Hazard ratio: How much more likely is it?

In statistics, calculating the hazard ratio is one of the four main ways of quantifying association. The other three are odds ratio, risk ratio, and absolute risk reduction. Conceptually these are all very similar: you determine the probability or odds of an outcome with and without a variable and then divide one by the other. If the answer is 1, your variable has nothing to do with your outcome. If it isn't 1, there is an association. Note that an association is NOT a causative relationship. Hazard ratio is a "fussy" version of the risk ratio that takes time into consideration. Because of that we will not be calculating any hazard ratios.

First, if you haven't done so, read up on risk ratio.

Hazard ratio is the limit of the risk ratio as the experiment time approaches zero. You can think of it like this (borrowing from the example given in the risk ratio explanation): the risk ratio expresses the increased probability that at some point in a time frame, a person will own a cat having had a childhood cat. The hazard ratio is the increased probability that said person is out adopting a cat right now. This is not an academically rigorous explanation, but thinking about it like this will very much give you the right idea.

Hazard ratio is at it's most useful when it is presented as a graph. A commonly seen form of this is a survival curve, such as for people diagnosed with cancer. This allows researchers to express how the probability of death (or other outcome) changes as time goes on. Using the cancer survival curve example, the hazard ratio (diagnosed with cancer vs. healthy) may start out very high, but as time goes on it drops to closer to 1. Such a pattern would suggest that shortly after diagnosis many people die, but those that survive for a few years are likely to go on surviving (maybe because they have a less aggressive form of the cancer).

Hazard ratio is better than risk ratio in certain cases. It is often used when the outcome being measured is something that will remove a study participant from the study (i.e. death or drop out), or when each participant is in the study for a different length of time. We've seen a lot of studies on this blog where the latter is true because the researchers pull medical records from a span of several years, so some records are older than others. A simplified version of hazard ratio is sometimes used where the researchers correct risk ratio for time by expressing probability as outcomes (deaths, cat adoptions, etc.) per person per year (or other fixed time period); this is called incidence rate.