Agricultural Science Digest

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Agricultural Science Digest, volume 43 issue 5 (october 2023) : 587-592

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

Nisha Datt1, Chetan2,*
1Department of Statistics, Maharshi Dayanand University, Rohtak-124 001, Haryana, India.
2Department of Statistics, Sri Venkateswara College, New Delhi-110 021, India.
Cite article:- Datt Nisha, Chetan (2023). Stability Variance and its Bayesian Estimation for Genotype × Environment Interaction in m × m Gracro-latin Square Layout . Agricultural Science Digest. 43(5): 587-592. doi: 10.18805/ag.D-5305.
Background: In agricultural, the field experimenters have been bona fide way of verifying the test result when researchers have developed new cultivating method in order to improve the overall crop production. To meet this requirement experimenters have continuously be measured in thinking the new and/or modified techniques and methodologies. 

Methods: To estimate stability variance with two treatments through Graceo-latin square layout apart from prevailing Latin square layout with one treatment. The maximal data information prior and asymptotically locally invariant prior are also studied for the obtained stability variance in such layout.

Result: The Graceo-latin square design layout with two different treatments has been proposed to study genotype × environment (G × E)  interaction. Statistical model for the design and ANOVA is also developed and simultaneous methods are proposed to diagrammatize the respective stability variance.
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 yjkr (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.
di is the additive effect of ith genotype.
ej is the effect of jth environment.
tk is the effect due to kth treatment on the ith genotype in the jth environment.
yr is the effect due to rth treatment on the ith genotype in jth environment.
gijkr is the genotype environment interaction receiving kth and  rth treatments.
eijkr 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 δj, δk and δr, 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 (SSEn), 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 (SSEr). 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. 𝛔i2 = 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, Zi) is a function of Zi and independent of µ, therefore
 







 
or


 
Hence,

Genotype × Environment (G×E) interaction play a key role identifying adaptability and stability for any genotype for obtaining high yield over a wide range of diverse environmental conditions around the world. The variability between the environments is smaller genotype is considered to be more stable. In plant breeding programmes, (G×E) interaction and its analyses along with its interpretation are still an intensive and extensive focused area. Numerous stability measures with different approaches are present in the literature. 

  1. Bhushan, A. and Samnotra, R.K. (2017). Stability studies for yield and quality trails in brinjal (Solanum melongena L.). Indian Journal of Agricultural Research. 51(4): 315-379.

  2. Birla, D. and Ramgiry S.R. (2015). AMMI analysis to comprehend genotype-by-environment (G × E) interaction in rainfed grown soybean [Glycine max (L.) Merrill]. Indian Journal of Agricultural Research. 49(1): 39-45.

  3. Chetan Laxmi, R.R. and Nisha (2019). General Bayesian Estima tion for Genotype × Environment Interaction in m × m Latin Square Layout. International Journal of Scientific and Technology. 8(11): 3480-3481.

  4. Chetan, Laxmi, R.R. and Sandhya (2018). Stability variance of Geno type × Environment Interaction in m × m Latin Square layout. International Journal of Research. 5(12): 4643- 4648. 

  5. Cotes, J.M., Crossa, A, Sanches, P.L. and Cornelius, A. (2006). Baye sian approach for assessing the stability of genotypes. Crop Sci. 46: 2654-2665.

  6. Elizalde, S.P., Jarquin, D. and Jose Crossa (2011). A General Baye sian Estimation Method of Linear-Bilinear Models Applied to Plant Breeding trials with Genotype × Environment Interaction. Journal of Agricultural, Biological and Envi ronmental Statistics. 17(1): 15-37. 

  7. Jode W. Edwards and Jean-Luc Jannink (2006). Bayesian Model ing of Heterogeneous Error and Genotype × Environment Interaction Variances. Crop Sci. 46: 820-833.

  8. Mamata, Hooda, B.K. and Hooda, E. (2019). A new index for evalu ation of G × E interaction in pearl millet using AMMI and GGE bilot analysis. Indian Journal of Agricultural Research. 53(5): 529-535.

  9. Shukla, G.K. (1972). Some Statistical aspects of partitioning geno type-environmental components of variability. Heredity. 29: 237-245.

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