211 156 B. Estimating the effect of the intervention on the transition from CKD 1/2 to CKD 3/4 Kanno et al report that the hazard ratio for the progression to CKD 3/4 over a period of 10 years, comparing individuals with and without hypertension, is 1.83 [1.34, 2.48]; the model was fully adjusted for age, sex, smoking, drinking, obesity, cardiovascular disease, diabetes mellitus, hypercholesterolemia, anti-hypertensive treatment, baseline eGFR, number of follow-up examinations, and year of baseline examination.2 The outcome probabilities in this study are bounded by the interval [0.015,0.30]. Applying the approximation in (2) to the results from Kanno et al, the optimal bias-ratio minimax conversion for the hazard ratio of 1.83 is approximately estimated as a risk ratio of 1.76, and the lower and upper limit of the hazard ratio confidence interval can be similarly transformed to obtain a risk ratio interval of [1.31,2.34]. Since these are biased approximations of the limits of the risk ratios, we need to correct these limits by estimating the maximum bias associated with this approximation, a correction that secures 95% coverage over repeated samples of the true risk ratio. A numeric grid search shows that for outcome probabilities in the range [0.015,0.30], the bias ratio is always less than 5.5%. Hence, a confidence interval will be sure to have at least 95% coverage of the true risk ratio, provided that the square-root transformation of the lower limit of the hazard ratio confidence interval is divided by 1.055 and the square-root transformation of the upper limit of the hazard ratio confidence interval is multiplied by 1.055. We therefore estimate that the hazard ratio of 1.83 [1.34, 2.48] can be transformed to a risk ratio of 1.76 and a 95% confidence interval of [1.24, 2.47]. Given this information, we can use the formula (1) above to estimate the risk ratio , = 0.81, with a 95% confidence interval of [0.71, 0.92]. The calculations are demonstrated in the R script below: ############################################# ## Effect of intervention on hypertension ## ########################################### num_treat <- 149 + 523 pb_a_treat <- 149 / (149 + 523) # p(b|intervention) num_ctrl <- 361 + 256 pb_a_ctrl <- 361 / (361 + 256) # p(b|~intervention) ##################### ## Functions used ## ################### cumulative_p <- function( p, n ) { 1 - (1 - p)^n }
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