Showing papers in "Journal of the American Statistical Association in 1949"

The Monte Carlo method.

Abstract: We shall present here the motivation and a general description of a method dealing with a class of problems in mathematical physics. The method is, essentially, a statistical approach to the study of differential equations, or more generally, of integro-differential equations that occur in various branches of the natural sciences.

Application of Least Squares Regression to Relationships Containing Auto-Correlated Error Terms

Abstract: We point out that autocorrelated error terms require modification of the usual methods of estimation and prediction; and we present evidence showing that the error terms involved in most current formulations of economic relations are highly positively autocorrelated In doing this we demonstrate that when estimates of autoregressive properties of error terms are based on calculated residuals there is a large bias towards randomness We demonstrate how much efficiency may be lost by current methods of estimation and prediction; and we give a tentative method of procedure for regaining the lost efficiency * We wish to express our thanks for the considerable assistance we have received from Richard Stone

A Procedure for Objective Respondent Selection within the Household

Abstract: In modern survey methods growing emphasis is placed on the objective selection of the sample. For surveys of the general population, increasing use is made of area sampling to obtain probability samples of households. Heretofore, scant attention has been given to the question of how to make an objective selection among the members of the household. A procedure for selecting objectively one member of the household is given as used in four surveys of the adult population. Demographic data as found in the sample are compared with outside sources for available factors. * Presented at the 107th Annual Meeting of the American Statistical Association, New York City, December 30, 1947.

On a Method of Estimating Birth and Death Rates and the Extent of Registration

Abstract: A mathematical theory is presented which when applied to a comparison of the registrars lists of births and deaths with a list obtained in a house-to-house canvass gives an estimate of the total number of events over an area in a specified period. It also gives the extent of registration. Allowance is made for the fact that the chance of an event being missed on 1 list may not be independent of its chance of being missed on the other list. The theory is applied to an enquiry that was conducted in 1947 over an area known as the Singur Health Center near Calcutta covering the years 1945 and 1946 separately. It is found that the estimated total number of events for the area is usually greater when the estimate is built up by summing the totals for individual groups than when it is computed at once for the aggregated population. This observation according to the theory confirms positive dependence and indicates that the greater figure is closer to the truth. The annual number of births and deaths in the Singur Health Center (total population 64000) is estimated subject to a standard error of from 1-3%. The registration is estimated to vary from about 40-70% with a standard error of about 3%. Higher estimates of the number of deaths were obtained under 3 different approaches to subdividing the population into homogeneous groups. Separate calculations were prepared for each of the 4 political subdivisions comprising the Center for each sex separately and for sex and age groups. In each case the separate estimates obtained for the subpopulations were combined to obtain 3 estimates of the number of deaths for the population of the Center as a whole. The largest estimated number of deaths was obtained using the 3rd approach. This suggests that the subdivision into sex and age groups minimized the correlation of events missed in the 2 data collection systems. (summaries in FRE SPA)

Traffic performance at urban street intersections

Abstract: A PRELIMINARY EXPLORATION WAS CONDUCTED INTO THE FUNDAMENTAL RELATIONSHIPS WHICH EXIST BETWEEN TRAFFIC UNITS. METHODS AND THE MORE COMMONLY OBSERVED TRAFFIC PHENOMENA OF ACCELERATION AND DECELERATION WERE OBSERVED. THE SEVERAL COMPONENTS OF THE PERFORMANCE OF VEHICLE STARTING AT URBAN INTERSECTIONS WERE INVESTIGATED. REACTION TIMES TO THE LIGHT CHANGE AND BETWEEN VEHICLES WERE FOUND TO BE IN SOME MEASURE DEPENDENT ON PEDESTRIAN MOVEMENT. SEVERAL METHODS OF PRESENTING STARTING-TIME-DISTANCE PERFORMANCE WERE USED FOR THE SAKE OF CLARITY. THE TIME-DISTANCE RELATIONSHIPS AT START PROVED TO BE SIMILAR AT THE DIFFERENT INTERSECTIONS STUDIED. TWO METHODS FOR DETERMINING THEORETICAL INTERSECTION CAPACITY WERE PRESENTED AND THEIR PRACTICAL APPLICATION DISCUSSED. IT IS CONCLUDED THAT A VERY SMALL PERCENTAGE OF DECELERATIONS IN ACTUAL PRACTICE REACH THE POSSIBLE MAXIMUM. THERE ARE TWO TYPES OF DECELERATION, THE FUNCTIONAL, AND WHAT HAS BEEN CALLED CHRONOTROPIC. NO TIME IS LOST DUE TO NOT TAKING ADVANTAGE OF MAXIMUM DECELERATION IN THE LATTER TYPE. THE TRAFFIC DELAY IN URBAN AREAS IS NOT CLOSELY RELATED TO THE RATES OF DECELERATION EMPLOYED SINCE MOST STOPS ARE OF A CHRONOTROPIC NATURE. A TECHNIQUE OF TIME-SPACING ANALYSIS IS APPLIED TO TRAFFIC BEHAVIOR AT NON- CONTROLLED INTERSECTIONS. PHYSICAL CHARACTERISTICS OF THE INTERSECTION AFFECT THE BEHAVIOR OF THE DRIVERS INSOFAR AS THEIR ABILITY TO MAKE EARLY DECISIONS AND AVOID DELAY IN CROSSING IS CONCERNED. THERE IS A TENDENCY AT THE NON- CONTROLLED INTERSECTIONS FOR THE MAJOR STREAM OF TRAFFIC TO BE AGGRESSIVE AND TO FORCE THE MINOR TO GIVE WAY. IN GENERAL, TIME-SPACE RELATIONSHIPS GOVERN THE BEHAVIOR OF MOTORISTS AT UNSIGNALIZED INTERSECTIONS. ROAD TRAFFIC CONFORMS SO WELL TO POISSON'S RANDOM SERIES THAT ITS APPLICATION LEADS TO RESULTS OF PRACTICAL IMPORTANCE. IT IS POSSIBLE TO DETERMINE DELAY AT SIGNALIZED INTERSECTIONS THROUGH THE USE OF POISSON'S FORMULA. A THEORETICALLY CORRECT METHOD HAS BEEN DEVELOPED. THE THEORY, IN REGARD TO SIGNALS WHICH FAIL TO CLEAR SOME VEHICLES RETARDED DURING THE RED PERIOD, IS DEVELOPED SINCE POISSON'S FORMULA IS APPLICABLE TO THIS PROBLEM AS WELL. THE TRAFFIC PROBLEM TREATMENT BY PROBABILITY THEORY AND OTHER PRINCIPALS DEVELOPED IN THIS STUDY ARE ONLY EXEMPLARY OF A VARIETY OF APPLICATIONS.

The Uses and Usefulness of Binomial Probability Paper

Abstract: This article describes certain uses of Binomial Probability Paper. This graph paper was designed to facilitate the employment of R. A. Fisher's inverse sine transformation for proportions. The transformation itself is designed to adjust binomially distributed data so that the variance will not depend on the true value of the proportion p, but only on the sample size n. In addition, binomial data so transformed more closely approximate normality than the raw data. The usefulness of plotting binomial data in rectangular coordinates, using a square-root scale for the number observed in each category, was first pointed out by Fisher and Mather [10]. The graph paper under discussion 1 is specially ruled to make this mode of plotting both simple and rapid. A graduated quadrant makes the angular transformation (p = cos2 φ or p = sin2 φ) easily available at the same time. Most tests of counted data can be made quickly, easily and with what is usually adequate accuracy with this paper. Some 22 examples ar...

An Attempt to Get the “Not at Homes” into the Sample Without Callbacks

Abstract: This paper describes a plan for eliminating the need for callbacks. Each person in the sample is visited only once. From each person interviewed information is obtained as to whether or not he was at home on specific instances, including the instance of the interview, which permits an estimate of the proportion of time he is at home during the interviewing hours. Questionnaires are divided into e.g. 6 groups according to the estimated proportion of time persons in each group are at home, viz., 1/6, 2/6, …, 6/6 of the time. The sample estimate, for any variable under study, is produced by weighting the results for each group by the reciprocal of the estimated per cent that persons are at home. It is shown that under certain conditions this estimate is unbiased and the variance of the estimate is obtained. A numerical comparison is made between this plan and the usual method of calling back.

On Estimating the Mean and Standard Deviation of Truncated Normal Distributions

Abstract: The problem considered is that of estimating the mean and standard deviation of a normally distributed population from a truncated sample when neither count nor measurements of variates in the omitted portion of the sample is known. Formulas are developed whereby certain special functions required-in solutions given for this problem by Karl Pearson and Alice Lee and by R. A. Fisher may be readily evaluated with the aid of an ordinary table of the areas and ordinates of the normal curve. A method of successive approximations is illustrated which, with the aid of the above formulas permits the utilization of either the Pearson-Lee or the equivalent Fisher method to obtain the desired estimates with an improvement in accuracy regardless of whether or not the special tables ordinarily required by these two methods are available. * A portion of this paper was presented before the Southeastern Section of the Mathematical Association of America at Tuscaloosa, Alabama, March 19, 1949.

A Sampling Study of the Merits of Auto-Regressive and Reduced form Transformations in Regression Analysis

Abstract: This paper is concerned with some aspects of regression analysis when the error terms are autocorrelated and there exists more than one relationship between the variables. In particular, we investigate the merits of autoregressive transformations and the reduced form transformation in dealing with these complications. An important result is that, unless it is possible to specify something about the intercorrelation of the error terms in a set of relations and to choose approximately the correct autoregressive transformation, a certain amount of scepticism is justified concerning the possibility of estimating structural parameters from aggregative time series of only twenty observations.

Minimum X2 and Maximum Likelihood Solution in Terms of a Linear Transform, with Particular Reference to Bio-Assay

Abstract: An old and commonly used device for fitting a curve y = F(x, α, β) is to find a function of y which is linearly related to x, or a function of x which is linearly related to y, or functions of y and x which are linearly related to each other The linearly related functions are plotted against one another, and a line is fitted to the points, usually by eye, sometimes by “least squares” in terms of the transforms Texts dealing with empirical curve-fitting characteristically use this scheme [6] Before Bliss and Fisher [2], no attempt was made to adjust the transforms systematically in order to achieve a fit that fulfilled defined criteria in terms of the original measures y and x, nor was it known that such adjustments were possible These authors presented for the bio-assay experiment a method in terms of probits, which as shown by Garwood, [5] accomplished a maximum likelihood estimate of the integrated normal curve, when the observations were distributed binomially Following the procedures use

Applications of some significance tests for the median which are valid under very general conditions.

Abstract: In two other papers ([1] and [2]) order statistics were used to derive some tests for the population median which have significance levels which are either exact or bounded under some very general conditions. These order statistic tests were found to be very efficient for small samples from a normal population; also they can be applied with very little computation. This paper contains applications of these tests to several well known statistical problems. Also a graphical method of applying the tests of [1] is outlined. * The results presented in this paper were obtained in the course of research conducted under the sponsorship of the Office of Naval Research. This research was performed while the author was at Princeton University.

Statistics of the Kinsey report.

Abstract: * Alfred c. Kinsey, Wardell b. Pomeroy, and Clyde e. Martin, Sexual Behavior in the Human Male. W.B. Saunders Company, Philadelphia and London, 1948. Pp. xv +804. $6.50. The book is commonly referred to simply as “the Kinsey report,” since Kinsey initiated the project in 1938 (p. 10), and did about four-sevenths of all the interviewing that had been done up to the date of publication (p. 11). Similarly, it is common to write “Kinsey” where “Kinsey, Pomeroy, and Martin” or “the authors” would be correct. These abbreviated forms of reference are used throughout this paper.

On Some Mathematical Problems Arising in the Development of Mendelian Genetics

Abstract: In this paper some basic problems of theoretical Mendelian genetics are presented. The general situation of an arbitrary number of linked loci for “diploids” (normal organisms) as well as for “autopolyploids” is considered, under random mating. Particular stress is laid upon formulating the problems and results within the framework of probability theory, expressing them in terms of three basic distributions, the distributions of genotypes and of gametes, and the segregation distribution. The task of the mathematical theory consists in establishing recurrence relations for the distributions of gametes and of genotypes, in integrating them and in investigating the limit behaviour of those distribution as n, the number of discrete non overlapping generations tends toward infinity. * Lecture delivered January 1948 at the University of Chicago.

On the “Information” Lost by Using a t-Test when the Population Variance is Known

Abstract: This note calls attention to the use of the power function as a means of determining how much “information” is lost by using some other test in place of the most powerful test of a given hypothesis. As an example of the method, the case of using a t-test for the mean of a normal population with known variance is analyzed.

The decomposition of a series of observations composed of a trend, a periodic movement and a stochastic variable

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Unemployment and migration in the depression, 1930-1935.

Abstract: This is a study of the reciprocal relationship between migration and unemployment in Michigan during the depression period 1930 to 1935. Specifically, it is concerned with two questions: By the use of matched control groups it is found that the differential in unemployment rates occurs after migration, not before. The results are consistent with the hypothesis that in a depression migrants tend to be at a disadvantage in the new labor market to which they move. 1. Do the migrants during a depression have a poor employment history as compared with non-migrants of similar characteristics? 2. Do the migrants have a better employment experience after migration than non-migrants of similar characteristics? * This study was made possible by a grant from the Faculty Research Funds of the Horace Rack-ham School of Graduate Studies of the University of Michigan.

Evaluation of Parameters in the Gompertz and Makeham Equations

Abstract: This paper describes a technique for determining the mortality characteristics of physical property through the use of accounting records of plant balances and yearly additions. Where time is not available for extensive actuarial research, the method produces results within tolerable limits of accuracy. Its limitations are pointed out.

The Use of Sampling in Great Britain

Abstract: The object of this article is to survey the use of sampling methods in Great Britain. The most important sample surveys undertaken by government departments, research organisations and commercial agencies are described with particular reference to their aims and sampling methods. Deficiencies in British sampling practice are discussed and suggestions for possible future developments are made. * My thanks are due to statisticians in Government departments and elsewhere for making available to me much of the information on which this article is based.

Confidence Limits in the Non-Parametric Case

Abstract: The purpose of this article is to give a survey 1 of certain methods available for finding confidence limits when nothing is assumed about the population from which a sample has been drawn except possibly that it has a continuous distribution. The following three cases are treated: a confidence band for the unknown cumulative distribution function, a confidence interval for the proportion of a population for which the variate is smaller (or larger) than a given value, and confidence intervals for quantiles. The results have been known for many years, but have often been accessible only to those who were able to follow rather involved mathematical arguments. It is the purpose of this paper to state certain important results without referring to any mathematical proofs. 1I should like to thank Professor Wolfowits for pointing out the need for such a survey.

Teaching Statistical Quality Control for Town and Gown

Abstract: During the next five years, in manufacturing plants and in the engineering schools, there will be many new programs initiated to meet the demands for education in Quality Control by Statistical Methods. This paper has been written with the hope that it will be of some help in the planning and executing of such programs. Beginning with a description of types of courses and a discussion of possible content, it touches on the questions of who is to do the teaching and how the subject matter might be presented and motivated. The importance of follow-up work is stressed. The latter part of the paper discusses the choice and utilization of instructional aids and describes some of the materials now available. * Presented at Section on Training of Statisticians of the American Statistical Association joint with American Society for Quality Control and Institute of Mathematical Statistics at Cleveland, Ohio, on December 27, 1948. Parts I and II were presented by Lloyd A. Knowler, Part III, under a separat...

On the best choice of sample sizes for a t-test when the ratio of variances is known.

Abstract: The situation considered is that of testing the difference of the means of two normal populations on the basis of a sample from each population, where the ratio of the population variances is known. The choice of sample sizes has been restricted to certain pairs which are equally preferable from the viewpoint of practical considerations (cost, difficulty of obtaining sample values, etc.). This note presents an easily applied method of determining which of these pairs of sample sizes yield the most powerful one-sided and symmetrical tests.

The current status of state and local population estimates in the Census Bureau.

Abstract: * Adapted from a paper read by the first-named author before a Customer Administrative School for Public Health Executives and Vital Statistics Registrars at the International Business Machine Corporation, Endicott, N. Y., July 2, 1947.