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The relationship between treatment effect and the hazard ratio is given as . A statistically important, but practically insignificant effect can produce a large hazard ratio, e.g. a treatment increasing the number of one-year survivors in a population from one in 10,000 to one in 1,000 has a hazard ratio of 10.
It is defined as the inverse of the absolute risk increase, and computed as / (), where is the incidence in the treated (exposed) group, and is the incidence in the control (unexposed) group. [1] Intuitively, the lower the number needed to harm, the worse the risk factor, with 1 meaning that every exposed person is harmed.
Hazard ratio, such as "an increase in both total and invasive breast cancers in women randomized to receive estrogen and progestin for an average of 5 years, with a hazard ratio of 1.24 compared to controls." [10]
The summary output also gives upper and lower 95% confidence intervals for the hazard ratio: lower 95% bound = 1.15; upper 95% bound = 3.26. Finally, the output gives p-values for three alternative tests for overall significance of the model:
The hazard ratio is the quantity (), which is = in the above example. From the last calculation above, an interpretation of this is as the ratio of hazards between two "subjects" that have their variables differ by one unit: if P i = P j + 1 {\displaystyle P_{i}=P_{j}+1} , then exp ( β 1 ( P i − P j ) = exp ( β 1 ( 1 ...
The absolute risk reduction (ARR), however, was much smaller, because the study group did not have a very high rate of cardiovascular events over the study period: 2.67% in the control group, compared to 1.65% in the treatment group. [15] Taking atorvastatin for 3.3 years, therefore, would lead to an ARR of only 1.02% (2.67% minus 1.65%).
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This maximum likelihood maximization depends on the specification of the baseline hazard functions. These specifications include fully parametric models, piece-wise-constant proportional hazard models, or partial likelihood approaches that estimate the baseline hazard as a nuisance function. [4]