3.2 The Form of Bivariate Analysis


The calculations for a bivariate analysis are formally identical with those required for covariance analysis. The difference is that, whereas in covariance analysis there is a major variable and a secondary variable whose purpose is to improve the precision of comparisons of mean values of the major variable, in a bivariate analysis the two variables are treated symmetrically. Bivariate analysis of variance consists of an analysis of variance for X I, analysis of variance for X2, and a third analysis (of covariance) for the products of X1 and X2. Computationally this third analysis of sums of products is most easily achieved by performing three analyses of variance for X 1, X2, and Z =X I + X2. The covariance terms are then calculated by substracting corresponding SS for X I and for X, from that for Z and dividing by 2. The bivariate analysis including the intermediate analysis of variance for Z are given in Table 3 for the maize/cowpea experiment discussed earlier.

The bivariate analysis of variance, like the analysis of variance, provides a structure for interpretation. In addition to the sums of squares and products for each component of the design. the table includes an error mean square line which provides a basis for assessing the importance of the various component sums of squares and products. The general interpretation of this analysis is quite clear and is essentially similar to the pattern of analysis of cowpea yield. There are large differences attributable to the different maize varieties and to the variation of nitrogen level; there is also a suggestion that there may be an interaction between cowpea variety and nitrogen level.

Table 3. Bivariate Analysis of Variance for Maize/Cowpea Yield Data (0.001 kg/ha) in Intercrop Trial


                     Maize SS       Cowpea SS         SS for       Sum of
   Source        df     (X1)           (X2)         (XI + X2)     products       F     Correlation


Blocks            2      0.29         0.0730          0.247        -0.058       1.75       -0.40
M variety         2     17.52         0.4094         12.665        - 2. 632    11.90       -0.98
C variety         1      0.03         0.0060          0.062         0.013      0.44         1.00
Nitrogen          3    28.50          0.1131         25.081        -1.766      10.59       -0.98
MxC               2      1.11         0.0099          0.922        -0.099       0.82       -0.95
Mx N              6      1.25         0,0676          0.920        -0.199       0.64        0.93
CxN               3      0.24         0. 1724         0.152        -0.130       2.40       -0.64
MxCxN             6      1.28         0.1354          1.349        -0.033       1.40       -0.08
Error            46     15.90         0.5993         13.671        -1.414                  -0.46
  (MS)                  (0.346)       (0.0130)                    (-0.031)
Total            71    66.13           1.5861        55.080        -6.318


Note: See Table 1

3.3 Diagrammatic Presentation

We have argued earlier that interpreting the patterns of variation in maize and cowpea yields without allowing for the background pattern of random variation can be misleading. The primary advantage of the bivariate analysis is that it leads to a simple form of graphic presentation of the mean yields for the pair of crops making an appropriate allowance for the background correlation pattern. The graphic presentation uses skew axes for the two yields instead of the usual perpendicular axes. If the yields are plotted on skew axes with the angle between the axes determined by the error correlation, and if, in addition. the scales of the two axes are appropriately chosen, then the resulting plot, such as Fig. 2. has the standard error for