There are a few things to consider with a radome over a GPS antenna.
- the reflection that occurs at the interfaces with air.
- the loss inside the radome.
- the near-field effect the radome has on the tuning of the antenna.
Dielectric effects and reflection
The radome is a sheet of dielectric, which has a permeability different to that of air. There will be a reflection of the wave as it enters the material, and as it leaves. In a thin radome, much less than 1/4 wavelength, these two effects can be combined.
The reflection depends on the thickness of the material, and the dielectric constant. Most plastics have an \$\epsilon_r\$ of about 2.1, and glass is about 4.0 (tempering doesn't matter). For a thin skin, the effect is approximately linear with thickness and dielectric, so a 1 mm radome has half the impact of a 2 mm radome, and a 5 mm plastic radome has the same effect as a 2.5 mm plate glass radome.
I can't find the equations online or in my textbooks, so you'll have to go with my memory for now, which is that for normal incidence, a 5 mm sheet of plastic should not cause a significant transmission loss at GPS frequencies.
But that's not all. At increasing angles of incidence, the plate becomes "thicker" to the incident wave, the reflection becomes stronger and the transmission weaker. Also, the two polarisation components have different transmission coefficients, so the circular polarisation of the antenna will be degraded. A GPS antenna needs to operate with signals from as low as 10 degrees above the horizon, 80 degrees from normal incidence, and at this angle, even a thin sheet of plastic has a significant effect.
The best solution is to have a dome or even hemispherical radome, made from the best thin dielectric. The definition of best is the highest \$strength / (thickness * \epsilon_r) \$.
2. Losses in the material
If the material is lossy, that will reduce the signal transmitted. This should drive your choice of materials towards pure plastics, or high-glass-fraction composites. Again, in proportion to the thickness of the material - a glass-loaded plastic can be made much thinner than a pure plastic, for the same strength, so it may be even less lossy.
RF Cafe has a table of dielectric constants and loss tangents
The big thing to watch out for is the effect of additives in the plastic. Black plastic is usually bad as it is often coloured with carbon black. Cheap regrind plastic has a particularly high percentage of this black pigment. UV-stabilisers can also increase the loss. Translucent pigments that colour the plastic will have no effect.
Nylon is a common strong plastic that is unfortunately not ideal for radomes, as it absorbs a significant amount of moisture, which raises the dielectric constant, and increases the loss.
3. dielectric loading of the GPS antenna
The dielectric in the radome has a third effect - by interacting with the near fields of the antenna, it affects the tuning of the antenna, decreasing its sensitivity in the GPS band.
For a patch antenna, it is probably sufficient to keep the radome at least 20 mm away, for minimum effect. If you need to get closer, some manufacturers sell test kits, with a range of patches designed for different radome thicknesses. You can evaluate all of the options and then specify the module with the best tuning in your exact situation.
Testing what you have built
You can evaluate the radome solution you have built by recording some detailed information about the GPS signal strength for all satellites, with the antenna in the open air, and in the box. Plot the signal strength of satellites against elevation angle, and observe the change in performance at the zenith, which should be small, and the effects near the horizon, which will be bigger.
My opinion is that tempered glass is not a good radome material, unless you have other very pressing requirements for it. It's transparent so the sun will heat up the electronics. It's strong, but for its dielectric impact, not as good as glass-filled plastic. It is probably only available from 5 mm thickness, which is quite thick for a GPS radome. It would be better to choose a glass-filled plastic.