“Every $1 the government invests in XYZ science/research endeavor, returns $XYZ.” How are claims like these calculated?

Physicist here!

I run into statements like this all the time: "For every $1.00 the US Government invested into X Scientific Endeavor (e.g. Apollo Missions, Medical Research, Natural Sciences Research, etc…), it generated $Y worth of (value/return/economic activity/GDP contribution/growth, etc…).

How are these values determined?

Please forgive me for not being specific to a specific program/initiative or a specific outcome! I'm leaving this generalized on purpose because I suspect that these are the variables that matter most (e.g. the finer the resolution of input and output, the more precise the measurement, right?)

So, I'm super curious how that factors in? What inputs are meaningful and what outputs are meaningful? How are these output values determined and how precise are they?

We use these statements all the time in science communication/outreach as a short hand for "investing in scientific research is worth the money", but rarely stop and think about how the values we're reporting to the public are determined.