A Monte Carlo study of the effects of correlated method variance in moderated multiple regression analysis

It has been suggested that hierarchical regression analysis provides an unambiguous conclusion with regard to the existence of moderator effects (Arnold & Evans, 1979). This paper examines the impact of correlated error among the dependent and independent variables in order to explore whether or not artificial interaction terms can be generated. A Monte Carlo study was performed to investigate the effects of correlated error on noninteraction and interaction models. The results are clear-cut. Artifactual interaction cannot be created; true interactions can be attentuated. Some practical suggestions are provided for drawing inferences from hierarchical regression analysis.     

Analysis of variance with Fourier analysis of coherent data

Like orthogonal polynomials, the sine and cosine components used in Fourier analysis are orthogonal under summation over equally spaced points. Therefore, they can be used in the same way as orthogonal polynomials are used with analysis of variance. They are usually used on records consisting of sequences of observations taken equally spaced in time. Analysis of variance can be performed on replicated records. The most straightforward cases occur under the following conditions: (1) The phenomenon is periodic, (2) the phenomenon is coherent (synchronized) over the various replicates, (3) the record length is an integral multiple of the fundamental period, and (4) the phenomenon is not buried too deeply in noise. Two numerical examples are examined that satisfy these conditions. Planning is illustrated for a case that violates Conditions 3 and 4.     

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.     

Analysis of variance of one-, two-, and three-treatment designs for a PDP-8

Operating instructions for a series of analysis of variance programs for one-, two-, and three-treatment experimental designs is described. The emphasis is on versatility, speed, accuracy, and sufficiency of output. The on-line aspect of FOCAL allows extensive transformations of raw data. Procedures and terminology conform to Kirk (1968) to provide information for pooling error terms for various models, mean comparisons, and trends analysis. Patches are given for data tape input.     

Synthesis of variance

The distribution of a linear combination of two statistics distributed as is Chi-square is studied. The degree of approximation involved in assuming a Chi-square distribution is illustrated for several representative cases. It is concluded that the approximation is sufficiently accurate to use in many practical applications. Illustrations are given of its use in extending the Chi-square, the Student t and the Fisher z tests to a wider range of problems.     

A critique of the correlated trait-correlated method and correlated uniqueness models for multitrait-multimethod data

The correlated trait-correlated method (CT-CM) and correlated uniqueness (CU) confirmatory factor analysis models for multitrait-multimethod data are critiqued. Although the CU model often returns convergent and admissible factor solutions when the CT-CM model does not, the CU model is shown to have theoretical and substantive shortcomings. On the basis of this critique, the authors recommend that the CT-CM model be regarded as the generally preferred model and that the CU model be invoked only when the CT-CM model fails.     

A criterion for significant common factor variance

Up to the present only empirical methods have been available for determining the number of factors to be extracted from a matrix of correlations. The problem has been confused by the implicit attitude that a matrix of intercorrelations between psychological variables has a rank which is determinable. A table of residuals always contains error variance and common factor variance. The extraction of successive factors increases the proportion of error variance remaining to common factor variance remaining, and a point is reached where the extraction of more dimensions would contain so much error variance that the common factor variance would be overshadowed. The critical value for this point is determined by probability theory and does not take into account the size of the residuals. Interpretation of the criterion is discussed.