Since the other answer is just questioning the errors with a belief statement,
we thus believe the measurement errors of both the geodetic effect and the frame-dragging effect should be close to each other.
one should point out that science is not about beliefs but about measurements.
The functional dependence of the two quantities whose errors differ by a factor of 2.5 could easily explain the difference if we had the formulae.From the linked announcement:
"We measured a geodetic precession of 6.600 plus or minus 0.017 arcseconds and a frame dragging effect of 0.039 plus or minus 0.007 arcseconds," says Everitt.
For readers who are not experts in relativity: Geodetic precession is the amount of wobble caused by the static mass of the Earth (the dimple in spacetime) and the frame dragging effect is the amount of wobble caused by the spin of the Earth (the twist in spacetime). Both values are in precise accord with Einstein's predictions.
We see that the values themselves have a large difference and the geodetic precession is measured much more accurately than the frame dragging .
The gyroscopes will certainly have been tested that they showed no deflection when in lab conditions. There are extensive descriptions of the construction of the gyroscopes and the experiment.
From the question:
If they didn't measure a negative result too, how does this the experiment prove the theory correct?
I am answering this to make clear that a theory can never be proven correct. It can only be proven wrong. A theory is continually tested by new data and one can say "the new data confirm the predictions of the theory", but confirmation is not proof. Even one good measurement that contradicts a theory falsifies it, i.e. proves that it is wrong at some part of its domain of definition, if the measurement is repeatable.
Measurement errors are a part of what defines a domain of definition.
The domain of definition has to be examined and the limits found. Example: Newtonian mechanics and General Relativity, where Newtonian mechanics are a limiting case of General relativity, valid for specific domains.
Any alternate theories could be toyed with within the errors, but that does not mean that the confirmation of GR is false because there are experimental errors