A goal model family is a set of related goal models that conform to the same metamodel, with commonalities and variabilities between models. Goal model families stem from the evolution of initial models into several versions over time and/or the variation of models over the space dimension (e.g., products). In contexts where there are several versions/variations of a goal model, analyzing a set of related models with typical similarities, one model at a time, often involves redundant computations and may require repeated user assistance (e.g., for interactive analysis) and laborious activities. This paper proposes the use of union models as first-class artifacts to analyze families of goal models, in order to improve performance of language-specific analysis procedures. The paper empirically evaluates the performance gain resulting from adapting (or lifting) an existing analysis technique specific to the Goal-oriented Requirement Language (GRL) on a family of GRL models, all at once using a union model, compared to analyzing individual models, one model at a time. Our experiments demonstrate, based on the use of the IBM CPLEX optimizer, the usefulness and performance gains of using union models to perform a computationally expensive analysis, namely quantitative backward propagation, on a family of GRL models.