IN selection with probability proportional to size x from within strata without replacement, the usual method of selection gives rise to bias in the estimate of the total of a variate y derived by weighting the units by weights proportional to i]x, By means of numerical examples it is shown that the amount of this bias is usually quite trivial. If, however, unbiased estimates are required, the true total probabilities of selection of the different units can be calculated easily for samples of 2, and with considerably more labour for samples of 3.The bias in the ordinary formula for the estimation of error is also investigated, and the formula is shown to be reasonably accurate. Horvitz and Thompson have given an unbiased estimator of the error variance, but this is shown to be inefficient and a new unbiased estimator is given.A method of revising the size measures so that with the usual method of selection the true total probabilities of selection are proportional to the original size measures is given for samples of 2. Horvitz and Thompson's solution of this problem does not appear to give satisfactory approximations in the cases met with in practice.The selection of successive members of a sample with arbitrary sets of probabilities chosen solely so that the total probabilities shall be proportional to the original size measures, which has been advocated in various quarters, is criticized.* Throughout the paper the formulae are givenfor one stratum only. In practice, of course, summation over all strata will be required.t In Section 7. 16 it is stated that the formulae for the estimation of error are approximate because the probability of selection is not strictly proportional to size. As will be seen from the present paper there are other reasons why the formulae are approximate.

SummaryThe molecular pathogenesis of type 1 von Willebrand disease (VWD) is uncertain in most patients. We examined 30 type 1 VWD families in the UK Haemophilia Centre Doctors' Organization study. Heterozygosity for Y/C1584 was present in eight of 30 (27%) families and 19 of 76 (25%) individuals with type 1 VWD recruited into the study. Eighteen (95%) of these 19 individuals were blood group O. C1584 did not co-segregate with VWD in four families, and co-segregated in one family; the results were equivocal in three families. In all families increased susceptibility of von Willebrand factor (VWF) to a disintegrin and metalloprotease with thrombospondin motifs (ADAMTS) 13 proteolysis co-segregated with C1584 in affected and unaffected individuals. These data show that C1584, associated with blood group O, is prevalent among patients with type 1 VWD but not necessarily causative of disease and should not be used in isolation to diagnose VWD. Increased susceptibility of C1584 VWF to ADAMTS13 proteolysis may be physiologically significant and increase an individual's risk of bleeding and presenting with VWD.

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WHEN a new process is suggested for use in agriculture or technology it is usually necessary to carry out experiments to estimate the increase in output that would result if the new process replaced that in current use. If the cost of experimentation and the scale of potential application of the new process are known, one method of arriving at an optimum amount of experimentation is to minimize the total risk given by the sum of the cost of the experiment plus the expected loss due to wrong decisions. The main difficulty is that the last quantity depends on the true increase in output due to introducing the new treatment, which can only be estimated from the results of the experiments. The paper discusses the case where an initial experiment is carried out, and it is required to decide whether to accept or reject the new treatment at once or else to carry out further experimentation. In the latter case the optimum amount of further experimentation has also to be decided. A solution is provided by minimizing the risk' after eliminating the unknown parameter by averaging over its fiducial distribution based on the evidence from the initial experiment. Means for applying the resulting decision rule in practice are provided, and its performance under various circumstances is discussed.

A disjunctive combination of a finite set of two-person games Γ1, Γ2, …, Γk may be defined thus: The players play alternately, each in turn making a move in one and only one of the individual games. If, in addition, the conditions are imposed that(i) a player loses if unable to move (in any game),(ii) the games are impartial, i.e. the allowable moves from any position do not depend on which player is about to play (or on the previous moves, though these can be ‘built in’ to the position if necessary),(iii) the games are of bounded play, i.e. for each game Γi corresponding to any initial position Pj there is an integer bij such that the game must terminate after at most bij moves, then Grundy (6) has shown that there is a function G(P) (called by him Ω(P)) of the positions P with the following properties:(1a) G(P) = 0 for a terminal position, from which no move is possible; for other positions G(P) is the smallest non-negative integer different from all values of G(Qi), where there is a permissible move from P to Qi,(1b) for a disjunctive combination of games, G for the combined position is the nim-sum of the G's the individual positions. By the nim-sum, we mean that each separate G is to be written in the scale of 2, as Σar 2r, and then in forming the sum, the ar's are to be added mod 2 for each value of r, as in the theory of Nim((1), (7), (8), pp. 36–8).