R code for the adjustment of the Wald confidence interval for a
difference of proportions, with matched pairs.
This is the interval called Wald+2 in Agresti and Min, Statistics
in Medicine, 2005, which adds
0.5 to each cell before constructing the Wald CI. The CI is truncated
when it overshoots the boundary.
-------------------------------------------------------------------------- diffpropci <- function(b,c,n,conflev) { z <- qnorm(1-(1-conflev)/2) diff <- (c-b)/(n+2) sd <- sqrt((b+c+1)-(c-b)^2/(n+2))/(n+2) ll <- diff - z*sd ul <- diff + z*sd if(ll < -1) ll = -1 if(ul > 1) ul = 1 c(ll, ul) } # Adjusted Wald interval for difference of proportions with matched pairs # "conflev"=confidence coefficient, n=sample size, b,c = off-diag counts ---------------------------------------------------------------------------
R code from and article by Yang, Sun, and Hardin for a non-iterative R
function for finding Tango's score confidence interval for a
difference of proportions with matched pairs (Statistics in
Medicine 1998), described in Agresti and Min (Statistics in
Medicine, 2005); for details, see https://works.bepress.com/zyang/1/
------------------------------------------------------------------------------------------------- ### R code to calculate the Tango's CI using a non-iterative method; ### "A non-iterative implementation of Tango's score confidence interval for a paired difference of proportions"; ### Statistics in Medicine; ### by Zhao Yang, Xuezheng Sun, James W. Hardin; ### Version Date: 12Jul2012 ### b is the cell count in the (Success,Failure) cell, and c the cell count in the (Failure,Success) cell TangoCI <- function(N, b, c, alpha = 0.05) { g <- qnorm(1 - alpha/2)^2 m0 <- N^2*(N/g + 1)^2 m1 <- -N*(b - c)*(N/g + 1)*(4*N/g + 1) m2 <- 2*N*(b - c)^2*(3*N/g + 2)/g - N^2*((b + c)/g + 1) m3 <- N*(b - c)*(2*(b + c)/g + 1) - (b - c)^3*(4*N/g + 1)/g m4 <- (b - c)^2*((b - c)^2/g - (b + c) )/g u1 <- m1/m0; u2 <- m2/m0; u3 <- m3/m0; u4 <- m4/m0; if (b != c){ nu1 <- -u2 nu2 <- u1*u3 - 4*u4 nu3 <- -(u3^2 + u1^2*u4 - 4*u2*u4) d1 <- nu2 - nu1^2/3 d2 <- 2*nu1^3/27 - nu1*nu2/3 + nu3 CritQ <- d2^2/4 + d1^3/27 y1A <- h1 <- h2 <- h3 <- h4 <- root1 <- root2 <- root3 <- root4 <- c() if (CritQ > 0) { ## keep one real root; BigA <- -d2/2 + sqrt(CritQ) BigB <- -d2/2 - sqrt(CritQ) x1 <- sign(BigA)*abs(BigA)^(1/3) + sign(BigB)*abs(BigB)^(1/3) y1A <- x1 - nu1/3 } if (CritQ == 0) { ## keep two real roots; BigA <- -d2/2 + sqrt(CritQ) BigB <- -d2/2 - sqrt(CritQ) Omega <- complex(real = -1/2, imaginary = sqrt(3)/2) Omega2 <- complex(real = -1/2, imaginary = -sqrt(3)/2) x1 <- sign(BigA)*abs(BigA)^(1/3) + sign(BigB)*abs(BigB)^(1/3) x2 <- Omega*sign(BigA)*abs(BigA)^(1/3) + Omega2*sign(BigB)*abs(BigB)^(1/3) y1A[1] <- x1 - nu1/3 y1A[2] <- x2 - nu1/3 } if (CritQ <0) { ## keep three real roots; BigA <- -d2/2 + sqrt(as.complex(CritQ)) BigB <- -d2/2 - sqrt(as.complex(CritQ)) Omega <- complex(real = -1/2, imaginary = sqrt(3)/2) Omega2 <- complex(real = -1/2, imaginary = -sqrt(3)/2) x1 <- BigA^(1/3) + BigB^(1/3) x2 <- Omega*BigA^(1/3) + Omega2*BigB^(1/3) x3 <- Omega2*BigA^(1/3) + Omega*BigB^(1/3) y1A[1] <- x1 - nu1/3 y1A[2] <- x2 - nu1/3 y1A[3] <- x3 - nu1/3 } y1 <- Re(y1A) #keep the real part; ny <- length(y1) for (i in 1:ny){ h1[i] <- sqrt(u1^2/4 - u2 + y1[i]) h2[i] <- (u1 * y1[i]/2 - u3)/(2*h1[i]) h3[i] <- (u1/2 + h1[i])^2 - 4*(y1[i]/2 + h2[i]) h4[i] <- (h1[i] - u1/2)^2 - 4*(y1[i]/2 - h2[i]) if (h3[i] >= 0){ root1[i] <- ( - (u1/2 + h1[i]) + sqrt( h3[i] ) )/2 root2[i] <- ( - (u1/2 + h1[i]) - sqrt( h3[i] ) )/2 } if (h4[i] >= 0){ root3[i] <- ( (h1[i] - u1/2) + sqrt( h4[i] ) )/2 root4[i] <- ( (h1[i] - u1/2) - sqrt( h4[i] ) )/2 } } lower <- max(-1,min(root1, root2, root3, root4, na.rm = TRUE)) upper <- min(max(root1, root2, root3, root4, na.rm = TRUE), 1) if (b == N & c == 0) root <- c(lower, 1) else if (b == 0 & c == N) root <- c(-1, upper) else root <- c(lower, upper) } if (b == c){ root1 <- -sqrt(-u2) root2 <- sqrt(-u2) root <- c(root1, root2) } return(root) } TangoCI(44,0,1) -------------------------------------------------------------------------------------------------
This is older R code (by Yongyi Min) for Tango's score confidence
interval for a difference of proportions with matched pairs,
(Statistics in Medicine 1998),
described in Agresti and Min (Statistics
in Medicine, 2005)
----------------------------------------------------------------------------- scoreci <- function(b,c,n,conflev) { pa = 2*n z = qnorm(1-(1-conflev)/2) if(c == n) {ul = 1} else{ proot = (c-b)/n dp = 1-proot niter = 1 while(niter <= 50){ dp = 0.5*dp up2 = proot+dp pb = - b - c + (2*n-c+b)*up2 pc = -b*up2*(1-up2) q21 = (sqrt(pb^2-4*pa*pc)-pb)/(2*pa) score = (c-b-n*up2)/sqrt(n*(2*q21+up2*(1-up2))) if(abs(score)<z){ proot = up2 } niter=niter+1 if((dp<0.0000001) || (abs(z-score)<.000001)){ niter=51 ul=up2 } } } if(b == n) {ll = -1} else{ proot = (c-b)/n dp = 1+proot niter = 1 while(niter <= 50){ dp = 0.5*dp low2 = proot-dp pb = - b - c + (2*n-c+b)*low2 pc = -b*low2*(1-low2) q21 = (sqrt(pb^2-4*pa*pc)-pb)/(2*pa) score = (c-b-n*low2)/sqrt(n*(2*q21+low2*(1-low2))) if(abs(score) < z){proot = low2} niter = niter+1 if((dp<0.0000001) || (abs(z-score)<.000001)){ ll = low2 niter = 51 } } } c(ll,ul) } # Tango score interval for difference of proportions with matched # pairs, confidence coefficient = conflev, n = sample size, # b,c = off-diagonal counts -----------------------------------------------------------------------------
R code for the adapted binomial score confidence interval for the
subject-specific odds ratio,
with matched pairs, described in Agresti and Min, Statistics in
Medicine, 2004.
This uses the Wilson score CI for a binomial parameter with the
off-diagonal counts.
----------------------------------------------------------------------------- oddsratioci <- function(b,c,conflev) { z <- qchisq(conflev,1) A <- b + c + z B <- 2*c + z C <- c^2/(b+c) l <- (B - sqrt(B^2-4*A*C))/(2*A) u <- (B + sqrt(B^2-4*A*C))/(2*A) ll <- l/(1-l) ul <- u/(1-u) c(ll,ul) } # adapts Wilson binomial score CI to form CI for subject-specific # odds ratio with matched pairs data # "conflev"=confidence coefficient, b,c = off-diag counts -----------------------------------------------------------------------------