of axes which is part of the instruction of the diagram the standard error per observation is I (measured in the units of yl and Y2). The standard error of a mean of n observations is therefore 14n and the standard error of a difference between two points is 4I(21n.)



                                       Nitrate Levels


1200


7000


5000


Cowpea


Maize


3000


Figure 3. Bivariate plot of pairs of mean yields for four nitrogen levels (0, 40, 80. and 120 kg/ha). Maize and cowpea yields are in kilograms per hectare. (Data from Table 1).

Confidence regions for individual treatment means can be constructed as circles with radius 'I(2FIn), where F is the appropriate percentage point of the F distribution on 2 and e degrees of freedom (e is the error degrees of freedom).

The analogue of the F test in a univariate analysis of variance is also an F test. The basic concept on which the test is based is the determinant constructed from the two sums of squares and the sum of products. Suppose that the error SSP are E,1, E2, and El 2, then the determinant is


                                            El x E2 - E~2

and it reflects both the sizes of El and E2 and the strength of the linear relationship between x 1 and x2. To asses the treatment variation for a treatment SSP with values TI, P), and T12 we calculate a statistic, L, which compares the determinant of treatment plus error with that for error

                               L =(TI + E1)(T2, + ED) - (T12 + E12)' E1E2 -E~2
The test of significance then involves comparing

                                           F =(VE
                                                             t