore, the following model may be considered for such experiment.

 

 

Where, 

µ is the general mean effect.

d_i is the additive effect of ith genotype.

e_j is the effect of jth environment.

t_k is the effect due to kth treatment on the ith genotype in the jth environment.

y_r is the effect due to rth treatment on the ith genotype in jth environment.

g_ijkr is the genotype environment interaction receiving kth and  rth treatments.

e_ijkr is the random error term which is assumed to be NID (0, 𝛔²).

If δ represent the set of m² values, then (i, j, k, r)∈  δ; di is the possible pair (j, k, r) associated with a fixed value of i and similarly δ_j, δ_k and δ_r, then parameters involved in the above model may be obtained by minimizing the sum of square error.

 

and

If it is assumed that 

 

then

on solving above equations simultaneously

Since the algebraic sum of deviation of a set of observation about their mean is zero and other product term also zero. Therefore, the analysis of variance may be given as:

Or may be portioned as:

 

 

 

Where,

The total sum of square (TSS) into its component parts, the sum of square for genotype (SSG), sum of square for environment (SSE_n), sum of square for treatment  (SST₁), the sum of square for another treatment  (SST₂), the sum of square for interaction (SSI) and the sum of square for error (SSE_r). The ANOVA for such layout may be framed as:

A stable genotype possesses an unchanged performance regardless of any variation of environmental conditions. The stability is termed as stability variance and can be estimated as:

By using the estimated values

or

is the stability variation for the ith genotype under experimental conditions. If it is assume that stability variance is equal to the within environment at variance 𝛔² i.e. 𝛔_i² = 0, then it can be a stable genotype. The variance of the estimated parameters for the proposed model may be given by 

 

Similarly, 

 

Maximal sata information prior (MDIP) for the conditions under consideration may be given by

It is well known that

 

then
 

 

is the data information averaged over the value (µ, si) with g(µ, si) prior this average may aso given by 

 
 

The average information in the data density minus the information in the prior density therefore, 

 

Lagrangian expression is

 

According to the Euler-Lagrange equation

 

 

Gives

 

Asymptotically locally invariant (ALI) prior for the conditions under consideration may be given by

 

Since the conditions of ALI prior are satisfied. Therefore, ALI prior  for (µ, Zi) may be estimated by
 

 

Where,

 

Since the g(µ, Z_i) is a function of Z_i and independent of µ, therefore
 

 

or

 

Hence,