We develop the mathematical tools necessary to describe the interaction between a resonant pole and a threshold energy. Using these tools, we analyze the properties an opening threshold has on the resonant pole mass (the "cusp effect"), leading to an effect called "pole-dragging." We consider two models for resonances: a molecular, mesonic model, and a color-nonsinglet diquark plus antidiquark model. Then, we compare the pole-dragging effect due to these models on the masses of the f0(980), the X(3872), and the Zb(10610) and compare the effect's magnitude. We find that, while for lower masses, such as the f 0 (980), the pole-dragging effect that arises from the molecular model is more significant, the diquark model's pole-dragging effect becomes dominant at higher masses such as those of the X(3872) and the Z b (10610). This indicates that for lower threshold energies, diquark models may have less significant effects on predicted resonant masses than mesonic models, but for higher threshold energies, it is necessary to include the pole-dragging effect due to a diquark threshold in high-precision QCD calculations.