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The present study was directed at determining the extent to which the Type I Error rate is affected by violations in the basic assumptions of the q statistic. Monte Carlo methods were employed, and a variety of departures from the assumptions were examined. (Author)

Statistics are an essential tool for making proper judgement decisions. It is concerned with probability distribution models, testing of hypotheses, significance tests and other means of determining the correctness of deductions and the most likely outcome of decisions. Measures of central tendency include the mean, median and mode. A second important class of descriptive statistics is measures of variability described by the standard deviation. Inferential statistics estimates what a total set of measures would be like on the basis of a sample. The standard error of the mean is a statistic that indicates an estimate of the reliability of the sample mean. The t-ratio is the ratio of the expected difference in a set of scores to the obtained difference. This can also be interpreted in relation to the normal curve. The coefficient of correlation is a measure showing the relationship of measurements on one set of variables to the measurements on another set of variables. Two variables can be highly correlated without being causally connected. (DJ)

The harmonic mean and Kramer (1956) unequal n forms of the Tukey multiple comparison statistic were investigated for Monte Carlo Type I and Type II errors under conditions of assumption violations. The two major questions concerning the sensitivity of multiple comparison statistics for different types of pairwise contrasts and the affect of increasing the number of treatment levels are discussed. Neither procedure consistently out-performs the other; the choice of test depends upon the population condition(s) and the pattern of unequal cell frequencies. (Author)

The logical structure of the Campbell and Fiske multitrait-multimethod approach is applied to the problem of studying growth and its determinants. The resulting model is a special case of Joreskog's general model for the analysis of covariance structures. The relationships of traditional psychometric formulations to this model are detailed. (Author)

This paper, a revision of the original document, "Correcting Partial, Multiple, and Canonical Correlations for Attenuation" (see TM 000 535), presents the formula for correcting coefficients of partial correlation for attenuation due to errors of measurement. In addition, the correction for attenuation formulas for multiple and cannonical correlations are discussed. The attenuating effects of measurement error are examined algebraically for a special case of partial correlation and multiple correlation. Finally, the formula for the corrected partial r is related to recent work on the measurement of change. (Author/AG)

The accuracy of estimating test means for groups of twelfth-grade students by the item-sampling technique was examined. The subjects were from 35 twelfth-grade schools participating in the National Longitudinal Study of Mathematical Abilities. Half of the students in each school were assigned to a treatment condition where they took a complete 24-item mathematics test on the first day of testing and took item-sampled versions of the same test on the second day of testing. A second random group of students within each of the schools took the item-sampled version of the mathematics test on day 2 but did not take the complete version of the mathematics test on day 1. There was no evidence to indicate that taking the complete 24-item mathematics test influenced the performance on the item-sampled version on the second day of testing. Reasonably close estimates of mean performance were obtained from the item-sampling situation as compared to the means estimated from the conventional type of testing. The differences between the means estimated from conventional type testing and from item-sampling testing were found to diminish as a function of the number of students tested in the school (square root transformation). (Author/PR)

Three simplifying conditions are given for obtaining least squares (LS) estimates for a nonlinear submodel of a linear model. If these are satisfied, and if the subset of nonlinear parameters may be LS fit to the corresponding LS estimates of the linear model, then one attains the desired LS estimates for the entire submodel. Two illustrative analyses employing this method are given, each involving an Eckart-Young (LS) decomposition of a matrix of linear LS estimates. In each case the factors provide an LS fit of the nonlinear submodel to the original data. The minimum error sum of squares for this fit is the error sum of squares for the corresponding linear model plus a function of the eigenvalues involved in the factorization. An Eckart-Young factorization, however, is only a special case of an LS decomposition of LS estimates. The present method is more generally applicable (under the three simplifying conditions) whenever any LS procedure may be found for fitting certain parameters of a nonlinear submodel to the corresponding LS estimates of a linear model. (Author)

Joreskog's general method for analysis of covariance structures was developed for estimating a model involving structures of a very general form on means, variances, and covariances of multivariate observations. This method achieves a great deal of generality and flexibility, in that it is capable of handling most standard statistical models as well as many nonstandard and complicated ones. This paper describes a computer program for the method. When the variance-covariance matrix of the observed variables is unconstrained, the method may be used to estimate location parameters and to test linear hypotheses about them. For example, the program may be used to handle such standard problems as multivariate regression, ANOVA, and MANOVA. A unique feature of the method is that it can also be used when the variance-covariance matrix is constrained to be of a certain form. Various other models involving correlated errors or errors of measurement can also be handled. An illustration of how input data is set up, and what the printout looks like for two small sets of data, are included. (Author/CK)

This study examines the attitudes which black high school students hold toward other blacks when they speak standard English and when they speak black English. Tenth-grade black students listened to recordings of black speakers, each using both standard English and black English for the same text. The students were asked to judge the speakers on several personality characteristics, such as friendly, honest, unselfish, intelligent, good looking, lucky, etc. The statistical results provide a measurement of attitudes that the students hold toward the dialects used in the test. According to the results, there is an overwhelming preference for speakers of standard English. The context of the experiment must be considered along with the results, and reasons for the results should be questioned. Black students and teachers need to be taught what linguists have discovered--that black English is a valid and systematic language and that it is an imminently suitable medium of expression. (Author/VM)

Guttman's "rank-reduction" theorem is employed to demonstrate how most of the algebra and computational work for standard correlational and analysis of variance methods can be examined within a single matrix system. Analyses of both correlational and experimental data are illustrated with an easy-to-use computer program. This application and the corresponding discussion emphasize distinctions and qualifications needed for appropriate interpretations of both types of data. Since this theorem was initially presented within the framework of factor analysis, it is intended that this paper will contribute to a better understanding of relationships among factor analytic, general correlational, and experimental design methods. (Author/CK)

A general method for estimating the unknown coefficients in a set of linear structural equations is described. In its most general form the method allows for both errors in equations (residuals, disturbances) and errors in variables (errors of measurement, observational errors) and yields estimates of the residual variance-covariance matrix and the measurement error variances as well as estimates of the unknown coefficients in the structural equations, provided all these parameters are identified. Two special cases of this general method are discussed separately. One is when there are errors in equations but no errors in variables. The other is when there are errors in variables but no errors in equations. The methods are applied and illustrated using artificial, economic and psychological data. (Author)

In this paper problems caused by the existence of errors of measurement are identified for factor analysis, regression analysis, ANOVA, and ANCOVA. At least one detrimental effect is shown to exist for each type of analysis. When a researcher's interest is with infallible variables, he runs the risk of biased results from all of the procedures except ANOVA. The estimates of parameters in all four procedures suffer from inflated error variance. Some partial solutions are indicated, but clearly more work is needed on several of the problems. Most statistical procedures have been developed for models where variables are assumed to be free from errors of measurement. Since almost all educational research involves use of fallible variables, it is important that the effects of errors of measurement for the various models be understood and that the understanding be reflected in current research practice. (Author/GS)

This paper relates educational and psychological statistics to certain "Research Statistical Tools" (RSTs) necessary to accomplish and understand general research in the behavioral sciences. Emphasis is placed on acquiring an effective understanding of the RSTs and to this end they are are ordered to a continuum scale in terms of individual precision, power, and robustness. The Precision-Power-Gradient theory is suggested as a means of structuring an introductory graduate course in statistics. Twelve RSTs are considered, ranging from normative comparisons (at the simplest level), through parametric/non-parametric statistics, analysis of variance and covariance, correlations, factor analysis, to canonical correlation analysis, and, briefly, other advanced techniques. Equally important is a working knowledge of the computer, which enables the RSTs to be used with greater facility and accuracy. Computer orientation should include a basic knowledge of FORTRAN programming, nature and availability of computer-based statistical libraries, the ability to utilize such libraries, with a general understanding of computer output. (Author/AE)

The suppressor variable, a variable wholly uncorrelated with a criterion, but which nevertheless improves prediction because of its relationship with a predictor, is critically examined. For a suppressor so defined, formal identities are shown with part, partial, and multiple correlational procedures. It is demonstrated that if maximum prediction is the primary goal, the use of the suppressor approach is appropriate if, and only if, the suppressor-criterion correlation is zero. Otherwise multiple correlation will always yield a higher validity. Mathematical constraints are shown to be operating on suppressors with a resultant effect of producing minimal incremental validity for suppressors within the range of typical validity coefficients. It is also demonstrated that additional predictors will yield a greater increment in prediction in comparison with suppressors. On the other hand, because of its conceptual advantages, the formally equivalent part correlation is recommended over the multiple correlation approach for suppressor variables which enter into theoretical relationships and for the measurement of latent constructs. (Author/DG)

This volume contains the five major addresses and subsequent discussion from the Symposium on the General Linear Models Approach to the Analysis of Experimental Data in Educational Research, which was held in 1967 in Athens, Georgia. The symposium was designed to produce systematic information, including new methodology, for dissemination to the educational research community to (1) promote wider use of sound methodology and (2) provide caveats regarding limitations of this powerful approach. The authors and their papers are: (1) Graybill, Franklin A., "Introduction to the Use of General Linear Models in the Analysis of Experimental Data," (2) Ward, Joe H., Jr., "Synthesizing Regression Models--An Aid to Learning Effective Problem Analysis," (3) Winer, B. J., "Problems in the Use of General Linear Model Methods," (4) Bargmann, Rolf E., "A Survey of Appropriate Methods of Analysis of Factorial Designs," and (5) Bock, R. Darrell, "Remarks on Analysis of Variance and Analysis of Regression." The papers and ensuing discussion elucidate strengths and limitations of the general linear models approach, discuss procedures for handling computations, and present the independent views of major authorities on theory and of established practical authorities on the use and usability of methods. (HW)

THIS PAPER DEALS WITH SOME TESTS OF HYPOTHESIS FREQUENTLY ENCOUNTERED IN THE ANALYSIS OF MULTIVARIATE DATA. THE TYPE OF HYPOTHESIS CONSIDERED IS THAT WHICH THE STATISTICIAN CAN ANSWER IN THE NEGATIVE OR AFFIRMATIVE. THE DOOLITTLE METHOD MAKES IT POSSIBLE TO EVALUATE THE DETERMINANT OF A MATRIX OF HIGH ORDER, TO SOLVE A MATRIX EQUATION, OR TO INVERT A MATRIX OF HIGH ORDER. HYPOTHESES WHICH, IN THE UNIVARIATE CASE, WOULD LEAD TO A USE OF THE STUDENT T-DISTRIBUTION ARE ALSO PRESENTED. THE ANALYSIS OF DATA COLLECTED FROM A "DESIGNED EXPERIMENT" IS DISCUSSED, INCLUDING THE LATIN SQUARE, RANDOMIZED BLOCK, CROSS-CLASSIFICATION, AND FACTORIAL DESIGNS. AN ANALYSIS OF REGRESSION IS PRESENTED ALONG WITH A TEST OF HYPOTHESIS ON THE DISPERSION MATRIX. (HW)

AN ANALYSIS OF THE NONUNIQUE PORTION OF EXPLAINED VARIATION WHICH CAN BE ATTRIBUTED TO THE REGRESSOR VARIABLES AS A GROUP IN REGRESSION ANALYSIS IS DISCUSSED. THE UNIQUE SUMS OF SQUARES IS PRESENTED AS A BASIS FOR UNDERSTANDING THE PROCEDURE FOR PARTITIONING THE NONUNIQUE PART. A METHOD OF PARTITIONING NONUNIQUE VARIATION IS DEFINED IN TERMS OF A THREE VARIABLE CASE--DEFINITION OF COMMONALITIES, REDUCED FORM OF THE COMMONALITIES, AND PARTITIONING THE SUMS OF SQUARES. TABLES AND AN APPENDIX ARE INCLUDED. (BK)