Showing papers in "Journal of the royal statistical society series b-methodological in 1966"

A General Class of Coefficients of Divergence of One Distribution from Another

Abstract: Let P1 and P2 be two probability measures on the same space and let 0 be the generalized Radon-Nikodym derivative of P2 with respect to P1 If C is a continuous convex function of a real variable such that the Pl-expectation (generalized as in Section 3) of C(+) provides a reasonable coefficient of the Pl-dispersion of 0, then this expectation has basic properties which it is natural to demand of a coefficient of divergence of P2 from P1 A general class of coefficients of divergence is generated in this way and it is shown that various available measures of divergence, distance, discriminatory information, etc, are members of this class

A Generalized Least‐Squares Approach to Linear Functional Relationships

Abstract: SUMMARY A linear functional relationship between mathematical variables is postulated. The problem considered is the estimation of this relationship given observed random variate values, various assumptions being made about the distribution of the departures of these from the corresponding mathematical variable values. The least-squares principle is generalized to deal with cases in which these departures are correlated. The bivariate case is considered in detail. Extension to p variates involves no new principle if only a single linear relationship is postulated. A simple application in growth studies is discussed.

The Chi‐Square Test for Heterogeneity of Proportions, after Adjustment for Stratification

Abstract: SUMMARY In testing differences between proportions or rates of occurrence of events in different categories, expected numbers may be calculated separately within certain strata and subsequently summed over strata. For the Poisson case an approximate X2-statistic is derived as a simple modification of the usual test statistic. Other approaches and situations are discussed more briefly. IN many situations in which a number of proportions are to be compared the familiar x2-heterogeneity test needs to be modified to take account of some natural blocking or stratification of the observations. For example, to compare the proportions of individuals in m occupational groups who exhibit certain symptoms, one could apply the usual X2-test on m -1 degrees of freedom. It would, however, be sensible to take account of the age distributions of the individuals in the m groups because the prevalence of symptoms may be markedly affected by age. The individuals may therefore be subdivided into 1 age groups. A X2-test on mr-1 degrees of freedom could be carried out in each age group, but it would be useful to have a composite test, also on m -1 degrees of freedom, to assess the systematic differences between occupational groups within age groups. In vital statistics such problems are traditionally handled by methods of standardization, and these methods are widely used also in epidemiological survey work in problems similar to that referred to above. Standardization is, however, often performed without reference to sampling variation. In this paper I discuss methods of obtaining an asymptotically valid X2-test on mr-I degrees of freedom, with particular reference to the method of indirect standardization. Suppose that the main comparison is between m categories, denoting a typical category as the jth, and that the secondary subdivision is into 1 strata, of which a typical one is the ith. The observed proportion in the (i,j)th cell is N?>/t?, where Ni, is the number offailures and tij is the number of cases. The notation is shown in Table 1, in which marginal totals are denoted in the usual way except that we use n and t rather than n and t.. The distinction between capital and lower-case letters is adopted intentionally because in the statistical treatment only the Nij and Nj are regarded as random variables. In many epidemiological and vital statistical applications the NP4 may be regarded, apart from linear restraints, as Poisson rather than binomial variables. This may be either because the basic situation is as shown in Table 1, with all the failure-rates

A Derivation of the Probabilistic Explication of Information

Abstract: SUMMARY The quantitative explication of I(H: EJ G), the amount of information concerning H provided by E given G, is obtained by assuming that I(H: E. F i G) is some function of I(H: E G I ) and I(H: F I E. G). This assumption is less restrictive than the more customary assumption of strict additivity. LET E, F, G and H be propositions, and let I(H: EJ G) denote the amount of information concerning H provided by E given G. We shall consider the explication of I(H: El G) as a real number in terms of probabilities. We shall not discuss the theory of communication (Shannon and Weaver, 1949) in which expected amounts of information are relevant. We write E. F for the logical conjunction of two propositions E and F. Our conclusion, based on some very reasonable desiderata (called "axioms"), is that I(H: El G) should be defined as a continuous monotonic function of the association factor, P(H.EJ G)/{P(HJ G)P(EJ G)} or P(EJ H. G)/P(EJ G),

Cyclic Incomplete Block Designs

Abstract: SUMMARY It is shown that cyclic incomplete block designs for a given number of treatments (t) may be effectively reduced in number in view of equivalences which exist among designs of the same size. Designs are enumerated for 5 tS 15. 1. DEFINITION

A Bayesian Approach to Classification

Abstract: SUMMARY In this paper a branch of Bayesian statistical decision theory is investigated. A wide range of prediction problems in regression situations is considered within one statistical framework. A model is derived and applied to problems of classification.