If you think hazard ratios are some special being in the Universe, is it possible that you are struggling with the concept of person-time?
‘One person in a row’ has limitations
In traditional clinical (or epidemiological?) training in statistics, we always have this default structure of one person in a row. If you imagine an excel sheet then each person (or patient) will occupy a row, with each column their characteristics. This default approach has not been true for other fields, for example lab science, psychology, or economics.
Hazard ratios
So when we speak of hazard ratios as a special being, we would like to emphasise it is not relative risk. Two common arguments that I have heard of are that:
(1) hazard ratios are the ratios of instantaneous risk at a snapshot time point between exposure and non-exposure groups, whilst relative risk refers to the cumulative over a period of time.
(2) hazard ratios are from models that estimate ‘time to event’, whilst relative risks are from models that estimate what ever the rate of event over a period of time.
I have no intention nor energy to argue whether these sentences are correct or incorrect. I would like to point out that just because you write two sentences with different words, it does not mean the two things are different to the extent that they are mutually exclusive.
AER% and the time scale relativity
The concept is simple but let us be amused with how the banks calculate interests and how they explain to us in the UK. At the time this is written the interests rate in the UK is as high as 4.8% annually, but the interests is given monthly. Because 4.8% is the AER, annual equivalent rate, you are not expected to get 0.4% monthly from your saving, but slightly less than 0.4%. Actually I have worked out for you and this is 0.3991%. This is because of the cumulative rate, and you only expect to get 4.8% if you keep the interest in the pot for it to generate interests.
Here you would say 4.8% is the cumulative rate, but would you say 0.39991% is the instantaneous rate? Why not?
You would say so because you think relatively it is an instantaneous rate. You would not say so because obviously it is not instantaneous, and it is just monthly.
Hazard ratio is just a type of relative risk
There is a time scale for data collection. You may follow up patient annually to get the outcome. You could link to their hospital records to get the outcome dates. Each research has its challenges, and we would not usually get angry just because you could not follow up patients instantaneously, especially in scenarios where it does not really matter clinically.
A relative risk does not automatically become instantaneous when you use Cox regression. It is the timescale of risk that we need to specify when there is a possibility of misleading.
