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Back to the forum  Query: 2017-12-11 12:18 CET (UTC+1h)
 

CV from CI, pooling, sample size [R for BE/BA]

posted by Helmut Homepage - Vienna, Austria, 2016-07-27 14:09  - Posting: # 16521
Views: 1,983

Dear all,

following a question at LinkedIn’s non-public user-group “BA/BE professionals”, code which combines some of PowerTOST’s functions to###################
# your data below #
###################
design <- c("3x6x3", "2x2x2", "2x2x4")
N      <- c(17, 34, 33)             # (total) sample sizes
lower  <- c(0.8666, 1.0211, 0.9750) # lower CL
upper  <- c(1.2845, 1.2665, 1.1390) # upper CL
CI     <- c(95, 90, 90)             # level of the CI in %
theta0 <- 1.10                      # expected T/R-ratio
target <- 0.80                      # target power
des    <- "2x2x4"                   # planned design
RSABE  <- FALSE                     # reference-scaling desired?
reg    <- "EMA"                     # any of "EMA", "FDA", "HC"
adjust <- FALSE                     # iteratively adjust alpha to
                                    # prevent inflation of the TIE?
##########################################################
# don't change below unless you know what your are doing #
##########################################################
library(PowerTOST)
alpha  <- (1-CI/100)/2
digits <- max(nchar(as.character(c(lower, upper))))-2
if (RSABE) RSABE.des <- "yes" else RSABE.des <- "no"
PE     <- round(sqrt(lower*upper), digits)
CL     <- df <- calc <- bal <- CVs <- CLlo <- CLhi <-vector()
# calculate the CVs and their confidence intervals
for (j in seq_along(N)) {
  CL[j]   <- paste0(CI[j], "%")
  n       <- N[j]
  df[j]   <- eval(parse(text=known.designs()[which(known.designs()[["design"]] == design[j]), "df"],
                        srcfile=NULL))
  calc[j] <- known.designs()[which(known.designs()[["design"]] == design[j]), "df"]
  steps   <- known.designs()[which(known.designs()[["design"]] == design[j]), "steps"]
  if (n %% steps == 0) bal[j] <- "yes" else bal[j] <- "no"
  CVs[j]  <- CVfromCI(upper=lower[j], lower=upper[j], n=N[j], design=design[j], alpha=alpha[j])
  CLlo[j] <- CVCL(CV=CVs[j], df=df[j], side="2-sided", alpha=0.05)[[1]]
  CLhi[j] <- CVCL(CV=CVs[j], df=df[j], side="2-sided", alpha=0.05)[[2]]
}
CVdata <- data.frame(PE, CL, lower, upper, CVs,  CLlo, CLhi, N, design,
                     paste("study", 1:length(N)), bal, calc, paste("=", df))
names(CVdata) <- c("PE", "CI", "lo", "hi", "CV", "lower CL", "upper CL",
                   "n", "design", "source", "balanced", "df", "")
# calculate the pooled CV and the 80% upper one-sided CL
CVpooled <- CVpooled(CVdata[, c(5, 8:10)], alpha=0.2)
CVdata[, 5:7] <- round(CVdata[, 5:7], digits) # precision like input
# estimate sample size and expected power for the pooled CV
# and its upper CL
alpha.1  <- rep(0.05, 2)
if (RSABE) { # reference-scaling
  CL.lo <- CVCL(CV=CVpooled[["CV"]], df=CVpooled[["df"]],
                side="lower", alpha=0.2)[["lower CL"]]
  CL.hi <- CVCL(CV=CVpooled[["CV"]], df=CVpooled[["df"]],
                side="upper", alpha=0.2)[["upper CL"]]
  txt   <- paste("Pooled CV =", round(CVpooled[["CV"]], digits),
                 "with", CVpooled[["df"]], "degrees of freedom")
  if (reg == "FDA") { # the FDA's RSABE
    if (des == "2x3x3" | des == "2x2x4" | des == "2x2x3") {
      tmp <- sampleN.RSABE(CV=CVpooled[["CV"]], theta0=theta0,
                           targetpower=target, design=des,
                           details=FALSE, print=FALSE)
      n.est1   <- tmp[["Sample size"]]
      pwr.est1 <- round(tmp[["Achieved power"]], digits)
      L.1      <- scABEL(CV=CVpooled[["CV"]], regulator=reg)[["lower"]]
      U.1      <- scABEL(CV=CVpooled[["CV"]], regulator=reg)[["upper"]]
      tmp1 <- sampleN.RSABE(CV=CL.lo, theta0=theta0,
                            targetpower=target, design=des,
                            details=FALSE, print=FALSE)
      tmp2 <- sampleN.RSABE(CV=CL.hi, theta0=theta0,
                            targetpower=target, design=des,
                            details=FALSE, print=FALSE)
      if (tmp2[["Sample size"]] >= tmp1[["Sample size"]]) { # critical: upper CL
        n.est2   <- tmp2[["Sample size"]]
        pwr.est2 <- round(tmp2[["Achieved power"]], digits)
        L.2      <- scABEL(CV=CL.hi, regulator=reg)[["lower"]]
        U.2      <- scABEL(CV=CL.hi, regulator=reg)[["upper"]]
        txt      <- paste(txt, "\nUpper 80% confidence limit of the CV =",
                          round(CL.hi, digits), "\n")
      } else { # critical: lower CL
        n.est2   <- tmp1[["Sample size"]]
        pwr.est2 <- round(tmp1[["Achieved power"]], digits)
        L.2      <- scABEL(CV=CL.lo, regulator=reg)[["lower"]]
        U.2      <- scABEL(CV=CL.lo, regulator=reg)[["upper"]]
        txt      <- paste(txt, "\nUpper 80% confidence limit of the CV =",
                          round(CL.lo, digits), "\n")
      }
    } else {
      stop("Design not implemented.")
    }
  } else { # the EMA's and Health Canada's ABEL
    if (des == "2x3x3" | des == "2x2x4" | des == "2x2x3") {
      if (!adjust) { # according to the GL: alpha 0.05
        tmp <- sampleN.scABEL(CV=CVpooled[["CV"]], theta0=theta0,
                              targetpower=target, design=des,
                              regulator=reg, details=FALSE, print=FALSE)
        n.est1   <- tmp[["Sample size"]]
        pwr.est1 <- round(tmp[["Achieved power"]], digits)
        L.1      <- scABEL(CV=CVpooled[["CV"]], regulator=reg)[["lower"]]
        U.1      <- scABEL(CV=CVpooled[["CV"]], regulator=reg)[["upper"]]
        tmp1 <- sampleN.scABEL(CV=CL.lo, theta0=theta0,
                               targetpower=target, design=des,
                               regulator=reg, details=FALSE, print=FALSE)
        tmp2 <- sampleN.scABEL(CV=CL.hi, theta0=theta0,
                               targetpower=target, design=des,
                               regulator=reg, details=FALSE, print=FALSE)
        if (tmp2[["Sample size"]] >= tmp1[["Sample size"]]) { # critical: upper CL
          n.est2   <- tmp2[["Sample size"]]
          pwr.est2 <- round(tmp2[["Achieved power"]], digits)
          L.2      <- scABEL(CV=CL.hi, regulator=reg)[["lower"]]
          U.2      <- scABEL(CV=CL.hi, regulator=reg)[["upper"]]
          txt      <- paste(txt, "\nUpper 80% confidence limit of the CV =",
                            round(CL.hi, digits), "\n")
        } else { # critical: lower CL
          n.est2   <- tmp1[["Sample size"]]
          pwr.est2 <- round(tmp1[["Achieved power"]], digits)
          L.2      <- scABEL(CV=CL.lo, regulator=reg)[["lower"]]
          U.2      <- scABEL(CV=CL.lo, regulator=reg)[["upper"]]
          txt      <- paste(txt, "\nUpper 80% confidence limit of the CV =",
                            round(CL.lo, digits), "\n")
        }
      } else { # iteratively adjusted alpha
        tmp <- sampleN.scABEL.ad(CV=CVpooled[["CV"]], theta0=theta0,
                                 targetpower=target, design=des,
                                 regulator=reg, details=FALSE, print=FALSE)
        n.est1   <- tmp[["Sample size"]]
        pwr.est1 <- round(tmp[["Achieved power"]], digits)
        L.1      <- scABEL(CV=CVpooled[["CV"]], regulator=reg)[["lower"]]
        U.1      <- scABEL(CV=CVpooled[["CV"]], regulator=reg)[["upper"]]
        if (!is.na(tmp[["adj. alpha"]])) alpha.1[1] <- tmp[["adj. alpha"]]
        tmp1 <- sampleN.scABEL.ad(CV=CL.lo, theta0=theta0,
                                 targetpower=target, design=des,
                                 regulator=reg, details=FALSE, print=FALSE)
        tmp1 <- sampleN.scABEL.ad(CV=CL.hi, theta0=theta0,
                                 targetpower=target, design=des,
                                 regulator=reg, details=FALSE, print=FALSE)
        if (tmp2[["Sample size"]] >= tmp1[["Sample size"]]) { # critical: upper CL
          n.est2   <- tmp[["Sample size"]]
          pwr.est2 <- round(tmp[["Achieved power"]], digits)
          L.2      <- scABEL(CV=CL.hi, regulator=reg)[["lower"]]
          U.2      <- scABEL(CV=CL.hi, regulator=reg)[["upper"]]
        } else { # critical: lower CL
          n.est2   <- tmp1[["Sample size"]]
          pwr.est2 <- round(tmp1[["Achieved power"]], digits)
          L.2      <- scABEL(CV=CL.lo, regulator=reg)[["lower"]]
          U.2      <- scABEL(CV=CL.lo, regulator=reg)[["upper"]]
        }
        if (!is.na(tmp[["adj. alpha"]])) alpha.1[2] <- tmp[["adj. alpha"]]
      }
    } else {
      stop("Design not implemented.")
    }
  }
} else { # conventional ABE
  tmp <- sampleN.TOST(CV=CVpooled[["CV"]], theta0=theta0,
                      targetpower=target, design=des, print=FALSE)
  n.est1   <- tmp[["Sample size"]]
  pwr.est1 <- round(tmp[["Achieved power"]], digits)
  tmp <- sampleN.TOST(CV=CVpooled[["CVupper"]], theta0=theta0,
                      targetpower=target, design=des, print=FALSE)
  n.est2   <- tmp[["Sample size"]]
  pwr.est2 <- round(tmp[["Achieved power"]], digits)
  L.2 <- L.1 <- 0.8
  U.2 <- U.1 <- 1.25
  txt <- paste("Pooled CV =", round(CVpooled[["CV"]], digits), "with",
               CVpooled[["df"]], "degrees of freedom",
               "\nUpper 80% confidence limit of the CV =",
               round(CVpooled[["CVupper"]], digits), "\n")
}
Nest <- data.frame(rep(des, 2), rep(RSABE.des, 2), rep(reg, 2),
                   rep(sprintf("%.4f", theta0), 2),
                   sprintf("%.4f", c(L.1, L.2)),
                   sprintf("%.4f", c(U.1, U.2)), alpha.1,
                   round(c(CVpooled[["CV"]], CVpooled[["CVupper"]]), digits),
                   c(n.est1, n.est2), c(pwr.est1, pwr.est2))
names(Nest) <- c("Design", "RSABE desired", "regulator", "theta0",
                 "theta1", "theta2", "alpha", "CV", "n", "Achieved power")
if (length(unique(design)) > 1 & "parallel" %in% design)
  warning("Mixing crossover and parallel studies = nonsense!")
print(as.data.frame(CVdata), row.names=F);cat(txt);print(as.data.frame(Nest), row.names=F)


Which gives:
     PE  CI     lo     hi     CV lower CL upper CL  n design  source balanced    df     
 1.0551 95% 0.8666 1.2845 0.2832   0.2247   0.3844 17  3x6x3 study 1       no 2*n-4 = 30
 1.1372 90% 1.0211 1.2665 0.2667   0.2132   0.3574 34  2x2x2 study 2      yes   n-2 = 32
 1.0538 90% 0.9750 1.1390 0.2736   0.2386   0.3210 33  2x2x4 study 3       no 3*n-4 = 95
Pooled CV = 0.2741 with 157 degrees of freedom
Upper 80% confidence limit of CV = 0.2888
 Design RSABE desired regulator theta0 theta1 theta2 alpha     CV  n Achieved power
  2x2x4            no       EMA 1.1000 0.8000 1.2500  0.05 0.2741 28         0.8015
  2x2x4            no       EMA 1.1000 0.8000 1.2500  0.05 0.2888 32         0.8135


Doing the math is just the first step. Before you pool CV I would suggest to inspect whether the confidence intervals of the CVs overlap (columns 6–7). If not, try to find out why (different CROs, populations, bioanalytical methods, …). Use common sense to decide which CVs are reliable enough to pool.

Another question at LinkedIn was whether one should use the highest CV of several studies for conservatism. Maybe, maybe not. If the sample sizes were very close and the CVs not too different, why not. Otherwise (or if different designs were used) I prefer the upper CL of the pooled CV.

Note:

[image]Regards,
Helmut Schütz 
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