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The asymptotically distributed free (ADF) method is often used to estimate parameters or test models without a normal distribution assumptions on variables, both in covariance structure analysis and in correlation structure analysis. However, little has been done to study the differences in behaviors of the ADF method in covariance structure analysis and correlation structure analysis. In this thesis the behaviors of the ADF method in covariance structure analysis and correlation structure analysis were studied for three test statistics, chi-square test TAGLS and its small-sample improvements TYB and TF(AGLS). Results showed that the ADF method in correlation structure analysis with test statistic TAGLS performs much better at small sample sizes than the corresponding test for covariance structures. However, test statistics TYB and TF(AGLS) under the same conditions generally perform better with covariance structures than with correlation structures. Results also showed that condition numbers of weight matrices are systematically increased with substantial increase in variance as sample size decreases. Implications for research and practice are discussed.