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Hazard ratios are often used to assess the differences observed in survival curves in clinical trials.
In this lesson, we will learn about “hazard ratios.”
Interpretation of hazard ratios
The hazard ratio expresses how many times the time to occur the outcome, when the explanatory variable increases by 1 unit.
The hazard ratio is based on “1.” That is, even if the value of the explanatory variable changes, there is no change in the time to outcome occurrence.
If the hazard ratio exceeds “1,” it means that the risk of death or disease progression is XX% higher than in the control group. Conversely, if the hazard ratio does not exceed “1,” it means that the risk of death or disease progression is XX% lower than that of the control group.
Try to calculate the hazard ratio
Let us apply the Cox proportional hazards model with the objective variable (outcome) as “1: death, 0: censored” and the explanatory variables as “prescription drug” and “smoking status,” and find the hazard ratios for prescription drug and smoking status.
The objective variable (outcome) is 1 for risk factors, such as death (recurrence), and 0 for others.
If the explanatory variables are two categories of categorical data, the one that assumes a lower mortality rate is set to 1 and the other, to 0. In this case, product A and non-smoking are assumed to be 1.
[Table 1] Basic data of the case
Table 2 shows the regression coefficients and hazard ratios of the equations.
[Table 2] Regression coefficients and hazard ratios for equations
The hazard ratio is a value determined by the following formula.
Hazard ratio = e regression coefficient
However, e is the base of the natural logarithm, 2.7183….
[Calculation example]
Drug hazard ratio = e-0.950 = 0.387
Excel function = EXP (-0.950)
When applying the Cox proportional hazards model to solve for objectives, hazard ratios are used instead of regression coefficients. The hazard ratio enables us to examine how much the explanatory variables contribute to the outcome.
If the explanatory variables are “treatment P, treatment Q,” the hazard ratio can indicate how many times higher the outcome probability (mortality rate) is for P than for Q.
Also, suppose that the hazard ratio of the outcome is “1.5” for “1: death, 0: censored” during the 3-year observation period. In this case, the interpretation is that treatment P is 1.5 times more likely to result in death over 3 years than treatment Q. In other words, treatment P is worse than Q.
The hazard ratio for this example is 0.387, which is less than 1. In this case, the interpretation is that the outcome probability (mortality) is 0.387 times higher for product A than for a placebo. In other words, product A reduced the mortality rate by 61.3% (=1-0.387) compared to a placebo, and it is interpreted that product A had a life-prolonging effect.
From Table 2 we know that the hazard ratio for smoking status is 0.480, which can be interpreted as a 52.0% reduction in mortality for non-smoking as compared to smoking.
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