An alternative formulation of the multigroup common factor model with minimal uniqueness constraints is considered. This alternative formulation is based on a simple identification constraint that is related to the standard maximum likelihood constraint used in single-group common factor analysis. It is argued that the alternative formulation leads to less technical difficulties in applications than earlier formulations of this multigroup common factor model. Furthermore, associated tests for various measurement invariance constraints across groups are proposed, such as an omnibus test for the absence of uniform bias. By means of an empirical example, the fitting of several multigroup common factor models with minimal uniqueness constraints and the testing for measurement invariance over groups are demonstrated. The nesting of multigroup confirmatory factor models under the multigroup common factor model with minimal uniqueness constraints is also discussed. Finally, a small study is performed to investigate the drop in power to detect uniform bias in using the multigroup common factor model with minimal uniqueness constrains instead of a confirmatory special case. The results of the study show a small drop in power under all research conditions.