##### Abstract

The statistical procedure of analysis of variance
was invented by R.A. Fisher during his stay as
statistician at Rothamsted Experimental Station. His
first, more or less tentative, discussion of the theory
was set forth in a paper published in 1923 (11), and this
was quickly followed up by the more assured and much more
complete exposition in his book "Statistical Methods
for Research Workers" (12), which revolutionised previous
ideas on the principles of scientific experiment.
Little additional work was published on the sllbject
until 1933, but since then many workers, among whom
may be mentioned M.S. Bartlett, W.G. Cochran, J. Wishart,
and above all F. Yates, the present chief statistician
at Rothamsted, have developed the theory on the lines
laid down by Fisher. impetus was given to this
development especially by the publications of Yates
and of Fisher himself (13), in which the new methods
of factorial design, confounding, and covariance
introduced at Rothamsted were first made more generally
known. Fisher's theories met with spasmodic opposition
from statisticians such as "Student ", Neyman, and others,
but have triumphed over all opposition and today are the
basis of almost all scientific experimental work amenable
to statistical treatment.

Nevertheless one would look in vain throughout
the literature for any rigorous and at the same time
reasonably simple mathematical treatment of the theory
of analysis of variance. Fisher's own exposition is
for the most part seemingly intuitive, being designed
for the non- mathematical reader, as are for the most
part the papers of Yates. Modern text -books such as
Snedecor's "Statistical Methods" (28) present the
methods without the theory behind them and appeal to
the intuition of the reader. Where proofs are
attempted, vital points are usually glossed over or
assumed, as being beyond the scope of an elementary
book. Among the very few British mathematical papers
on analysis of variance are those of Irwin (15,16), but
his treatment is complicated and unwieldy. Cochran (6)
realised the advantages of matrix notation in a subject
of this sort, and many of his theorems are equivalent
to the lemmas of this thesis, but Cochran left the
application of his method undeveloped.

The present thesis constitutes an attempt to put
forward a progressive mathematical theory of analysis
of variance as applied to the various situations met
with in agricultural research in particular, but the
applications are, of course, perfectly general. Matrix
notation has been used throughout to simplify a subject
which would otherwise prove rather unwieldy for
mathematical treatment. The basic theories are those
of Fisher, Yates, etc., and are now so generally
accepted as to require no special references.
Acknowledgment by reference is therefore made only in
the case of specific points where this has seemed
necessary.