Alvin D. Wiggins

Cut-off points for aggreate herd testing in the presence of disease clustering and correlation of test errors

Abstract In order to test if disease is present in a large herd, an investigator will often subject only a small sample of animals to a fallible diagnostic test. The herd is declared positive for disease if the number of test-positive animals is greater than or equal to a previously chosen cut-off value. Such a test, called an aggregate test, has a sensitivity and specificity that depends on the sample size, the cut-off point and the sensitivity and specificity of the individual test. It also depends on the distribution of the disease among the herds being tested and on the fact that factors such as herd-level seropositivity may cause some herds to be more prone to testing errors than others. In this paper, we use the beta-binomial distribution to model all these factors and thereby calculate and tabulate aggregate test sensitivities and specificities under a variety of conditions. Receiver operating characteristic (ROC) curve methodology permits the choice of optimum sample sizes and cut-off values. We also investigate the situation in which an investigator may be willing to miss detecting the disease if the prevalence in the herd is low. A compiled FORTRAN program for the calculation of aggregate test cut-off point properties, including positive and negative predictive values, is available from the authors.

A mathematical model relating the power law to the exponential law in biological turnover studies

Abstract Suppose that a research worker performs an experiment involving one dependent variable and one independent variable, say time. He may then plot his data on cartesian coordinate paper as part of a search for a simple relationship, hopefully linear. If the relationship is not sensibly linear in cartesian coordinates, he may decide to replot on semilog paper or even log-log paper. If the semilog plot is satisfactory he may then stop and argue for an “exponential law” relating his two variables. However, another researcher in similar circumstances may decide that his data do not “linearize” until the log-log plot, in which case he may argue for a “power law” relationship between the same two variables. The present paper derives a mathematical model of radioisotope retention, considered as a Markov process, to suggest that these two research workers may merely be looking at two different aspects of the same underlying phenomenon. Some remarks are made on the notion of the “biological half time”.

An inverse sampling scheme using blocks

Abstract A block of B animals, sampled from a herd of size H ⩾ B , is called a “reactor block” if at least one among the B animals reacts positively to a diagnostic test. To estimate r , the number of reactors in the herd, an inverse sampling scheme is proposed, in which, not one, but B (⩾ 1) animals at a time are tested simultaneously. The specimens from each of the B animals are not pooled as in some mass screening schemes. Rather, each of the B specimens retains its identity throughout the laboratory processng. For a finite population and a preassigned positive integer n , the joint probability law of K i , the number of blocks sampled from reactor block i − 1 up to and including reactor block i , and X i , the number of reactors in the i th reactor block, i = 1,…, n , is derived. The Neyman factorization theorem is invoked to demonstrate joint sufficiency of the statistics Σ n i =1 K i and Σ n i =1 X i . A dual interpretation of the problem, applicable to estimation of wild animal populations, is indicated. Sampling from an infinite population is considered, the appropriate joint probability law is derived, and joint sufficiency is shown to persist. Both maximum likelihood and Bayes estimates are derived for both the finite and infinite population cases. Taking into account cost considerations, an optimal choice of the block size B is derived.

Distribution-free confidence bounds for pr y< when fx and gy = fx-6 are continuous and symmetric

For and continuous and symmetric and differing at most by a shift parameter, distribution-free confidence intervals for are obtained by means of the Chebyshev inequality and an upper bound for the variance of the Mann-Whitney statistic. The (two-sided) intervals are reliable for small samples and about 20 to 30 per cent shorter than those obtained by Ury for and completely unknown for equal sample sizes, with larger savings otherwise. They are also shorter than the upper bounds obtained by Birnbaum and McCarty (1958) when the confidence coefficient does not exceed 0.95.

A reliability algebra of four-state safety devices

Abstract A “four-state device” can be exemplified by an electrical relay (two states, “energized” and “de-energized”, for the relay coil, times two states, “open” and “closed”, for the relay contacts). Such a device can be represented, abstractly, by a two-by-two Boolean matrix. By using these matrices as elements and by defining some binary operations by abstract representations of certain circuit configurations, an algebra is constructed. It is shown that this algebra contains Boolean algebra as a special case. A probability metric for a single four-state device is introduced by way of a simple random walk on four lattice points in the plane, which yields Markov transition probabilities. A simple example of a “network” is chosen, and the “reliability” (strictly speaking, the time-dependent probability of being in any of the four possible states) of the network is calculated. It is thus seen to be possible, at least in principle, to calculate the reliability of a relay contact network, taking into account the reliability of the coils which activate those contacts.

On the use of the directional derivative in obtaining multivariate extreme values

Current use of the directional derivative appears, with notable exceptions such as Whittle (1971, 1973) and Vainberg (1973), to be limited largely to textbooks on advanced calculus, and to spaces of at most three dimensions. The present paper develops a calculus of the directional derivative for arbitrary finite dimensional vector spaces. Applications are made to classical maximum likelihood estimation in the case of the multivariate normal density and to other multivariate problems involving stationary points.