Stability analysis

SHUKLA'S STABILITY VARIANCE AND KANG'S

This procedure calculates the stability variations as well as the statistics of selection for the yield and the stability. The averages of the genotype through the different environment repetitions are required for the calculations. The mean square error must be calculated from the joint variance analysis

Examples:.

• Analisis de estabilidad parametrico: Caso 13 genotipos y 12 localidades. Experimento con 4 repeticiones, donde la variancia estimada del error es 1.8
  Rendimiento promedio por geneotipo en cada localidad
  
  stability.par:
  (i) determines the contribution of each genotype to GE interaction by calculating var(i);
  (ii) assigns ranks to genotypes from highest to lowest yield receiving the rank of 1;
  (iii) calculates protected LSD for mean yield comparisons;
  (iv) adjusts yield rank according to LSD (the adjusted rank labeled Y);
  (v) determines significance of var(i) usign an aproximate F-test;
  (vi) assigns stability rating (S) as follows: -8, -4 and -2 for var(i) significant at the 0.01, 0.05 and 0.10 probability levels, and 0 for nonsignificant var(i)
  ( the higher the var(i), the less stable the genotype);
  (vii) sums adjusted yield rank, Y, and stability rating, S, for each genotype to determine YS(i) statistic; and
  (viii) calculates mean YS(i) and identifies genotypes (selection) with YS(i) > mean YS(i).

  > library(agricolae)
  > # yield averages of 13 genotypes in localities
  > V1 <- c(10.2, 8.8, 8.8, 9.3, 9.6, 7.2, 8.4, 9.6, 7.9, 10, 9.3, 8.0, 10.1)
  > V2 <- c(7, 7.8, 7.0, 6.9, 7, 8.3, 7.4, 6.5, 6.8, 7.9, 7.3, 6.8, 8.1)
  > V3 <- c(5.3, 4.4, 5.3, 4.4, 5.6, 4.6, 6.2, 6.0, 6.5, 5.3, 5.7, 4.4, 4.2)
  > V4 <- c(7.8, 5.9, 7.3, 5.9, 7.8, 6.3, 7.9, 7.5, 7.6, 5.4, 5.6, 7.8, 6.5)
  > V5 <- c(9, 9.2, 8.8, 10.6, 8.3, 9.3, 9.6, 8.8, 7.9, 9.1, 7.7, 9.5, 9.4)
  > V6 <- c(6.9, 7.7, 7.9, 7.9, 7, 8.9, 9.4, 7.9, 6.5, 7.2, 5.4, 6.2, 7.2)
  > V7 <- c(4.9, 2.5, 3.4, 2.5, 3,2.5, 3.6, 5.6,3.8, 3.9, 3.0, 3.0, 2.5)
  > V8 <- c(6.4, 6.4, 8.1, 7.2, 7.5, 6.6, 7.7, 7.6, 7.8, 7.5, 6.0, 7.2, 6.8)
  > V9 <- c(8.4, 6.1, 6.8, 6.1, 8.2, 6.9, 6.9, 9.1, 9.2, 7.7, 6.7, 7.8, 6.5)
  > V10 <-c(8.7, 9.4, 8.8, 7.9, 7.8, 7.8, 11.4, 9.9, 8.6, 8.5, 8.0, 8.3, 9.1)
  > V11 <-c(5.4, 5.2, 5.6, 4.6, 4.8, 5.7, 6.6, 6.8, 5.2, 4.8, 4.9, 5.4, 4.5)
  > V12 <-c(8.6, 8.0, 9.2, 8.1, 8.3, 8.9, 8.6, 9.6, 9.5, 7.7, 7.6, 8.3, 6.6)
  > data<-data.frame(V1,V2,V3,V4,V5,V6,V7,V8,V9,V10,V11,V12)
  > rownames(data)<-LETTERS[1:13]
  > stability.par(data, rep=4, MSerror=1.8, alpha=0.1, main="Genotype")
  
   

   INTERACTIVE PROGRAM FOR CALCULATING SHUKLA'S STABILITY VARIANCE AND KANG'S
                          YIELD - STABILITY (YSi) STATISTICS

   Genotype
   Environmental index  - covariate

   Analysis of Variance
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Source         d.f.   Sum of Squares  Mean Squares         F  p.value
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   TOTAL           155     2121.2544                 
   GENOTYPES        12      101.0877         8.4240         3.31 <0.001
   ENVIRONMENTS     11     1684.3067       153.1188        85.07 <0.001
   INTERACTION     132      335.8600         2.5444         1.41  0.005
   HETEROGENEITY    12       34.7256         2.8938         1.15  0.325
   RESIDUAL        120      301.1344         2.5095         1.39  0.0089
   POOLED ERROR    432                       1.8000                      
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

   Stability statistics
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  Genotype    MEANS       Sigma-square      s-square      Ecovalence
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   1       7.383333        2.134311 ns     2.188123 ns     22.018462
   2       6.783333        1.672824 ns     1.407468 ns     17.723077
   3       7.250000        0.805606 ns     0.908578 ns      9.651282
   4       6.783333        2.919766 ns     2.346256 ns     29.329231
   5       7.075000        1.604036 ns     1.788301 ns     17.082821
   6       6.916667        3.924945 *      4.339373 **     38.685128
   7       7.808333        4.043485 *      4.474131 **     39.788462
   8       7.908333        2.899022 ns     2.358281 ns     29.136154
   9       7.275000        4.251970 **     4.044379 *      41.728974
   10      7.083333        1.853320 ns     2.064596 ns     19.403077
   11      6.433333        2.167039 ns     2.237111 ns     22.323077
   12      6.891667        1.692631 ns     1.884965 ns     17.907436
   13      6.791667        3.108168 ns     2.581331 ns     31.082821

  Signif. codes:  0 '**' 0.01 '*' 0.05 'ns' 1

  Simultaneous selection for yield and stability  (++)
  
     Genotype    Yield Rank Adj.rank Adjusted Stab.var Stab.rating YSi ...
  1         A 7.383333   11        1       12 2.134311           0  12   +
  2         B 6.783333    2       -1        1 1.672824           0   1   
  3         C 7.250000    9        1       10 0.805606           0  10   +
  4         D 6.783333    2       -1        1 2.919766          -2  -1   
  5         E 7.075000    7       -1        6 1.604036           0   6   +
  6         F 6.916667    6       -1        5 3.924945          -4   1   
  7         G 7.808333   12        2       14 4.043485          -4  10   +
  8         H 7.908333   13        2       15 2.899022          -2  13   +
  9         I 7.275000   10        1       11 4.251970          -8   3   
  10        J 7.083333    8       -1        7 1.853320           0   7   +
  11        K 6.433333    1       -2       -1 2.167039           0  -1   
  12        L 6.891667    5       -1        4 1.692631           0   4   
  13        M 6.791667    4       -1        3 3.108168          -2   1   

   Yield Mean: 7.10641`
   YS    Mean: 5.076923
   LSD (0.05): 0.4514298
   - - - - - - - - - - -
   +   selected genotype
   ++  Reference: Kang, M. S. 1993. Simultaneous selection for yield
   and stability: Consequences for growers. Agron. J. 85:754-757.
   

• Analisis de estabilidad con una covariable (precipitacion)
  
  > precipitation<- c(1000,1100,1200,1300,1400,1500,1600,1700,1800,1900,2000,2100)
  > stability.par(data, rep=4, MSerror=1.8, alpha=0.1, main="Genotype", cova=TRUE, name.cov="Precipitation", file.cov=precipitation)

   INTERACTIVE PROGRAM FOR CALCULATING SHUKLA'S STABILITY VARIANCE AND KANG'S
                          YIELD - STABILITY (YSi) STATISTICS
   Genotype
   Precipitation  - covariate

    Analysis of Variance
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   Source         d.f.   Sum of Squares  Mean Squares         F  p.value
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   TOTAL           155     2121.2544                 
   GENOTYPES        12      101.0877         8.4240         3.31 <0.001
   ENVIRONMENTS     11     1684.3067       153.1188        85.07 <0.001
   INTERACTION     132      335.8600         2.5444         1.41  0.005
   HETEROGENEITY    12        2.2053         0.1838         0.07  1
   RESIDUAL        120      333.6547         2.7805         1.54 <0.001
   POOLED ERROR    432                       1.8000                      
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  
   Stability statistics
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  Genotype    MEANS       Sigma-square      s-square      Ecovalence
   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   1       7.383333        2.134311 ns     2.346535 ns     22.018462
   2       6.783333        1.672824 ns     1.841329 ns     17.723077
   3       7.250000        0.805606 ns     0.880597 ns      9.651282
   4       6.783333        2.919766 ns     3.201336 ns     29.329231
   5       7.075000        1.604036 ns     1.755238 ns     17.082821
   6       6.916667        3.924945 *      4.312894 **     38.685128
   7       7.808333        4.043485 *      4.431759 **     39.788462
   8       7.908333        2.899022 ns     3.132659 ns     29.136154
   9       7.275000        4.251970 **     4.645273 **     41.728974
   10      7.083333        1.853320 ns     2.023124 ns     19.403077
   11      6.433333        2.167039 ns     2.370126 ns     22.323077
   12      6.891667        1.692631 ns     1.855561 ns     17.907436
   13      6.791667        3.108168 ns     3.349497 *      31.082821

  Signif. codes:  0 '**' 0.01 '*' 0.05 'ns' 1

  Simultaneous selection for yield and stability  (++)

     Genotype    Yield Rank Adj.rank Adjusted Stab.var Stab.rating YSi ...
  1         A 7.383333   11        1       12 2.134311           0  12   +
  2         B 6.783333    2       -1        1 1.672824           0   1   
  3         C 7.250000    9        1       10 0.805606           0  10   +
  4         D 6.783333    2       -1        1 2.919766          -2  -1   
  5         E 7.075000    7       -1        6 1.604036           0   6   +
  6         F 6.916667    6       -1        5 3.924945          -4   1   
  7         G 7.808333   12        2       14 4.043485          -4  10   +
  8         H 7.908333   13        2       15 2.899022          -2  13   +
  9         I 7.275000   10        1       11 4.251970          -8   3   
  10        J 7.083333    8       -1        7 1.853320           0   7   +
  11        K 6.433333    1       -2       -1 2.167039           0  -1   
  12        L 6.891667    5       -1        4 1.692631           0   4   
  13        M 6.791667    4       -1        3 3.108168          -2   1   

   Yield Mean: 7.10641
   YS    Mean: 5.076923
   LSD (0.05): 0.4514298

   - - - - - - - - - - -
   +   selected genotype
   ++  Reference: Kang, M. S. 1993. Simultaneous selection for yield
   and stability: Consequences for growers. Agron. J. 85:754-757. 
   

• Nonparametric stability analysis
  
  A method based on the statistical ranges of the study variable per environment for the stability analysis
  References: Haynes K G, Lambert D H, Christ B J, Weingartner D P, Douches D S, Backlund J E, Fry W and Stevenson W. 1998.
  Phenotypic stability of resistance to late blight in potato clones evaluated at eight sites in the United States American Journal Potato Research 75, pag 211-217.
  
  Examples:.

• Analisis de estabilidad, ranking=TRUE
  
  > library(agricolae)
  > data(haynes)
  > stability.nonpar(haynes,"AUDPC",ranking=TRUE)
  
  Nonparametric Method for Stability Analysis
  -------------------------------------------
  Estimation and test of nonparametric measures
  Variable: AUDPC

  Ranking...
             FL MI ME MN ND   NY PA WI
  A84118-3    7 11 11 14  8 14.0 12 11
  AO80432-1   6  9 13 13 12 12.0 15 14
  AO84275-3  10 10 12  8  9  7.0 11 12
  AWN86514-2  3  3  3  1  3  3.0  2  1
  B0692-4     1  8  4  3  2  2.0  1  3
  B0718-3     2  1  2  2  4  4.0  3  4
  B0749-2F   15 12 16 10 13 13.0 13  8
  B0767-2     4  2  1  4  1  1.0  4  2
  Bertita     8  7  7  6 10  8.5 10  9
  Bzura       9  4  5  5  6  5.0  5  5
  C0083008-1 11 15 14 15 15 15.0 14 15
  Elba       13 13 10  9 14 11.0  7 13
  Greta       5  5  8  7  5  6.0  6  7
  Krantz     14 16 15 16 16 16.0 16 16
  Libertas   12  6  6 12  7  8.5  8  6
  Stobrawa   16 14  9 11 11 10.0  9 10

  Statistics...
                Mean Rank   s1   Z1    s2   Z2
  A84118-3    741.62   13 4.82 0.22 16.70 0.34
  AO80432-1   734.38   12 6.21 0.73 26.57 0.47
  AO84275-3   635.88    9 6.20 0.70 28.53 0.87
  AWN86514-2  176.88    2 5.71 0.15 23.64 0.09
  B0692-4     224.50    4 3.11 4.37  7.12 3.28
  B0718-3     192.50    3 6.64 1.59 30.57 1.43
  B0749-2F    772.88   14 5.07 0.05 19.14 0.07
  B0767-2     153.00    1 5.79 0.20 23.07 0.05
  Bertita     502.12    8 3.57 2.73  9.43 2.30
  Bzura       331.75    5 6.04 0.47 26.55 0.46
  C0083008-1 1022.12   15 7.11 2.90 38.84 5.09
  Elba        719.00   11 6.57 1.43 29.71 1.18
  Greta       412.88    6 4.59 0.47 14.71 0.70
  Krantz     1169.62   16 7.04 2.67 42.84 7.67
  Libertas    500.88    7 4.50 0.59 14.00 0.87
  Stobrawa    693.38   10 4.36 0.82 13.64 0.95
  ------------------------
  Sum of Z1:  20.08986
  Sum of Z2:  25.84532
  ------------------------

  Test...
  The Z-statistics are measures of stability. The test for the significance
  of the sum of Z1 or Z2 are compared to a Chi-Square value of chi.sum.
  individual Z1 or Z2 are compared to a Chi-square value of chi.ind.

        MEAN    es1   es2      vs1      vs2  chi.ind  chi.sum
  1 561.4609 5.3125 21.25 1.111905 60.75223 8.733011 26.29623
  ---
  expectation and variance: es1, es2, vs1, vs2
   

• Analisis de estabilidad, ranking=FALSE
  
  > stability.nonpar(haynes,"AUDPC",ranking=FALSE)
  
  Nonparametric Method for Stability Analysis
  -------------------------------------------
  
  Estimation and test of nonparametric measures
  Variable: AUDPC
  
  Statistics...
                Mean Rank   s1   Z1    s2   Z2
  A84118-3    741.62   13 4.82 0.22 16.70 0.34
  AO80432-1   734.38   12 6.21 0.73 26.57 0.47
  AO84275-3   635.88    9 6.20 0.70 28.53 0.87
  AWN86514-2  176.88    2 5.71 0.15 23.64 0.09
  B0692-4     224.50    4 3.11 4.37  7.12 3.28
  B0718-3     192.50    3 6.64 1.59 30.57 1.43
  B0749-2F    772.88   14 5.07 0.05 19.14 0.07
  B0767-2     153.00    1 5.79 0.20 23.07 0.05
  Bertita     502.12    8 3.57 2.73  9.43 2.30
  Bzura       331.75    5 6.04 0.47 26.55 0.46
  C0083008-1 1022.12   15 7.11 2.90 38.84 5.09
  Elba        719.00   11 6.57 1.43 29.71 1.18
  Greta       412.88    6 4.59 0.47 14.71 0.70
  Krantz     1169.62   16 7.04 2.67 42.84 7.67
  Libertas    500.88    7 4.50 0.59 14.00 0.87
  Stobrawa    693.38   10 4.36 0.82 13.64 0.95
  ------------------------
  Sum of Z1: 20.08986
  Sum of Z2: 25.84532
  ------------------------
  
  Test...
  The Z-statistics are measures of stability. The test for the significance
  of the sum of Z1 or Z2 are compared to a Chi-Square value of chi.sum.
  individual Z1 or Z2 are compared to a Chi-square value of chi.ind.
  
  MEAN es1 es2 vs1 vs2 chi.ind chi.sum
  1 561.4609 5.3125 21.25 1.111905 60.75223 8.733011 26.29623
  ---
  expectation and variance: es1, es2, vs1, vs2
  >