International Journal of Statistics and Probability; Vol. 6, No. 6; November 2017 ISSN 1927-7032 E-ISSN 1927-7040 Published by Canadian Center of Science and Education 35 Can Variances of Latent Variables be Scaled in Such a Way That They Correspond to Eigenvalues? Karl Schweizer 1 , Stefan Troche 2 , Siegbert Reiß 1 1 Department of Psychology, Goethe University Frankfurt, Germany 2 Department of Psychology and Psychotherapy, University of Witten/Herdecke, Witten Germany Correspondence: Karl Schweizer, Department of Psychology, Goethe University Frankfurt, Theodor-W.-Adorno-Platz 6, 60323 Frankfurt a. M., Germany. E-mail: K.Schweizer@psych.uni-frankfurt.de Received: August 1, 2017 Accepted: August 18, 2017 Online Published: September 15, 2017 doi:10.5539/ijsp.v6n6p35 URL: https://doi.org/10.5539/ijsp.v6n6p35 Abstract The paper reports an investigation of whether sums of squared factor loadings obtained in confirmatory factor analysis correspond to eigenvalues of exploratory factor analysis. The sum of squared factor loadings reflects the variance of the corresponding latent variable if the variance parameter of the confirmatory factor model is set equal to one. Hence, the computation of the sum implies a specific type of scaling of the variance. While the investigation of the theoretical foundations suggested the expected correspondence between sums of squared factor loadings and eigenvalues, the necessity of procedural specifications in the application, as for example the estimation method, revealed external influences on the outcome. A simulation study was conducted that demonstrated the possibility of exact correspondence if the same estimation method was applied. However, in the majority of realized specifications the estimates showed similar sizes but no correspondence. Keywords: latent variance, eigenvalue, confirmatory factor analysis, scaling, sum of squared factor loadings 1. Introduction In both confirmatory and exploratory factor analysis (CFA and EFA) the sum of squared factor loadings can be computed. In EFA, this sum is considered an estimate of the eigenvalue that reflects the amount of variability explained by the factor (Vogt, 2005). In the present essay, it is investigated whether the sum of squared factor loadings of CFA corresponds to the sum of squared factor loadings of EFA. We explore whether the sum of squared factor loadings can be used as a measure of the variance of the latent variable (=factor) in CFA in analogy to the eigenvalue in EFA also known as characteristic root and latent root (Marcus & Minc, 1988, p. 144). An investigation is reported in which simulated data were used constructed to represent different underlying structures. 1.1 The Role of the Variance in Confirmatory Factor Analysis and its Scaling As a descriptive statistic, the variance characterizes distributions of random variables. It can be computed in different ways depending on the scale of the random variable. If the random variable is a latent variable (i.e. a factor), the variance additionally depends on the model from which the latent variable originates. The characteristics of this model and the method employed for estimating the parameters of the model influence the variance. The model of the covariance matrix (Jöreskog, 1970) includes observed variances, variances of the latent variables and error variances. The estimation method determines how the observed variance is subdivided into latent variance that is also addressed as true variance and error variance. The variance of a latent variable reflects the impact of the source underlying the latent variable. A stronger impact leads to a larger variance. However, the variance of the latent variable has not played a major role in CFA although it is a kind of effect-size indicator similar to the effect size in experimentation (Cohen, 1988; Rosenthal, 1994; Rosenthal & Rubin, 2003). In experimentation, the focus was for a long time on significance testing. Depending on the sample size, however, very small effects can yield statistical significance even if they are trivial and negligible for practical reasons. As a consequence, nowadays the report of effect sizes is required besides the significance level in the presentation of experimental results. Like means, the scale of variances depends on the units of measurement. For example, the variance of a set of processing times is considerably larger when the processing times are measured in milliseconds rather than in seconds. If the scale of the data is known, the difference may be regarded as a triviality. However, if the distributions of different random