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This paper compares the Pearson Chi-Square and Kolmogorov goodness-of-fit tests with respect to validity under the following conditions: (1) the N independent observations are tabulated and arranged into k mutually exclusive groups that are equally probable under the hypothesis to be tested; and (2) both N and k are "small"; i.e., not greater than 50. A random sampling experiment was performed, and the results show that in general for the conditions considered, the Pearson test is more valid than the Kolmogorov test. [ABSTRACT FROM AUTHOR]

This paper investigates the small sample properties of minimum chi-square estimates of the parameters of stochastic brand choice models. It also describes and evaluates a statistical test which is appropriate for discriminating between two stochastic brand choice models when one is a constrained version of the other. [ABSTRACT FROM AUTHOR]

This paper presents a follow-up records' study of 773 former marriage counseling clients of 21 marriage counselors. The clients' primary treatment form was compared with six categories of counseling outcome. A chi-square statistical treatment demonstrated that the six forms of counseling resulted in different outcomes significant beyond the .001 level. A rank ordering of the six forms for effectiveness suggested conjoint interviews and conjoint groups to be the most effective forms. Different forms may be helpful to those who divorce as opposed to those remaining married, and some findings are quite different from current supposition. [ABSTRACT FROM AUTHOR]

This paper applies the exponential distribution to stock price reactions to determine, at three confidence levels, the critical percentage price reaction beyond which a reaction constitutes a strong likelihood of a major reversal or halt in the stock's present general price trend. We show that these critical values can be used to determine stop losses, so that a trader's position is closed when the probability of a major reversal or halt against his position is large, and an open position is kept open when the probability that the reaction is a major one is small or moderate. The distribution fit on ten stocks on daily and weekly bases for over a year's duration each, tested using the chi-squared statistic, was found to be good. Further, three stops, corresponding to three confidence levels of the distribution were tested for each stock against a subsequent six-month period of price action. The percentage of successful tests of these stops for each of the stocks corresponded very closely to expected results ascertained by using data from the previous period. [ABSTRACT FROM AUTHOR]

This article is concerned with some problems that occur in certain tactical considerations: how should one place k circles [spheres] in the plane [3- space] so that their union has the greatest standard normal probability measure, that is, so as to maximize the probability that a random normal point will fall in one or more of the circles [spheres]. For k>3 the problem seems hopeless, (except for certain special situations); the case for k=3 is still unresolved and is being worked on by a number of investigators, and the case for k=2 is solved completely in this paper. The results for k=2 have some practical value when applied to actual problems arising in tactical considerations, and some theoretical value, as a method of attacking the problem for k≥3. [ABSTRACT FROM AUTHOR]

A method suitable for hand computation is presented for determining the probability of killing a point target when (a) the lethal volume is a sphere centered on the burst point of the weapon, and (b) the distribution of the weapon's burst points is described by a three-dimensional multivariate normal distribution about the target. [ABSTRACT FROM AUTHOR]

This article presents a discussion on "segregation indices" which deals with issues like the way to define the amount of Negro segregation in a city. It is to define measures of association for a contingency table with only two columns, that is, where one of the attributes, like Negro and non-Negro, is dichotomous. Instead of a minority and dominant group, one may, for instance, compare men and women, an experimental and control group, and so forth. Likewise, the manifold attribute need not be location. If there is anything special about the residential segregation problem, it is that there are a large number of census tracts roughly equal in population, that is, that the row totals of the contingency table are approximate. To convert chi square for a table with two columns into an index with the range o to 1, one merely divide it by the total population. The quotient is called the mean square contingency, or phi square. Two of the proposed indices are nothing but special applications of phi, in one case for a particular fourfold table, in the other by k table. Furthermore, the familiar approach by k table is more easily generalized.

This article presents a reply to a response by Donald Morrison about the article "Adaptive Behavior in Automotive Brand Choices," from the February 1969 issue of "Journal of Marketing Research." The author believes that Morrison is mistaken in his conclusion that the original research is erroneous. He says that Morrison is actually testing a different hypothesis with the same set of data. Morrison's hypothesis is that individuals are not homogenous, as the author stated in his original paper. The author concludes by showing a chi-square test where the results favor his belief that past purchases of a particular make influence the probability that the consumer will again purchase that make.

When an experiment is run with a factorial layout and some of the factors are random effects, there may not be an exact test for some effect of interest. This paper considers three statistics that could be used to test that hypothesis using a ratio of linear combinations of independent Chi-square statistics. The common case, utilizing four Chi-square statistics, is examined for a number of configurations of nuisance parameters. Both the power and the probability of a type I error are used in the comparison. Two statistics appear to be equally good over a large region and, in certain situations, the statistics involving the subtraction of Chi-square statistics is shown to be more stable. [ABSTRACT FROM AUTHOR]

This paper deals with a multivariate extension of Friedman's chi[sup 2, sub tau]-test. A rank permutation distribution and the large sample properties of the criterion are studied. The asymptotic relative efficiency (A.R.E.) for a sequence of translation alternatives is studied and bounds are given for certain special cases. It is shown that, under specified conditions, the A.R.E. of this test with respect to the likelihood ratio test is largest, when the block dispersion matrices differ and can be greater than unity when the differences are large. [ABSTRACT FROM AUTHOR]

Ellison has shown that the Wald-Wolfowitz tolerance limits for a normal distribution, x + lambda s, are good only to 0(n/N[sup 2]), rather than to 0(1/N[sup 2]). Here x is distributed normally with mean mu and variance sigma[sup 2]/N while s[sup 2]/sigma[sup 2] is distributed as chi[sup 2, sub n]/n independently of x. Thus, for n much greater than N[sup 2] the usual values of lambda are incorrect; Ellison has proposed an alternative in this case. This paper derives new lambda's which have two advantages over the Wald-Wolfowitz and the Ellison limits. First, they are shown to be better approximations. Secondly, they are easily calculated in the sense that only tables of the normal and chi[sup 2] distributions are required and the solution of a non-linear equation is not required. [ABSTRACT FROM AUTHOR]

The role of the extended family as a source of information and as the basis for mutual assistance and social contact is studied among rural and urban samples in central Brazil. Chi-square tests are used to determine the presence of association between length of residence of migrants and occupational status with the extent of mutual assistance and social contact. While migrants have fewer contacts than do natives, the extent of contact tends to increase, particularly with relatives of the husband, with length of residence of the migrants. Occupational status is positively associated with extended family contacts of all types in both rural and urban samples. [ABSTRACT FROM AUTHOR]

A "blowup factor" is defined for the measurement of the effect on forecast error variance of two types of misspecification which may be implicit in the choice of a particular adaptive forecasting scheme: (1) misspecification of the number of non-zero parameters of the stationary linear stochastic process generating the observed time series, and (2) misspecification arising from postulating stationarity when in fact the generating process is non-stationary in mean. [ABSTRACT FROM AUTHOR]

Several authors have considered point and interval estimation of the exponential scale parameter, σ, on the basis of subsets of the order statistics. In this paper, we suggest a procedure for finding approximate confidence intervals for σ in large samples, using k sample quantiles of a random sample of size n. The procedure is based on a simple chi-square approximation to the distribution of the asymptotically best linear estimate of σ. We compare this procedure with the one given by Ogawa (1962) based on an approximating t-distribution. We find that the interval based on the chi-square approximarion is easier to calculate, and performs better when k is small and n is large. [ABSTRACT FROM AUTHOR]

Recently the U. S. Water Resources Council has proposed standardization of the analysis of peak flood discharges by fitting a Pearson Type III distribution to the logarithms of the data. This action has served to draw attention to the inadequacy of available tables of percentage points of the Pearson Type III distribution and the need for better tables. Many tables of percentage points of the related chi-square distribution are available in the literature, perhaps the most comprehensive being those published by the author in 1964. These could be used to obtain percentage points of the Pearson Type III distribution, but it would be much more convenient to have a table from which percentage points of the latter distribution could be read directly for uniformly spaced values of the skewness coefficient. The author has therefore, by a modification of the programs used to compute his 1964 tables of percentage points of the chi-square distribution, obtained percentage points, corresponding to cumula. tire probability P = .0001, .0005, .001, .005, .01, .02, .025, .04, .05, .1(.1).9, .95, .96, .975, .98, .99, .995, .999, .9995, .9999, of the standardized Pearson Type III distribution with skewness α[sub 3] ≡ γ[sub 1] = 0.0(0.1)9.0. This paper includes the five-decimal-place table, accurate to within a unit in the last place, together with a description of the method of computation and a discussion of possible applications, including the estimation of the return periods of floods. Space has been conserved by giving results only at intervals of 0.2 in &alpha[sub 3] for &alpha[sub 3] > 4.8; the intermediate values, if needed, can be obtained by interpolation. [ABSTRACT FROM AUTHOR]

This paper is concerned primarily with the method of moments in dissecting a mixture of two normal distributions. In the general case, with two means, two standard deviations, and a proportionality factor to be estimated, the first five sample moments are required, and it becomes necessary to find a particular solution of a ninth degree polynomial equation that was originally derived by Karl Pearson [10]. A procedure which circumvents solution of the nonic equation and thereby considerably reduces the total computational effort otherwise required, is presented. Estimates obtained in the simpler special case in which the two standard deviations are assumed to be equal, are employed as first approximations in an iterative method for simultaneously solving the basic system of moment equations applicable in the more general case in which the two standard deviations are unequal. Conditional maximum likelihood and conditional minimum chi-square estimation subject to having the first four sample moments equated to corresponding population moments, are also considered. An illustrative example is included. [ABSTRACT FROM AUTHOR]

The concepts of alpha error, beta error, power and alternative hypotheses are first developed for a simple percentage problem in market research. These concepts are then used to explain the method of calculating simple size in the multi-cell case using the non-central Chi-Square distribution. The development is by means of an advertising research problem for delineating "different" geographical trading areas and the sample size necessary to detect differences of given magnitudes. [ABSTRACT FROM AUTHOR]

The article reports on sampling problems in trade statistics in the field of marketing. The author focuses on the success rate and reliability of patented sampling techniques utilized by the U.S. Bureau of Foreign and Domestic Commerce. Government sampling techniques relevant to the retail and wholesale trade are criticized. A case study focusing on grocery wholesales located in the U.S. is discussed in relation to the 1935 Census of Distribution. Applications of the Chi Square test are also examined.

This article present an approach to measure association such that a single basic logic is used regardless of the measurement level of the variables being associated. As data progress from nominal to ordinal to equal interval in mathematical character, association measures should simply incorporate the added properties into the format used at the lower levels. The logic is adapted to the types of measurement present in the independent and dependent variables so that a complete range of association measures is available. The approach requires little new at the interval-measurement level (dependent variable) and only modest modification of existing procedures for ordinal dependent variables. The two basic approaches to association measures appear to be assessing departure from randomness and determining extent of predictive accuracy. Departure from randomness is characteristic of the chi-squared based measures, which typically suffer from normalizing procedures that admit no easy operational interpretation.

The manifest goals of this research were to: (a) devise a means of measuring the relative importance for youth of two potential types of referents-parents or peers and (b) obtain information on behavioral correlates of parent and peer orientations. Tests of relationship were made relative to parent or peer orientation and: the age-grade level, sex, the perceived source of need satisfaction, and the socioeconomic class of respondents. Ordinal type scales were used where applicable to gain measurements of the variables. Chi-square tests and rank-order correlations provided the measures of association between type of orientation and the independent variables. All the associations were found to be significant, except socioeconomic class. [ABSTRACT FROM AUTHOR]

Examines a sample of college students from entrepreneurial and bureaucratic backgrounds using the Traditional Family Ideology Scale. Devices and procedure; Result of the chi square test.

The article presents a study on 238 first-grade children's perceptions of interactions, as reported by questionnaire-interviews; and actual teacher-student interaction, and measured by a Reading Observation Record, to determine whether teachers discriminated against boys. Obtained data were treated by use of chi-square and analysis of variance procedures. In addition, reading achievement scores of boys and girls were compared. Results indicated that children perceived that teachers discriminated against boys and favored girls in the situation of reading instruction. On the other hand, an analysis of observations of actual teacher-pupil interaction revealed no differential treatment. Boys' and girls' achievement did not differ significantly.

This discussion of the hypothesis, evidence, and inference presented in an American Educational Research Journal article has attempted to highlight the strengths and weaknesses of the material presented. The purpose here is not to question the competency of the principal investigator, for undoubtedly many of the questions raised herein could be answered by referring to the detailed report of the study that was submitted to the U. S. Office of Education. This discussion is presented to call the attention of the readers to strengths and weaknesses of the study, to encourage their consideration of them in reaching conclusions, and to encourage others to engage in the process of critical evaluation of publisbed research articles. [ABSTRACT FROM AUTHOR]

The multiserver queue with Poisson arrivals and Erlangian, gamma or chi-square service time is solved for the steady state case. A general procedure, adaptable to computer programming, for writing the equations for the state probabilities is given. Generating functions for the time invariant state probabilities are found. A method of computing numerical solutions is described. [ABSTRACT FROM AUTHOR]

A technique previously developed for interpreting measures of association for fourfold tables based on markovian assumptions is here extended to derive measures of second order association. Transition rates between the states represented by [ABSTRACT FROM AUTHOR]

This article focuses on the measurement of similarity in social profiles. The comparison of states with respect to a series of characteristics simultaneously, when interest lies in the similarity between profiles of the series values rather than between composite indexes representative of each series, introduces the need for other measures of similarity. By similarity in profile is meant here similarity in the relative sizes of all values in a series. The method which proved most satisfactory in measuring similarity in profiles was based on the well-known fact that the process of squaring values in a series increases the relative weight of the larger values. Accordingly, two profiles with a given total of differences between corresponding component values in their series would be considered most similar when this total difference was distributed more evenly among the individual component value differences rather than occurring as a lump difference between corresponding values of only two of the components. Distributed evenly, the total difference would be broken down into smaller differences, the sum of whose squares would be less than the square of the few lumped differences would be if the total difference were concentrated in two or three pairs of components instead of being distributed in small amounts among all the component pairs. The chi square measure of dissimilarity between an observed and expected distribution seemed to offer a possibly appropriate inverse measure of similarity.

In this article the authors described a method of testing relative merits of area and quota sampling by drawing various samples from a known universe, which is made up of records based on a house-to-house survey in Iowa City. Stratification simply means dividing the universe into groups which are presumed to be more homogeneous than the entire universe. This is merely a technique for the reduction of the variance of a sample estimate. The differentiation between quota and area samples is made by the way the respondents are selected within strata, not through the presence of strata or their choice. A statistical method, chi-square test was developed by Karl Pearson to test the hypothesis that an observed distribution is a random sample from a proposed universe. The hypothesis is rejected when the probability of securing a value of chi-square as large as, or larger than, the value obtained is small, usually less than 5 percent. It can be seen from the given formula that chi-square is defined in absolute frequencies, not relative frequencies.

Explores the use of statistical tests for decision-making in regional geographical construction in the United States. Chi square; Variance analysis; Relationship between regional constructs and the areal dispersion of selected traits within regional frames.

A response by Enrico T. Federighi to a letter to the editor about his article "The Use of Chi-Square in Small Samples" in the December 1950 issue is presented.

Looks on the use of Chi square for testing regional geographical boundaries. Absolute frequency of a geographic area; Validity of the statistical test; Use of acres as a unit of mensuration.