and ranges


                                          y10 = klxO

                                          yll =  klxl      )




                                   Y2o = k, X20 -  VL2)



      Plot the four pairs of y values O(ylo. y2o), B(y1o, y21) and Cyl. y2) on standard rectangular axes,
      using the same scale for yI and for Y2. The xI axis is constructed by joining the points 0 and A. the x2
      axis by joining O and B. The xi scale is defined by O(xl - xlo) and A(x! = x; I); the x2 scale is
      defined by (0x2 = x20) and B(x2 = x2I). Further points on both axes may be marked using a ruler and
      the two defining points. The rotation of the x! and x2 axes to achieve symmetry can be performed
      subjectively or by simple trigonometry. Individual points for pairs of mean yields may be plotted by
      first measuring x1 along the xi axis, and x2 parallel to the x2 axis. More details of the diagram
      construction are given by Dear and Mead (1983, 1984).

The interpretation of the diagrams is extremely straightforward. The results in Fig. 2 show that the differences among the three maize varieties are important for both maize and cowpea yields, with the difference between varieties 2 and 3 clearly less than between either variety and variety 1. There is a clear consistency through the sequence of varieties I to 2 to 3, with the increase in maize yield being directly reflected in a decrease in cowpea yield. The three points fall nearly on a line illustrating the strong relation between the two crop yields over the three varieties. (Note also that the correlation for maize varieties, shown in Table 3, is -0.98). Remember that random correlation between the two yields has been allowed for by the skewness of the axes and the displayed pattern is additional tot he background correlation pattern.

The results for nitrogen main effects and the interaction of cowpea variety with nitrogen are shown in Figs. 3 and 4. The four nitrogen levels produce four pairs of mean yields in an almost straight line. The dominant effect is on the yield of maize which increases consistently with increasing nitrogen. In addition there is a clear pattern of compensation between the two crop yields with cowpea yield decreasing as maize yield increases. The pattern of yields for the cowpea variety/nitrogen interaction emphasizes the two effects of yield increase for one crop and compensation between crops. For variety A the effect of increasing nitrogen is simply an increases of maize yield, the "line" of the nitrogen level means being almost exactly parallel to the maize yield axis. In contrast for variety B the dominant effect is the change in the balance of maize/cowpea yields with the maize yield increasing consistently with increasing nitrogen and the cowpea yield showing a corresponding decline.

3.4 Significance Testing

There are two forms of test that are useful in bivariate analysis, and these correspond to the t and F tests used in the analysis of a single variate. We have already mentioned in the discussion of the skew axes plot that the standard error of a difference is the same in all directions in these diagrams. because of the scaling