CHAPTER-V GENOTYPE-ENVIRONMENT INTERACTION: ANALYSIS FOR NEIGHBOUR EFFECTS

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1 CHAPTER-V GENOTYPE-ENVIRONMENT INTERACTION: ANALYSIS FOR NEIGHBOUR EFFECTS 5.1 Introduction Designs of experiments deal with planning, conducting, analyzing and interpreting tests to evaluate the factors that control the value of a parameter or group of parameters. As first step of experiments is planning, construction of design in statistical terminology. Hence, previous two chapters are devoted to the construction of neighbour design using different methods of construction. Then one shall be interested to analyze the data for the significance. Analysis of data shall be discussed in this chapter to obtain stability parameters for neighbour designs. Here is considered the earlier part i.e. analysis of the design, later one shall be discussed in the next chapter. Although plant breeders were aware of the differential responses of improved genotypes to changes in environments long back but statistical analyses were developed in early sixties. 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, when these effects were measured on the same scale as the genotypic effects. The regression technique of Finlay and Wilkinson was improved upon by Eberhart and Russell (1966) by adding another stability parameter, viz. the deviation from regression and provided a fresh approach to GE interaction analysis. The approaches given by Finlay and Wilkinson (1963) and Eberhart and Russell (1966) are purely statistical and the components of this analysis have not been related to parameters in a biometrical genetical model. The second approach, based on fitting of models which specify contributions of genetic, environmental and genotype-environment interactions to generation means, variances, etc., was developed by Bucio Alanis (1966). 106

2 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. The approach has come to be 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. Some major weakness of the statistical theory of regression approach developed by these workers were pointed out by raising the following basic issues, (i) the fundamental statistical assumptions were not usually satisfied and that the choice of the sums of squares and degrees of freedom from which the regression components were subtracted was not proper, (ii) the use of mean yield under each environment as the environmental value of the particular set of genotypes as an independent variable which was, in fact, not independent of the phenotypic variable regressed onto it. With these considerations, Freeman and Perkins (1971) developed an approach, known as modified approach. Digby (1979) developed a modified regression approach wherein he improved upon the adjustment of Patterson by introducing a sensitivity parameter. Finney(1980), Patterson and Silver(1980) & Patterson(1980) further extended the approach for such cases. Laxmi (1992) found a relationship between different regression models and different components in a model by a graphic method and had shown how these are interlinked with each other. She also developed a new line of stability analysis using indicator variable so as to judge adaptability of varieties to meet potential yield when some observations are missing. Finlay and Wilkinson (1963) worked on classical block model where assumptions are that the response on a plot to a particular treatment does not affect the response on the neighbouring plots and the fertility associated with plots in a block is constant. However, in many fields of agricultural research, the treatment applied to one experimental plot in a block may affect the response on the neighbouring plots if the blocks are linear with no 107

3 guard areas between the plots. The estimates of treatment differences may therefore deviate because of interference or competition from neighbouring units. So after 20 years, Wilkinson et al. (1983) proposed a method called nearest neighbour (NN) analysis in which local trends are removed by taking second differences or partial differences. B. L. Misra, Bhagwandas and Nutan (1991) gave several methods of construction of generalized neighbour design of equal and unequal block sizes and an outline of analysis of generalized neighbour design has been suggested. Azais et al. (1993) defined the block design for competition effects to be balanced and he also gave an outline of analysis for models with no neighbour effect, one-sided neighbour effect and two-sided neighbour effects. Bailey (2003) considered analysis of only one-side neighbour effects. Jaggi Gupta and Ashraf (2006) suggested analysis of complete block designs partially balanced for neighbouring competition effects and have also been obtained efficiency of neighbour effects. Pateria et al.(2007) considered a series of block designs by putting N-1 MOLS with N treatments one below another and had obtained designs for N5 & N7 with necessary analytical methods. Pateria, Jaggi and Varghese(2009) suggested analysis for self-neighboured strongly balanced block designs and have given analysis for the designs which are variance balanced for estimating the contrast pertaining to direct effects and also variance balanced for estimating the contrast pertaining to neighbour (left and right) effects. Though Wilkinson et al. (1983) proposed a method called nearest neighbour (NN) analysis in which local trends are removed by taking second differences or partial differences but they do not consider the competition effects or interference effects for finding out the stability measures of variety. Here emphasis is given on finding out the stability measures for competition i.e. neighbour designs. In this chapter analysis is considered for neighbour design for GE interaction. 108

4 5.2 Eberhart and Russell (1966) Model Before discussing the model given by Eberhart and Russell (1966) let us present the approach to GE interaction analysis given by Finlay and Wilkinson (1963). In this technique, the mean yield of all genotypes for each location is considered for quantitative grading of the environments and the linear regression of the mean values for each genotype onto the mean values for environments is estimated. The model is defined as y ij μ i + B i I j + δ ij + e ij (5.2.1) where y ij is the ith genotype mean in the jth environment, i1,2,...,v; j1,2,...,b; μ i is the overall mean of the ith genotype; B i is the regression coefficient that measures the response of the genotype of varying environments, δ ij is the deviation from regression of the ith genotype at the jth environment; e ij is the error such that E(e ij ) 0 and Var (e ij ) V and I j is the environmental index defined by I j - (5.2.2) such that 0 Hence, the regression coefficients, B i may be estimated as B i / (5.2.3) and it is considered as the first stability statistic. (The analysis of variance (ANOVA) of the model has been presented in table 5.2.1) 109

5 Table Analysis of Variance (Finlay and Wilkinson, 1963 Model) Source of Degrees of Sum of Square Mean Sum of Variation Freedom Square Replication with b(n-1) Environments pooled _ environments over Genotypes(G) (v-1) CF M G Environments (E) (b-1) ( ) 2 / b M E GE interaction (GE) (v-1)(b-1) - M GE Regressions (v-1) / M R Deviation from regression (v-1)(b-2) GE SS Regression SS M d Error b(v-1)(n-1) Pooled M e On the basis of the empirical findings Eberhart and Russell (1966) developed their regression model on similar lines that there exists a linear relationship between phenotype 110

6 and environment and measured its effect on the character under study. However, they adopted another approach to obtain the phenotypic regression (b i ) of the value y ij on the environment E j, as against to genotypic regression (B i ) of g ij on E j as formulated by Finlay and Wilkinson. The model considered by Eberhart and Russell may be written as y ijk μ i + b i E j + δ ij + e ijk (5.2.4) where y ijk 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), μ i is the mean of the ith genotype over all the environments, b i is the regression coefficient that measures the response of ith genotype to the varying environments, E j is the environmental index obtained as - such that 0, δ ij is the deviation from regression of the ith genotype in the jth environment, and is the random component. The ANOVA for this model is shown in table With this model, the sums of the squares due to environments and GE interaction are partitioned into environments (linear), GE interaction (linear) and deviations from the regression model. Table Analysis of Variance (Eberhart and Russell, 1966 Model) Source of Degrees of Sum of Square Mean Sum Variation Freedom of Square Genotypes(G) (v-1) CF M G Environments (E) (b-1) CF M E GE interaction (GE) (v-1)(b-1) CF M GE 111

7 Environment (linear) (El) 1 / M El GE (linear) (GEl) (v-1) / - SS Env. (linear) M GEl Pooled Deviation (d) v(b-2) M d Deviation due to genotype i (v-2) - ( / M g Pooled Error b(v-1)(n-1) Pooled over environments M e With this approach, the first stability parameter is a regression coefficient, b i which can be estimated by b i / By using the estimates of the parameters, performance of each genotype can be predicted as ij + b i E j, where is an estimate of the µ i. The deviation δ ij [ (y ij - i)]can be squared and summed to provide an estimate of another stability parameter ( ) as [ / (b-2)] / n (5.2.5) where / n is the estimate of the pooled error, and [ ij - ] - ( / (5.2.6) two parts: The model provides a means of partitioning the GE interaction of each genotype into 112

8 (i) (ii) the variation due to the response of genotype to varying environmental indices (sums of square due to regression), and the unexplainable deviation from regression on the environmental indices. Further, they define that the stable variety will be one with b i 1.0 and 0; and the null hypothesis H 0 : µ 1 µ 2 µ m can be tested by the F-test (approximately) F M G / M d with homogeneous deviation mean squares, being M d the pooled deviations. The hypothesis that there are no genetic differences among phenotypes for their regression on the environmental index H 0 : β 1 β 2 β m can be tested by the F-test F M EI / M d The deviations from regression for each genotype can be further tested by F [ / ( b-2] / Pooled error. Thus, in this approach one can see that two measures of sensitivity of the genotype to changes on environment are worked out: (i) the linear sensitivity measure in terms of the linear regression coefficient, b i, of the ith genotype to the environmental change, and (ii) the non linear sensitivity measure in terms of the deviation from regression mean square, i.e. / (b-2). Hence, the second stability parameter ( played a very important roleas the estimated variance,, is a function of the number of environments, so several environments with minimum replications per environment are necessary to obtain reliable estimates of. The pooled deviations were partitioned genotype wise accordingly so to examine their stability. 113

9 5.3 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 y ij μ + d i + E j + g ij + e ij (5.3.1) where μ is the general mean over all genotypes and environments; d i is the additive effect of the ith genotype ; E j is the effect of the jth environment; g ij is the GE interaction of the ith genotype with jth environment; and e ij is the error terms. Since d i, E j and g ij are fixed effects over the jth environment and therefore, 0; 0 and 0 (5.3.2) If the ith genotype is regressed onto the jth environment, one can write g ij B i E j + δ ij (5.3.3) where B i is the linear regression coefficient for the ith genotype and δ ij is the deviation from regression for the ith genotype in the jth environment. Hence the model in (5.3.1) can be written as y ij μ + d i + (1 + B i ) E j + δ ij + e ij (5.3.4) By following the approach, two aspects of phenotypes are recognized: (i) linear sensitivity to change in environment as measured by the regression coefficient, B i and (ii) non linear sensitivity to environmental changes which is expressed by δ ij. The ANOVA for the model has been presented in table

10 Table Analysis of Variance (Perkins and Jinks, 1968 Model) Source of Degrees of Sum of Square Mean Variation Freedom Sum of Square Genotypes(G) (v-1) b M G Environments (E) (b-1) v M E GE interaction (GE) (v-1)(b-1) - SSG - SSE M GE Heterogeneity M H between regression (v-1) (H) Reminder (v-1)(b-2) M d (d) Error b(v-1)(n-1) Pooled M e It may be seen from the ANOVA table that the analysis of variance consists of two parts: 1. A conventional analysis of variance so as to check whether GE interaction is significant, and 115

11 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) (ii) heterogeneity between regression sum of squares, and the reminder sum of squares. The two components of GE interaction can be tested for their significance against M e, 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. However, in case of significance of heterogeneity mean square compared with the remainder mean square, the GE interaction may be useful for prediction and shows its considerable importance in such cases. In this approach, two measures of sensitivity of the genotype to changes on environment are calculated: (i) the linear regression coefficient, B i, of the ith genotype to the environmental measure giving as a measure of the linear sensitivity, and (ii) the deviation from regression mean square, / (b-2) a measure of the nonlinear sensitivity. 116

12 If model of Eberhart and Russell (1966) and model of Jinks and Perkins (1968) are compared then the following relationships hold between the parameters: μ i (μ + d i ); b i (1 + B i ) and δ ij. δ ij (5.3.5) The relationship between two regression coefficients, B i and b i under these models occurs because in model the GE component, g ij, is regressed onto E j values and the regression of E j on itself has got already a slope of unity, while in model only the GE component, g ij, is regressed onto E j. The estimates of the various parameters can be obtained as (5.3.6) where is the mean yield of all the genotypes in all environments; d i can be obtained as the deviation from the mean yield for the ith genotype, i.e. - (5.3.7) and E j environmental index can be obtained as the deviation from the mean yield for the jth environment, i.e. E j - (5.3.8) 5.4 Freeman and Perkins (1971) Model Some statistical shortcomings have been brought out against the regression models given by Eberhart and Russell (1966) and Jinks and Perkins (1968a) as described earlier. One may discuss the major two as under: 1. The environment (linear) sum of squares which has the same sum of squares for environments with (b-1) degrees of freedom in Jinks and Perkins (1968a) model has been allocated only one degree of freedom for the same sum of squares in Eberhart and Russell (1966) model, which amounts to an over-estimation of the 117

13 corresponding mean square. According to the statistical rules, any sum of squares has a unique number of degrees of freedom and these cannot vary if a different partitioning scheme is adopted. 2. The use of non-independent measure of environment, that is, the mean of all varieties in that environment, leads to a statistical invalidity of the model. With these observations, Freeman and Perkins (1971) developed a new approach having rectification to the shortcomings. The first short-coming which relates to analysis part was removed by a modification in the method of analysis. The second relates to the model and it was met by introducing an independent measure of the environments. Freeman and Perkins (1971) considered the model as y ijk μ + d i + E j + g ij + e ijk (5.4.1) where y ijk stands for the performance of the ith genotype in the jth environment and in the kth replicate, and the remaining stands as specified in (5.3.1). Using the convention that y ij. refers to the total over the suffix replaced by a dot, whereas refers to the mean over the suffix omitted; more than one suffix can be replaced or omitted. Further, μ is estimated by, d i by (..), E j by (.. ) and g ij by ( +.). The ANOVA has been presented in Table The sums of squares can also be replaced by appropriate summations of their respective estimates, i.e. the sum of squares being expressible either as - or as

14 Table Analysis of Variance (Freeman and Perkins, 1971 Model) Source of Variation Degrees of Freedom Sum of Square Genotypes(G) (v-1) - Environments (E) (b-1) - Combined regression 1 - Residual b-2 Env. SS Reg. SS. GE interaction (GE) (v-1)(b-1) - + Heterogeneity of regression (v-1) - / Reminder (v-1)(b-2) GE SS Het. Reg. SS Error vb(n-1) - / n 119

15 The total variation within genotypes is obtained by summing up the E and GE components. Partitioning GE interaction into m groups is not possible as (v-1)(b-1) df are not divisible by v. As all the components are orthogonal, partitioning of GE sum of squares into (v-1) components attributable to various genotypes has been considered by following up the approach differently, which is discussed as under: Let x j be the measure of the environment and it is the same for all the genotypes. In the foregoing works it was taken x j to be or ( ), but it is not valid for x j to be linear function of because the linear regression of on x j is considered. This may be expressed as: B i (x j ) + error term (5.4.2) where is the mean of x j. Assuming (x j ) as Z j, one can estimate B i by b i, where b i n C xy / n C xx (say). After some adjustments one may have the sum of squares for regression, with one df, by n /. Subtracting this expression from gives the deviation from regression within the ith genotype as a sum of squares with (b-2) df. The combined regression line over all genotypes has slope / r m / r m C xx (say) (5.4.3) The overall regression sum of squares is vn 2 C xx, i.e. v n /, the square of a linear function of, and thus part of the environment sum of squares. 5.5 Laxmi (1992) Model From discussions of the methods given by Eberhart and Russell (1966), Perkins and Jinks (1963), Freemen and Perkins (1971) and several other workers, it is quite apparent that phenotypic expression, y ij of ith genotype in the jth environment depends on three genotypic properties; a mean expression, a linear response to environment and residual 120

16 deviation from regression. Here one may remember that the estimates of various parameters are given as, d i., E j. and +. (5.5.1) with the assumption that E (e ij ) 0 and Var (e ij ) constant. Generally, there is observed heterogeneity of variance due to environments, which may arise due to their differences in nature. Let us assume that Var (e ij ) V j for jth environment and hence, Laxmi(1992) gave the model by B i (x j ) + e ij B i Z j + e ij (5.5.2) where Z j (x j ), x j is a measure on the environment j, is the mean of x j, B i the regression coefficient and e ij the error term. Now we may consider a set of positive numbers, W j, to be used as weighting factor for a weighted linear regression, and let W j V -1 j.without loss of generality, we may assume that the Z j have been so chosen that 0 (5.5.3) After considering residuals and applying the method of least squares, we have / and / (5.5.4) Relations remain valid even when the W j are an arbitrary set of positive numbers, rather than the reciprocals of variances, V j and will always provide unbiased estimates corresponding to population parameter. Anscombe and Turkey (1963) have examined such solutions and pointed out that these estimates would not enjoy the maximum variance 121

17 property of the least squares analysis. The same holds true when arbitrary W j are taken into consideration. However, the situation is entirely analogous to the estimation of components of variance from an analysis of variance. The weighted analysis of variance for the model is shown in Table If the error variances are very different from each other then all the analyses made by regression approach may fail, with this approach it is possible to bring out a solution to the problem and can go for further analyses of GE interaction and stability. Table Analysis of Variance (Laxmi, 1992 Model) Source of Degrees of Sum of Square Mean Sum Variation Freedom of Square Genotypes(G) (v-1) M G Environments (E) (b-1) M E GE interaction (GE) Linear Regression (v-1)(b-1) (TSS SSG SSE) SSGE M GE (v-1) SSLR M LR (LR) Residual (v-1)(b-2) (SSGE SSLR) M r (r) Error b(v-1)(n-1) Pooled M e 122

18 If a genotype is included for which B i 0, i.e. it is completely insensitive to the environmental effect then the data for the genotype should not be used to estimate environmental effects in ordinary analysis. Such data are not to be excluded from the present analysis and one should proceed as usual with the modified methods. These were the models discussed by authors for GE analysis. To include the competition effect one has to consider models with neighbour effects. 5.6 Proposed Block Model for GE Interaction with Two-Sided Neighbour Effects In an experiment on competition studies, the effect of a genotype/treatment applied to a plot can be written as the total sum of direct effect due to the treatment applied to a given plot, left neighbour effect due to the competition with the treatment applied to the immediate left neighbour plot and right neighbour effect due to the competition with the treatment applied to the immediate right neighbour plot. Therefore, the competition treatments constitute an ordered triplet. Blocks of the competition designs are the sets of these triplets arranged in sequence. Several authors had worked on neighbouring competition effects for the analysis of genotype/treatment effects but no reference till date is available on analysis of GE interaction considering neighbour effects. The model with differentiated two- sided neighbour effects for GE interaction is: Y ijl μ + τ (i, j) + δ (i - 1, j) + ρ (i + 1, j) + E j + g ij + e i jl (5.6.1) where Y ijl is the response from the ith plot in the jth block ( i 1,2,...,k; j 1,2,..., b; l 1,2,,n), μ is the general mean effect, τ (i, j) is the direct effect of the genotype/treatment in the ith plot of jth block, E j is the effect of jth environment/block, δ (i - 1, j) is the left neighbour effect due to the genotype/treatment in the (i 1)th plot of jth block, ρ (i+1,j) is the right neighbour effect due to the treatment/ genotype in the (i + 1)th plot of jth block, g ij is the GE interaction of the ith genotype with jth environment and e ijl are error terms independently and normally distributed with mean zero and variance σ 2. Since τ i,j, δ (i 1),j, ρ (i + 1),j, E j and g ij are fixed effects and therefore, 123

19 0; 0 and 0 (5.6.2) If the ith genotype is regressed onto the jth environment, one can write g ij B i E j + η ij (5.6.3) where B i is the linear regression coefficient for the ith genotype and η ij is the deviation from regression for the ith genotype in the jth environment. Considering this set-up, the joint information matrix for model (5.6.1) for estimating different effects by generalized least squares is obtained as follows: C where M 1 is the v x v incidence matrix pertaining to direct versus left treatments, M 2 is the v x v incidence matrix pertaining to direct versus right treatments, M 3 is the v x v incidence matrix pertaining to left versus right treatments, N 1 is the v x b incidence matrix pertaining to direct treatments versus blocks, N 2 is the v x b incidence matrix pertaining to left treatments versus blocks, N 3 is the v x b incidence matrix pertaining to right treatments versus blocks. Further R τ diagonal (r 1, r 2,, r v ); r i (i 1, 2, v) being the number of times the i-th treatment appears in the design, 124

20 R δ diagonal (r 11, r 12,...,r 1v ) being the number of times the treatments in the design has i-th treatment as left neighbour, R ρ diagonal (r 21, r 22, r 2v ) being the number of times the treatments in the design has i-th treatment as right neighbour. K diagonal (k 1, k 2, k b ) The above 3v x 3v information matrix (C) for estimating the different effects in a block design setting are non-negative definite with row and column sums equal to zero. For the multi-environment experiments which are conducted in incomplete design for v genotypes in b environments with border plots as discussed in chapter III and IV with parameters vbs 2 +s+1, rks+1, λ1, we obtained R τ R δ R ρ ri; and [(r-1) I+J] where I is an identity matrix of order v x v and J is a matrix of order v x v with all elements unity. Further, for neighbour design of OS2 series M 1 J rj, J 2 vj, M 1 I M 1, J rj, I,, and r k. Therefore, the joint information matrix for estimating different effects for neighbour designs of OS2 series, reduces to C 1 (5.6.4) 125

21 using the result given by Dey(1986, p.93) which shows that for a BIBD R NK -1 N [I v -1 J ] (5.6.5) Then matrix C 1 becomes C 1 (5.6.6) Again this 3v x 3v matrix i.e. C 1 is non-negative definite with zero row and column sums. The information matrix for estimating the direct effects of treatments i.e. C τ can be obtained as given below: C τ A 1 B 1 D 1 (5.6.7) where A 1 [I v -1 J ], (5.6.8) B 1 (5.6.9) D 1 (5.6.10) As D 1 is a singular matrix, therefore its g-inverse has been found out. So D 1 126

22 (5.6.11) Now, consider B 1 D 1 (5.6.12) (5.6.13) (5.6.14) (5.6.15) ( because M 1 J rj, M 1 I M 1, and r k) On post multiplying by, we get B 1 D 1 x (5.6.16) (5.6.17) (5.6.18) (5.6.19) On simplification, we obtain B 1 D 1 (5.6.20) Finally, C τ so obtained is 127

23 C τ A 1 B 1 D 1 [I v -1 J ] - (5.6.21) I - (5.6.22) Thus C τ τ 1 Q 1 (5.6.23) are the reduced normal equations for estimating the direct treatment effects and solutions of these equations provide, where (,,, )' and Q 1 T 1 N 1 K -1 B (5.6.24) further T 1 is the vector of total of direct treatment effects and N 1 is the v x b incidence matrix pertaining to direct treatments versus blocks. Now consider the joint information matrix, for estimating different effects for neighbour designs of OS2 series, C 2, which is obtained by interchanging first & second rows and then interchanging first & second columns of C 1 matrix. Therefore, C 2 obtained is C 2 (5.6.25) using the result given by Dey(1986), the matrix C 2 becomes C 2 (5.6.26) 128

24 Again this 3v x 3v matrix i.e. C 2 is non-negative definite with zero row and column sums. The information matrix for estimating the left treatment effects i.e. C δ can be obtained as given below: C δ A 2 B 2 D 2 (5.6.27) where A 2, (5.6.28) B 2 (5.6.29) D 2 (5.6.30) As D 2 is a singular matrix, therefore, its g-inverse has been found out. So D 2 (5.6.31) (5.6.32) Now, consider B 2 D 2 (5.6.33) (5.6.34) (5.6.35) (5.6.36) 129

25 ( because J rj, vj, I, and r k) On post multiplying by, we get B 2 D 2 x (5.6.37) (5.6.38) (5.6.39) (5.6.40) On simplification, we obtain B 2 D 2 (5.6.41) Finally, C δ so obtained is C δ A 2 B 2 D 2 [I v -1 J ] - (5.6.42) I - (5.6.43) Then C δ τ 2 Q 2 are the reduced normal equations for estimating the left treatment effects and solutions of these equations provide, i.e. vector of estimates of left treatment effects and Q 2 T 2 N 2 K -1 B where T 2 is the vector of total of left neighbours and N 2 is the v x b incidence matrix pertaining to left treatments versus blocks 130

26 Now again consider the joint information matrix C 1 for estimating different effects for neighbour designs of OS2 series C 1 (5.6.44) which is partitioned in such a way that A 3, (5.6.45) B 3 (5.6.46) D 3 (5.6.47) The information matrix for estimating the right effects of treatments i.e. C ρ can be obtained as given below: C ρ A 3 B 3 D 3 (5.6.48) As D 3 is a singular matrix, therefore, its g-inverse has been found out. So D 3 (5.6.49) (5.6.50) Now, consider B 3 D 3 (5.6.51) 131

27 (5.6.52) (because ) (5.6.53) (5.6.54) (5.6.55) On post multiplying by we get B 3 D 3 x (because ) Therefore, B 3 D 3 x (5.6.56) (5.6.57) (5.6.58) (5.6.59) On simplification, we obtain B 3 D 3 (5.6.60) Finally, C ρ so obtained is C ρ A 3 B 3 D 3 132

28 [I v -1 J ] - (5.6.61) I - (5.6.62) Then C ρ τ 3 Q 3 are the reduced normal equations for estimating the right treatment effects and solutions of these equations provide, i.e. vector of estimates of right treatment effects and Q 3 T 3 N 3 K -1 B where T 3 is the vector of total of right neighbours and N 3 is the v x b incidence matrix pertaining to right treatments versus blocks Putting back these estimates of direct treatment effects, left neighbour effects and right neighbour effects, separated above, in the model, the total sum of squares is partitioned into different components of variation constituting analysis of variance. Then ANOVA for GE Interaction with neighbouring effect is shown in Table With this model, the sum of squares due to environments and GE interaction are partitioned into environments (linear), GE interaction (linear) and deviations from the regression model. Table Analysis of Variance (GE interaction with Neighbouring Effect, Model) Source of Degrees of Sum of Square Mean Sum of Variation Freedom Square Environments (E) s 2 +s CF M E Genotypes(G) s M G Left Effect(L) s M L 133

29 Right Effect (R) s M R GE interaction (GE) s( s 2 +s ) CF M GE Environment (linear) (El) GE (linear) (GEl) 1 / s / - SS Env. (linear) M El M GEl Pooled Deviation (d) s(s 2 +s-1) M d Deviation due to genotype i s 2 +s-1 ( - M g Pooled Error s(s 2 +s+1)n Pooled over environments * n is the number of times the whole BIBD is repeated. Here is the adjusted treatment sum of square for direct treatments is the adjusted treatment sum of square for left neighbours and is the adjusted treatment sum of square for right neighbours For calculation purpose, these equations further can be replaced by sum of squares with summations (5.6.63) 134

30 (5.6.64) (5.6.65) If model proposed here is compared with the model given by Eberhart and Russell (1966), then the following relationship holds μ i μ + τ (i, j) + δ (i - 1, j) + ρ (i + 1, j) ; b i (1 + B i ) and δ ij. η ij (5.6.66) It may be seen from the ANOVA table that sum of squares of direct treatment effects is different from the one obtained by Eberhart and Russell as the earlier consists of the effects of neighbour treatments. It further results in differentiating sum of squares due to GE interaction considering neighbour effects in the proposed model, from that of considering only direct treatment effects, discussed by Eberhart and Russell (1966). Therefore, regression of genotype on environments and deviation due to genotype may be highly significant when tested against which is the pooled mean square error. This gives an idea for measure of stability parameters considering neighbour effects, which is going to be discussed in the following chapter. References Anscombe, F.J. and Tukey, J.W. (1963): The examination and analysis of residuals.,technometrics, 5, Azais, J.M., Bailey, R.A. & Monod, H.(1993): A catalogue of efficient neighbour designs with border plots. Biometrics, 49, Bailey, R.A. (2003): Designs for one- sided neighbour effects. Journal of the Indian Society Agriculture Statistics,56(3), Bucio Alanis, L. (1966). Environmental and genotype-environment components of variability I-Inbred Lines. Heredity, 31,

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