Summary The paper describes the symbolic notation and syntax for specifying factorial models for analysis of variance in the control language of the genstat statistical program system at Rothamsted. The notation generalizes that of Nelder (1965). Algorithm AS 65 (Rogers, 1973) converts factorial model formulae in this notation to a list of model terms represented as binary integers. A further extension of the syntax is discussed for specifying models generally (including non‐linear forms).

The paper is in two parts. Part I presents results of a Monte Carlo randomization study of Papadakis's covariance method of NN analysis which show that (i) a non-iterated Papadakis analysis tends to be conservatively biassed; (ii) iteration of the analysis as suggested by Bartlett (1978) leads to substantial positive bias in the treatment F ratio; (iii) the method is very inefficient when there are substantial trend effects in the data. A theoretical explanation of these results is given.Part II describes a new method of NN analysis discovered by the first author and developed in collaboration with the co-authors. The method is essentially a "movingblock" analogue of classical forms of analysis for "fixed" blocks (or rows, columns). It avoids the defects of Papadakis's method and leads to approximately unbiassed analyses. It is nearly always and often substantially more efficient on average than classical analyses of complete or incomplete block experiments, and also more efficient than standard analyses of Latin or lattice square designs if there are appreciable row X column interactions in the data. New criteria of design for NN balance are described. Validity of the new method under randomization is demonstrated empirically with Monte Carlo studies.

The controversy concerning the fundamental principles of statistics still remains unresolved. It is suggested that one key to resolving the conflict lies in recognizing that inferential probability derived from observational data is inherently noncoherent, in the sense that their inferential implications cannot be represented by a single probability distribution on the parameter space (except in the Objective Bayesian case). More precisely, for a parameter space RI, the class of all functions of the parameter comprise equivalence classes of invertibly related functions, and to each such class a logically distinct inferential probability distribution pertains. (There is an additional cross-coherence requirement for simultaneous inference.) The noncoherence of these distributions flows from the nonequivalence of the relevant components of the data for each.Noncoherence is mathematically inherent in confidence and fiducial theory, and provides a basis for reconciling the Fisherian and Neyman-Pearsonian viewpoints. A unified theory of confidence-based inferential probability is presented, and the fundamental incompatibility of this with Subjective Bayesian theory is discussed.

No abstract

No abstract