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The article presents an appendix note on a statistical question raised in the preceding paper.

The article presents a discussion about dialectic materialism and modern science. It discusses the conditions in which dialectic materialism creates for scientists and for scientific pursuits. Dialectic materialism holds that internal contradiction are inherent in all things and phenomena of nature, and treats reality being in constant flux and change. The relationship and the contributions of dialectic materialism in physics, theory of relativity, mathematics, and quantum theory are also discussed.

The Bradley-Terry model for a paired-comparison experiment with t treatments postulates a set of t 'true' treatment ratings π1, π2, ..., π1 such that π1 is ≥ 0, σpi;i = 1 and the probability for preferring treatment i to treatment j is π2 (π2 + πj)-1. Thus, according to this model, every comparison of two treatments results in a definite preference for one of the two. This is an unrealistic restriction since when there is no difference between the responses due to two treatments, any method of expressing preference for one over the other is somewhat arbitrary. This paper considers a modification of the Bradley-Terry model by introducing an additional parameter, called threshold parameter, into the model. This permits 'ties' in the model. The problem of estimation and tests of hypotheses for the parameters of the modified model is also dealt with in the paper. [ABSTRACT FROM AUTHOR]

The article describes a process for determining migration probability. A stochastic process may be defined as a process which develops over time and where events at time 2 are not independent of events at time 1. This lack of independence may be viewed in probabilistic terms. Since the application of stochastic probability processes to problems of predicting individual behavior requires data obtained from case studies where each individual can be classified with respect to a given set of categories at successive points in time, the projected analysis required a set of individual longitudinal migration histories. Consequences of the basic mechanism of the model utilized in the paper are also detailed.

Merton points out that "the most precise significance of the word function is of course found in mathematics …. " But, says Martindale, "The work of most of the recent functionalists has been at the opposite pole from this meaning." Consequently, assertions made by functionalists have not always been received with the degree of confidence that their authors might consider appropriate. It is the purpose of this paper to suggest a way in which qualitatively formulated propositions of a functional kind may be translated into a language which points to empirical referents and thus, if all goes well, leads in the end to the formulation of mathematical equations. [ABSTRACT FROM AUTHOR]

The article presents information on several books. Some of the books are "Problems of Historical Psychology," by Zevedie Barbu and "Developments in Mathematical Psychology. The book consists of studies of three mathematical formulations that have proved of interest to psychologists during the past fifteen jean: Information theory, in which an attempt is made to characterize certain summary properties of time invariant choices; Mathematical learning theories, in which the goal is to describe the systematic changes of choices that result from environmental responses to them; Quasi-linear theories of tracking, in which a person who is trying to follow closely a continuously moving target is treated as a slightly noisy linear system. The author delineate the problem areas being treated and Introduce the basic mathematical ideas used, cite some of the major mathematical results, and survey and criticize the more important papers in which the theories are confronted with experimental data. Another book is "Industrial Productivity and Motivation."

It has been known for some time that male and female reproduction rates may give widely divergent results, even when calculated from the same population. In this paper the author examines the mathematical relationships between the male and female reproduction rates in the stable population and obtains the conditions under which the same true rate of natural increase would be yielded both by the male and female rates. He shows that these differences may ultimately be reduced to differences in the nuptiality of the two sexes and illustrates his theory by an arithmetical example. [ABSTRACT FROM AUTHOR]

A multiple-choice item is called ideal if all its alternatives are equally attractive to every person who cannot answer the item correctly without guessing. This paper discusses reasons for attempting to construct ideal items, develops methods for testing whether items are ideal or at least more nearly ideal than other items, and presents examples of both ideal and nonideal items. [ABSTRACT FROM AUTHOR]

The purpose of this paper is the presentation of formulae for obtaining matrix derivatives of the second order to use in making tests for maxima and minima. The theory of such second order derivatives is presented. These formulae require the rearrangment of the parameter elements in vector form and the transformed results feature Kronecker products, which have certain desirable properties. Application is made to several types of problems. [ABSTRACT FROM AUTHOR]

In this paper the first two moments of the Statistic (treatment sum of squares)/(treatment +error sums of squares) over all possible random assignments of treatments to the experimental plots are obtained for two associate PBIB designs. They are compared with the corresponding moments of a central beta distribution to study the extent to which the normal theory test may serve as an approximation to the randomization test. It is found that the approximation is reasonable for some classes of PBIB designs. [ABSTRACT FROM AUTHOR]

In many scientific investigations, it is desired to bring together, or link, two or more documents which represent the same individual, even though these documents do not contain a unique identifier and were derived from different sources. In medical and public health research and elsewhere, this problem is known as the document linkage problem. This paper considers some aspects of classifying pairs of documents into one of two populations when their items are identifying information, where each item of information can take on three distinct values correct, incorrect or missing. Section I identifies three document linkage problems. Sections 2 and 3 deal with the mathematical formulation of the multivariate document linkage problem. Section 4 gives the classification procedure and Section 5 deals with the application of the theory to a problem in the field of public health. [ABSTRACT FROM AUTHOR]

Economic statisticians do not enjoy an untarnished reputation for accurate forecasting. We have managed, over the years, to come up with some memorable failures. While we have also had our share of successes, they are not as well remembered nor as numerous as we should like. Recently, however, we have begun to pay more attention to the record, and a substantial body of evidence on forecasting performance has accumulated. In this paper I propose to review this record, try to arrive at a balanced appraisal, and offer some suggestions for improvement. [ABSTRACT FROM AUTHOR]

This paper deals with methods of "cluster analysis". In particular we attack the problem of exploring the structure of multivariate data in search of "clusters". The approach taken is to use a computer procedure to obtain the "best" partition of n objects into g groups. A number of mathematical criteria for "best" are discussed and related to statistical theory. A procedure for optimizing the criteria is outlined. Some of the criteria are compared with respect to their behavior on actual data. Results of data analysis are presented and discussed. [ABSTRACT FROM AUTHOR]

This paper describes certain quota sampling procedures and attempts to show that they are very close to traditional probability sampling. Quotas are shown to depend on availability for interviewing and evidence is presented to show that sex, age, and employment status are reasonable predictors of availability. Quota sampling methods are not unbiased but data are presented which suggest that the bias is generally of the order of 3 to 5 per cent. It is shown, however, that the cost differentials between these quota samples and call-back samples are small. The major advantage of this new procedure may well be the speed with which interviewing may be completed during crises such as the Kennedy assassination. [ABSTRACT FROM AUTHOR]

In this paper we review the literature on the problem of handling multivariate data with observations missing on some or all of the variables under study. We examine the ways that statisticians have devised to estimate means, variances, correlations and linear regression functions from such data and refer to specific computer programs for carrying out the estimation. We show how the estimation problems can be simplified if the missing data follows certain patterns. Finally, we outline the statistical properties of the various estimators. [ABSTRACT FROM AUTHOR]

In this paper confidence bands are given for simple linear model. The confidence bands that are given are straight lines rather than curves and are either (1) parallel or (2) trapezoidal. The confidence bands are over a finite length interval. [ABSTRACT FROM AUTHOR]

A common problem in many areas is to achieve a combination of control variables which will simultaneously maximize all of the responses which are functions of the control variables. The fact that in general we can not simultaneously maximize all of the responses has led to the formulation of compromise solutions to this vector maximum problem. A point in the control variable space is better than a second point if each response at the first point is greater than or equal to the corresponding response at the second point. If a point is such that there is no better point, it is said to be admissible, or efficient. A set of points is complete if given any point not in the set there is a point in the set which is better. In this paper this formulation is used to consider the allocation of sample size to several strata when there are several characteristics of interest. [ABSTRACT FROM AUTHOR]

This paper considers some problems in the probabilistic analysis of physical evidence in criminal investigations. Two basic assumptions are made: (1) That the number of persons or objects possessing a particular set of properties can be considered as a random variable, and (2) that it is possible to estimate the probability function of this random variable. Two models (one with and one without the assumption that the suspect is a random selection from the set of possible suspects) which arc applicable to the evaluation of partial transfer evidence--an important category of physical evidence that is found in most criminal investigations--are developed. A detailed analysis of the models and an example are presented for the case when the estimated probability distribution is binomial with an expected value less than 1. As the expected value becomes smaller, the assumption of randomness in the selection of the suspect becomes immaterial to the evaluation of the evidential significance. [ABSTRACT FROM AUTHOR]

Focuses on the numerical value symbols on maps. Way of giving number identification to the stimulus; Selection of value screens for map use; Cartographic aspect of the test result.

Presents a response to Frank W. Gery's paper "Mathematics and the Understanding of Economic Concepts," published in the September 1970 issue of the "Journal of Economic Education." Discussion on the impact of mathematics training on economics achievements; Explanation for the coefficient of mathematics training; Arguments on the use of mathematical equations; Estimation of biases in the mathematical coefficients.

The article outlines the significant contributions of Russian mathematicians throughout history and the leading roles they have always played in the advance of mathematical theory. Among these scientists were Leonard Euler and Nicholas Lobachevsky, discoverer of non-Euclidian geometry. The author contends that Russian mathematics is untainted with any political and socio-economic bias. Most of their mathematicians were active in all fields of mathematical research, specifically applied mathematics. They also liked to publish books and mathematical journals. Nevertheless, they did not have enough freedom in making personal contacts outside Soviet Union.

This article offers a practical guide to goodness-of-fit tests using statistics based on the empirical distribution function (EDF). Five of the leading statistics are examined-those often labelled D, W², V, U², A²-and three important situations: where the hypothesized distribution F(x) is completely specified and where F(x) represents the normal or exponential distribution with one or more parameters to be estimated from the data. EDF statistics are easily calculated, and the tests require only one line of significance points for each situation. They are also shown to be competitive in terms of power. [ABSTRACT FROM AUTHOR]

This article documents the biases in using sample arithmetic or geometric means of one-period returns to assess long-run expected rates ot return. The formulas developed are applicable to other compound growth processes. For types of distributions of one-period returns likely to be encountered for bonds and stocks, numerical values for these biases are given. Four unbiased estimators of long-run expected rates of return are developed and their relative efficiency examined. [ABSTRACT FROM AUTHOR]

A number of attempts have been made to estimate the mean fecundability of samples of women, but all seem to be subject to reservations. In the U.S. 1965 Fertility Survey, a carefully selected representative sample of U.S. women of childbearing age were interviewed. They were questioned about their contraceptive history and about the dates of their pregnancies. It is reasonable to assume that of these pre-nuptially pregnant respondents falsely claiming a nuptial conception, the great majority would claim that they had conceived immediately after marriage, viz, in the first month. Hence, one would estimate that the mean delay between first exposure and conception in the case of women genuinely non-pregnant at marriage was not 10.4 but 11.4 months. Arithmetic mean fecundabiity at marriage of U.S. non-sterile couples is estimated at about 0.15; it is suggested that this value declines roughly linearly to about 0.04 at age 40. The decline is due to a decline in coital rates and to an increase in unrecognized spontaneous abortion.

Where Youden's extreme rank sum test is used, an early decision is often possible by performing the trials in sequence with the following method. After each trial is completed, the greatest lower bound and least upper bound of both extreme rank sums are calculated for all remaining trials. If the critical regions cover either or both intervals of the statistics or the intervals lie completely outside the critical regions, the round-robin test is terminated with the acceptance of the proper hypothesis. [ABSTRACT FROM AUTHOR]

This article discusses some mathematical elements in social sciences. The mathematical and statistical literature which can be classified as being of interest to social scientists is continually increasing - and increasing progressively. Business economics had a comprehensive literature about conflicting interests between firms, and sociology had an equally comprehensive literature about conflicts between groups and between persons in a group, finally psychology had a treatment of the individual decision-maker's conflict state in a situation in which a decision is to be made in choosing among several uncertain alternatives. The numerical solution is a combination of available information and rules of calculation resulting in a concrete alternative. In mathematics, tradition has formerly left the latter solution in the cold, but in recent years it has returned to favor in connection with the development of the simulation method, which may enable social science to make experiments to a much higher degree and in so doing get a better approximation to the optimal situation wanted.

This article highlights the limitations of Philip W.Porter's point on minimum travel. Porter had proposed that for any distribution, the point of minimum aggregate travel would lie on a line bisecting the acute angle formed by the two regression lines. The actual distance of Porter's point form the mean point is some fraction of the distance of the orthogonal median line from the mean point. However, his point is not the median or point of minimum aggregate travel, because it depends on the two linear regressions, which in turn depend on the choice of coordinate axes.

Analyzes the value of interrupting a map projection. Maximum angular deformation; Mean deformation for the land areas; Requirements for the interruption of oval equal-area projections.

Conventions exist for carrying out the arithmetic of population projection, finding the intrinsic rate of natural increase, and other demographic calculations. These are good approximations only for populations which are stationary or increasing very slowly. For rapidly increasing populations, they can be readily improved. The aim of this article is to review present practices and suggest some improvements in finite approximations on population. The improved formulae take account of the way in which the increase shifts the distribution of population within the five-year age group. A number of examples are given of the derivation of corrections to existing formulae.

Results are developed for play-the-winner sampling for choosing the better of two binomial populations when the maximum number of tests is specified. It is shown that for a fixed number of tests the probability of correct selection with alternating assignment exceeds that for play- the-winner sampling. However, when the probability of correct selection is fixed, neither sampling method is uniformly better than the other as measured by the expected number of tests on the poorer population, or on the total expected number of tests. [ABSTRACT FROM AUTHOR]

The article presents a list of books received by the journal as of November 1953. Included are "The Federal Loyalty-Security Program," by Eleanor Bontecou, "Mathematical Aspects of the Quantum Theory of Fields," by K. O. Friedrichs and "Signal, Noise and Resolution in Nuclear Counter Amplifiers," by A. B. Gillespie.