Most methods for structural equation modeling (SEM) focused on the analysis of covariance matrices. However, "Historically, interesting psychological theories have been phrased in terms of correlation coefficients." This might be because data in social and behavioral sciences typically do not have predefined metrics. While proper methods for conducting correlation structure analysis have been developed, they emphasized on either how to get consistent standard errors of parameter estimates or how to ensure that the model-implied matrix remains to be a correlation matrix. Motivated by the fundamental needs for more efficient/accurate parameter estimates and greater power in conducting statistical tests, this article explores advantages of correlation structure analysis over its conventional covariance counterpart. Issues related to reparameterization and placement of parameters are discussed. A new concept is introduced for comparing efficiency/accuracy of parameter estimates that are not on the same scale. Via the analysis of many real datasets, meta results show that correlation structure analysis yields uniformly more accurate parameter estimates and more powerful statistical tests than its covariance-structure-analysis counterpart on parameters that are of substantive interests. The same pattern of results between the two model parameterizations is also found by Monte Carlo simulation. Issues related to correlation structure analysis and substantive elaboration of models that are not scale-invariant are discussed as well. The results are expected to promote technical and software developments of correlation structure analysis as well as its adoption in data analysis. [This is the online version of an article published in "Structural Equation Modeling: A Multidisciplinary Journal."]