Probability and Statistics - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 44

of variance. Most of this pioneering work was performed while he was at the
Rothamsted Experimental Station in the U.K. (1919-1933).
Fisher's methods entered most statistical textbooks during his lifetime and
continue to be taught widely. Degrees of freedom were a subject of disagreement
during the 1920s, mainly between Pearson and Fisher. This debate was decidedly
won by Fisher. For example, he established that in a chi-square test for goodness of
fit the number of degrees of freedom should be decreased by one for every statistical
parameter estimated. The mathematical proof of this simple rule is not at all simple.
Fisher illustrated the validity of his new result in a 1926 coup de grace administered
on the basis of 12,000 (2 2) contingency tables obtained under random sampling
conditions by E.S. Pearson, son of Karl Pearson. Using these data, Fisher calculated
that, on average, the chi-square for a 2

2 table contingency table has only one
degree of freedom instead of the three previously assumed by the Pearsons
( cf. Fisher Box 1978 ). These disputes in the 1920s illustrate that the formulae
derived by Fisher are not at all that easy to understand. However, many textbooks
on applications of statistics in science and engineering are easy to read because they
do not contain the formulae underlying the significance tests but only instructions
on how to test hypotheses by means of statistical tables such as those for the t- ,
chi-square and F - distributions. The idea of teaching simple rules only is that
practitioners should not be sidetracked by the underlying mathematics. There also
exist easy-to-read geostatistical topics such as those written by Isobel Clark ( 1970 )
and Isaaks and Srivastava ( 1989 ) but circulation and acceptance of these 3-D based
statistical ideas has been more limited.
It should be kept in mind that most of Fisher's techniques are applicable only if the
observations are independent. This requirement was well known to Fisher and other
mathematical statisticians including Kolmogorov ( 1931 ) who established the axioms
of probability calculus. Another consideration, which is easier to understand, is
that the assumption of normality (Gaussian frequency distribution curve) for the
numbers treated in significance tests has to be approximately satisfied. This is because
the tables for Student's t -test, the chi-square test for goodness of fit, analysis of
variance and several other well-known significance tests are based on random
variables that have frequency distributions derived from the normal distribution.
Rothhamsted continues to be an important research center for statistical research
and applications. The methods of Georges Matheron are now used in agricultural
research in addition to Fisher's methods; for example, Richard Webster ( 2001 ),
BAB Rothhamsted Research, published a widely read topic on “Geostatistics for
Environmental Scientists”. This post-Fisher development can be regarded as an
extension of traditional statistical theory based on random variables satisfying the
axioms set out by Kolmogorov ( 1931 ).

2.1.2 Spatial Statistics

Danie Krige ( 1951 ) in South Africa first advocated the use of regression analysis
to extrapolate from known gold assays to estimate mining block averages