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Hi Jenny


Hazard ratios are usually associated with Survival Rates but they can be used with any event. It is probably easier to understand if they are converted into odds or probabilities. You can use the Hazard Ratio to calculate the odds that a treated group will experience an event before the control group.  If the odds are 50% (i.e. 1:1), then the groups are the same. The probabilities range from zero to one, and are often represented as a percentage (0 % to 100%).  The formula for translating a hazard ratio to a probability is:
                         probability = (hazard ratio) / (1 + hazard ratio).

eg if the probability that at treated patient will heal before an untreated patient is:
                        probability = (3)/(1+3) = ¾ = 0.75
So there is a 75% chance that the the treated patient will heal before the control patients [Shore 2007].  For more on the practical applications of hazard ratios, please see the section “Questions Asked By Patients” on page 2791 of the paper “Hazard Ratio in Clinical Trials” by S.L. Spruance et al. http://aac.asm.org/content/48/8/2787.full

Hazard ratios are often used in medical studies because they describe nicely the results of clinical trials.  Understanding that these are relative measures of treatment versus control and how to translate it into odds is essential to applying the results. 
More help found here:

http://www.medicine.ox.ac.uk/bandolier/painres/download/whatis/what_are_haz_ratios.pdf


Here is a simple definition from GraphPad

Definition of the hazard ratio
Hazard is defined as the slope of the survival curve — a measure of how rapidly subjects are dying.
The hazard ratio compares two treatments. If the hazard ratio is 2.0, then the rate of deaths in one treatment group is twice the rate in the other group.
As part of the survival analysis of two data sets, Prism reports the hazard ratio with its 95% confidence interval.
Interpreting the hazard ratio
The hazard ratio is not computed at any one time point, but includes all the data in the survival curve. 
Since there is only one hazard ratio reported, it can can only be interpreted if you assume that the population hazard ratio is consistent over time, and that any differences are due to random sampling. 
If the hazard ratio is not consistent over time, the value that Prism reports for the hazard ratio will not be useful. If two survival curves cross, the hazard ratios are certainly not consistent (unless they cross at late time points, when there are few subjects still being followed so there is a lot of uncertainty in the true position of the survival curves). 
Note that a hazard ratio of two does not mean that the median survival time is doubled (or halved). A hazard ratio of two means a patient in one treatment group who has not died (or progressed, or whatever end point is tracked) at a certain time point has twice the probability of having died (or progressed...) by the next time point compared to a patient in the other treatment group.
For other cautions about interpreting hazard ratios, see these two review papers:
How the hazard ratio is computed
There are two very similar ways of doing survival calculations: log-rank, and Mantel-Haenszel. Both are explained in chapter 3 of Machin, Cheung and Parmar,Survival Analysis (details below).
The Mantel Haneszel approach uses these steps:
  1. Compute the total variance, V, as explained on page 38-40 of a handout by Michael Vaeth. Note that he calls the test "log-rank" but in a note explains that this is the more accurate test, and also gives the equation for the simpler approximation that we call log-rank.
  2. Compute K = (O1 - E1) / V, where O1 - is the total observed number of events in group1 E1 - is the total expected number of events in group1. You'd get the same value of K if you used the other group.
  3. The hazard ratio equals EXP(K)
  4. The lower 95% confidence limit of the hazard ratio equals:
      EXP(K - 1.96/sqrt(V))
  5. The upper 95% confidence limit equals:
        EXP(K + 1.96/sqrt(V))
The logrank approach uses these steps:
  1.  As part of the Kaplan-Meier calculations, compute the number of observed events (deaths, usually) in each group (Oa, and Ob), and the number of expected events assuming a null hypothesis of no difference in survival (Ea and Eb).
  2. The hazard ratio then is:
    HR= (Oa/Ea)/(Ob/Eb)
  3. The standard error of the natural logarithm of the hazard ratio is: sqrt(1/Ea + 1/Eb)
  4. The lower 95% confidence limit of the hazard ratio equals:
    EXP( (Oa-Ea)/V - 1.96*sqrt(1/Ea + 1/Eb))
  5. The upper 95% confidence limit equals:
    EXP( (Oa-Ea)/V + 1.96*sqrt(1/Ea + 1/Eb))  http://www.graphpad.com/faq/viewfaq.cfm?faq=1226


this is also useful
http://statpages.org/

Jo           

From: Jenny Morris <[log in to unmask]>
To: [log in to unmask]
Sent: Wednesday, 14 March 2012, 9:08
Subject: Interpretation of hazard ratios

Can anyone give me a easy to understand definition of hazard ratios and how they should be interpreted.  This is a for a group of pre-registration nursing students.
 
Thanks.
 
Jenny
 
Dr Jenny Morris
Associate Professor (Senior Lecturer) in Health Studies
Faculty of Health, Education and Society
University of Plymouth
Knowledge Spa
Treliske
Truro TR1 3HD
Cornwall
 
Tel: 01872 256461
Web: http://www.plymouth.ac.uk/faculties/health
 
‘High quality education for high quality care’