Bioequivalence and Bioavailability Forum • Pocock’s “natural constant”

Pocock’s “natural constant” [Two-Stage / GS Designs]

posted by Helmut – Vienna, Austria, 2014-10-13 16:53 (3837 d 21:46 ago) – Posting: # 13692
Views: 11,443

Dear Detlew,

have you every wondered where the magick 0.0294 comes from? In R we can do better, of course:

require(mvtnorm)

alpha   <- 0.05

mu      <- c(0, 0)

sigma   <- diag(2)

sigma[sigma == 0] <- 1/sqrt(2)

z       <- qmvnorm(1-0.05002333, tail="both.tails", mean=mu, sigma=sigma)$quantile

p0      <- pmvnorm(lower=rep(-z, 2), upper=rep(z, 2), mean=mu, sigma=sigma)[1]


C       <- qmvnorm(1-alpha, tail="both.tails", mean=mu, sigma=sigma)$quantile

p1      <- pmvnorm(lower=rep(-C, 2), upper=rep(C, 2), mean=mu, sigma=sigma)[1]


alpha1 <- 2*(1-pnorm(C))

head   <- "%s"

body   <- "%s  %-10.7g  %-8.7g  %-8.7g  %+.7f%s"

title1 <- "\nPocock (1977) Table 1. \u03b1 = 0.05??\n"

title2 <- "N    \u03b1\u2019          z         \u03b1            RSE\n"

cat(paste0(

  sprintf(head, title1),

  sprintf(head, title2),

  sprintf(body, "2", 0.0294, 2.178, 1-p0, 100*(1-p0-0.05)/0.05, "%\n"),

  sprintf(body, "2", alpha1, C, 1-p1, 100*(1-p1-0.05)/0.05, "%\n")))



Pocock (1977) Table 1. α = 0.05??

N    α’          z         α            RSE

2  0.0294      2.178     0.05002322  +0.0464379%

2  0.02938572  2.178273  0.04999989  -0.0002221%

Nitpicking as usual. Compare at the location of maximum inflation …

Method C: alpha0= 0.05, alpha (s1/s2)= 0.02938572 0.02938572 

Futility criterion Nmax= Inf

CV= 0.2; n(stage 1)= 12; GMR= 0.95

BE margins = 0.8 ... 1.25

GMR= 0.95 and mse of stage 1 in sample size est. used



1e+06 sims at theta0= 1.25 (p(BE)='alpha').

p(BE)   = 0.051252

p(BE) s1= 0.035768

pct studies in stage 2= 78.86%



Distribution of n(total)

- mean (range)= 22.9 (12 ... 98)

- percentiles

 5% 50% 95%

 12  22  40

… with the usual stuff:

Method C: alpha0= 0.05, alpha (s1/s2)= 0.0294 0.0294 

Futility criterion Nmax= Inf

CV= 0.2; n(stage 1)= 12; GMR= 0.95

BE margins = 0.8 ... 1.25

GMR= 0.95 and mse of stage 1 in sample size est. used



1e+06 sims at theta0= 1.25 (p(BE)='alpha').

p(BE)   = 0.051247

p(BE) s1= 0.035777

pct studies in stage 2= 78.86%



Distribution of n(total)

- mean (range)= 22.9 (12 ... 98)

- percentiles

 5% 50% 95% 

 12  22  40

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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Complete thread:

How to find a suitable adjusted α? Helmut 2014-05-31 18:20
- Suitable code for suitable adjusted α d_labes 2014-06-02 08:28
  - Handling inflation Helmut 2014-06-02 14:44
  - Potvin revis(it)ed Helmut 2014-06-02 22:07
    - Potvin revis(it)ed d_labes 2014-06-03 08:47
      - Montague revis(it)ed Helmut 2014-06-03 13:49
      - Pocock’s “natural constant”Helmut 2014-10-13 14:53
        
        Pocock’s “natural constant” ElMaestro 2014-10-13 15:30
        Another “natural constant”? d_labes 2014-10-14 08:56
        
        Λ Helmut 2014-10-14 13:36