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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 article analyzes the second-order interaction in multidimensional contingency tables in marketing research. THe author examines the introduction of chi-square analysis in marketing research as well as the use of chi-square to study three-dimensional contingency tables. The author also examines the association of the three attributes of multidimensional contingency tables. The author suggests that these complex techniques are needed to examine the more complicated relationships that exist between marketing and consumers.

In this article we propose two minimum chi-square estimators for a bivariate probit model. We call one estimator the Full Information and the other Limited Information Minimum Chi-Square because the first takes account of all the a priori information while the second does not. Both estimators are shown to be consistent. Moreover, the first is shown to be asymptotically as efficient as the maximum likelihood estimator and yet is computationally much simpler. For illustration, estimates are computed for the data used by Ashford and Sowden [1970.] [ABSTRACT FROM AUTHOR]

The use of Patnaik's [28] chi-square approximation to the noncentral chi-square distribution and the Wilson-Hilferty [321 transformation of chi-square to approximate normality are explored as a simple, efficient means of finding k-order-statistic confidence bounds on parameters and reliability of the one- and two-parameter negative exponential distributions. An important implication of the results leads to obtaining simple closed-form approximations to percentiles of the beta distribution either with integer or noninteger parameters. The general methodology is also used to approximate confidence or prediction intervals for the time of the kth exponential failure based on r

Mean square errors associated with median response estimates are compared for the normal and logistic distributions, based on both maximum likelihood and minimum chi square analyses. In broad perspective there is little difference among the mean square errors for these four estimates, regardless of sample size and stimulus level spacing. The first three of the median response estimates require an iterative numerical solution, whereas the minimum chi square analysis for the logistic distribution yields an explicit expression for the associated median response estimate. This explicit estimation expression may be particularly convenient when nonuniform spacing of stimulus levels is involved in testing. [ABSTRACT FROM AUTHOR]

Departures from statistical independence are conjoined with an assessment of predictive accuracy' in a coefficient of association for nominal-level 2 x 2 contingency tables which is both interpretable as a PRE measure and consistent with research hypotheses of manifold forms. Measurement assumptions and operating characteristics o[ the measure are delineated; definitional and computational formulate are derived from classical probability theory; comparisons with other relevant statistics are made' and The test of significance is shown to be the traditional chi-square test. [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]

Cette note essaye de répondre au sujet de plainte suivant à savoir que les mains de bridge distribuées par ordinateur, dans les tournois, semblent se répartir de façon inhabituelle et que la répartition est biaisée en faveur de mains à longues suites. Les résultats de quelques études empiriques sur les mains de bridge distribuées par ordinateur et par une personne, sont exposés dans cette note. [ABSTRACT FROM AUTHOR]

The asymptotic non-null distribution is obtained for the modified form of the Pearson chi-square statistic studied by Dahiya and Gurland [3]. By utilizing this result the power is obtained for specific alternative distributions in testing for normality, This enables recommendations to be made as to the number of class intervals to be employed in applying the aforementioned modification of the Pearson chi-square test of normality. [ABSTRACT FROM AUTHOR]

Two tables of modest size and three relatively simple formulas provide all that is required to obtain percentage points of the Chi[sup 2] distribution at 25 significance levels with relative accuracy of at least .99995; and to obtain Poisson confidence limits of such accuracy at 13 confidence levels. For a small number of degrees of freedom or observed number of events, where the formulas are weak, the tables give exact Chi[sup 2] values or Poisson limits. [ABSTRACT FROM AUTHOR]

This article shows that information partitioning in contingency tables is a very general property of exponential families, One partitioning scheme occurs when the hypotheses of interest are stated as linear expressions in the natural parameters, A second partitioning scheme is possible when the hypotheses are stated as linear in the parameter which is the expectation of the statistic in the exponent. [ABSTRACT FROM AUTHOR]

Use of the Bonferroni inequality for controlling family Type I error rate is discussed and new tables of t and chi square are provided. The tabled percentage points provide the capability for unequal Type I error allocation over a family of statistical tests. Use of these tables is illustrated. [ABSTRACT FROM AUTHOR]

Consider sampling in two stages for testing a simple hypothesis about a population whose elements may be classified into a finite number of classes. The joint asymptotic distribution of the two Pearson chi[sup 2] statistics based on the first and combined samples is derived and used to construct a two-stage test. The advantage of this test relative to the conventional one-stage test is discussed in terms of its asymptotic power. Tables of critical values for 1 percent and 5 percent level tests for 1-10 degrees of freedom ore given as well as modifications for cases involving unknown parameters. [ABSTRACT FROM AUTHOR]

A step-by-step method is presented herein far partitioning a certain kind of hypothesis H about the m-way contingency table into (a) a series of hypotheses about marginal tables formed from the m-way table by ignoring one or more of table's m dimensions; and (b) a hypothesis about independence, conditional independence, or conditional equiprobability in the m-way table. This step-by-step method facilitates both the testing of H and the calculation of F, the estimated expected frequencies in the m-way table under FI. The method introduced herein for calculating F is easier to apply than the usual iterative-scaling method in many cases. [ABSTRACT FROM AUTHOR]

This article reports a Monte Carlo evaluation of the levels of significance of chi-square tests for the order of two- and three-state Markov chains. Results are consistent with asymptotic theory only when samples are quite large. [ABSTRACT FROM AUTHOR]

A sample of size N is taken from an equiprobable t-category multinomial distribution. The precise values of Pearson's X[sup 2] and the likelihood ratio statistic A were computed for t = 3(1)6, N = 3(1)12; t = 6(1)14, N = 6(1)2t; t = 15(1)18, N = 6(1)28. The logarithms of the tail areas were smoothed and fitted by means of quadratic expressions. Much information is given concerning the accuracy of the chi-squared fits to the distributions of X[sup 2] and A. A table of -log[sub 10] P(x[sup 2] > a) is given, for a = 1(1)50, and d.f. = 1(1)20. The distribution of the total number of repeats within ceils is also discussed. [ABSTRACT FROM AUTHOR]

The article focuses on an inequality for a class of bivariate chi-square distributions. A recurrent problem in simultaneous inference is discussed in the article. While the joint distributions of test statistics may be known in form, these frequently depend on parameters, values of which are unknown. In such cases it is not possible to give simultaneous procedures, which hold exactly with specified probability; instead, it is expedient to give approximate procedures for which useful probability bounds can be established. Considerable effort has been expended in this direction, particularly in connection with simultaneous inference regarding means. The study is concerned with the construction of two-sided simultaneous confidence limits for variances in the bivariate case. In the p-variate normal case, researcher C.G. Khatri has given a lower bound to the simultaneous confidence coefficient associated with one-sided lower confidence limits and with additional assumptions regarding the covariance matrix, Khatri has similarly bounded the simultaneous confidence coefficient in the case of one-sided upper limits; however, two-sided limits was not examined.

Given a sample of n vector observations from a multivariate normal population, Anderson and Roy-Gnanadesikan have given for the variances a set of confidence intervals which are approximate in that a lower bound only is known for the joint confidence coefficient. In the present study, exact procedures are developed in terms of multivariate Chi-square distributions, and more general approximate procedures are given via Bonferroni's inequality. A numerical investigation suggests that the Bonferroni lower bound is fairly sharp for a variety of parameter values, and it always is superior to the Roy-Gnanadesikan procedure in the bivariate case examined. A lower bound in terms of independent statistics further is examined far a special class of one-sided intervals. [ABSTRACT FROM AUTHOR]

The article discusses the analysis of cross-classified data. There are two different kinds of contingency tables, that is, truncated contingency tables in which the entries in some of the cells of the table are omitted from the analysis and the more usual contingency tables in which none of the cells of the table are omitted. In the usual non-truncated contingency table, one need to calculate statistician Karl Pearson's chi-square statistic for testing the null hypothesis of independence in the table, using the appropriate number of degrees of freedom. The concept of "quasi-independence" is useful in analyzing tables for which some of the cells are truncated and also in analyzing the more usual kinds of cross-classification tables since it leads to methods that focus attention in turn on various subsets of the entire table, making possible a more detailed analysis of the association between the row and column classifications in the table. For the non-truncated cross-classification table, if the rows and columns are not independent, it will often be of interest to estimate what are the kinds of interaction or association present in the table and what are their magnitudes.

The standard tables used for the Kolmogorov-Smirnov test are valid when testing whether a set of observations are from a completely specified continuous distribution. If one or more parameters must be estimated from the sample then the tables are no longer valid. A table is given in this note for use with the Kolmogorov-Smirnov statistic for testing whether a set of observations is from a normal population when the mean and variance are not specified but must be estimated from the sample. The table is obtained from a Monte Carlo calculation. A brief Monte Carlo investigation is made of the power of the test. [ABSTRACT FROM AUTHOR]

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]

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.

The article focuses on prestige patterns in scholarship and science. To throw some light on this question a questionnaire was recently sent to 1,275 individual chosen at random from the membership rosters of twelve different learned societies. The homogeneity of the returns was determined by the Chi square test. The sample was checked for homogeneity with respect to academic rank and section where the respondent was employed at the time he made out his questionnaire and at the time he was listed in the roster of his society. It is Interesting to note that history and economics are the only two fields in which one individual achieved a ranking frequency considerably above that of any other man listed. In every field individuals were listed prominently for their administrative ability, personal leadership in establishing schools and stimulating followers, for important offices held and for other characteristics not directly relevant to research in the limited sense.

This article presents information regarding parole prediction variables and the time factor in violations by burglars. Parole prediction studies have shown that it is possible to determine, before an inmate is released from prison under parole supervision, what are his relative chances for making a successful extramural adjustment. The original data were set up in 23 contingency tables with the number of months before violation as the independent variable in each case and one of the 23 parole prediction factors as the dependent variable. The Chi-square test was undertaken to test the probability that differences between the actual and the theoretical frequencies would occur by chance. In these instances where the probability that the differences were due to chance was very small the coefficient of mean square contingency was determined in order to measure the amount of the relationship that exists between the variables. Whether the relationship is a positive or a negative, one was determined by inspection of the data and the signs of the differences.

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.

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 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 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.

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]

The article presents information on coalition formation as a function of differences in resources. The initial partner preferences and the coalitions formed are contrasted with the various theoretical predictions. The vast majority of subjects in each of the resources positions preferred the weaker of the other two players. Chi-square tests on the preference results, with an equal division of subjects as the expected values, revealed that these preferences were significant. With respect to coalitions formed, the 60-30 union occurred in all but one case. A chi-square test, with the expected frequencies of the three possible coalitions equal, indicated that the preponderance of 60-30 coalitions was significant. When the predicted frequencies of certain theories are tested against the observed frequencies by a chi-square goodness-of-fit test, a problem is encountered. For example, scholar W.A. Gamson's minimum winning coalition theory and scholar J.M. Chertkoff's theory that people expect rewards to be divided at a point midway between equality and parity are represented as predicting invariant choices. Gamson's theory includes no rationale for why the non-predicted choice ever occurs, but Chertkoff suggested that his theory might ultimately have to include some low finite likelihood for the occurrence of the less preferred choice. Since no specific proposal was made about what the probability of the less preferred choice would be, for the moment it has been treated as zero.

The implications of replacing the assumption of normal distribution of the parameters (A, b, c) by chi-square and other related nonnegative distributions are discussed here in the framework of chance-constrained linear programming. [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]

Answers a question concerning the degrees of freedom for chi-square test. Rule for computing the degrees of freedom for the chi-square goodness; Selection of fitted parameters.

This article considers the problem of testing a completely specified continuous symmetric distribution against alternatives which are also symmetric about the same point. The symmetry is utilized in obtaining a new distribution-free statistic of the Kolomogorov-Smirnov type which can be used to halve the width of the Kolomogorov-Smirnov confidence band for the unknown distribution function. [ABSTRACT FROM AUTHOR]

If the parameters are estmated independently of the sample, the chi-square test statistic for a goodness of fit test has a limiting distribution that is stochastically larger than that of the test of fit for a completely specified distribution. Thus, if the critical values for the test of fit for a completely specified distribution are incorrectly used, the probability that we will reject the null hypothesis when it is true is greater than the desired level of significance. [ABSTRACT FROM AUTHOR]

Offers a nomogram or an alignment chart for chi square. Probability values given by the nomogram; Applications of the nomogram. [ABSTRACT FROM AUTHOR]

Empirical studies by sociologists rely frequently upon Chi-Square analysis to indicate existence and nature of association between variables. A cursory glance through the 50 issues of the 1949 issues of the "The American Journal of Sociology" and the journal "American Sociological Review," discloses at least half a dozen reports utilizing this tool. As an analytical tool, Chi-Square has much to recommend it. It is readily understood, is easily calculated, conveniently summarizes many categories of data simultaneously and is not so high powered as to outreach the refinement of most of the data to which it is applied. Certain difficulties are encountered, however, in determining independence. When each of the expected cell frequencies exceeds ten this test gives quite acceptable results. But if one or more of the expected cell frequencies is less than ten, the Chi-Square test tends to underestimate the small probabilities with the result that many hypotheses of independence are rejected, and relationship assumed, at a particular level which would not have been so if the true probability had been known.

Information on the social history of a psychiatric patient is most useful immediately on admission to a hospital and for some days thereafter. In the psychiatric services of military hospitals, the usual way of gathering this information is by interviewing members of the patient's family face to face or speaking with them on the telephone. However, respondents are often difficult to reach in this way, and difficulties lead to delays. Several studies have shown that useful information about psychiatric patients can be obtained through the use of self-administered questionnaires. Social history information has been collected from various sources-fathers, mothers, wives, and peers, but there seems to be no generally accepted position on who can provide the most useful information. In this study, the level of agreement between the card-sort and the interview was 79 percent for fathers, 77 percent for mothers and 81 percent for wives. Using the chi -square test of independence, no significant differences were found between the three groups of relatives.

Periodically, the uses and misuses of probability statistics in social and behavioral science research have been reviewed. The frequently used chi-square test, and the interpretations given to data analyzed by this statistic will serve to illustrate the problem. To avoid these errors of confusing significance with strength of association and of misinterpretating form of relationship, two elementary safeguards can be exercised in reporting results. One is routinely to compute and report a measure of degree of association in addition to the statistical test whenever this is possible. The second safeguard is the introduction of care and caution in the verbal interpretation of data tables and the inferred association of variables.

The article presents a note on the correlation of attributes. Sociologists, who have to work a great deal with unmeasured factors, or attributes, often find themselves with data having high frequencies. A data of this kind, if the frequencies are large, is generally used like an actuarial table to infer, say, that if a prisoner is a murderer he has about 85 chances in 100 of succeeding on parole, whereas if he is a swindler he has only 62 chances. The above procedure is considered legitimate if a Chi-square test shows a significant value of Chi-square, and worthwhile if some measure of association applied to the data is found to be reasonably high. The usual inference is then that there must be correlation between the type of crime committed and recidivism. This interpretation presented in the article suggests that somewhat more clarification is needed than is furnished by most statistical texts as to just what is meant by correlation in such a case. In case a sociologist has reason to investigate the correlation between, say, the rates of recidivism of two types of criminals, it will be necessary for him to get more data than are usually available, namely, the percentage of recidivists among, say, murderers in various prisons, and the corresponding percentages of recidivists among, say, sex offenders.

The article focuses on comments given by the author on topics related to 2 X 2 contingency tables. In the article two ideas are proposed. The first is that the Yates continuity correction should not be used in 2 X 2 contingency tables unless row and column totals are nonrandom. The second idea is that if row totals, or column totals, are not fixed in advance of the experiment, the Yates correction should not be used for two reasons. First, it provides poor estimates of the desired probabilities. Second, it sometimes exchanges points inside the critical region for points outside the critical region, thus providing a different test. Statistician Olli S. Miettinen points out that much of the controversy over use of a continuity correction is due to a lack of distinguishing between two distinct situations. The first situation involves estimating the exceedance probability associated with a known possible value of a discrete random variable. The second situation involves estimating the exceedance probability associated with an arbitrary real number, such as a quantile from the chi-square' distribution.

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.

The author makes some corrections to, and comments on, the article by C.R. Miller and E.W. Butler, "Anomia and Eunomia: A Methodological Evaluation of Srole's Anomia Scale," published in the periodical "American Sociological Review." These corrections are primarily concerned with certain statistical tests that are reported in the article. First of all, the chi-square analysis reported in Table 4 should have 20 degrees of freedom rather than 31, the reason is that 11 independent parameters have been estimated from these data. As the authors point out, however, their estimating procedure is not efficient. Briefly, this means that there are probably estimates of the latent parameters which would give a lower chi-square value, a "better fit. The chi-square test is thus conservative as a test of the model, and it is sure that the actual P level would be larger than 0.33. A second point has to do with the significance test reported in footnote 29, comparing the sizes of the anomic classes in the urban and suburban cities.