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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]

Exact small sample behavior in two-way contingency tables is investigated for Pearson's chi-square statistic (X²), Light and Margolin's C statistic and its related R² measure of association, Kuliback's minimum discrimination information statistic (2&lcirc;), and Goodman and Kruskal's Lambda. R² is shown to be identical to Goodman and Kruskal's tb, leading to a test for independence based on 4. In small samples from a product of multinomials model, the null distribution of C is better approximated by a x² distribution than is the null distribution of X²; both are considerably better approximated by a x² distribution than is the null distribution of 2&lcirc;. It is proved for tables with two columns and any number of rows that if the column totals are equal, then X² ≤ 2&lcirc;; thus, X² is more conservative than 2&lcirc;. Hence, use of 2&lcirc; should be avoided in testing independence in tables with small samples. [ABSTRACT FROM AUTHOR]

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 presents a data analysis approach consisting of a synthesis of experimental design methods and spectral methods of time series analysis which is appropriate in exploratory situations where the re- cording process generates a long sequence of correlated observations. [ABSTRACT FROM AUTHOR]

Following the results of Bhat and Nagnur [2], the locally asymptotically most stringent test of the hypothesis of no preference in paired comparison experiments is derived for the models of Rao and Kupper [9] and Davidson [3]. It is shown that the asymptotic relative efficiency of these tests when compared to the test based on the likelihood ratio statistic is unity. [ABSTRACT FROM AUTHOR]

A method is given for obtaining sequential tests of composite hypotheses for a certain class of distributions. Approximate properties of the tests are derived and the method illustrated. [ABSTRACT FROM AUTHOR]

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]

In this article the problem of testing for homogeneity of k (k > 3) independent negative binomial distributed random variables with parameters r[sub i], p (for i = 1, ..., k) is studied. The alternative hypotheses which are considered are a p-mixture of negative binomial distributions and a positive trend of the p's. Several cases may be distinguished: (1)p known, (2) p unknown and r[sub 1] =...= r[sub k],(3)p unknown and r[sub 1], ..., r[sub kappa] possibly unequal. The proposed tests are based on explicit optimality criteria. [ABSTRACT FROM AUTHOR]

A test for the hypothesis of zero second-order interaction is proposed. The test is obtained by making use of conditional reference sets which are derived in Zelen [17]. This permits the derivation of an exact conditional test of significance for testing the hypothesis of zero second order interaction. The techniques used, in approximating the large sample distribution of this test, are extensions of those due to Gart [5] and Zelen [17]. The test is applied to a numerical example. [ABSTRACT FROM AUTHOR]

Hypothesis test procedures for tests of separate families of hypotheses are defined. These procedures are then compared using Monte Carlo samples as data for five pairs of invariant distributions. The comparisons are made by calculating an approximate relative efficiency for each procedure with respect to the likelihood ratio procedure, which for three pairs of distributions considered is uniformly most powerful among the class of invariant procedures, [ABSTRACT FROM AUTHOR]

The asymptotic method described in an earlier article (Bowman and Shenton [2]) is applied to derive the power of the square root of b[sub 1] test of normality against skewed alternatives. In particular, normal mixtures of two components and Type A distributions, each depending on four parameters, are considered. Comparisons of the assessment of percentage points of square root of b[sub 1] using Cornish-Fisher expansions based on the normal and Monte Carlo simulations are given. [ABSTRACT FROM AUTHOR]

Cochran's Q-test, proposed to test the equality of three proportions for correlated observations, may have a larger asymptotic significance level than the nominal one, unless the admissible family of distributions is restricted to ensure that under the hypothesis the correlations between the observations are also equal. For a large class of such restricted families, optimal C[sup 2](alpha) tests of the hypothesis are obtained, and the family for which the Q-test is an optimal C[sub 2](alpha) test is identified. The use of Q for testing a related hypothesis, suggested by Madansky, is also discussed: it is shown that the Q-test is not consistent against all alternatives to Madansky's hypothesis, while the classical chi[sup 2]-test provides an optimal C[sub 4](alpha) test for it. [ABSTRACT FROM AUTHOR]

Procedures for discriminating between models from separate families of hypotheses are examined, with principal emphasis on procedures invariant under location and scale transformations. The discrimination procedures considered are compared using Monte Carlo samples as data for five pairs of invariant distributions. These comparisons are made with respect to the best invariant procedure, a procedure requiring no knowledge of sampling distributions, using approximate relative efficiencies calculated from the Monte Carlo results. On the basis of these efficiencies and the computational complexities of the procedures, suggestions are made for their use. [ABSTRACT FROM AUTHOR]

We consider the problem of drawing inferences on the difference of the population means when an incomplete sample from a bivariate normal population is available. Whereas Mehta and Gurland [7] have considered this problem from the sampling theory point of view, we tackle here the same problem from a Bayesian approach. [ABSTRACT FROM AUTHOR]

A Monte Carlo study of confidence limits for the slope of the major axis of a bivariate normal distribution confirms that, when imaginary limits are interpreted as corresponding to an infinite interval covering all possible values of the parameter, the confidence interval behaves satisfactorily under repeated sampling. The excess of the actual over the nominal significance level is negligible even if samples are small and correlation is moderate. The probability of detecting a relationship correctly is never smaller for the major axis than for ordinary regression. imaginary and exclusive confidence limits do not create problems in practice. [ABSTRACT FROM AUTHOR]

The standard problem of testing hypotheses begins with the formulation of a model containing one or more unknown parameters. The hypothesis is expressed as some restriction on the possible values of the parameters. In common economic applications, a linear regression model may be specified and the hypothesis usually is that one or more regression coefficients are zero. The standard theory is not applicable if the null and alternative hypotheses are disparate or nonnested. The purpose of this paper is to suggest a general purpose method for coping with this type of problem and to compare it with some leading alternatives. Section II of the article is devoted to some theoretical considerations. Some illustrative computations are discussed in section III. Section IV contains some brief conclusions. Three basic procedures have been considered for testing nonnested hypotheses. First, the development of D.R. Cox's generalized likelihood ratio statistic. Second, Cox's exponential weighting procedure and third, a linear weighting procedure.

This paper presents a comparison between similar types of American and Canadian forecasts of annual gross national product for an identical period (1960-69) using the same error measures. It was concluded that on mean, mean absolute, and mean square error bases, American forecasts were significantly more accurate than were the Canadian ones in predicting relative changes in their respective annual gross national products during the period 1960-69. Forecasters in both the United States and Canada were likely to have made the same types of errors on average in forecasting any given year. The general conclusion from the analysis of the errors seemed to suggest that a bias toward underestimating change may be less evident in the American gross national product forecasts than in those made in Canada. The negative mean errors were smaller, the percentage of underestimates was smaller, the covariance components of the mean square errors were larger, and the regression and test statistics were less suggestive. One of the more important points is that when "adjustments" of the predictions for differences in the basic nature of the economies are made, the conclusions as to differences in the "accuracy" of aggregate forecasts made in various countries may change a great deal. That is, conclusions as to differences in forecasting ability based solely on the comparison of "absolute accuracy" could be seriously misleading if the economies are not of approximately similar natures. The method of "adjustment" used here assumes that information as to the nature of the economies is contained in the past history of the variables to be forecast, so that a more worthwhile comparison is obtained when the respective judgmental sets are first "adjusted" by extrapolative predictions from these past actuals. Comparisons of the "relative" accuracy of forecasts showed that the American predictions of annual gross national product percentage changes had larger relative mean square errors than did the Canadian sets. That is, the American sets were only about as accurate as, and probably less accurate then, relative to extrapolative predictions. the forecasts made in Canada. The predictive efficiency of the Canadian gross national product forecasts relative to that of the extrapolative predictions was generally higher than that of the American predictions; the amount of autonomous information and available extrapolative information included was also somewhat higher. However, the extrapolative information actually used in forecasting was quite low in both countries, probably because of the essentially trendless nature of percentage changes in gross national product. [ABSTRACT FROM AUTHOR]

The two-sided bivariate t-test for simultaneous comparison of two treatments with a control is studied. The problem of allocating a given number of observations to the treatments and the control is discussed, and it is concluded that, in the absence of preferences, a general purpose rule would be to choose the sample sizes equal. A table is presented which gives the sample size required to ensure that the null hypothesis will be rejected with at least a prescribed probability, given the significance level and the standardized differences between the treatments and the control. Similar tables are provided for use when the object is to ensure that both treatments means will be found significantly different from the control mean with at least a given probability, or to ensure that a specified treatment will be found significant with at least a given probability. [ABSTRACT FROM AUTHOR]

Modifications of the two-sided one-sample Kolmogorov-Smirnov "goodness-of-fit" test, for use with censored and truncated samples, are suggested. Tables of the distributions of the modified statistics are given. Applications to life testing and reliability estimation problems are discussed. [ABSTRACT FROM AUTHOR]

A family of confidence intervals with endpoints that are simple averages of sample order statistics is defined. In this family the interval with shortest expected length is selected and compared to other confidence intervals for the same parameter. [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]

Analogous conditional and unconditional confidence interval procedures for the two parameter Weibull distribution are discussed. Results of an empirical study are given which suggest that, in many practical situations, the two procedures considered lead to near-equivalent results. Ramifications of the empirical results are also considered. [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]

Theory of testing hypotheses (see [1]) derives sufficient conditions for the existence of optimum tests. These conditions often require that the class of distributions in question be ordered in some way, e.g., stochastically ordered, monotone likelihood ratio, Polya of some order, or exponential class. This article gives four simple examples which show that the sufficient ordering restrictions cannot be weakened to less stringent ordering restrictions and which thus give some insight into the structure and use of such orderings. A fifth example shows that ordering properties are not necessary for the existence of optimal tests. [ABSTRACT FROM AUTHOR]

Procedures are developed for specifying confidence intervals, from possibly censored data, for any characteristic of a two-parameter Weibull population. Although the methods are formally Bayesian, they have exact relative frequency interpretation for uncensored and Type II censored data. Some simulation results are given to indicate relative frequency behavior for Type I censoring, and for comparison to other methods. [ABSTRACT FROM AUTHOR]

The small sample properties of several distributed lag estimators are examined under conditions of misspecified lag structure, such as fitting a geometric estimator to a second order lag distribution. Evaluation includes analyses of the estimated average lags and the adjustment rates. Lag structure misspecification is found to yield large biases in the average lag and adjustment rate, especially compared to the biases resulting from a correctly specified but poorly designed estimator. The test statistics are found to be unreliable and often misleading indicators of lag structure misspecification. [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]

Following Riedwyl [8] we generalize some one-sample statistics of the Cramer-von Mises type so that they can be used to test grouped data for goodness of fit. We prove that under the null hypothesis the asymptotic distributions of these statistics, when suitably standardized, coincide with the corresponding classical statistics if the ratio of the sample size to the number of groups (k) remains constant. For k fixed the asymptotic distributions are given under the null hypothesis, and it is shown how to obtain them under any alternative. Some results for finite sample sizes are also derived, [ABSTRACT FROM AUTHOR]

This article tests the hypothesis of a variable lag of shipments of capital goods behind new orders, applied to three major Canadian industries producing mainly to order. This hypothesis, confirmed by earlier studies on American data, is corroborated on the Canadian data. Several supplementary conclusions are also drawn, including a suggestion that constraining the weights in the Almon lag distribution to zero at the beginning and end of the distribution may introduce distortion into the resulting estimates. [ABSTRACT FROM AUTHOR]

The combined use of direct estimation (in saturated models) and indirect testing (of unsaturated models), proposed in [7, 8] for the analysis of a given multidimensional contingency table, is extended here to the analysis of a set of T multidimensional tables (T is greater than or equal to 2). For this set of tables, a three-fold classification of models is introduced: (1)models that assume "complete homogeneity" among the tables; (2) models that allow "complete heterogeneity" among the tables; and (3) models that allow "partial heterogeneity." Stepwise procedures proposed in [8] for model selection are extended. "Guided" and "unguided" selection methods and "multidirectional" methods are introduced. For illustrative purposes, a set of two 4-way tables is analyzed. [ABSTRACT FROM AUTHOR]

In paired comparisons of a treatment and control, it frequently happens that the two members of each pair can be classified into distinguishable types. The completely randomized design, which assigns the members of each pair at random to treatment and control, then may by chance assign the treatment primarily to subjects of one type and thereby confound treatment and type. This difficulty can be avoided by restricting the randomization. We find that such restriction is desirable by analyzing several of the standard tests (Wilcoxon, t, and tests for dichotomous response) for efficiency and deficiency. [ABSTRACT FROM AUTHOR]

This study is concerned with deriving the properties of the preliminary test estimator for the general linear normal regression model, ascertaining the characteristics of the risk functions over the parameter space, and determining the conditions necessary for the risk of this estimator to exceed or be less than the conventional one under squared error loss. A test procedure and the problem of choosing an optimal level of significance for the test are discussed. Some theorems and lemmas used in evaluating the risk and some properties of functions of the non-central F distribution are developed in the appendices. [ABSTRACT FROM AUTHOR]

Two methods, one based upon grouping, are proposed for fitting o regression line to bivariate data when the means of the observations have known order along the line. The two procedures are compared with respect to their limiting distributions and asymptotic efficiency. Criteria are given for choosing one of these procedures to fit the line. [ABSTRACT FROM AUTHOR]

Suppose a population containing items classified either as defectives or as non-defectives. This article presents a new group testing method to detect all defectives in the population. The proposed method applies either when the actual number of defectives {or an upper bound) is known or when its probability distribution is known. The method is surprisingly simple yet compares favorably with other existing methods by the number of tests required to detect all defectives. [ABSTRACT FROM AUTHOR]

The Kolmogorov goodness-of-fit test is known to be conservative when the hypothesized distribution function is not continuous. A method for finding the exact critical level (approximate in the two-sided case) and the power in such cases is derived. Thus the Kolmogorov test may be used as an exact goodness-of-fit test for all completely specified distribution functions, whether continuous or not continuous. Several examples of the application of this extension of the Kolmogorov test are also included. [ABSTRACT FROM AUTHOR]

The probabilities of the Type I errors of the Welch approximate-t test and the Aspin-Welch test for the Behrens-Fisher problem have been calculated for selected sets of degrees of freedom and nominal significance levels. The results show satisfactory agreement to the desired level. Thus, in practice, one can just use the usual t-table to carry out the Welch approximate t-test without much loss of accuracy. [ABSTRACT FROM AUTHOR]

A measure of variation for categorical data is discussed. We develop an analysis of variance for a one-way table, where the response variable is categorical. The data can be viewed alternatively as falling in a two-dimensional contingency table with one margin fixed. Components of variation are derived, and their properties are investigated under a common multinomial model. Using these components~ we propose a measure of the variation in the response variable explained by the grouping variable. A test statistic is constructed on the basis of these properties, and its asymptotic behavior under the null hypothesis of independence is studied. Empirical sampling results confirming the asymptotic behavior and investigating power are included. [ABSTRACT FROM AUTHOR]

In this article we study the exact finite sample distribution function of a t-statistic that can be used to test hypotheses and construct confidence intervals for structural coefficients in simultaneous equations models. It is shown that the exact distribution function is equal to Student's t-distribution when one of its parameters is zero and converges to Student's t-distribution function as another of its parameters increases without bound. Computations of the exact distribution function and its first three moments indicate the error that might result if Student's t-distribution function is used as an approximation to the exact distribution function. [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]

As is true of any statistical application, testing econometric statistical hypotheses involves constructing critical regions with prescribed probabilities of Type I error and determining probabilities of Type II error under alternative hypotheses. Such constructions and determinations presuppose knowledge of exact statistical distribution functions of econometric estimators and test statistics either directly, or indirectly, as in the case where a statistical distribution function that approximates the exact distribution with known margin of error is used. If this requisite information is not available, econometric statistical inference remains guesswork. Partly because of the complicated nature of systems of simultaneous econometric structural equations and the large numbers of structural constants required to characterize distribution functions of econometric statistics, however, only a few exact marginal distributions of econometric estimators and test statistics have been extracted so far. This article is intended as a contribution to the growing literature of econometric distribution theory and to assist the efforts of econometric statisticians in providing a rational basis for econometric statistical inference. [ABSTRACT FROM AUTHOR]

The principle of minimum discrimination information estimation is described and used to generate estimates for tests of hypotheses concerning various interactions and effects in the analysis of multidimensional contingency tables. All classical hypotheses for contingency tables can be generated by the use of this principle when certain marginals are considered as fixed. Analysis of information tables are given for a four-way contingency table. [ABSTRACT FROM AUTHOR]

Let t[sub I]= x[sub I]/s, I = 1, 2 be noncentral Student t-variates with noncentrality parameters delta[sub I], where x[sub 1], x[sub 2] and s are independent. The simultaneous distribution of t[sub 1] and t[sub 2] is studied and the power function of the studentized maximum and minimum modulus tests are tabulated for significance level 0.05, degrees of freedom 1, 2, 5, 10, 20, 50, infinity and delta[sub 1], delta[sub 2] = 0(1)5. Asymptotically exact approximations to these power functions are discussed. The marginal power of the maximum modulus test is also tabulated. [ABSTRACT FROM AUTHOR]

Suppose that independent samples are available from two normal populations with means mu[sub 1] and mu[sub 2] and known variances. It is desired to test H: mu[sub 1] = 0. A possible preliminary test procedure in this case consists of first testing mu[sub 1] = mu[sub 2] and then testing H using either the sample mean from the first population or the pooled sample mean depending on the outcome of the test regarding equality of mu[sub 1] and mu[sub 2]. This procedure is shown to be biased. A power comparison of this procedure with the generalized likelihood ratio test of H indicates that the latter test is more powerful in a fairly extensive parametric region. It is noted that biasedness is frequently encountered in preliminary test procedures. In light of the example, it is recommended that power comparisons be made between such procedures and any available unbiased tests to ascertain whether extensive power loss regions exist or not. [ABSTRACT FROM AUTHOR]

The powers of several one-sided Kolmogorov tests for a shift in the location of a normal distribution with known variance are computed and compared. An accurate approximation to the power of the ordinary Kolmogorov test for this problem is also given. [ABSTRACT FROM AUTHOR]

Under a reasonable hypothesis about the distribution of trade shipments among a group of K countries, or of non-self-directed acts among any group of K individuals, the likelihood of obtaining a given table of observed frequencies n[sub ij] is [Multiple line equation(s) cannot be represented in ASCII text] where n[sub I]=n[sub i1] + ... + n[sub iK] and m[sub I] = n[sub 1i] + ... + n[sub K I] are the total number of shipments made and received, respectively, by country I, N is the total number of shipments altogether, and Pi and Q[sub I] are the theoretical tendencies of country I to ship and to receive shipment respectively and satisfy SIGMA P[sub I] = SIGMA Q[sub I] = 1. In the attempt to test the correctness of the hypothesis, a critical problem is that of finding the maximum-likelihood estimates of the P's and Q's, that is, the values making L as large as possible. We give all solutions and show that, for the usual table of observations, the solution is unique and amounts to finding the value of a for which K-2 -R[sub 1] ... - R[sub K] is 0 - [Multiple line equation(s) cannot be represented in ASCII text] [ABSTRACT FROM AUTHOR]

The lack of memory property of the exponential distribution plays an important part in the branch of applied probability. This property assumes the information regarding the probability distribution. In the present paper we give a characteristic property of the exponential distribution based on the means of the conditional distributions. Considering a random variable T with finite mean and such that P(T > O) > 0, and denoting by y a positive number such that P(T > y) > 0, we show that T has an exponential distribution if and only if the mean of the conditional distribution, given T > y, exceeds the mean of the unconditional distribution by the quantity y for all y. Also given in the paper is a similar characterization for the geometric distribution. Since the information concerning the expected values is easily accessible, we expect these properties to be useful in dealing with the practical problems. [ABSTRACT FROM AUTHOR]

This article suggests a distribution-free signed rank test for the parallelism of two regression lines. Pitman efficiency comparisons of the signed rank test and a normal theory t-test are presented. Monte Carlo sampling is used to compare the signed rank test with a nonparametric test proposed by Potthoff [7]. [ABSTRACT FROM AUTHOR]

There is not much in the literature on tolerance limits for discrete distributions, since generally one will find it difficult to derive explicit expressions for these limits. Distribution-free tolerance limits for discrete cases were discussed. If the form of the distribution function is known, one should try to determine the formula of the tolerance limits for the particular family of distributions to which it belongs. Otherwise, an important information is not utilized. In applications for sampling inspection reliability, inventory control, the observed random variables are very often assumed to have a binomial, negative-binomial or Poisson distribution. In these cases one can obtain explicit expressions for tolerance limits, which have desirable optimum properties. In the present article we derive uniformly most accurate upper tolerance limits for families of distributions with the monotone likelihood ratio property. For these families there exist uniformly most accurate upper confidence limits. Furthermore, the distribution functions which belong to monotone likelihood ratio families are stochastically ordered. Thus, monotone relationship exists in these cases between tolerance limits and confidence limits. This relationship is utilized to derive the formulae of uniformly most accurate tolerance limits. The general theory is presented in Section 2. Section 3 is devoted to uniformly most accurate upper confidence limits.

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]