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DiM_Bretthorst_UMS.r
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DiM_Bretthorst_UMS.r
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### (C) 2005-2023 by Leo Guertler
### R-code supplement
### to the book
###
### "Subjektive Ansichten und objektive Betrachtungen"
###
### written by Gürtler & Huber (2023)
###
### All R-code is published under the GPL v3 license:
###
### https://www.gnu.org/licenses/gpl-3.0.en.html
###
### except for 'borrowed' code - see links and references.
### For this R-code the original license of the respective
### authors is valid.
###
### R-code published on
###
### https://osdn.net/projects/mixedmethod-rcode
### https://github.com/abcnorio/mixedmethod-rcode
################################################################################
# --------------------------------------------------
# ON THE DIFFERENCE IN MEANS
# --------------------------------------------------
################################################################################
# success rates and integration bounds based on successes and total N
SucRatesIntBounds <- function(Si, Ni, Sii, Nii, smin, snames=c("sample1","sample2"))
{
# --------------------------------------------------
# Success rates and integration bounds in the case
# of the (conservative) Bayes-Laplace prior
# --------------------------------------------------
# necessary variables:
# N, S(uccesses) for sample 1 (i) and 2 (ii)
# defintion of constants
# based on frequencies (successes + failures)
# Di = mean group 1
# si = standard deviation group 1
# Dii = mean group 2
# sii = standard deviation group 2
Di <- (Si + 1) / (Ni + 2)
si <- sqrt(Di * (1 - Di) / (Ni + 3))
Dii <- (Sii + 1) / (Nii + 2)
sii <- sqrt(Dii * (1 - Dii) / (Nii + 3))
# calculation boundaries
Nmax <- max(Ni, Nii)
Nmin <- min(Ni, Nii)
# boundaries standard deviations
sL <- floor(1000 * sqrt((Nmax+1) / (Nmax+3)) / (Nmax + 2)) / 1000
sH <- ceiling(1000 / (2 * sqrt(Nmin + 3))) / 1000
# boundaries means
L <- floor(100 * (smin + 1) / (Nmax + 2)) / 100
H <- 1 - L
res <- list(Si=Si, Ni=Ni, Sii=Sii, Nii=Nii, smin=smin,
Di=Di, si=si, Dii=Dii, sii=sii,
L=L, H=H, sL=sL, sH=sH,
snames=snames)
attr(res,"typ") <- c("SRIB")
return(res)
}
# call:
# res.SIB <- SucRatesIntBounds(Si=11, Ni=15, Sii=10, Nii=16, smin=0, snames=c("voluntary","non-voluntary"))
################################################################################
#########################################################################
# DIFFERENCE IN MEANS INTEGRALS
#
### PMV - NORMAL
#
#low = lownum / (2*s^2)
#up = upnum / (2*s^2)
PMV.hypo <- function(NN,dd,upnum,lownum,sL,sH)
{
integpmv <- function(s)
{
1 / (s^NN) * exp(-dd / (2 * s^2)) *
( pgamma(upnum/(2*s^2),1/2)*gamma(1/2) + pgamma(lownum/(2*s^2),1/2)*gamma(1/2) )
}
pmv <- integrate(integpmv, lower=sL, upper=sH)$value
#cat("\npmv =\t",pmv,"\n")
PMV <- pmv / sqrt(2*NN)
return(PMV)
}
# call:
# PMV <- PMV.hypo(NN=NN, dd=dd, upnum=upnum, lownum=lownum, sL=sL, sH=sH)
### PMV - LOG
PMV.hypo.log <- function(NN,dd,upnum,lownum,sL,sH, Nsteps=100, ...)
{
half <- 1/2
integpmv.log <- function(s, ...)
{
# 1 / (s^NN) * exp(-dd / (2 * s^2)) *
# ( pgamma(upnum/(2*s^2),1/2)*gamma(1/2) + pgamma(lownum/(2*s^2),1/2)*gamma(1/2) )
sqrtpi.log <- lgamma(1/2)
# logs <- c(pgamma(upnum/(2*s^2),1/2,log=TRUE) + sqrtpi.log,
# pgamma(lownum/(2*s^2),1/2,log=TRUE) + sqrtpi.log
# )
#pmv <- integrate(integpmv, lower=sL, upper=sH)$value
#PMV <- pmv / sqrt(2*NN)
#res <- log(1) - log(s)*NN + (-dd / (2*s^2)) + logSumExp(logs)
s2.2 <- 2*s^2
# v <- c( pgamma(upnum/(2*s^2),1/2,log=TRUE,lower=TRUE) + sqrtpi.log,
# pgamma(lownum/(2*s^2),1/2,log=TRUE,lower=TRUE) + sqrtpi.log
v <- c( pgamma(upnum/s2.2,half,log=TRUE,lower=TRUE) + sqrtpi.log,
pgamma(lownum/s2.2,half,log=TRUE,lower=TRUE) + sqrtpi.log
)
#v
-log(s)*NN + (-dd / (2*s^2)) + .llog.2add.short(v)
#.llog.2add.short(pgamma(upnum/(2*s^2),1/2,log=TRUE,lower=TRUE) + sqrtpi.log,
# pgamma(lownum/(2*s^2),1/2,log=TRUE,lower=TRUE) + sqrtpi.log)
#logSumExp(logs)
}
pmv.log <- simpsonrule.nlb(integpmv.log, sL, sH,
method="log",
Nsteps=Nsteps,
half=half)
PMV.log <- pmv.log - log( sqrt(2*NN) )
return(PMV.log)
}
# call:
# PMV.log <- PMV.hypo.log(NN=NN, dd=dd, upnum=upnum, lownum=lownum, sL=sL, sH=sH)
### PMV - BROB
PMV.hypo.brob <- function(NN,dd,upnum,lownum,sL,sH, Nsteps=100, ...)
{
integpmv.brob <- function(s, ...)
{
1 / (as.brob(s)^NN) * exp(as.brob(-dd) / (2 * s^2)) *
( gamma_inc_gen(a=1/2,z0=0,z1=upnum/(2*s^2),log=TRUE)*sqrt(pi) +
gamma_inc_gen(a=1/2,z0=0,z1=lownum/(2*s^2),log=TRUE)*sqrt(pi) )
}
pmv.brob <- simpsonrule.nlb(integpmv.brob, sL, sH, method="brob",Nsteps=Nsteps)
PMV.brob <- pmv.brob / sqrt(2*NN)
return(PMV.brob)
}
# call:
# PMV.brob <- PMV.hypo.brob(NN=NN, dd=dd, upnum=upnum, lownum=lownum, sL=sL, sH=sH, Nsteps=100)
# debug calls:
# PMV <- PMV.hypo(NN=NN, dd=dd, upnum=upnum, lownum=lownum, sL=sL, sH=sH)
# PMV.log <- PMV.hypo.log(NN=NN, dd=dd, upnum=upnum, lownum=lownum, sL=sL, sH=sH)
# PMV.brob <- PMV.hypo.brob(NN=NN, dd=dd, upnum=upnum, lownum=lownum, sL=sL, sH=sH)
#
# log(PMV)
# PMV.log
# PMV.brob@x
### PMbarV - NORMAL
#
# Bretthorst, p.6f.
# sign in inner brackets between the two pgamma*gamma's
# sign + in case if U1H, H1L are of different sign
# sign - in case if U1H, U1L are of same sign
# same for U2H, U2L
PMbarV.hypo <- function(NN,zz,dd,upinum,lowinum,sL,sH,H,L,Ni,Nii)
{
integpmbarv <- function(s)
{
#(* + if (L-DD) < 0 *)
1/(s^(NN-1)) * exp(-zz/(2*s^2)) *
( pgamma(upinum/(2*s^2),1/2)*gamma(1/2) + pgamma(lowinum/(2*s^2),1/2)*gamma(1/2) ) *
( pgamma(upiinum/(2*s^2),1/2)*gamma(1/2) + pgamma(lowiinum/(2*s^2),1/2)* gamma(1/2) )
}
pmbarv <- integrate(integpmbarv, lower=sL, upper=sH)$value
PMbarV <- pmbarv / (2*(H-L) * sqrt(Ni*Nii))
return(PMbarV)
}
# call:
# PMbarV <- PMbarV.hypo(NN=NN, zz=zz, dd=dd, upinum=upinum, lowinum=lowinum, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
### PMbarV - LOG
PMbarV.hypo.log <- function(NN,zz,dd,upinum,lowinum,sL,sH,H,L,Ni,Nii, Nsteps=100, ...)
{
sqrtpi.log <- lgamma(1/2)
half <- 1/2
integpmbarv.log <- function(s, ...)
{
#(* + if (L-DD) < 0 *)
#1/(s^(NN-1)) * exp(-zz/(2*s^2)) *
# ( pgamma(upinum/(2*s^2),1/2)*gamma(1/2) + pgamma(lowinum/(2*s^2),1/2)*gamma(1/2) ) *
# ( pgamma(upiinum/(2*s^2),1/2)*gamma(1/2) + pgamma(lowiinum/(2*s^2),1/2)* gamma(1/2) )
s2.2 <- 2*s^2
# v1 <- c( pgamma(upinum/(2*s^2),1/2, log=TRUE) + sqrtpi.log,
# pgamma(lowinum/(2*s^2),1/2, log=TRUE) + sqrtpi.log )
# v2 <- c( pgamma(upiinum/(2*s^2),1/2, log=TRUE) + sqrtpi.log,
# pgamma(lowiinum/(2*s^2),1/2, log=TRUE) + sqrtpi.log )
v1 <- c( pgamma(upinum/s2.2,half, log=TRUE) + sqrtpi.log,
pgamma(lowinum/s2.2,half, log=TRUE) + sqrtpi.log )
v2 <- c( pgamma(upiinum/s2.2,half, log=TRUE) + sqrtpi.log,
pgamma(lowiinum/s2.2,half, log=TRUE) + sqrtpi.log )
# log(1) == 0
log(1) - ( (NN-1)*log(s) ) +
(-zz/(2*s^2)) +
.llog.2add.short( v1 ) +
.llog.2add.short( v2 )
}
#pmbarv <- integrate(integpmbarv, lower=sL, upper=sH)$value
pmbarv.log <- simpsonrule.nlb(integpmbarv.log, sL, sH,
sqrtpi.log=sqrtpi.log,
half=half,
method="log", Nsteps=Nsteps)
#PMbarV <- pmbarv / (2*(H-L) * sqrt(Ni*Nii))
PMbarV.log <- pmbarv.log - ( log(2*(H-L)) +
(1/2)*( log(Ni) + log(Nii)) ) #log(sqrt(Ni*Nii)) )
return(PMbarV.log)
}
# call:
# PMbarV <- PMbarV.hypo.loog(NN=NN, zz=zz, dd=dd, upinum=upinum, lowinum=lowinum, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
### PMbarV - BROB
PMbarV.hypo.brob <- function(NN,zz,dd,upinum,lowinum,sL,sH,H,L,Ni,Nii, Nsteps=100, ...)
{
integpmbarv.brob <- function(s)
{
1/(as.brob(s)^(NN-1)) * exp(as.brob(-zz)/(2*s^2)) *
( pgamma(upinum/(2*s^2),1/2)*gamma(1/2) + pgamma(lowinum/(2*s^2),1/2)*gamma(1/2) ) *
( pgamma(upiinum/(2*s^2),1/2)*gamma(1/2) + pgamma(lowiinum/(2*s^2),1/2)*gamma(1/2) )
}
# pmbarv.brob <- simpsonrule.brob(fx=integpmbarv.brob, sL=sL, sH=sH, Nsteps=Nsteps)
pmbarv.brob <- simpsonrule.nlb(fx=integpmbarv.brob, sL, sH, method="brob",Nsteps=Nsteps)
PMbarV.brob <- pmbarv.brob / (2*(H-L) * sqrt(Ni*Nii))
return(PMbarV.brob)
}
# call:
# PMbarV.brob <- PMbarV.hypo.brob(NN=NN, zz=zz, dd=dd, upinum=upinum, lowinum=lowinum, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii, Nsteps=100)
# debug calls:
# PMbarV <- PMbarV.hypo(NN=NN, zz=zz, dd=dd, upinum=upinum, lowinum=lowinum, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
# PMbarV.log <- PMbarV.hypo.log(NN=NN, zz=zz, dd=dd, upinum=upinum, lowinum=lowinum, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
# PMbarV.brob <- PMbarV.hypo.brob(NN=NN, zz=zz, dd=dd, upinum=upinum, lowinum=lowinum, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
#
# log(PMbarV)
# PMbarV.log
# PMbarV.brob
### PMVbar - NORMAL
# note Mathematica: gamma[a,z0,z1] = gamma[a,z1] - gamma[a,z0]
UiA <- function(A) Ni*(Dsi-2*Di*A+A^2)/2
UiiA <- function(A) Nii*(Dsii-2*Dii*A+A^2)/2
PMVbar.hypo <- function(Dsi,Dsii,sL,sH,H,L,Ni,Nii)
{
integpmvbar <- function(A)
{
1/UiA(A)^(Ni/2) *
1/UiiA(A)^(Nii/2) *
( pgamma(UiA(A)/(sH^2),Ni/2)*gamma(Ni/2) - pgamma(UiA(A)/(sL^2),Ni/2)*gamma(Ni/2) ) *
( pgamma(UiiA(A)/(sH^2),Nii/2)*gamma(Nii/2) - pgamma(UiiA(A)/(sL^2),Nii/2)*gamma(Nii/2) )
}
pmvbar <- integrate(integpmvbar, lower=L, upper=H)$value
PMVbar <- pmvbar / (4*log(sH/sL))
return(PMVbar)
}
# call:
# PMVbar <- PMVbar.hypo(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
### PMVbar - LOG
#UiA <- function(A) Ni*(Dsi-2*Di*A+A^2)/2
#UiiA <- function(A) Nii*(Dsii-2*Dii*A+A^2)/2
UiA.log <- function(A) log(Ni) + log(Dsi-2*Di*A+A^2) -log(2) #ok
UiiA.log <- function(A) log(Nii) + log(Dsii-2*Dii*A+A^2) -log(2) #ok
PMVbar.hypo.log <- function(Dsi,Dsii,sL,sH,H,L,Ni,Nii, Nsteps=100, ...)
{
#const.
lg.Ni.2 <- lgamma(Ni/2)
lg.Nii.2 <- lgamma(Nii/2)
Ni.2 <- Ni/2
Nii.2 <- Nii/2
sH.2 <- sH^2
sL.2 <- sL^2
g.Ni.2 <- gamma(Ni/2)
g.Nii.2 <- gamma(Nii/2)
integpmvbar.log <- function(A, ...)
{
#1/UiA(A)^(Ni/2) *
# 1/UiiA(A)^(Nii/2) *
# ( pgamma(UiA(A)/(sH^2),Ni/2)*gamma(Ni/2) - pgamma(UiA(A)/(sL^2),Ni/2)*gamma(Ni/2) ) *
# ( pgamma(UiiA(A)/(sH^2),Nii/2)*gamma(Nii/2) - pgamma(UiiA(A)/(sL^2),Nii/2)*gamma(Nii/2) )
UiA.A <- UiA(A)
UiiA.A <- UiiA(A)
#Ni.2 <- Ni/2
#Nii.2 <- Nii/2
v1 <- c( pgamma( UiA.A/sH.2, Ni.2, log=TRUE,lower=FALSE ) + lg.Ni.2, #a1
pgamma( UiA.A/sL.2, Ni.2, log=TRUE,lower=FALSE ) + lg.Ni.2
)
v2 <- c( pgamma( UiiA.A/sH.2, Nii.2, log=TRUE,lower=FALSE ) + lg.Nii.2, #a2
pgamma( UiiA.A/sL.2, Nii.2, log=TRUE,lower=FALSE ) + lg.Nii.2
)
log(1) - ( Ni.2 * log(UiA.A) ) + #ok
log(1) - ( Nii.2 * log(UiiA.A) ) + #ok
.llog.2sub.short(v1) + .llog.2sub.short(v2)
}
pmvbar.log <- simpsonrule.nlb(integpmvbar.log,
L, H,
sL.2=sL.2, sH.2=sH.2,
lg.Ni.2=lg.Ni.2, lg.Nii.2=lg.Nii.2,
Ni.2=Ni.2, Nii.2=Nii.2,
g.Ni.2=g.Ni.2,
g.Ni.2=g.Nii.2,
method="log", Nsteps=Nsteps)
PMVbar.log <- pmvbar.log - ( log(4) + log( log(sH/sL) ) )
return(PMVbar.log)
}
# call:
# PMVbar.log <- PMVbar.hypo.log(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
### PMVbar - BROB
# not possible for pgamma due to its definition without brob numbers
UiA.brob <- function(A) as.brob( Ni*(Dsi-2*Di*A+A^2)/2 )
UiiA.brob <- function(A) as.brob( Nii*(Dsii-2*Dii*A+A^2)/2 )
PMVbar.hypo.brob <- function(Dsi,Dsii,sL,sH,H,L,Ni,Nii, Nsteps=100, ...)
{
integpmvbar.brob <- function(A)
{
1/as.brob(UiA.brob(as.brob(A))^(as.brob(Ni)/2)) *
1/as.brob(UiiA.brob(as.brob(A))^(as.brob(Nii)/2)) *
( brob( (pgamma(UiA(A)/(sH^2),Ni/2,log.p=TRUE,lower=FALSE)+lgamma(Ni/2)) ) -
brob( (pgamma(UiA(A)/(sL^2),Ni/2,log.p=TRUE,lower=FALSE)+lgamma(Ni/2)) ) ) *
( brob( (pgamma(UiiA(A)/(sH^2),Nii/2,log.p=TRUE,lower=FALSE)+lgamma(Nii/2)) ) -
brob( (pgamma(UiiA(A)/(sL^2),Nii/2,log.p=TRUE,lower=FALSE)+lgamma(Nii/2)) )
)
}
#pmvbar.brob <- simpsonrule.brob(fx=integpmvbar.brob, sL=L, sH=H, Nsteps=Nsteps)
pmvbar.brob <- simpsonrule.nlb(fx=integpmvbar.brob, L, H, method="brob", Nsteps=Nsteps)
PMVbar.brob <- pmvbar.brob / (4*log(sH/sL))
return(PMVbar.brob)
}
# call:
# PMVbar.brob <- PMVbar.hypo.brob(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii, Nsteps=100)
# debug calls:
# PMVbar <- PMVbar.hypo(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
# PMVbar.log <- PMVbar.hypo.log(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
# PMVbar.brob <- PMVbar.hypo.brob(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
#
# log(PMVbar)
# PMVbar.log
# PMVbar.brob
### PMbarVbar - NORMAL
# Bretthorst, p.11
# sign: minus if U1H, U1L have same sign
# sign: plus if U1H, U1L have different sign
# same true for U2H, U2L
UiiB <- function(B) Nii*(Dsii-2*Dii*B+B^2)/2
PMbarVbar.hypo <- function(Dsi,Dsii,sL,sH,H,L,Ni,Nii)
{
integpmbarvbar1 <- function(A)
{
1/(UiA(A)^(Ni/2)) *
#lower=FALSE for integral z^inf
#the absolute value is ok, only the sign would differ
#the multiplication of both integrals if handled identical
#will lead to positive values
#and with prob distributions we do not have negative probs!
( pgamma(UiA(A)/(sH^2),Ni/2,lower=FALSE)*gamma(Ni/2) -
pgamma(UiA(A)/(sL^2),Ni/2,lower=FALSE)*gamma(Ni/2) )
}
integpmbarvbar2 <- function(B)
{
1/(UiiB(B)^(Nii/2)) *
#lower=FALSE for integral z^inf
( pgamma(UiiB(B)/(sH^2),Nii/2,lower=FALSE)*gamma(Nii/2) -
pgamma(UiiB(B)/(sL^2),Nii/2,lower=FALSE)*gamma(Nii/2) )
}
pmbarvbar1 <- integrate(integpmbarvbar1, lower=L, upper=H)$value
pmbarvbar2 <- integrate(integpmbarvbar2, lower=L, upper=H)$value
pmbarvbar <- pmbarvbar1 * pmbarvbar2
PMbarVbar <- pmbarvbar / (4*(H-L) * log(sH/sL))
return(PMbarVbar)
}
# call:
# PMbarVbar <- PMbarVbar.hypo(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
### PMbarVbar - LOG
# Bretthorst, p.11
# sign: minus if U1H, U1L have same sign
# sign: plus if U1H, U1L have different sign
# same true for U2H, U2L
#UiiB <- function(B) Nii*(Dsii-2*Dii*B+B^2)/2
# UiiB.log <- function(B, ...)
# {
# #a <- log(Nii) - log(2)
# #b <- log(Dsii)
# #ce <- log(2) + log(Dii) + log(B)
# #d <- 2*log(B)
# v1 <- c(log(Dsii), 2*log(B))
# v2 <- log(2) + log(Dii) + log(B)
# #return( a + .llog.2sub.short( c( .llog.2add.short(c(b,d)),ce) ) )
# return( log(Nii) - log(2) + .llog.2sub.short( c( .llog.2add.short(v1),v2) ) )
# }
UiiB.log <- function(B, ...)
{
v1 <- c(log(Dsii), 2*log(B))
v2 <- log(2) + log(Dii) + log(B)
return( log(Nii) - log(2) + .llog.2sub.short( c( .llog.2add.short(v1),v2) ) )
}
# call:
# UiiB.log(B)
# log(UiiB(B))
PMbarVbar.hypo.log <- function(Dsi,Dsii,sL,sH,H,L,Ni,Nii, Nsteps=100, ...)
{
lg.Ni.2 <- lgamma(Ni/2)
lg.Nii.2 <- lgamma(Nii/2)
Ni.2 <- Ni/2
Nii.2 <- Nii/2
sH.2 <- sH^2
sL.2 <- sL^2
integpmbarvbar1.log <- function(A, ...)
{
#1/(UiA(A)^(Ni/2)) *
#lower=FALSE for integral z^inf
#the absolute value is ok, only the sign would differ
#the multiplication of both integrals if handled identical
#will lead to positive values
#and with prob distributions we do not have negative probs!
# ( pgamma(UiA(A)/(sH^2),Ni/2,lower=FALSE)*gamma(Ni/2) -
# pgamma(UiA(A)/(sL^2),Ni/2,lower=FALSE)*gamma(Ni/2) )
#v1 <- c( pgamma(UiA(A)/sH.2, Ni.2, lower=FALSE, log=TRUE) + lg.Ni.2,
# pgamma(UiA(A)/sL.2, Ni.2, lower=FALSE, log=TRUE) + lg.Ni.2 )
UiA.A <- UiA(A)
v1 <- c( pgamma(UiA.A/sH.2, Ni.2, lower=FALSE, log=TRUE) + lg.Ni.2,
pgamma(UiA.A/sL.2, Ni.2, lower=FALSE, log=TRUE) + lg.Ni.2 )
#log(1) - ( Ni.2 * log(UiA(A)) ) + .llog.2sub.short(v1) #todo check which one is greater! -> put into function!!!!!!!!!!!!!
log(1) - ( Ni.2 * log(UiA.A) ) + .llog.2sub.short(v1) #todo check which one is greater! -> put into function!!!!!!!!!!!!!
}
integpmbarvbar2.log <- function(B, ...)
{
#ok#1/(UiiB(B)^(Nii/2)) *
# #lower=FALSE for integral z^inf
#ok# ( pgamma(UiiB(B)/(sH^2),Nii/2,lower=FALSE)*gamma(Nii/2) -
#ok# pgamma(UiiB(B)/(sL^2),Nii/2,lower=FALSE)*gamma(Nii/2) )
UiiB.B <- UiiB(B)
# v2 <- c( pgamma(UiiB(B)/sH.2, Nii.2, lower=FALSE, log=TRUE) + lg.Nii.2,
# pgamma(UiiB(B)/sL.2, Nii.2, lower=FALSE, log=TRUE) + lg.Nii.2 )
v2 <- c( pgamma(UiiB.B/sH.2, Nii.2, lower=FALSE, log=TRUE) + lg.Nii.2,
pgamma(UiiB.B/sL.2, Nii.2, lower=FALSE, log=TRUE) + lg.Nii.2 )
log(1) - ( Nii.2 * log(UiiB(B)) ) + .llog.2sub.short(v2)
}
pmbarvbar1.log <- simpsonrule.nlb(integpmbarvbar1.log,
lower=L, upper=H,
lg.Ni.2=lg.Ni.2,
Ni.2=Ni.2,
sL.2=sL.2, sH.2=sH.2,
sL=sL, sH=sH,
method="log", Nsteps=Nsteps)
pmbarvbar2.log <- simpsonrule.nlb(integpmbarvbar2.log,
lower=L, upper=H,
lg.Ni.2=lg.Ni.2,
Ni.2=Ni.2,
sL.2=sL.2, sH.2=sH.2,
sL=sL, sH=sH,
method="log", Nsteps=Nsteps)
#pmbarvbar <- pmbarvbar1 * pmbarvbar2
#PMbarVbar <- pmbarvbar / (4*(H-L) * log(sH/sL))
v1 <- c(log(H), log(L))
v2 <- log(log(sH)-log(sL))
pmbarvbar.log <- pmbarvbar1.log + pmbarvbar2.log
PMbarVbar.log <- pmbarvbar.log -
( log(4) + .llog.2sub.short(v1) + v2 )
return(PMbarVbar.log)
}
# call:
# PMbarVbar.log <- PMbarVbar.hypo.log(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
### PMbarVbar - BROB
UiiB.brob <- function(B) as.brob( Nii*(Dsii-2*Dii*B+B^2)/2 )
PMbarVbar.hypo.brob <- function(Dsi,Dsii,sL,sH,H,L,Ni,Nii, Nsteps=100, ...)
{
#first integral
#lower=FALSE for new simsponrule.nlb!!!
integpmbarvbar1.brob <- function(A)
{
1/( UiA(as.brob(A))^as.brob(Ni/2) ) *
( brob( pgamma(UiA(A)/(sH^2),Ni/2,log.p=TRUE,lower=FALSE)+lgamma(Ni/2) ) -
brob( pgamma(UiA(A)/(sL^2),Ni/2,log.p=TRUE,lower=FALSE)+lgamma(Ni/2) ) )
}
#second integral
#lower=FALSE for new simsponrule.nlb!!!
integpmbarvbar2.brob <- function(B)
{
1/( UiiB.brob(B)^(Nii/2)) *
( brob(pgamma(UiiB(B)/(sH^2),Nii/2,log.p=TRUE,lower=FALSE)+lgamma(Nii/2)) -
brob(pgamma(UiiB(B)/(sL^2),Nii/2,log.p=TRUE,lower=FALSE)+lgamma(Nii/2)) )
}
#pmbarvbar1.brob <- simpsonrule.brob(fx=integpmbarvbar1.brob, sL=L, sH=H, Nsteps=Nsteps)
#pmbarvbar2.brob <- simpsonrule.brob(fx=integpmbarvbar2.brob, sL=L, sH=H, Nsteps=Nsteps)
pmbarvbar1.brob <- simpsonrule.nlb(fx=integpmbarvbar1.brob, L, H, method="brob", Nsteps=Nsteps)
pmbarvbar2.brob <- simpsonrule.nlb(fx=integpmbarvbar2.brob, L, H, method="brob", Nsteps=Nsteps)
pmbarvbar.brob <- pmbarvbar1.brob * pmbarvbar2.brob
PMbarVbar.brob <- pmbarvbar.brob / (4*(H-L) * log(sH/sL))
return(PMbarVbar.brob)
}
# call:
# PMbarVbar.brob <- PMbarVbar.hypo.brob(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii, Nsteps=100)
# debug calls:
# PMbarVbar <- PMbarVbar.hypo(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
# PMbarVbar.log <- PMbarVbar.hypo.log(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
# PMbarVbar.brob <- PMbarVbar.hypo.brob(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
#
# log(PMbarVbar)
# PMbarVbar.log
# PMbarVbar.brob
################################################################################
################################################################################
# create constants from initial values
.DiM_ccs <- function(inval=NULL, smin=0)
{
# create constsns
# extract values for calculations from inputv object (class "UMS")
# general
Di <- inval[["Di"]]
Dii <- inval[["Dii"]]
si <- inval[["si"]]
sii <- inval[["sii"]]
Ni <- inval[["Ni"]]
Nii <- inval[["Nii"]]
L <- inval[["L"]]
H <- inval[["H"]]
sL <- inval[["sL"]]
sH <- inval[["sH"]]
snames <- inval[["snames"]]
smin <- inval[["smin"]]
# defintion of constants
NN <- Ni+Nii
DD <- (Ni * Di + Nii * Dii) / NN
Dsi <- (Ni-1) / Ni * si^2 + Di^2
Dsii <- (Nii-1) / Nii * sii^2 + Dii^2
DsD <- (Ni * Dsi + Nii * Dsii) / NN
ss <- sqrt(NN * (DsD - DD^2) / (NN-1))
cat("\nL - Mean_comb < 0 : ",L < DD," [comparison L < DD]\n")
if(L < DD)
{
cat("'+'-sign between Gamma-factors is ok\n\n")
} else cat("'+'-sign between Gamma-factors is false!\n\n")
# PMV
dd <- NN * (DsD - DD^2)
lownum <- NN * (L-DD)^2
upnum <- NN * (H-DD)^2
# PMbarV
zz <- Ni * (Dsi - Di^2) + Nii * (Dsii - Dii^2)
lowinum <- Ni * (L - Di)^2
upinum <- Ni * (H - Di)^2
lowiinum <- Nii * (L - Dii)^2
upiinum <- Nii * (H - Dii)^2
# compile input results independent from method
constants <- data.frame(sample1=snames[1],sample2=snames[2],
Di,Dii,Ni,si,Nii,sii,
NN,DD,ss,L,H,sL,sH,Dsi,Dsii,
DsD,dd,
lownum,upnum,
lowinum,upinum,
lowiinum,upiinum,
smin,
zz)
# resulting table/ dataframe with input values
input.df <- data.frame("No." = c(Ni,Nii,NN),
"Standard Deviation" = c(si,sii,ss),
"Mean" = c(Di,Dii,DD),
"Data set" = c(snames,"combined"),
check.names = FALSE)
# table/ dataframe with prior values/ information
prior.df <- data.frame("Numerical Example" = c("Prior Mean lower bound","Prior Mean upper bound",
"Prior Standard Deviation lower bound","Prior Standard Deviation upper bound"),
"Value" = c(L,H,sL,sH), check.names = FALSE)
res <- list(constants=constants,
input.df=input.df,
prior.df=prior.df
)
return(res)
}
# call:
#
################################################################################
################################################################################
# calculate the four integrals
# PMV, PMbarV, PMVbar, PMbarVbar
.DiM_callintegrals <- function(DiM.ccs,
method="normal",
Nsteps=100,
parallel=FALSE
)
{
# general note
# necessary variables in the function
# NN, DD, Dsi, Dsii, DsD, ss,
# dd, lownum, upnum, low, up, pmv, PMV,
# zz, lowinum, upinum, lowiinum, upiinum, pmbarv, PMbarV,
# pmvbar, PMVbar, pmbarvbar, PMbarVbar, cc,
# mv, mbarv, mvbar, mbarvbar,
# samemeans, diffmeans, samevars, diffvars, diffsets
constants <- DiM.ccs[["constants"]]
attach(constants)
# notes
# Bretthorst, p.6f.
# sign in inner brackets between the two pgamma*gamma's
# sign + in case if U1H, H1L are of different sign
# sign - in case if U1H, U1L are of same sign
# same for U2H, U2L
# PMVbar
# no constants
# PMbarVbar
# no constants
# Bretthorst, p.11
# sign: minus if U1H, U1L have same sign
# sign: plus if U1H, U1L have different sign
# same true for U2H, U2L
# integpmbarvbar1
# lower=FALSE for integral z^inf
# the absolute value is ok, only the sign would differ
# the multiplication of both integrals if handled identical
# will lead to positive values
# and with prob distributions we do not have negative probs!
#
# brob version:
# first and second integral with
# lower=FALSE for new simpsonrule.nlb!!!
# calculate integrals
if(method=="normal")
{
cat("\nWork based on normal values\n")
cat("\nCalculate PMV\n")
PMV <- PMV.hypo(NN=NN, dd=dd, upnum=upnum, lownum=lownum, sL=sL, sH=sH)
cat("Calculate PMbarV\n")
PMbarV <- PMbarV.hypo(NN=NN, zz=zz, dd=dd, upinum=upinum, lowinum=lowinum, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
cat("Calculate PMVbar\n")
PMVbar <- PMVbar.hypo(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
cat("Calculate PMbarVbar\n")
PMbarVbar <- PMbarVbar.hypo(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii)
} else if(method=="log")
{
cat("\nWork based on log values\n")
if(parallel) require(parallel)
cat("\nCalculate PMV\n")
PMV <- PMV.hypo.log(NN=NN, dd=dd, upnum=upnum, lownum=lownum, sL=sL, sH=sH, Nsteps=Nsteps)
cat("Calculate PMbarV\n")
PMbarV <- PMbarV.hypo.log(NN=NN, zz=zz, dd=dd, upinum=upinum, lowinum=lowinum, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii, Nsteps=Nsteps)
cat("Calculate PMVbar\n")
PMVbar <- PMVbar.hypo.log(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii, Nsteps=Nsteps)
cat("Calculate PMbarVbar\n")
PMbarVbar <- PMbarVbar.hypo.log(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii, Nsteps=Nsteps)
} else if(method=="brob")
{
cat("\nWork based on Brobdingnag values\n")
require(Brobdingnag)
cat("\nCalculate PMV\n")
PMV <- PMV.hypo.brob(NN=NN, dd=dd, upnum=upnum, lownum=lownum, sL=sL, sH=sH, Nsteps=Nsteps)
cat("Calculate PMbarV\n")
PMbarV <- PMbarV.hypo.brob(NN=NN, zz=zz, dd=dd, upinum=upinum, lowinum=lowinum, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii, Nsteps=Nsteps)
cat("Calculate PMVbar\n")
PMVbar <- PMVbar.hypo.brob(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii, Nsteps=Nsteps)
cat("Calculate PMbarVbar\n")
PMbarVbar <- PMbarVbar.hypo.brob(Dsi=Dsi, Dsii=Dsii, sL=sL, sH=sH, H=H, L=L, Ni=Ni, Nii=Nii, Nsteps=Nsteps)
}
detach(constants)
DiM.integrals <- c(PMV, PMbarV, PMVbar, PMbarVbar)
names(DiM.integrals) <- c("PMV","PMbarV","PMVbar","PMbarVbar")
return(DiM.integrals)
}
# call:
#
################################################################################
################################################################################
# compile results bsed on the four integrals
.DiM_prepres <- function(DiM.integrals,
method=NA,
convback=FALSE, # convert probs back from log-scale
percfac=1, # not 100% but sum(ps) = 1
dig=5
)
{
stopifnot(!is.na(method))
if(method=="normal" && convback==TRUE)
{
cat("\nTYPE=normal & convback=TRUE -> no conversion\nwill be done for values on the normal-scale.\n\n")
convback <- FALSE
}
if(method=="normal")
{
denom <- sum(DiM.integrals) # PMV, PMbarV, PMVbar, PMbarVbar
cc <- 1 / denom
# output not as p*100%, just p*1
mv <- percfac * cc * DiM.integrals["PMV"]
mbarv <- percfac * cc * DiM.integrals["PMbarV"]
mvbar <- percfac * cc * DiM.integrals["PMVbar"]
mbarvbar <- percfac * cc * DiM.integrals["PMbarVbar"]
samemeans <- mv + mvbar
diffmeans <- mbarv + mbarvbar
samevars <- mv + mbarv
diffvars <- mvbar + mbarvbar
diffsets <- mvbar + mbarv + mbarvbar
samesets <- 1 - diffsets
OR.diffmeans <- diffmeans/samemeans
OR.samemeans <- samemeans/diffmeans
OR.diffvars <- diffvars/samevars
OR.samevars <- samevars/diffvars
OR.diffsets <- diffsets/mv
OR.samesets <- mv/diffsets
} else if(method=="log")
{
denom <- .llog.2add.short( DiM.integrals )
#.llog.2add.short(c(PMV, PMbarV, PMVbar, PMbarVbar))
cc <- - denom #log(1)
# no *100%, just *1
percfac.l <- log(percfac)
mv <- percfac.l + cc + DiM.integrals["PMV"]
mbarv <- percfac.l + cc + DiM.integrals["PMbarV"]
mvbar <- percfac.l + cc + DiM.integrals["PMVbar"]
mbarvbar <- percfac.l + cc + DiM.integrals["PMbarVbar"]
samemeans <- .llog.2add.short( c(mv, mvbar) )
diffmeans <- .llog.2add.short( c(mbarv, mbarvbar) )
samevars <- .llog.2add.short( c(mv, mbarv) )
diffvars <- .llog.2add.short( c(mvbar, mbarvbar) )
diffsets <- .llog.2add.short( c(mvbar, mbarv, mbarvbar) )
samesets <- .llog.2sub.short( c(log(1),diffsets) )
OR.diffmeans <- diffmeans - samemeans
OR.samemeans <- samemeans - diffmeans
OR.diffvars <- diffvars - samevars
OR.samevars <- samevars - diffvars
OR.diffsets <- diffsets - mv
OR.samesets <- mv - diffsets
} else if(method=="brob")
{
# calculate total probability (denominator of Bayes' Theorem)
# sum of all four hypotheses (probabilities)
denom <- sum(list2vec.brob(DiM.integrals))
cc <- 1 / denom
# output not as p*100%, just p*1
mv <- percfac * cc * DiM.integrals[["PMV"]]
mbarv <- percfac * cc * DiM.integrals[["PMbarV"]]
mvbar <- percfac * cc * DiM.integrals[["PMVbar"]]
mbarvbar <- percfac * cc * DiM.integrals[["PMbarVbar"]]
samemeans <- mv + mvbar
diffmeans <- mbarv + mbarvbar
samevars <- mv + mbarv
diffvars <- mvbar + mbarvbar
diffsets <- mvbar + mbarv + mbarvbar
samesets <- 1 - diffsets
OR.diffmeans <- diffmeans/samemeans
OR.samemeans <- samemeans/diffmeans
OR.diffvars <- diffvars/samevars
OR.samevars <- samevars/diffvars
OR.diffsets <- diffsets/mv
OR.samesets <- mv/diffsets
}
# unnormalized + total prob
DiM.integrals.ext <- c(DiM.integrals, cc, denom)
names(DiM.integrals.ext)[c(5,6)] <- c("cc","denom")
# normalized + Odds Ratios (OR)
DiM.probsOR.res <- c(mv, mbarv, mvbar, mbarvbar,
samemeans, diffmeans, samevars, diffvars,
samesets, diffsets,
OR.diffmeans, OR.samemeans,
OR.diffvars, OR.samevars,
OR.diffsets, OR.samesets
)
names(DiM.probsOR.res) <- c("mv", "mbarv", "mvbar", "mbarvbar",
"samemeans", "diffmeans", "samevars", "diffvars",
"samesets", "diffsets",
"OR.diffmeans", "OR.samemeans",
"OR.diffvars", "OR.samevars",
"OR.diffsets", "OR.samesets"
)
# probs = 1:10
# ORs = 11:16
# diffmeans vs samemeans
# diffvars vs samevars
# diffsets vs samesets
diff.v <- c("diffmeans","diffvars","diffsets")
same.v <- c("samemeans","samevars","mv")
diff.terms <- c(rep("different",2),"different means and/ or")
same.terms <- c(rep("the same",2), "the same means and")
diffvssame.v <- c("diffvssamemeans","diffvssamevars","diffvssamesets")
samevsdiff.v <- c("samevsdiffmeans","samevsdiffvars","samevsdiffsets")
diffsame.terms <- diff.terms
names(diffsame.terms) <- diff.v
# diff greater/ same for means/vars/sets
if(method=="normal")
{
diffgreaterthansame <- DiM.probsOR.res[ diff.v ] > DiM.probsOR.res[ same.v ]
diffbysame <- DiM.probsOR.res[ diff.v ] / DiM.probsOR.res[ same.v ]
diffbysame.ID <- which(diffgreaterthansame == FALSE)
if(length(diffbysame.ID) > 0) diffbysame[ diffbysame.ID ] <- 1 / diffbysame[ diffbysame.ID ]
} else if(method=="log")
{
diffgreaterthansame <- DiM.probsOR.res[ diff.v ] > DiM.probsOR.res[ same.v ]
diffbysame <- DiM.probsOR.res[ diff.v ] - DiM.probsOR.res[ same.v ]
diffbysame.ID <- which(diffgreaterthansame == FALSE)
if(length(diffbysame.ID) > 0) diffbysame[ diffbysame.ID ] <- log(1) - diffbysame[ diffbysame.ID ]
} else if(method=="brob")
{
diffbysame <- sapply(seq_along(1:length(diff.v)), function(i)
{
DiM.probsOR.res[diff.v[i]][[1]] / DiM.probsOR.res[same.v[i]][[1]]
})
names(diffbysame) <- diff.v
diffgreaterthansame <- sapply(seq_along(1:length(diff.v)), function(i)
{
DiM.probsOR.res[diff.v[i]][[1]] > DiM.probsOR.res[same.v[i]][[1]]
})
names(diffgreaterthansame) <- diff.v
diffbysame.ID <- which(diffgreaterthansame == FALSE)
if(length(diffbysame.ID) > 0)
{
diffbysame[ diffbysame.ID ][[1]] <- 1 / diffbysame[ diffbysame.ID ][[1]]
}
}
diff.t <- c("Different vs. same means","Different vs. same variances","Different vs. same sets")
same.t <- c("Same vs. different means","Same vs. different variances","Same vs. different sets")
if(length(diffbysame.ID) > 0)
{
diffsame.terms[ diffbysame.ID ] <- same.terms[ diffbysame.ID ]
diffvssame.v[ diffbysame.ID ] <- samevsdiff.v [ diffbysame.ID ]
diff.t[ diffbysame.ID ] <- same.t[ diffbysame.ID ]
}
DiM.probsOR.orig <- c(DiM.integrals.ext, DiM.probsOR.res, diffbysame)
# convert back to normal scale (log, brob) - but not case normal
if(convback==FALSE || method=="normal")
{
cat("\nMethod = ",method,"\nconvback = ",convback,"\n\nProbabilities and Odds Ratios (= OR) will not be\nconverted from log-scale back to non-log-scale.\n\n",sep="")
DiM.probsOR.exp <- NA
} else
{
if(method=="brob")
{
DiM.probsOR.exp <- sapply(DiM.probsOR.orig, as.numeric)
} else if(method=="log")
{
DiM.probsOR.exp <- exp(DiM.probsOR.orig)
}
}
# correct wrong names
names(diffgreaterthansame) <- names(DiM.probsOR.orig)[23:25] <- diffvssame.v
if(!is.na(DiM.probsOR.exp[1])) names(DiM.probsOR.exp) <- names(DiM.probsOR.orig)
# table/ dataframe with resulting probabilities
if(method=="brob")
{
p.H.D1D2I <- brobgiveback(DiM.probsOR.res[1:10])
ORs <- sapply(DiM.probsOR.res[11:16], Brobdingnag:::getX)
ORs <- brobgiveback(DiM.probsOR.res[11:16])
} else if(method %in% c("normal","log"))
{
p.H.D1D2I <- DiM.probsOR.res[1:10]
ORs <- DiM.probsOR.res[11:16]
} else
{
stop("Unknown method=",method)
}
# TRUE for log and brob
cat("\nMethod = ",method,"\nconvback = ",convback,"\n\n",sep="")
tempnams <- c(rep("integrals (unnormalized)",4),
c("cc","denom"),
rep("integrals (normalized)",4),
rep("hypothesis",6),
rep("Odds Ratio",6),
rep("Odds Ratio (diff vs same)",3)
)
# give out table to show conversion log/ brob back to normal
if(method=="brob")
{
proutvalues <-brobgiveback(DiM.probsOR.orig)
} else
{
proutvalues <- DiM.probsOR.orig
}
DiM.probs.res.df <- data.frame("Hypothesis (Probabilities)" = c(
"Same Mean, same Standard Deviation",
"Different Means, same Standard Deviation",
"Same Mean, different Standard Deviations",
"Different Means, different Standard Deviations",
"The Means are the same",
"The Means are different",
"The Standard Deviations are the same",
"The Standard Deviations are different",
"The Data Sets are the same",
"The Data Sets are different"),
"abbrev" = c("C&S","Cbar,S","C,Sbar","Cbar&Sbar","C","Cbar","S","Sbar","C&S","Cbar|Sbar"),
"p(H|D1,D2,I)"=p.H.D1D2I,
check.names=FALSE
)
# table/ dataframe with Odds Ratio results
DiM.OR.res.df <- data.frame("Odds Ratio - Hypothesis in favor of ..." = c(
"A difference in Means",
"The same Means",