Much discussions surrounds the issue of “analytic flexibility” seen in multi-analyst studies. In these, researchers crunch data following diverse plausible model structures as to produce some form of useful inference, only to find a “universe of uncertainty” when comparing their findings. Uncertainty in models, be those mathematical, statistical, or other, can arise from various sources, including parametric and structural factors. We discuss the issue of analytic flexibility in general, and how global sensitivity analysis can be usefully applied to tackle the problem. We illustrate the main concepts via a very simple example taken from sustainability science. Specifically we show with our example how uncertainty increases when transitioning from purely parametric variations to structural modifications in a simple three-factor model inspired by the Human Development Index. Global sensitivity analysis, performed using variance-based methods, reveals that structural uncertainty increases interaction effects, leading to a higher effective dimension and model complexity. These findings emphasize the importance of considering structural uncertainty in model assessments, including in applications where aggregation rules influence outcomes, such as for the case of many composite indicators for sustainability.