ANALYSIS OF RESPONSE SURFACES


 Statistical tests of response surfaces are based on
three measures of variations in the dependent variable
(frequently yield): 1) the variation explained by
regression, 2) unexplained variation, and 3) total
variation or the sum of 1) and 2). A surface relating the
response of rice to N can be described as:

 Y = f(NIP, K, water, soils, management, etc.),

where N is the variable input and those factors to the
right of the vertical line are fixed or non-variable.
In the generalized function, there are terms relating the
yield to quantities of N and there is an error term:

 Y = a + bi N + b2 N2 + e

Included in the error term (e) are errors in measurement
of the quantity of N applied (although this is assumed to
be precise); error in controlling levels of P, K,
irrigation, etc.; differences in management; and other
factors, such as incidence of pests that affect some plots
more than others. Also included in the error term is the
error which results because the form of the curve utilized
does not perfectly fit the curvature of the biological
phenomenon. In summary, explained variation is comprised
of a + blN + b2N2, and unexplained variation is contained
in the term e.
 Total variation in yield is measured by the sum of
the squared deviations of yield from the average yield,
designated asry2. The deviations from the mean value of
Y are shown in Fig. VI-6a. Explained variation (variation
due to regression) can be shown as:



 E9122 = E{bi(Zxiy)}

Unexplained variation (the sum of squared deviations from
the regression) is:

 Z(Y Y)2 = Zy2 9122