Using a general analysis of variance algorithm for covariance designs

Procedures are described which enable researchers to implement balanced covariance designs of from one to four independent variables. Use is made of three subroutines from IBM’s Scientific Subroutine Package which implement a general decomposition algorithm for balanced designs. FORTRAN instructions, illustrating the main calling program, are given.     

The variance error of the P50-discriminant

The P₅₀-discriminant has been reported elsewhere in connection with its use in predicting whether selective service registrants if inducted would become normal operative soldiers or would commit offenses causing their imprisonment. The standard error of the P₅₀-discriminant is a good measure to use in determining how far to the side of this statistic a particular case falls. The standard error formula itself has also been published elsewhere; but its derivation, as the variance error, is given here.The author gratefully acknowledges the very extensive assistance kindly given to him by Dr. Truman L. Kelley and Dr. Frederick Mosteller. This assistance was given without reference to the utility of the P₅₀-discriminant, upon which matter the author reports elsewhere and for which he takes full responsibility.     

The estimation of true score variance and error variance in the classical test theory model

The use of one-way analysis of variance tables for obtaining unbiased estimates of true score variance and error score variance in the classical test theory model is discussed. Attention is paid to both balanced (equal numbers of observations on each person) and unbalanced designs, and estimates provided for both homoscedastic (common error variance for all persons) and heteroscedastic cases. It is noted that optimality properties (minimum variance) can be claimed for estimates derived from analysis of variance tables only in the balanced, homoscedastic case, and that there they are essentially a reflection of the symmetry inherent in the situation. Estimates which might be preferable in other cases are discussed. An example is given where a natural analysis of variance table leads to estimates which cannot be derived from the set of statistics which is sufficient under normality assumptions. Reference is made to Bayesian studies which shed light on the difficulties encountered. Work on this paper was carried out at the headquarters of the American College Testing Program, Iowa City, Iowa, while the author was on leave from the University College of Wales.     

Correcting an analysis of variance for clustering

A great deal of educational and social data arises from cluster sampling designs where clusters involve schools, classrooms, or communities. A mistake that is sometimes encountered in the analysis of such data is to ignore the effect of clustering and analyse the data as if it were based on a simple random sample. This typically leads to an overstatement of the precision of results and too liberal conclusions about precision and statistical significance of mean differences. This paper gives simple corrections to the test statistics that would be computed in an analysis of variance if clustering were (incorrectly) ignored. The corrections are multiplicative factors depending on the total sample size, the cluster size, and the intraclass correlation structure. For example, the corrected F statistic has Fisher's F distribution with reduced degrees of freedom. The corrected statistic reduces to the F statistic computed by ignoring clustering when the intraclass correlations are zero. It reduces to the F statistic computed using cluster means when the intraclass correlations are unity, and it is in between otherwise. A similar adjustment to the usual statistic for testing a linear contrast among group means is described.     

Accounting for common method variance in cross-sectional research designs

Cross-sectional studies of attitude-behavior relationships are vulnerable to the inflation of correlations by common method variance (CMV). Here, a model is presented that allows partial correlation analysis to adjust the observed correlations for CMV contamination and determine if conclusions about the statistical and practical significance of a predictor have been influenced by the presence of CMV. This method also suggests procedures for designing questionnaires to increase the precision of this adjustment.     

The variance of d' estimates obtained in yes-no and two-interval forced choice procedures

In studies of detection and discrimination, data are often obtained in the form of a 2 x 2 matrix and then converted to an estimate of d' based on the assumptions that the underlying decision distributions are Gaussian and equal in variance. The statistical properties of the estimate of d', d' are well understood for data obtained using the yes-no procedure, but less effort has been devoted to the more commonly used two-interval forced choice (2IFC) procedure. The variance associated with d' is a function of true d' in both procedures, but for small values of true d' the variance of d' obtained using the 2IFC procedure is predicted to be less than the variance of d' obtained using yes-no; for large values of true d', the variance of d' obtained using the 2IFC procedure is predicted to be greater than the variance of d' from yes-no. These results follow from standard assumptions about the relationship between the two procedures. The present paper reviews the statistical properties of d' obtained using the two standard procedures and compares estimates of the variance of d' as a function of true d' with the variance observed in values of d' obtained with a 2IFC procedure.     

A simple test for heterogeneity of variance in complex factorial designs

A simple procedure for testing heterogeneity of variance is developed which generalizes readily to complex, multi-factor experimental designs. Monte Carlo Studies indicate that the Z-variance test statistic presented here yields results equivalent to other familiar tests for heterogeneity of variance in simple one-way designs where comparisons are feasible. The primary advantage of the Z-variance test is in the analysis of factorial effects on sample variances in more complex designs. An example involving a three-way factorial design is presented.     

Robust techniques for testing heterogeneity of variance effects in factorial designs

Several ways of using the traditional analysis of variance to test heterogeneity of spread in factorial designs with equal or unequaln are compared using both theoretical and Monte Carlo results. Two types of spread variables, (1) the jackknife pseudovalues ofs ² and (2) the absolute deviations from the cell median, are shown to be robust and relatively powerful. These variables seem to be generally superior to the Z-variance and Box-Scheffé procedures.This research was sponsored by Public Health Service Training Grant MH-08258 from the National Institute of Mental Health. The author thanks Mark I. Appelbaum, Elliot M. Cramer, and Scott E. Maxwell for their helpful criticisms of this paper. An earlier version of this work was presented at the Annual Meeting of the Psychometric Society, Murray Hill, New Jersey, April, 1976.     

The role of correlation in analysis of variance

A test of the significance of a row or column agent in an analysis of variance may be expressed in the form of correlation between the agent and the variate. A test of the significance of interaction variance may be expressed in the form of correlation between the agents. These expressions are principally of theoretical interest in that the degree of significance in anF test or the value of a correlation coefficient may be controlled at will, or inadvertently, within certain limits.This study was made possible by the Bureau of Psychological Services, Institute for Human Adjustment, Horace H. Rackham School of Graduate Studies, University of Michigan.     

The analysis of variance and covariance techniques in relation to the conventional formulas for the standard error of a difference

In this paper it is demonstrated that the analysis of variance techniques yield results equivalent to the calculation oft by means of expressions based on the short or the long formula. It is also shown that the covariance technique gives results equivalent to those obtained by means of the formula fort which should be used with matched groups. The conditions necessary for equivalent results are such that the conventional formulas fort would normally be used rather than the variance or covariance techniques. However, a knowledge of the relationships described in this paper should contribute to one's understanding of the variance and covariance techniques.The relationships described in this paper were brought to the attention of the author by the excellent article of Eugene Shen (8). The proofs given here and the examples are the work of the present author.     

The analysis of variance and pairwise scaling

Variance analyses are presented for two data layouts—each corresponding to the class of all ordered pairs from a single finite set. The analysis of the dominance layout is in terms of a fixed effects linear model which includes parameters representing the scale values of the elements of the set, response bias, and pairwise interactions. A parallel parametrization is carried out for the composition layout for which corresponding point estimates and hypothesis tests are given. A joint treatment of concurrently observed dominance and composition layouts is suggested and illustrative data are presented.This research was supported in part by National Institutes of Health Grant MH-04439-06. The author would like to express his appreciation to Richard Beatty of the University of Toronto, James Baker and William Carroll of Oregon Research Institute, and J. E. Keith Smith of the University of Michigan, for their helpful comments concerning aspects of this work. Computing assistance was obtained from the Health Sciences Computing Facility, UCLA, sponsored by NIH Grant FR-3.     

The history of an error

Lukács has had a colorful career as a Communist theoretician. One of the strangest events is the fact that he continued to recant in 1967, when there was no longer the external pressure to do so. This may be due to the fact that his differences with Marx on subject-object, praxis, etc., are not those of a humanist.     

The choice of constraints in correspondence analysis

A discussion of alternative constraint systems has been lacking in the literature on correspondence analysis and related techniques. This paper reiterates earlier results that an explicit choice of constraints has to be made which can have important effects on the resulting scores. The paper also presents new results on dealing with missing data and probabilistic category assignment.I am most grateful to the following for their helpful comments. Arto Demirjian, Michael Greenacre, Michael Healy, Shizuhiko Nishisato, Roderick Mcdonald, and several anonymous referees.     

Some types of error in a choice response task

Errors from a serial response task involving single-finger responses to alphabetic stimuli are analysed and discussed in relation to findings which have been reported from tasks with more compatible stimulus-response relationships. Errors are divided into three distinguishable subsets and in each case found to have longer latencies than correct responses. Those which result from mirroring the required response about the centre of the hand are found to resist elimination during practice and their frequency seems to depend on the type of code used. In all cases error correction times are faster than the times to make a correct response but mirror errors and errors involving a finger adjacent to the correct response are corrected faster than other errors. The findings are discussed in relation to the theory of choice reaction time and error correction.     

Some thoughts regarding the sensitivity of t to error in estimated within-group variance

A. J. Riopelle (2003) has eloquently demonstrated that the null hypothesis assessed by the t test involves not only mean differences but also error in the estimation of the within-group standard deviation, s. He is correct in his conclusion that the precision of the interpretation of a significant t and the null hypothesis tested is complex, particularly when sample sizes are small. In this article, the author expands on Riopelle's thoughts by comparing t with some equivalent or closely related tests that make the reliance of t on the accurate estimation of error perhaps more salient and by providing a simulation that may address more directly the magnitude of the interpretational problem.     

Risky choice as a function of amount and variance in food supply.

In Experiment 1, 4 rats earned their daily food ration by choosing on a trials basis between a "risky" and a "riskless" lever. The risky lever produced either 15 45-mg food pellets or no pellets, and on average provided five pellets per choice. The riskless lever always produced three pellets. Across conditions, the number of trials per session was varied. Body weight and choice of the risky lever decreased as the number of trials per session decreased, even though body weight could only be defended by increased choice of the risky lever. In Experiment 2, trials per session were fixed, but the number of pellets delivered by the risky and riskless levers was either at the same level as in Experiment 1 or tripled from those levels. Now choice of the risky lever was inversely related to the size of reinforcement and to body weight. The results of these experiments show that risk aversion covaries with the amount of food available in a session and the daily variance in the amount of food earned.