This paper extends the scaled and adjusted corrections to goodness-of-fit test statistics, originally developed by Satorra and Bentler (1988a,b), to score, Wald, and difference test statistics in the context of multi-sample analysis of moment structures. The theory is framed within the general context of multi-sample analysis, under general conditions on the distribution of observable variables. The paper discusses computational issues and the relationship between scaled and corrected statistics and asymptotic robust ones. A Monte Carlo study illustrates the performance of corrected score test statistics in finite samples. The paper begins by introducing moment structure analysis, which is widely used in behavioral, social, and economic studies to analyze structural relationships between variables, some of which may be latent. Commercial software for structural equation modeling is available, such as LISREL and EQS. In multi-sample analysis, data from several samples are combined to test for across-group invariance of model parameters. Asymptotic distribution-free (ADF) methods are developed but may lack robustness to small and medium-sized samples. An alternative is to use normal-theory estimation with asymptotic robust standard errors and test statistics. Satorra and Bentler (1994) developed a family of corrected normal-theory test statistics that outperform asymptotic robust test statistics in small and medium-sized samples. The paper extends Satorra-Bentler (SB) corrections to score (Lagrange multiplier), difference, and Wald test statistics in the general context of multi-sample analysis. The theory relies on the notation and results of Satorra (1989) and Satorra and Bentler (1994, 1988a,b). The paper discusses the asymptotic distribution of goodness-of-fit test statistics, including the Wald, score, and difference tests, and their performance under normal and non-normal data. The paper also discusses the adjusted test statistic, which is a Satterthwaite-type test statistic. The adjusted test statistic is defined as $ \overline{\overline{T}} = \frac{d}{\text{tr}(U\Gamma)}T $, where $ d $ is the integer closest to $ d' = \frac{(\text{tr}(U\Gamma))^2}{\text{tr}((U\Gamma)^2)} $, and $ U $ is a matrix defined in the paper. The null distribution of the adjusted test statistic is taken to be a chi-square distribution with $ d $ degrees of freedom. The paper concludes with a Monte Carlo study illustrating the performance of the above statistics in finite samples. The study shows that the SB scaled statistic, $ \overline{T}_S^* $, outperforms the alternative robust test statistic $ T_S $ in small samples, while $ T_S $ outperforms other test statistics in large samples. The adjusted test statistic $ \overline{T}_S^* $ shows acceptable performance across various sampleThis paper extends the scaled and adjusted corrections to goodness-of-fit test statistics, originally developed by Satorra and Bentler (1988a,b), to score, Wald, and difference test statistics in the context of multi-sample analysis of moment structures. The theory is framed within the general context of multi-sample analysis, under general conditions on the distribution of observable variables. The paper discusses computational issues and the relationship between scaled and corrected statistics and asymptotic robust ones. A Monte Carlo study illustrates the performance of corrected score test statistics in finite samples. The paper begins by introducing moment structure analysis, which is widely used in behavioral, social, and economic studies to analyze structural relationships between variables, some of which may be latent. Commercial software for structural equation modeling is available, such as LISREL and EQS. In multi-sample analysis, data from several samples are combined to test for across-group invariance of model parameters. Asymptotic distribution-free (ADF) methods are developed but may lack robustness to small and medium-sized samples. An alternative is to use normal-theory estimation with asymptotic robust standard errors and test statistics. Satorra and Bentler (1994) developed a family of corrected normal-theory test statistics that outperform asymptotic robust test statistics in small and medium-sized samples. The paper extends Satorra-Bentler (SB) corrections to score (Lagrange multiplier), difference, and Wald test statistics in the general context of multi-sample analysis. The theory relies on the notation and results of Satorra (1989) and Satorra and Bentler (1994, 1988a,b). The paper discusses the asymptotic distribution of goodness-of-fit test statistics, including the Wald, score, and difference tests, and their performance under normal and non-normal data. The paper also discusses the adjusted test statistic, which is a Satterthwaite-type test statistic. The adjusted test statistic is defined as $ \overline{\overline{T}} = \frac{d}{\text{tr}(U\Gamma)}T $, where $ d $ is the integer closest to $ d' = \frac{(\text{tr}(U\Gamma))^2}{\text{tr}((U\Gamma)^2)} $, and $ U $ is a matrix defined in the paper. The null distribution of the adjusted test statistic is taken to be a chi-square distribution with $ d $ degrees of freedom. The paper concludes with a Monte Carlo study illustrating the performance of the above statistics in finite samples. The study shows that the SB scaled statistic, $ \overline{T}_S^* $, outperforms the alternative robust test statistic $ T_S $ in small samples, while $ T_S $ outperforms other test statistics in large samples. The adjusted test statistic $ \overline{T}_S^* $ shows acceptable performance across various sample