THIS paper attempts to give a systematic derivation of the principal results of the analysis of block experiments using matrix notation. It also shows how the problem of design can be formulated in an analytical form and illustrates the type of mathematical problem that arises in such a formulation.The systematic approach to the analysis of experiments enables methods to be evolved for performing these analyses on automatic calculating machines and to facilitate this the paper discusses the analysis of spoilt experiments and the iterative solution of the equations arising in the recovery of inter-block information.
TOCHER-The Design and Analysis of Block Experiments[No.1,
Matrices of this type constantly occur and we define the left-hand side of this equation as {Zn(A)}--l.Generally the suffix n can be suppressed. We have shown that {Zn(A)}-l = Zn(-A/I +n~). This can also be deduced from the results Zn(A)Zn([.l) = ZnCA + [.l + A[.ln) Zn(O) = lOrn].
1952]We also have TOCHER-The Design and Analysis 0/ Block Experiments 47 IXZ().) +~Z(IJ·) = (IX +~)Z«IX). +~f-l)/(IX + B».We shall require I Zn().) I. This can best be found by premultiplying Z by an n X n orthogonal matrix y with first row (I/Vn)l' and post-multiplying the result by y'. Then
SUMMARYThis paper describes some of the problems that arise when automatic computers are used for conducting sampling experiments. The generation of random elements is discussed in detail and some methods of producing random variables with common distributions are described. The use of sampling methods to evaluate multivariate integrals is discussed. Finally a programme is devised to conduct a special form of a restricted random walk.
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