## Relative and absolute measures of effect

The hazard ratio introduced in Section 9.3.4 is a relative measure of effect. Thus, when comparing two treatments, an experimental and control, a hazard ratio of 0.75 represents a 25 per cent reduction in the relative risk of an event on the experimental group compared to the control group at any given time. This is a useful scale on which to conduct and report analyses. In reporting results it is useful to report an absolute difference as well as this relative difference. It is not good practice, though, to read survival figures off the two Kaplan-Meier curves, as individual points on the curve are unreliable, even though the whole curve is a good summary of the overall experience of the two groups. One means of calculating an absolute difference at a particular timepoint is as follows:

Absolute difference at time t = exp{hazard ratio x loge[control survival at time t]}

This approach implicitly assumes that the hazards are proportional in the two groups, that is the relative risk of an event in the two groups is constant over time. Although this a slightly limiting assumption, it is generally preferable to reading off differences between the Kaplan-Meier curves at individual timepoints.

Example. In the BA06 bladder cancer trial [3] comparison of the duration of survival of the chemotherapy and no chemotherapy groups gave a hazard ratio of 0.85 with a 95 per cent confidence interval of 0.71-1.02. This estimate represents a 15 per cent reduction in the risk of death with chemotherapy. Reading from the Kaplan-Meier curves the 3-year survival in the no chemotherapy group is approximately 50 per cent (i.e.control survival is 0.5). Applying equation 9.25 gives an absolute difference of 5.5 per cent. Thus, we estimate that 3-year survival is improved to 55.5 per cent in the chemotherapy group. To obtain a 95 per cent confidence interval for this absolute difference, we can use equation 9.25 again, and this time use the upper and lower end of the confidence of the hazard ratio, i.e, exp{0.71xloge [0.50]}-0.50 = 11.0, and exp{ 1.02 xloge[0.50]}-0.50 = -0.7. Thus the 95 per cent confidence interval for the absolute difference of 5.5 per cent is given by (-0.7 per cent to 11.0 per cent).

To translate relative differences to differences in median survival we can use the following equation:

Median survival in = Median survival in control (9.27)

experimental group group/hazard ratio.

This approach assumes that the two Kaplan-Meier curves both follow approximately exponential distributions, which is a more limiting assumption than proportional hazards for the absolute difference calculations, but again is preferred to reading values off the survival curves. To continue the BA06 bladder cancer trial, median survival in the no chemotherapy group is approximately thirty-six months, thus the estimated median survival in the experimental group is given by 36/0.85 = 42.4 months, a difference of 6.4 months. Confidence intervals for this can be obtained by again inserting the confidence intervals of the hazard ratio into equation 9.25, giving an approximate confidence interval of the absolute difference in medians of approximately -0.7 months to 15 months.

It is important to note that the relative difference is always larger in magnitude terms than the absolute difference, thus it is important to report both. Further, the absolute difference varies as the underlying control group varies with a maximum absolute difference at 50 per cent reducing as we move away from 50 per cent in both directions. Thus, the same relative difference may have different interpretations depending on the underlying control group rate. This is discussed also in Section 11.5.7.

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