As an example of how the variances of difference measures are calculated, the derivation of approximate variance or SE for RR is shown as follows (uses multivariate delta method):
Because of the skewed distribution of the untransformed estimated risk ratio it was advantageous to use logarithm. To show this, let us show the coverage rates first for a study where only 50 subjects are available in each groups, with pi1 = .2 and pi2 = .1 respectively:
As we see, the coverage rates for actual RR estimates are lower than expected. Since logRR approximately follows normal, this was not a problem for it. Now, to actually achieve normality for estimates of RR, let us investigate how much sample size is required. Trying various sample sizes, it was decided to go for sample size as large as 8,000! Here is the code:
Now, following R codes can be used to compare coverage rates and mean confidence interval widths between the two methods for sample size 8,000:
Therefore, we see, why use of logRR is so popular, since under normality assumption due to its symmetry works out well even for small sample sizes.
Because of the skewed distribution of the untransformed estimated risk ratio it was advantageous to use logarithm. To show this, let us show the coverage rates first for a study where only 50 subjects are available in each groups, with pi1 = .2 and pi2 = .1 respectively:
> n1 = 50 > n2 = 50 > pi1 = .2 > pi2 = .1 > # true RR > t.pi = pi1/pi2 > > # generate sample RR and calculate LRR and their C.I.s > sim = 1000 > set.seed(124) > x1 = rbinom(n = sim, size = n1, prob = pi1) > p1 = x1/n1 > x2 = rbinom(n = sim, size = n2, prob = pi2) > p2 = x2/n2 > RR = p1/p2 > LRR = log(RR) > v.RR = (p1/p2)^2 * ( (1-p1)/(p1*n1) + (1-p2)/(p2*n2) ) > v.LRR = ( (1-p1)/(p1*n1) + (1-p2)/(p2*n2) ) > ci.RR = cbind(RR - qnorm(.975)*sqrt(v.RR), RR + qnorm(.975)*sqrt(v.RR)) > ci.LRR =cbind(LRR - qnorm(.975)*sqrt(v.LRR), LRR + qnorm(.975)*sqrt(v.LRR)) > > # Find the coverage rates for RR and Log(RR) > covr.RR = ci.RR[,1] <= t.pi & ci.RR[,2] >= t.pi > sum(covr.RR[!is.na(covr.RR)])/sim [1] 0.914 > covr.LRR = ci.LRR[,1] <= log(t.pi) & ci.LRR[,2] >= log(t.pi) > sum(covr.LRR[!is.na(covr.LRR)])/sim [1] 0.968 |
> n1 = n2 = 8000 > > # generate sample RR and calculate LRR and their C.I.s > sim = 1000 > set.seed(124) > x1 = rbinom(n = sim, size = n1, prob = pi1) > p1 = x1/n1 > x2 = rbinom(n = sim, size = n2, prob = pi2) > p2 = x2/n2 > RR = p1/p2 > hist(RR) > > require("fBasics") Loading required package: fBasics Loading required package: MASS Loading required package: timeDate Loading required package: timeSeries Attaching package: 'fBasics' The following object(s) are masked from 'package:base': norm > shapiroTest(RR) Title: Shapiro - Wilk Normality Test Test Results: STATISTIC: W: 0.9972 P VALUE: 0.07717 Description: Wed Jan 26 18:36:18 2011 by user: Ehsan > jarqueberaTest(RR) Title: Jarque - Bera Normalality Test Test Results: STATISTIC: X-squared: 5.9183 P VALUE: Asymptotic p Value: 0.05186 Description: Wed Jan 26 18:36:18 2011 by user: Ehsan > cvmTest(RR) Title: Cramer - von Mises Normality Test Test Results: STATISTIC: W: 0.0846 P VALUE: 0.1802 Description: Wed Jan 26 18:36:18 2011 by user: Ehsan > lillieTest(RR) Title: Lilliefors (KS) Normality Test Test Results: STATISTIC: D: 0.0232 P VALUE: 0.2121 Description: Wed Jan 26 18:36:18 2011 by user: Ehsan > pchiTest(RR) Title: Pearson Chi-Square Normality Test Test Results: PARAMETER: Number of Classes: 32 STATISTIC: P: 33.088 P VALUE: Adhusted: 0.2742 Not adjusted: 0.3655 Description: Wed Jan 26 18:36:18 2011 by user: Ehsan > sfTest(RR) Title: Shapiro - Francia Normality Test Test Results: STATISTIC: W: 0.9975 P VALUE: 0.1165 Description: Wed Jan 26 18:36:18 2011 by user: Ehsan > qqnorm(RR) |
> n1 = n2 = 8000 > > # generate sample RR and calculate LRR and their C.I.s > sim = 1000 > set.seed(124) > x1 = rbinom(n = sim, size = n1, prob = pi1) > p1 = x1/n1 > x2 = rbinom(n = sim, size = n2, prob = pi2) > p2 = x2/n2 > RR = p1/p2 > LRR = log(RR) > v.RR = (p1/p2)^2 * ( (1-p1)/(p1*n1) + (1-p2)/(p2*n2) ) > v.LRR = ( (1-p1)/(p1*n1) + (1-p2)/(p2*n2) ) > ci.RR = cbind(RR - qnorm(.975)*sqrt(v.RR), RR + qnorm(.975)*sqrt(v.RR)) > ci.LRR =cbind(LRR - qnorm(.975)*sqrt(v.LRR), LRR + qnorm(.975)*sqrt(v.LRR)) > > # Find the coverage rates for RR and Log(RR) > covr.RR = ci.RR[,1] <= t.pi & ci.RR[,2] >= t.pi > sum(covr.RR[!is.na(covr.RR)])/sim [1] 0.959 > covr.LRR = ci.LRR[,1] <= log(t.pi) & ci.LRR[,2] >= log(t.pi) > sum(covr.LRR[!is.na(covr.LRR)])/sim [1] 0.961 > > # mean confidence interval widths > mean(ci.RR[,2] - ci.RR[,1]) [1] 0.3159202 > mean(ci.LRR[,2] - ci.LRR[,1]) [1] 0.1580062 |
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