Search

When data are transformed to satisfy a spherical normal linear model, the mean θ of a variate in the original scale is a function of the mean μ and variance σ² of a normal variate. We consider several approximate confidence interval methods for θ, including a new method based on exact confidence intervals for linear functions of μ and σ² Monte Carlo estimates of coverage probabilities demonstrate the suitability of the new method for applications involving a wide range of data transformations, parameter values and sample sizes. [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]

Necessary and sufficient conditions for admissibility are given for test procedures based on a preliminary test of significance. Three types of problems are studied-testing the normal mean, fixed effects models of the analysis of variance and random effects models. Admissibility in this instance is equivalent to the intuitive and practical condition that acceptance regions of the procedures have convex sections in certain variables, while other variables are fixed, It is easy to check when the conditions hold. A discussion of optimality properties of these and other types of pooling procedures is 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]

Examines some statistical measures in marketing research design and hypothesis testing. Regression analysis; Simple correlation; Analysis of variance; Strategy implications.

If the research goal in applying classical discriminant analysis techniques is to test for differences in the general characteristics of groups, then standard tests from multivariate analysis of variance are appropriate. Statistically significant differences among group means and dispersions indicate that the groups arose from separate populations. If, however, the goal is to explore group structure in terms of group overlap and the locations of individual observations relative to each other and to the group means, then it can be shown that the significance tests provide incomplete and often misleading representations of the data. Similarly, in problems involving just more that just two groups, the null hypothesis can be rejected, when in fact only one of the several groups arose from a different population.

This study employed several statistical techniques in testing the hypothesis that classification according to (1) growth, (2) stable, and (3) cyclical characteristics represents a factor for grouping stocks. These techniques showed that the residuals obtained by removal of general market effects from a sample of 100 stocks displayed cross-sectional dependence conforming to four distinct stock categories, including an oil group as well as the three hypothesized groups. In addition, regression analysis results indicated that these stock groupings accounted for an avenge of 14 percent of the variance in rate of return of stocks in the sample in comparison to 31 percent represented by general market effects. It was thus considered appropriate to assign a factor to the explanation of the variance of returns of a common stock additional to market, industry, and company, and based upon a system of classification corresponding to (1) growth, (2) stable, (3) cyclical, and (4) oil stocks. Correspondingly, the presence of stock groupings among the stock return residuals implied a violation of the specification of cross-sectional independence of the residuals for the single-index model. An examination of the residual correlation matrices for the single- and four-index models confirmed the existence of a significant degree of dependence among the residuals from the single-index model, whereas the four-index model showed little indication of a violation of this specification. As a result of the superiority in accounting for systematic effects among securities, the four-index model provided a closer approximation to the true correlation matrix than was provided by the single-index model. Since these analyses indicated significant violation of the specification of cross-sectional independence of residuals, the form of the single-index model was examined for additional departures from model specifications. A test of the intercepts from regressions of the four stock groupings on the S & P market index showed that these parameters were significantly different from zero, thereby indicating that the returns of the indexes were not consistently proportional to their risk as measured by the slope coefficient from the regression equation. Risk and return measures developed in conjunction with the test of the form of the regression equation and a statistical test of these measures indicated a consistency as well as an independence of risk-return relationships among the four groups of stocks. [ABSTRACT FROM AUTHOR]

An F-test of linear hypotheses is compared with Bonferroni t-tests. The individual confidence intervals from Bonferroni t-tests are uniformly shorter than S-intervals implied by the F-test. Power curves are constructed for a few specific alternative hypotheses as functions of the correlation between regressors for the special case of two hypotheses and two regressors. The power of the two procedures is similar when the correlation is small. For highly correlated regressors, however, the power of the Bonferroni method is generally inferior. Thus, if regressors can be controlled to be uncorrelated, the Bonferroni method is clearly superior; otherwise neither method dominates. [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]

When two correlation coefficients are calculated from a single sample, rather than from two samples, they are not statistically independent, and the usual methods for testing equality of the population correlation coefficients no longer apply. This article considers tests to be made using a sample from o multivariate normal distribution. Small sample level of significance and power are obtained using Monte Carla methods for Hotelling's test of H[sub 0:912] = 913, Williams's modification of Hotelling's test, and for two tests of H[sub 0:912] = 913 and H[sub 0:912] = 914 based on Fisher's z transformation. [ABSTRACT FROM AUTHOR]

It is shown that a test suggested by Bechhofer, Kiefer, and Sobel [1], for selecting the better of two binomial populations, can be formulated and solved when a maximum number of tests is specified. If is assumed that the probability of correctly selecting the better population is specified to be better than P[sup *] when the difference in success probabilities exceeds a specified change[sup *]. The populations are sampled equally and it is assumed that the trial ends either when the difference in the number of successes exceeds a calculated value of s, or when N tests have been made, whichever is sooner. [ABSTRACT FROM AUTHOR]

This article studies the effect that serial correlation of the observations has on the distribution of the mean and two one-sample nonparametric tests, the sign test and the Wilcoxon test. It is shown that relatively slight dependence has a strong influence on the level of these tests. [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]

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]

For Bernoulli random variables sequential tests of the simple hypotheses p = p[SUB o] vs. p = p[sub 1] are considered. In particular a comparison is made between the performance of sequential probability ratio tests (SPRTs) and curtailed single sampling plans (CSSPs), when both tests have the same error probabilities. In [1] it is shown that the CSSP has a strong optimality property. Nevertheless, if one admits randomized SPRTs, the latter are better. Numerical examples are considered in detail for p[sub o] = 1/2 and the CSSP with n = 2. The method of finding the randomized SPRTs with the correct error probabilities is indicated. These rules have positive probability of deciding without taking any observations. [ABSTRACT FROM AUTHOR]

This study is concerned with the extension of the Bradley-Terry model for paired comparisons to situations which allow an expression of no preference. A new model is developed and its performance compared with a model proposed by Rao and Kupper. The maximum likelihood estimates of the parameters are found using an iterative procedure which, under a weak assumption, converges monotonically to the solution of the likelihood equations. It is noted that for a balanced paired comparison experiment the ranking obtained from the maximum likelihood estimates agrees with that obtained from a scoring system which allots two points for a win, one for a tie and zero for a loss. The likelihood ratio test of the hypothesis of equal preferences is shown to have the same asymptotic efficiency as that for the Rao-Kupper model. Two examples are presented, one of which introduces a set of data for an unbalanced paired comparison experiment. Initial applications of the test of goodness of fit suggest that the proposed model yields a reasonable representation of actual experimentation. [ABSTRACT FROM AUTHOR]

Cochran's Q test for differences between related-sample percentages or proportions has generally been incorrectly presented in secondary sources. The most common mistake results from failure to recognize that rows containing only 1's or only O's, i.e., only successes or only failures, do not affect the value of Q. The F test, however, is affected by such rows. The probabilities from the x[sup 2] and F approximations are compared with the exact probabilities in three sets of data. A rule of thumb, based on extensive study of the distribution of Q in small samples, is given as an aid in judging when the x[sup 2] approximation is satisfactory for practical purposes. [ABSTRACT FROM AUTHOR]

The article presents information on a nomogram for the "Student"-fisher t test. A nomogram is given for estimating the probability (P) for a given value of the "Student"-Fisher t test. W.S. Gosset, an employee of the Guiness brewing company in Dublin, published papers in 1908 in which he correctly solved three problems: the probable error of a mean, the distribution of the mean divided by its estimated standard deviation and the distribution of the estimated correlation coefficient between independent variates. Later "Student" and economist R.A. Fisher calculated tables of the relevant t distribution and Fisher gives a table of t and probabilities, corresponding to various degrees of freedom. Fisher and F. Yates, scholar provide in addition a column for P. It seemed that presentation of the P, degrees of freedom, t relationship in the form of a nomogram would be advantageous. It makes possible a fairly exact estimate of probabilities less than 0.0001 and makes it possible to get an estimate of P for any value of t from 1 to 65, instead merely of selected values.

Balls are drawn one after another from k cells C[sub 1], ..., C[sub k] according to the multinomial distribution. Suppose the ith cell initially contains N[sub I] balls, and sampling stops as soon as any of the k cells, say C[sub alpha], empties first. Let X[sub I] denote the number of balls taken from cell C[sub iota] (all I is not equal to alpha) at stopping time. The joint asymptotic (as N[sub I] arrow right Infinity) distribution of the X[sub iota] (I is not equal to alpha) is derived under the most general configuration of the multinomial cell probabilities p[sub 1], ..., p[sub k]. Conditions on p[sub iota] and N[sub I] are given under which the asymptotic distribution is shown to be either normal or truncated (restricted) normal. An application of the asymptotic distribution theory for N[sub iota] = N[sub 0] (I =1, ..., k) in setting up approximate tests and confidence intervals for the largest p[sub I] is also given. Under certain conditions on pi and N[sub I] it is shown that the asymptotic probability that C[sub I] empties first is equal to the probability content of a positive orthant under a multivariate normal distribution. [ABSTRACT FROM AUTHOR]

The problem of iterated tests of hypotheses has been treated from the standpoint of multiple decision theory. Under the assumption that there is a UMP test (not necessarily similar) for each component hypothesis, a procedure has been developed which ensures an upper bound for various probabilities of misclassification and which maximizes the probabilities of correct classification in some class. [ABSTRACT FROM AUTHOR]

The article presents a table of the values of the power of the F-test corresponding to the degrees of freedom. The use of F-test while using a classical analysis of variance model, rejects the null hypothesis at the significance level. The values of &b.beta;, the Type II error, are tabulated to three decimal places. In this article the values of &b.beta; are tabulated to four decimal places and to obtain the values of &b.beta; for odd values, a linear interpolation may be used which generally gives three decimal place accuracy. The computation was terminated when the values of &b.beta; no longer changed in the sixth decimal figure. These values agree with the values of the power-function of the non-central chi-square distribution.

In the case of moving average schemes or what are termed the m-dependent stochastic processes, the successive observations in the series are not stochastically independent. As a result, the usual sign test, the test based on the rank correlation tau (for randomness against trend alternatives), and the test proposed by Moore and Wallis [9] and further developed by Goodman and Grunfeld [2] (for the comovement between two time series) are not valid and require some modifications. Accordingly, the modified forms of these non-parametric tests are studied here in detail, and their applications considered. [ABSTRACT FROM AUTHOR]

Focuses on the development in statistical theory which offer immediate utility for psychologists. Utility of scaling method in measurement of income; Importance of factor analysis in test hypotheses; Significance of correlation co-efficient method, a statistical derivative, used by psychologists.

This article presents information on the robustness of the studentized range statistic to violations of the normality and homogeneity of variance assumptions. Multiple comparison procedures in recent years have earned a prominent role in the analysis and interpretation of experimental research in the behavioral sciences. Most of these procedures are designed either to test individual contrasts between means after the null hypothesis of no treatment differences in analysis of variance has been rejected or to test a selected set of mean contrasts which are of a priori interest to an investigator in an experiment. Three popular techniques which have primarily been employed for the first purpose are the J.W Tukey's WSD method, the D. Newman-M. Keuls test and the D. B. Duncan's multiple range test. All of these tests have as their parent statistic the studentized range statistic q. It is generally conceded that the q statistic is less powerful overall than the corresponding F statistic, but this finding assumes normal distributions with equal variances. The present study is directed at determining the extent to which Type I error rate is affected by violations in the basic assumptions of the test based on the q statistic. Monte Carlo methods were employed and a variety of departures from the assumptions were examined.

As part of their review of the estimation and use of "change" scores, Cronbach and Furby and O'Connor recommended procedures designed to increase the precision of estimators of individual true gain. The intent was to develop estimators which yielded smaller mean squares of (D∞ - ...∞), where D∞ is true gain. The present study was designed to examine the effects of three parameters upon this mean square, where the estimator of D∞ employed was one originally proposed by F. M. Lord. The three parameters studied were the correlation between the true score on the initial test and true gain (rξγ), the reliability of the initial test rxx, and sample size. Of principal interest were the effects of the initial test true score-true gain correlation. The ANOVA results indicated that rξγ has a pronounced effect upon the precision of ...∞, but this effect is moderated by rXX. In general, higher values of rξγ yielded smaller errors in ...∞. These results were considered in the discussion of the extended estimator proposed by Cronbach and Furby. [ABSTRACT FROM AUTHOR]

Focuses on the extraction of components and the applicability of the factor model. Problems caused by the use of principal component analysis with correlation matrices; Interpretation of the components of intercorrelations among random normal deviates; Use of test variables and random variables; Principal component analysis of a matrix of correlations.

The article discusses empirical results concerning the testing of the significance of a correlation matrix using the Barlett's test. The problem in a matrix is to determine the probability of rejecting the null hypothesis. Thus, the sample correlation matrix could be considered to have arisen when sampling from a multivariate population of ten variables. The power of Barlett's test appears to be quite high that future factor analysts will see fit to include the test as an automatic first step for sampling correlation matrix.

Fifty subjects were given a battery of eye dominance tests to investigate whether there are five types (Lederer, 1961) or two types (Walls, 1951) of ocular dominance, and to investigate the relation between these tests and handedness. There was no evidence in support of either classification, or of any correlation between the preferred eye and the preferred hand. An alternative hypothesis to those formulated by Lederer and by Walls is presented, which seems to account more adequately for the results. [ABSTRACT FROM AUTHOR]

The correlation of ranks introduced by Pearson Spearman is one of the oldest and best known of nonparametric procedures. The rank correlation coefficient, r8, is generally expressed as r8 =1-6 Σ d2/(n3-n) where n is the number of measurements in each of the two variates in the correlation. Early attempts at attacking the question of significance of r8 were met with prohibitively formidable computations [3, 5, 6]. E.G. Olds examined both the normal curve and the Pearson Type II curve as approximating functions to the exact distribution of Σd2. Although he concluded that "the normal curve is not as satisfactory as the Type II," especially in the tails of the distribution, he decided to "sacrifice accuracy to expedience" and utilized the former in tabulating critical values of Σ d2 for n as large as 30 [6, 7]. The exact frequency distribution of Σ d2 has appeared for various values of n [1, 2, 4, 5, 6], the most extensive tabulation being that of D.B. Owen, for n = 4(1)11.

The article presents a commentary on the article "Abuses of Factor Scores," by Gene V. Glass and Thomas O. Maguire, that was published in the November 1966 issue. Glass and Maguire are appreciated for their article on how to compute factor scores. They demonstrated that factor analysis can be used as a research tool to reveal some kind of innate structure. The factor analysis tests their hypotheses and suggests new hypotheses for them to test by carefully designing new batteries of measures for further factor analyses. Persons using factor scores are suggested to be aware of its content. This is because, most people who use factor analysis have a random and scattered items or measures. Their chief purpose for using factor analysis is to reduce their number of scales by grouping their items or measures into a few scales. For this, they use a principal components analysis. All the analyst wants is a score for the scale that he named on the basis of the items of basic measures with high loadings on it. The use of rough scale scores instead of optimal and independent factor scores may help the amateur factor analyst in reducing the chances of errors that they do commonly make.

Questions the points raised by W. Thomas et al on the criterion used to identify statistically significant terms in farm planning. Description of an efficient farm plan; Implications of a correlation coefficient; Effect of removing statistically insignificant covariance terms.

Alternative systems of negative binomial and binomial weights are proposed for measuring the permanent component of a series as a weighted average of the observed series. If the observed series is generated by a stable first-order autoregressive process it is shown that there are alternative sets of weights using either weighting scheme for which the permanent and transitory components are uncorrelated. [ABSTRACT FROM AUTHOR]

The article presents a comment on a previous article that appeared in the 1973 issue of the Journal of the American Statistical Association. According to the author comments by economists Arthur P. Dempster, D.A.S. Fraser, and John W. Pratt make interesting reading in that they reflect the committed attitudes of their individualistic approaches to statistical inference. The author find that he can react sympathetically to most of the points they have made and do not feel committed to any one approach sufficiently strongly to have any desire to reject out of hand the others. According to the author, he comments only on points to which he had some objection or to which he had something to add. Both Pratt and Dempster comment adversely on the type of significance test procedure. It was the author's intention to produce something like the usual test of significance, and what he suggested seems reasonable, according to him. The quoting of probabilities may be more informative. In any case, this is a side issue and one in which any difference of opinion could be easily settled.

In this article, the author presents his comments on a paper written by researcher K.C. Land on unmeasured variables in path analysis, published in an earlier issue of the journal "Social Forces," as of September 1971. Land does not show a disturbance term affecting the unservable variable Xa in Figures 1, 2 or 3 of his paper. Following Land's equation 1 and the usual graphic conventions, one can only assume that Xa is an exact linear function of X1 in Figure 1. Land's treatment of the model in Figure 3 is incorrect whether or not one defines Xa as an exact function of X1 and X2. In either case his equations 9-12 cannot be solved simultaneously for unique estimates of the path coefficients of the model in Figure 3. The proportionality in the correlations is an overidentifying restriction and can only be met within the limits of sampling error. If it is not met, there is no solution for equations 9-12. One way to achieve identification is to require that the unobservable explain the correlation between X1 and X4 as expressed in Land's equation 13. This solution requires the postulation of a random disturbance in Xa and use of equation 13 in estimating precludes its later use in testing the model.