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# RJOINT procedure

Does modified joint regression analysis for variety-by-environment data (P.W. Lane & K. Ryder).

### Options

`PRINT` = string tokens What to print (`model`, `summary`, `estimates`, `monitoring`, `graph`); default `mode`, `summ`, `esti` Overall title for graph Y-axis title for graph X-axis title for graph Convergence criterion; default 0.001 Maximum number of cycles; default 15 Save structure from `MODEL` statement defining the model; default is to use the structure from the latest `MODEL` statement

### Parameters

`ENVIRONMENT` = factors The environment factor; no default The variety factor; no default To store estimates of sensitivities; default `*` To store estimates of variety means; default `*` To store estimates of environment effects; default `*` To store estimates of environment means; default `*` To store s.e.s of sensitivities; default `*` To store s.e.s of variety means; default `*` To store s.e.s of environment effects; default `*` To store the residual deviance To store the residual d.f Exit status – set to 0 if the analysis converged, 1 otherwise

### Description

Procedure `RJOINT` performs a modified joint regression analysis of data classified by two factors. This analysis is motivated by the study of variety-by-environment interactions in agriculture, where the two factors are varieties of some crop and environments at which experiments were carried out. The environments may be different sites within the same year, different years for the same site, or, as is more common, a combination of the two with little interest in individual year and site contributions. The intention is to characterize the sensitivity (or, inversely, the stability) of each variety to environmental effects by fitting a regression of the environment means for a variety on the average environment means. The model is thus nonlinear, of the form

yij = vi + bi × ej + error

where vi are variety means, ej are environment effects (with ∑ej =0) and bi are the sensitivity parameters (with mean(bi )=1). Usually, an experimenter is looking for varieties with large means and small sensitivities, to ensure a reliable crop under variable conditions. In `RJOINT` the factors are specified using the parameters `VARIETY` and `ENVIRONMENT`.

The data may consist of one value of the response, such as yield, for each combination of variety and yield. More often, the data are incomplete because not all varieties are tested at each environment; also, there may be multiple measurements of varieties at some environments. If the response is a count or a proportion, such as 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 should be specified by giving a `MODEL` statement before calling `RJOINT`; 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 converger and one otherwise.

Output is controlled by the option `PRINT`. The setting `model` prints a description of the model. The setting `summary` displays an analysis of variance (or deviance for non-Normal distributions) showing the effects of Varieties, Environments and Sensitivities: the last is the effect of allowing different sensitivities for each variety. The setting `estimates` displays two tables, one classified by varieties and the other by environments. For varieties the columns are: unadjusted means, final estimates of means (on the scale of the link function, if relevant), standard errors of estimates, back-transformed means (using the inverse of the link function), sensitivities, and standard errors of sensitivities. For environments the columns are: 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 variety means), and back-transformed means (using the inverse of the link function). Finally, the setting `graph` plots the model. The `TITLE` option defines the onerall title for the graph; the default is Joint regression analysis. The `YTITLE` and `XTITLE` options define titles for the y- and x-axes, respectively; the default for the y-axix is the name of the y-variate, and the default for the x-axis is the name of the `ENVIRONMENT` factor.

The remaining parameters allow the following results to be saved: sensitivities, variety means, environment effects, environment means, and standard errors of sensitivities, variety means, environment effects, 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`, `TITLE`, `YTITLE`, `XTITLE`, `TOLERANCE`, `MAXCYCLE`, `SAVE`.

Parameters: `ENVIRONMENT`, `VARIETY`, `SENSITIVITIES`, `VARMEANS`, `ENVEFFECTS`, `ENVMEANS`, `SESENSITIVIT`, `SEVARMEANS`, `SEENVEFFECTS`, `DEVIANCE`, `DF`, `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 varieties and/or environments. Do not restrict the environment or variety 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.

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

Procedures: `AMMI`, `GESTABILITY`, `GGEBIPLOT`. `RFINLAYWILKINSON`.

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

### Example

```CAPTION  'RJOINT example'; STYLE=meta
VARIATE  [NVALUES=170] 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] variety
&        [LEVELS=17] site
GENERATE variety,site
MODEL    yield
RJOINT   ENVIRONMENT=site; VARIETY=variety
```
Updated on June 18, 2019