In any field of science, statistical measures of confidence in a model or hypothesis works by comparing observations to the predictions of that model vs. the predictions from a null hypothesis.
In this case, the standard method for testing the attribution of warming to anthropogenic causes is to run two types of climate models: one set which includes both natural and anthropogenic changes in Earth's energy balance (the radiative forcings), and the other set including only the natural forcings. Then the models run through the physics and climate dynamics to predict the change in Earth's temperature over time under those scenarios. Finally, compare those predictions to what we actually observe. What we observe is consistent with models including both natural and anthropogenic forcings, and strongly inconsistent with natural forcings alone.
We might wonder if internal climate dynamics throws a wrench in these conclusions. How can we be sure that the observed warming is not due to some other cause besides the anthropogenic forcings? There are several sources of confidence:
- The additional thermal energy from global warming can be traced through the whole system (most of it is being absorbed in the oceans). The signal of global warming in surface temperature alone can be tricky, because the atmosphere stores only a small fraction of the total energy and much energy is transferred between atmosphere oceans and cryosphere, but when we check the whole system we find it agrees with the change in Earth's energy balance and how the energy is transferred between subsystems.
- We can be confident that the models are mostly capturing the physics correctly, because they do a good job of retrodicting past climate changes caused by solar variability and volcanoes. When run backwards, they don't predict nonsense.
- The enhanced greenhouse effect is measurable by its altitude dependence. It warms the surface and lower atmosphere, but cools the upper atmosphere, and observations reflect this trend.