Performs Finlay and Wilkinson’s joint regression analysis of genotype-by-environment data (P.W. Lane & K. Ryder).

### Options

`PRINT` = string tokens |
What to print (`model` , `summary` , `estimates` , `sortedsensitivities` , `monitoring` ); default `mode` , `summ` , `esti` , `sort` |
---|---|

`PLOT` = string tokens |
What graphs to plot (`lines` , `trellislines` , `sensitivities` ); default `*` |

`NBEST` = scalar |
Number of best genotypes to print in table of sorted sensitivities; default `*` i.e. print all of them |

`DIRECTION` = string token |
Direction to sort table of sorted sensitivities (`ascending` , `descending` ); default `asce` |

`TOLERANCE` = scalar |
Convergence criterion; default 0.001 |

`MAXCYCLE` = scalar |
Maximum number of cycles; default 15 |

`SAVE` = regression save structure |
Save structure from `MODEL` statement defining the model; default is to use the structure from the latest `MODEL` statement |

### Parameters

`GENOTYPES` = factors |
The genotype factor; no default |
---|---|

`ENVIRONMENTS` = factors |
The environment factor; no default |

`SENSITIVITIES` = tables |
Saves the estimates of sensitivities; default `*` |

`GENMEANS` = tables |
Saves the estimates of genotype means; default `*` |

`ENVMEANS` = tables |
Saves the estimates of environment means; default `*` |

`ENVEFFECTS` = tables |
Saves the estimates of environment effects; default `*` |

`SESENSITIVITIES` = tables |
Saves the s.e.s of sensitivities; default `*` |

`SEGENMEANS` = tables |
Saves the s.e.s of genotype means; default `*` |

`SEENVEFFECTS` = tables |
Saves the s.e.s of environment effects; default `*` |

`MSDEVIATIONS` = tables |
Saves the mean square deviations about the line fitted to each genotype; default `*` |

`DEVIANCE` = scalar |
Saves the residual deviance |

`DF` = scalar |
Saves the residual d.f |

`TITLE` = text |
Overall title for the graphs |

`YTITLE` = text |
Y-axis title for the graph of the lines |

`XTITLE` = text |
X-axis title for the graph of the lines |

`EXIT` = scalar |
Exit status: set to 0 if the analysis converged, 1 otherwise |

### Description

Procedure `RFINLAYWILKINSON`

performs the analysis proposed by Finlay & Wilkinson (1963) and Yates & Cochran (1938) to investigate the interaction between two factors. It is an update of the procedure `RJOINT`

, with syntax and output conventions revised for compatibility with the new QTL procedures. `RJOINT`

, however, is retained to allow existing programs to continue to run.

The analysis is motivated by the study of genotype-by-environment interactions in agriculture. The two factors are then genotypes of a particular crop and environments in which some experiments have been carried out. The factors are specified using the parameters `GENOTYPES`

and `ENVIRONMENTS`

.

The environments may be different sites within the same year, different years for the same site, or a combination of the two with little interest in individual year and site contributions. The intention is to characterize the *sensitivity* of each genotype to environmental effects by fitting a regression of the environment means for each genotype on the average environment means. Sensitivity provides a way of assessing the *stability* of the genotypes. The responses of genotypes with low sensitivity values are more stable with respect to changes of environment. Eberhart & Russell (1966) suggested that it is also interesting to consider the means of the squared deviations of the observations about the line fitted for each genotype. The genotypes with smaller mean square deviations are giving more predictable responses.

The model to fit is nonlinear, with the form

*y _{ij}* =

*g*+

_{i}*b*×

_{i}*e*+ error

_{j}where *g _{i}* are genotype means,

*e*are environment effects (with ∑

_{j}*e*=0) and

_{j }*b*are the sensitivity parameters (with mean(

_{i}*b*)=1). Usually, the aim is to find genotypes with large means and small sensitivities, to ensure a reliable crop under variable conditions.

_{i }The data may consist of one value of the response (e.g. yield) for each combination of genotype and environment. More often, however, the data are incomplete because not all genotypes are tested at each environment. Also, there may be multiple measurements of genotypes at some environments. If the response is a count or a proportion, as for example when investigating disease resistance, it will be more appropriate to use a generalized linear model based on a Poisson or binomial distribution and a log or logit link function.

The model and response variate must be specified by giving a `MODEL`

statement before calling `RFINLAYWILKINSON`

. For example,

`MODEL yield`

You can choose to fit any generalized linear model by setting the `DISTRIBUTION`

and `LINK`

options of `MODEL`

: thus, to model proportions, you could give a statement like

`MODEL [DISTRIBUTION=binomial; LINK=logit; DISPERSION=*]\`

` prop; NBINOMIAL=100`

The iterative process used in the procedure is controlled by the options `TOLERANCE`

and `MAXCYCLE`

. At each iteration, the maximum difference between estimates of the sensitivity parameters in successive iterations is compared to the tolerance: the process ends when the differences are small enough, or when the maximum number of iterations is reached. The progress of the search can be followed by including the `monitoring`

setting of the `PRINT`

option, and the `EXIT`

parameter can save a scalar with the value zero if the analysis converged and one otherwise.

Output is controlled by the `PRINT`

option. The `model`

setting prints a description of the model. The `summary`

setting displays an analysis of variance (or deviance for non-Normal distributions) showing the effects of Varieties, Environments and Sensitivities (i.e. the effect of allowing different sensitivities for each genotype). The `estimates`

setting displays two tables. The first table is classified by genotypes and contains the unadjusted means, estimated means (on the scale of the link function, if relevant), standard errors of the estimated means, back-transformed means (if relevant), sensitivities, standard errors of sensitivities, mean square deviations and the ranks of the genotypes according to their sensitivities. (The genotype with rank 1 is the one that is least sensitive.) The second table is classified by environments, and contains estimates of effects (on the scale of the link function, if relevant), standard errors of estimates, means (formed from the effects and the mean of the genotype means), back-transformed means (if relevant), the ranks of the environments according to their means. The `sortedsensitivities`

setting displays a table classified by genotypes, containing sensitivities and estimated means with their standard errors, mean square deviations and back-transformed means (if relevant). The rows of the table are sorted into either ascending or descending order of sensitivities, according to the setting of the `DIRECTION`

option (default `descending`

). The `NBEST`

option can be set to control the number of genotypes that are included; by default they are all printed. The `monitoring`

setting produces monitoring information during the fit.

The `PLOT`

option controls the graphs that are plotted, with settings:

`lines` |
plots the fitted lines, all on the same graph, |
---|---|

`trellislines` |
plots the fitted lines in a trellis plot, classified by genotypes, and |

`sensitivities` |
produces a scatter-plot matrix displaying the sensitivities, the mean square deviations and the estimated means. |

The `TITLE`

parameter defines the overall title for plots of the fitted lines; the default is “Finlay & Wilkinson analysis”. The `YTITLE`

and `XTITLE`

parameters define titles for the y- and x-axes, respectively for the plots of the fitted lines; the default for the y-axis is the name of the y-variate, and the default for the x-axis is the name of the `ENVIRONMENTS`

factor.

The remaining parameters allow the results from the analysis to be saved: sensitivities, genotype means, environment effects, environment means, and standard errors of sensitivities, genotype means, environment effects, mean square deviations, residual deviance and degrees of freedom. After calling the procedure, you can use the `RKEEP`

directive to access fitted values and residuals. Other results from the fit, that can be accessed via `RKEEP`

or `RDISPLAY`

, may not be correct: for example, the number of residual d.f. shown by

`RDISPLAY [PRINT=summary]`

does not allow for the estimation of sensitivities.

Options: `PRINT`

, `PLOT`

, `NBEST`

, `DIRECTION`

, `TOLERANCE`

, `MAXCYCLE`

, `SAVE`

.

Parameters: `ENVIRONMENTS`

, `GENOTYPES`

, `SENSITIVITIES`

, `GENMEANS`

, `ENVMEANS`

, `ENVEFFECTS`

, `SESENSITIVITIES`

, `SEGENMEANS`

, `SEENVEFFECTS`

, `MSDEVIATIONS`

, `DEVIANCE`

, `DF`

, `TITLE`

, `YTITLE`

, `XTITLE`

, `EXIT`

.

### Method

The procedure uses iterative scheme (A) referred to in Digby (1979). The scheme has been generalized to deal with alternative distributions and link functions. First the environment effects are estimated with the sensitivity parameters set to 1, and then the procedure alternates between estimating the sensitivities with given environment effects and estimating environment effects with given sensitivities. Convergence is tested by comparing the maximum difference between old and new sensitivities against the criterion (default 0.001), but the maximum number of cycles (default 15) will not be exceeded. If the `MAXCYCLE`

option is set to 1, the result is an unmodified joint regression analysis; see Finlay & Wilkinson (1963).

### Action with `RESTRICT`

A restriction applied to the response variate will be taken into account. Residuals and fitted values will be formed only for the restricted subset of values. If levels of the factors are not represented in the restricted subset, then no results will be shown for those genotypes and/or environments. Do not restrict the environment or genotype factor differently to the response variate: results may then be incorrect.

### References

Digby, P.G.N. (1979). Modified joint regression analysis for incomplete variety × environment data. *Journal of Agricultural Science, Cambridge*, 93, 81-86.

Eberhart, S.A. & Russell, W.A. (1966). Stability Parameters for Comparing Varieties. *Crop Science*, 6, 36-40.

Finlay, K.W. & Wilkinson, G.N. (1963). The analysis of adaptation in a plant-breeding programme. *Australian Journal of Agricultural Research*, 14, 742-754.

Yates, F. & Cochran, W.G. (1938). The analysis of groups of experiments. *Journal of Agricultural Science, Cambridge*, 28,556-580.

### See also

Procedures: `AMMI`

, `GESTABILITY`

, `GGEBIPLOT`

, `RJOINT`

.

Commands for: Regression analysis, REML analysis of linear mixed models.

### Example

CAPTION 'RFINLAYWILKINSON example'; STYLE=meta VARIATE [NVALUES=170] yield READ yield 2.70 2.32 2.35 1.86 4.76 5.13 2.37 3.18 3.60 3.99 2.51 4.71 2.46 2.98 4.06 2.55 4.10 2.77 2.56 2.65 2.03 4.77 4.24 2.31 3.27 3.33 3.86 3.25 4.10 2.97 2.91 4.25 2.35 3.95 3.13 3.72 3.47 2.66 6.08 5.74 2.45 4.16 * 4.95 * * * * * * * 3.34 3.38 2.52 2.48 5.54 5.46 2.47 3.74 * 4.48 * * * * * * * 3.40 3.10 2.73 2.55 5.72 5.71 2.64 3.69 4.00 4.66 2.77 5.56 2.21 2.61 4.15 2.15 4.25 2.80 2.31 1.99 1.79 4.39 4.69 2.05 3.13 2.53 * 2.78 4.79 3.12 2.86 3.97 2.70 4.40 2.73 2.66 2.02 2.24 5.07 5.12 2.05 3.30 3.30 * 2.80 5.15 2.28 2.49 4.34 1.81 3.54 2.77 2.48 2.53 * * 4.93 2.37 * 3.00 * 2.72 * * * * * * 2.78 3.23 2.70 2.61 6.24 5.77 2.56 3.82 4.03 4.91 2.94 5.41 2.88 2.57 * 2.44 4.27 3.00 2.76 1.59 2.07 5.04 4.56 2.27 3.39 3.25 3.79 * * * * * * * : FACTOR [NVALUES=170; LEVELS=10] genotype & [LEVELS=17] site GENERATE genotype,site MODEL yield RFINLAYWILKINSON [PLOT=trellislines] GENOTYPE=genotype; ENVIRONMENT=site