The original SMVCIR algorithm was developed by Simon J. Sheather, Joseph W. McKean, and Kimberly Crimin. The dissertation first presents a new version of this algorithm that uses the scaling standardization rather than the Mahalanobis standardization. This algorithm takes grouped multivariate data as input and then outputs a new coordinate space that contrasts the groups in location, scale, and covariance. The central goal of research is to develop a method to determine the dimension of this space with statistical confidence. A dimensionality test is developed that can be used to make this determination. The new SMVCIR algorithm is compared with two other inverse regression algorithms, SAVE and SIR in the process of developing the dimensionality test and testing it. The dimensionality test is based on the singular values of the kernel of the spanning set of the vector space. The asymptotic distribution of the spanning set is found by using the central limit theorem, delta method, and finally Slutsky's Theorem with a permutation matrix. This yields a mean adjusted asymptotic distribution of the spanning set. Theory by Eaton, Tyler, and others is then used to show an equivalence between the singular values of the mean adjusted spanning set statistic and the singular values of the spanning set statistic. The test statistic is a sample size scaled sum of squared singular values of the spanning set. This statistic is asymptotically equivalent in distribution to that of a linear combination of independent 21 random variables. Simulations are performed to corroborate these theoretic findings. Additionally, based on work by Bentler and Xie, an approximation to the test statistic reference distribution is proposed and tested. This is also corroborated with simulations. Examples are performed that demonstrate how SMVCIR is used and how the developed tests for dimensionality are performed. Finally, further directions of research are hinted at for SMVCIR and the dimensionality test. One of the more interesting directions is explored by briefly examining how SMVCIR can be used to identify potentially complex functions that link predictors and a continuous response variable.

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