I'm sorry but I must disagree with using bad test results as the criteria for a water change. By that point it is too late in a sense, because the bad conditions have taken some toll on the fish. I'll expand a bit on this before I turn to the volume issue.
Taking nitrate for instance. If nitrate is normally 10ppm, and you test once a week, and one week it is 15ppm or 20ppm, the increase in nitrates has likely affected the fish. Some species are more sensitive to nitrates than others. As I mention in my article on bacteria, high levels of nitrate, above 40 ppm, have been shown to slow fish growth, suppress breeding, and depress the immune system making the fish much more susceptible to disease. Considering that all our fish occur in natural waters with near-zero nitrates--and I could cite test numbers for several Amazonian and SE Asian streams showing nitrate so low it is unmeasurable--an increase of 5 or 10ppm is significant. The whole aim of regular maintenance including partial water changes is to maintain stability and prevent any fluctuation. That is more likely to lead to better fish health.
And stability brings me to the volume issue. Obviously the replacement water should be close to the tank water in many respects. Hardness, pH and temperature are obvious, but there are many others, including bacteria, organics, nutrients. And as Mikaila correctly noted, some variation is not bad, in fact the opposite; there is no need to "prepare" identical water with respect to these parameters, but the variation should not be astronomical either. My tanks run at pH 5 to 6.4 depending upon the tank, and one is 7.2 for basic water fish. My tap is 7.0, so changing 60% of the tank means replacing pH 5 water with pH 7. Yet I have never had any indication of a problem, and largely due to the biological stability which "buffers" such changes so they are less; the tank pH rises maybe .3 or .4, that's it. And I almost always use cooler water, using the hand test someone mentioned earlier. And all of this is also perfectly natural; fish in Amazonia spawn when the water temperature drops significantly (several degrees) and there is a corresponding change in pH, as studies by Stanley Weitzman and others have proven.
It is a false assumption to assume that doing smaller volume changes will somehow create more stability. Chances are, it will be the exact opposite. If the conditions are allowed to deteriorate to the extent that our flimsy test kits actually register them, things are probably much worse in actual fact. Suddenly changing such water can sometimes cause other problems like ammonia poisoning, nitrite poisoning, pH shock, etc. The fact is that water which is changed more regularly in significant amounts will be more stable. The tank's biological balance will be determined by the fish, plants, additives, organics, and water changes. Maintaining a schedule of higher-volume and weekly changes ensures this true stability will more likely hold. There will be much less of a "pull" biologically if the water is replaced more often and/or in greater quantity. Refer back to my comment on the discus fry previously.
Those of us who change 50-60% weekly and have done so for 20+ years know there are benefits to the fish. My fish are always more active, colourful, more likely to spawn, and "playful" the day following a water change. But my water parameters do not ever fluctuate from week to week, so while testing pH, nitrate, ammonia, nitrite, CO2, whatever would show absolutely no variance, the water change clearly has done something. And it comes back to that "crud" I wrote about previously, that cannot be measured in any way, but is certainly present if there is a live fish in the tank. That crud is what we are removing and replacing, and it is essential to do so--if you want healthy fish.