ABIVARIATE procedure

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


Options

PRINT = strings
Controls printing of statistics from the bivariate analysis (error, treatment); default erro, trea

APRINT = strings
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
Type of graphical output (lineprinter, highresolution); default high

STYLE = string
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