Structural Equation Modeling (SEM) is a framework of statistical methods that allows us to represent complex relationships between variables. SEM is widely used in economics, genetics and the behavioral sciences (e.g. psychology, psychobiology, sociology and medicine). Model complexity is defined as a model’s ability to fit different data patterns and it plays an important role in model selection when applying SEM. As in linear regression, the number of free model parameters is typically used in traditional SEM model fit indices as a measure of the model complexity. However, only using number of free model parameters to indicate SEM model complexity is crude since other contributing factors, such as the type of constraint or functional form are ignored. To solve this problem, a special technique, Confirmatory Tetrad Analysis (CTA) is examined. A tetrad refers to the difference in the products of certain covariances (or correlations) among four random variables. A structural equation model often implies that some tetrads should be zero. These model implied zero tetrads are called vanishing tetrads. In CTA, the goodness of fit can be determined by testing the null hypothesis that the model implied vanishing tetrads are equal to zero. CTA can be helpful to improve model selection because different functional forms may affect the model implied vanishing tetrad number (t), and models not nested according to the traditional likelihood ratio test may be nested in terms of tetrads. In this dissertation, an R package was created to perform CTA, a two-step method was developed to determine SEM model complexity using simulated data, and it is demonstrated how the number of vanishing tetrads can be helpful to indicate SEM model complexity in some situations.