Why do some studies that utilize the Cox Proportional Hazards Model analysis use relative risks, others used odds ratios, and others used hazard ratios?
Sometimes it's because one of those is the usual thing to do in the field the researcher is working in. Or, they may want to contrast their results with one or several earlier studies. There are also trends to use each in particular types of studies (e.g. case-control vs. cohort etc.)
But the three answer somewhat different questions and the researcher may be interested in one or the other. All three measure the relationship between an exposure or condition or trait and an outcome. But how they measure it varies. This is a somewhat unusual case where the names (in statistical language) actually make sense (in ordinary English).
Odds ratio: This is a ratio of odds. The odds are just like odds in gambling.
Relative risk: This is the risk in one condition relative to another.
Hazard ratio: This is a little less intuitive. The hazard is the "instantaneous hazard" -- the risk of something happening ant any particular instant. To really get this, you need a little calculus (at least, the idea of a limit). The first two are just algebra.