Produces graphs and statistics for bivariate analysis of variance (R.F.A. Poultney).

### Options

`PRINT` = string tokens |
Controls printing of statistics from the bivariate analysis `(error` , `treatment` ); default `erro` , `trea` |
---|---|

`APRINT` = string tokens |
Controls output from the (univariate) `ANOVAs` of `Y1` and `Y2` (usual `ANOVA` print options); default `aovt` |

`TREATMENTSTRUCTURE` = formula |
Treatment terms to be fitted in the analysis of variance; this option must be set |

`BLOCKSTRUCTURE` = formula |
Block model defining the error terms in the analysis of variance; if unset, the design is assumed to be unstratified (i.e. to have a single error term) |

`TERM` = formula |
Single model term identifying the treatment term whose means are to be plotted |

`STRATUM` = formula |
Stratum from which to extract treatment information; default is to take the bottom stratum |

`FACTORIAL` = scalar |
Limit on number of factors in a treatment term; default 3 |

`PROBABILITY` = scalar |
Significance level to use in the calculation of the radius of the confidence region and the region of non-significance; default 0.95 |

`GRAPHICS` = string token |
Type of graphical output (`lineprinter` , `highresolution` ); default `high` |

`STYLE` = string token |
controls the style of axes in a high-resolution graph (`xy` , `none` ); default `xy` |

`LABELS` = factor or text |
Plotting symbols for the means; default is to take the letters A to Z, then a to z |

### Parameters

`Y1` = variates |
First variate for the bivariate analysis |
---|---|

`Y2` = variates |
Second variate for the bivariate analysis |

`TITLE` = texts |
Title for the graph |

### Description

`ABIVARIATE`

produces a bivariate analysis of variance with a graphical representation of the results, as described by Dear & Mead (1983, 1984). The procedure was developed from a Genstat 4 macro, further information about which is given by Poultney & Riley (1986), and is intended primarily for data from intercropping experiments. The variates to be analysed (specified by parameters `Y1`

and `Y2`

) are measurements, usually yields, taken on the two crops. The final parameter, `TITLE`

, defines a title for the graph.

The procedure will work for any of the designs that can be analysed by `ANOVA`

, except that there must be no pseudo-factors. Option `TREATMENTSTRUCTURE`

defines the treatment formulae for the analysis, and the block formula is defined by the `BLOCKSTRUCTURE`

option. `BLOCKSTRUCTURE`

can be omitted if there is a single error stratum (i.e. the analysis is of a completely randomized design). The `FACTORIAL`

option controls the number of factors in each treatment term, as in the `ANOVA`

directive.

First of all, `ABIVARIATE`

calculates a univariate analysis of variance for each of the variates `Y1`

and `Y2`

, with output controlled by the `APRINT`

option. The settings are the same as those in the `ANOVA`

directive; by default `APRINT=aovtable`

.

Output from the bivariate analysis of variance, which follows, is controlled by the `PRINT`

option. The setting `error`

generates the error summary statistics from the bivariate analysis: Error Sum of Products, Variances after Adjustment for Covariance, Correlation Coefficient between `Y1`

and `Y2`

, Radius of Standard Errors, Radius of Confidence Regions, and Radius of Non-Significance Regions. The setting `treatment`

produces the following statistics for each treatment term estimated within the specified error stratum: Treatment Sum of Products, Wilks’ Lambda, Bivariate F-Statistic.

The stratum from which the means (and other information) are to be taken is defined by `STRATUM`

option; if this is omitted, the lowest stratum is used. The significance level to use in the calculation of confidence regions is defined by the `PROBABILITY`

option; by default this is 0.95.

The `TERM`

option specifies a treatment term whose means are to be represented graphically. The means are plotted on axes transformed to allow for the variability in, and the correlation between, each crop variate. The plotting symbols can be defined as a factor or text using the option `LABELS`

. Alternatively they will be taken to be the first *n* values of the series A to Z, a to z where *n* is the number of means to be plotted. The graph can be either line printer or high resolution, the default being high resolution. The external axes of a high-resolution graph can be suppressed by setting `STYLE=none`

.

Problems arise in situations where the table of means to be plotted is incomplete; this can occur when a whole factor level is restricted out, or where the treatment structure is nested within a control. The length of the vector `LABELS`

is calculated as the number of cells in the table, including missing values. If `LABELS`

is declared, it must have length equal to the dimension of the table otherwise a fault will occur. Similarly, the calculation of the radius statistics is based on the assumption that the table of means is complete and has equal replication. These values, if printed, would be incorrect for a table with missing cells and so are suppressed. They can be calculated by hand as shown by Dear & Mead (1983).

Options: `PRINT`

, `APRINT`

, `TREATMENTSTRUCTURE`

, `BLOCKSTRUCTURE`

, `TERM`

, `STRATUM`

, `FACTORIAL`

, `PROBABILITY`

, `GRAPHICS`

, `STYLE`

, `LABELS`

.

Parameters: `Y1`

, `Y2`

, `TITLE`

.

### Method

(1) calculate the SSP matrix for all terms in the formula

(2) transform the variables such that the new set are uncorrelated and have unit error variance

(3) calculate new axes based on the maximum and minimum points of the transformed variables

(4) draw the graph of the transformed means with the axes rotated such that they are at the same angle to the vertical

### Action with `RESTRICT`

Variates `Y1`

and `Y2`

can be restricted, however this restriction must be identical for the two variates. Some problems may occur when whole levels of factors are restricted out leaving empty cells in the table of means to be plotted (see above).

### References

Dear, K.B.G. & Mead, R. (1983). The use of bivariate analysis techniques for the presentation, analysis and interpretation of data. *Statistics in Intercropping Technical Report No. 1.* Department of Applied Statistics, University of Reading, U.K.

Dear, K.B.G. & Mead, R. (1984). Testing assumptions and other topics in bivariate analysis. *Statistics in Intercropping Technical Report No. 2.* Department of Applied Statistics, University of Reading, U.K.

Poultney, R.F.A. & Riley, J. (1986). A Genstat Macro for the Bivariate Analysis of Intercropping Data. *Genstat Newsletter*, 17, 27-46

### See also

Directive: `ANOVA`

.

Commands for: Analysis of variance.

### Example

CAPTION 'ABIVARIATE example',\ !t('Data are from a melon and okra intercropping trial;',\ 'for further details see Poultney & Riley',\ '(1986, Genstat Newsletter 17).'); STYLE=meta,plain FACTOR [LABELS=!t(M1,M2); VALUES=8(1,2)3] Melon FACTOR [LABELS=!t(O1,O2); VALUES=4(1,2)6] Okra FACTOR [LABELS=!t(year1,year2,year3); VALUES=16(1,2,3)] Year FACTOR [LEVELS=4; VALUES=(1...4)12] Block FACTOR [LEVELS=12; VALUES=4(1...12)] Plot TEXT [VALUES='1','2','3','*','+','#'] Tsymbol VARIATE [NVALUES=48] MelonDat,OkraDat READ [SERIAL=yes] OkraDat,MelonDat 1.06 3.28 2.18 1.98 2.52 1.96 1.56 1.67 5.58 3.08 2.35 2.07 1.90 2.40 2.59 4.10 5.4 6.2 7.0 5.8 5.6 7.8 8.3 6.9 4.4 4.0 5.0 4.2 7.1 6.6 9.2 4.8 3.13 2.30 2.21 2.30 2.81 3.30 4.81 5.80 2.70 4.34 2.50 3.61 4.70 4.90 2.30 2.44 : 63.4 29.0 45.4 46.4 44.6 42.7 50.2 45.9 37.0 53.9 56.9 39.0 53.1 47.9 28.3 20.1 32.0 20.0 20.2 19.0 9.5 16.5 6.8 18.5 18.0 20.5 12.8 14.0 21.0 16.5 5.0 10.0 16.6 22.9 15.1 16.7 10.3 34.3 13.3 12.6 20.5 22.7 14.8 11.8 20.1 19.4 23.3 21.1 : ABIVARIATE [TREAT=Melon*Okra*Year; Block=Block/Plot; TERM=Okra.Year;\ LABEL=Tsymbol] OkraDat; MelonDat; TITLE='melon and okra data'