Multisample measured variable path analysis is used to test whether causal/structural relations among measured variables differ across populations. Several invariance testing approaches are available for assessing cross-group equality of such relations, but the associated test statistics may vary considerably across methods. This study is a population analysis, examining five different strategies for invariance testing using an illustrative measured variable path model. The results demonstrate how inferences about parameters across populations can depend greatly upon the invariance testing approach used, thereby potentially leading to improper inference regarding the true invariance status of relevant parameters. (Contains 1 figure and 3 tables.)