Welcome to the last installment of our series on ROC curves and related statistics.
The first statistic in this section is the Relative Risk (RR or risk ratio). Using the d-dimer study from before, the equation for RR is:
Pos 19 39
Neg 1 81
RR= a/(a+b) /c/(c+d) = a*(c+d) / c*(a+b)
= 19/(19+39) /1(1+81) = 27
Relative risk is a ratio of the probability of a high d-dimer occurring in the VTE group versus a non-VTE group. Note the difference in the math for the OR and the RR. They are quite similar, but the calculated values are strikingly different. RR has become one of the standard measures in biomedical research. It usually means the multiple of risk of the outcome in one group compared with another group and is expressed as the risk ratio in some studies and clinical trials. When the risk ratio cannot be obtained directly (such as in a cases and controls study), the odds ratio is calculated and often interpreted as if it was the risk ratio.
Subsequently, the term relative risk commonly refers to either the risk ratio or the odds ratio. However, only under certain conditions does the odds ratio approximate the risk ratio. When the incidence of an outcome of interest in the study population is low (<10%), the odds ratio is close to the risk ratio. However, the more frequent the outcome becomes, the more the odds ratio will overestimate the risk ratio when it is more than 1 or underestimate the risk ratio when it is less than 1. In our example, the OR was 49 vs. the RR of 27. You may find it helpful (if not necessary) when reading two papers of similar research, but with OR in one and RR in the other, to do the math for both OR and RR.
The last statistic is the hazard ratio (HR).
In some research, hazard ratios are often used to measure an event (e.g. survival) at any point in time in a group of patients who have been given a specific treatment compared to a control group given another treatment or a placebo. A hazard ratio of one means that there is no difference in survival between the two groups. A hazard ratio of greater than one or less than one means that survival was better in one of the groups.
Here is a graph of survival over time in four groups. These plots are called Kaplan-Meirer plots.1
Stated another way: for any randomly selected pair of patients, one from the treatment group and one from the control group, the hazard ratio is the odds that the time to healing is less in the patient from the treatment group than in the patient from the control group. Without going into the math, an HR of 2 corresponds to a 67% (2/3) chance of the treated patient's healing first, and a hazard ratio of 3 corresponds to a 75% (3/4) chance of healing first. Keep in mind that the idea of healing first is not a measure of how much first? It is a qualitative measurement. For example: in a horse race, a horse may win by a head or a length. In both cases, the horse "won."
The hazard ratio is used often in the context of survival analysis, where two groups are followed over time and the two curves are plotted. For example, imagine a patient is offered a new drug to treat a particular disease:
Q: Doctor, will this drug really work for me?
A: Yes, a clinical study has shown that the new drug promotes healing. (In other words, the hazard ratio must be >1. Otherwise the doctor would not say the drug aids healing. The higher the HR, the "better" the healing.)
Q: Doctor, what are the chances I will do better on this new drug compared to no treatment?
A: The odds are roughly 2:1 (the probability is 66% -- 2/3) that you will have an episode of shorter duration than someone who did not take the drug.
This brings us to the end of this series on ROCs, their statistics and associated statistics. We hope you have learned from these and that next time you encounter these statistics you will feel comfortable reading, studying and, as importantly, being able to explain them to your colleagues!
1. Int Heart J. 2013;54(5):311-7. Sato T1, Yamauchi H, Suzuki S, et al.