I had a further thought on the test that perhaps overcomes some of the statistical problems and means that even if brightness limits or redshift limits are used in selecting the sample, a sensible test can still be made.

If, however selected, a sample of galaxy-quasar observed to be close on a line of sight but with significantly different redshifts is used to make a scatter diagram of galaxy-redshift versus quasar-brightness, there can be a control test where the galaxies and quasars are shuffled to get different partners still in galaxy-quasar pairs but not together in the sky.

In the case of the big bang being correct, the shuffling should have no effect on the scatter diagram because they are all random associations to start with. In the Arp alternative case, the shuffling should produce a significantly different scatter diagram. So the difference between the these two scatter diagrams tells us which model is better. Very similar implies big bang is correct, very different implies Arp is correct.

I think that by this means many problems of how to select the sample are removed, because the argument is true however the sample is selected as long as it isn't hand picked by an Arp believer. It is still OK to use Arp logic to determine the sample selecting.