Table of contents

Notations for truncated distribution for peptides

\(Tr\) is the threshold/truncation, can be measured as cutoff in a standard normalized distribution.

Parameter Case Control
mean \(\mu_{cs}\): mean of cases \(\mu_{cl}\): mean of controls
sampling variance \(\sigma_{cs}\): observed cases after truncation \(\sigma_{cl}\): observed controls after truncation
counts \(n_{cs}\): observed cases after truncation \(n_{cl}\): observed controls after truncation
conditional mean \(\hat{\mu}_{cs}\): truncated mean for \(n_{cs}\) cases \(\hat{\mu}_{cl}\): truncated mean for \(n_{ctrl}\) control

Simulation data

##A toy example for truncated distribution
##Assuming normal distribution
pcut=0.1
cutoff=qnorm(1-pcut)

N_cs=700
N_ctrl=700

u_cs=qnorm(1-0.05)
u_ctrl=qnorm(1-0.2)

TechCS=0.5##measuring error for mean
csSimu=rnorm(n=N_cs, mean=u_cs, sd = TechCS)
TechCtrl=0.7
ctrlSimu=rnorm(n=N_ctrl, mean=u_ctrl, sd = TechCtrl)

layout(matrix(1:2, 2, 1))
plot(density(csSimu), bty="l", xlim=c(-2, 4), main=paste(N_cs, "Case"))
rug(csSimu, side = 1, col="red")
n1=length(which(csSimu>cutoff))
bar_u_cs=mean(csSimu[csSimu>cutoff])
abline(v=c(u_cs, cutoff), lty=c(2), lwd=c(1, 3), col=c("red","black"))
points(bar_u_cs, 0, pch=1, cex=2, col="red")

plot(density(ctrlSimu), bty="l", xlim=c(-2, 4), main=paste(N_ctrl, "Control"))
rug(ctrlSimu, side = 1, col="blue")
n2=length(which(ctrlSimu>cutoff))
bar_u_ctrl=mean(ctrlSimu[ctrlSimu>cutoff])
abline(v=c(u_ctrl, cutoff), lty=c(2), lwd=c(1, 3), col=c("blue","black"))
points(bar_u_ctrl, 0, pch=1, cex=2, col="blue")

Likelihood (for Zhang Peng)

We need a likelihood function here.