210 155 ⊥ | The resulting risk ratio would provide an upper limit on the relative risk of the outcome when comparing the two treatment groups. In other words, any residual effect of the intervention on the outcome that does not involve the status of hypertension would further decrease the relative risk. Assuming | , we can show that: ( | ) = ( | ) ( | ) + ( | ) ( | ) The relative risk , is therefore given in formula 1: , = ( | ) ( | ‾) = , ⋅ ( | )+ ( ‾| ) , ⋅ ( | ‾)+ ( ‾| ‾) (1) To calculate this , , we need the conditional probabilities for the transition and the risk ratio for the transition . For the calculation of , , we used studies from the literature.2,3 Both studies reported their results in terms of hazard ratios and not risk ratios, so we sought a way to estimate the risk ratios from the hazard ratios. The empirical research literature often treats hazard ratios as risk ratios, an interpretation that holds approximately when the outcome is rare. However, VanderWeele offers a more accurate approximation when an upper and lower limit can be determined for the probability of the incidence in both study arms.4 In this case, the optimal minimax transformation of the hazard ratio can be shown to approximate the risk ratio under a proportional hazard model where is the hazard ratio and , are the unknown probability of incidence for the treatment and the control group; both probabilities bounded by known limits , . If >1 and < < < , the optimal bias-ratio minimax conversion on the interval [ , ] of a hazard ratio is given by the following square-root transformation: � 1−(1− ) 1−(1− )1/ � 1/2 (2) Since this expression is an approximation, applying it directly to the lower and upper bounds of the hazard ratio confidence interval would not yield a 95% coverage over repeated samples of the true risk ratio. However, using a numeric grid search for the outcome probabilities , within a specified interval, it is possible to determine the maximum bias ratio for the approximation, and to use it to obtain a conservative confidence interval.
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