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PoissonCalib-7Apr2021.Rmd
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PoissonCalib-7Apr2021.Rmd
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---
title: "All gBlock data"
author: "copyright 2021 Mary Lesperance"
date: '`r Sys.Date()`'
output:
html_document:
number_sections: yes
toc: yes
toc_depth: 3
pdf_document:
toc: yes
toc_depth: 3
word_document:
toc: yes
toc_depth: 3
geometry: margin=.5in
fontsize: 11pt
---
<!-- Turned off numbering for Supplemental File -->
<!-- \pagenumbering{gobble} -->
# Instructions
This file recreates the analyses in the publication,
'A Statistical Model for Calibration and Computation of Detection
and Quantification Limits for Low Copy Number Environmental
DNA samples' by Lesperance, Allison, Bergman, Hocking, Helbing, *eDNA*, 2021.
* Create a folder called Outputs in your working directory.
* Put files in working directory: PoissonCalib-Functions-7April2021.R, GEDWG_LOD_DATA3.csv (Klymus data), AssaySummary-ML.csv (Klymus results)
* Data set csv file requirements. Columns: Target, Lab, Cq, Sq
* For nondetects, set Cq to be empty or NA value
* For negative controls, set Sq to be empty or 0 or NA value
* DO NOT DUPLICATE Target names over different Labs!!
* The script uses observations with nonempty data and where phat <1 (num detect<num technical replicates)
* Only the SQ's up to the first one with phat==1 are used.
* Allows for variable numbers of SQ levels per Target.
* Assumes SQs == NA are zero, i.e. are negative controls.
* Adds in Negative Control (ntc) zeroes (24/(48 for Monroe) technical replicates).
* The code uses the R function optim. A convergence code 0 indicates successful completion.
* Ignore warnings if results are sensible.
```{r Packages, include=FALSE}
# Packages used
packages = c("dplyr","ggplot2","knitr","kableExtra","RColorBrewer")
## Load or install&load
package.check <- lapply(packages,
FUN = function(x) {
if (!require(x, character.only = TRUE)) {
install.packages(x, dependencies = TRUE)
library(x, character.only = TRUE)
}
}
)
## Source the functions
source("PoissonCalib-Functions-7April2021.R")
```
```{r setup, include=FALSE}
#ML See EcoF-PoissonBlank-gBlockAll-17June2020.Rmd
knitr::opts_chunk$set(echo = TRUE, fig.height=7, fig.width=6.5)
knitr::opts_chunk$set(fig.pos = '!h')
sink.indicator <- FALSE #sinks results to files if TRUE
```
```{r READIN, include=FALSE}
## READIN paragraph adapted from Merkes, Christopher <[email protected]>
## Used only to read in the data.
## Read in your data file (MODIFY FILE NAME AS NEEDED):
DAT <- read.csv("GEDWG_LOD_DATA3.csv") #gBlock data
dim(DAT)
## Create an analysis log file:
write(paste0("Analysis started: ",date(),"\n\n"),file="Outputs\\Analysis Log.txt")
## Check the data:
if(sum(colnames(DAT)=="Target")!=1) { #Is there a "Target" column?
A <- grep("target", colnames(DAT), ignore.case=T)
if(length(A)==1) { colnames(DAT)[A] <- "Target" } #Rename target column if it is mispelled but can be identified and there is only 1.
if(length(A)!=1) { write("There is a problem with the 'Target' column.\n\n",file="Analysis Log.txt",append=T) } #Add error message to analysis log.
if(length(A)>1) { cat("ERROR: multiple 'Target' columns detected.",colnames(DAT)[A],sep="\n") }
if(length(A)==0) { print("ERROR: cannot detect 'Target' column.") }
}
if(sum(colnames(DAT)=="Lab")!=1) { #Is there a "Lab" column?
A <- grep("lab", colnames(DAT), ignore.case=T)
if(length(A)==1) { colnames(DAT)[A] <- "Lab" } #Rename Lab column if it is mispelled but can be identified and there is only 1.
if(length(A)!=1) { write("There is a problem with the 'Lab' column.\n\n",file="Analysis Log.txt",append=T) } #Add error message to analysis log.
if(length(A)>1) { cat("ERROR: multiple 'Lab' columns detected.",colnames(DAT)[A],sep="\n") }
if(length(A)==0) { print("ERROR: cannot detect 'Lab' column.") }
}
if(sum(colnames(DAT)=="Cq")!=1) { #Is there a "Cq" column?
A <- grep("cq|ct|cycle",colnames(DAT),ignore.case=T)
if(length(A)==1) { colnames(DAT)[A] <- "Cq" } #Rename cq column if it is mispelled but can be identified and there is only 1.
if(length(A)!=1) { write("There is a problem with the 'Cq' column.\n\n",file="Analysis Log.txt",append=T) } #Add error message to analysis log.
if(length(A)>1) { cat("ERROR: multiple 'Cq' columns detected.",colnames(DAT)[A],sep="\n") }
if(length(A)==0) { print("ERROR: cannot detect 'Cq' column.") }
}
if(sum(colnames(DAT)=="SQ")!=1) { #Is there a "SQ" column?
A <- grep("sq|copies|starting|quantity",colnames(DAT),ignore.case=T)
if(length(A)==1) { colnames(DAT)[A] <- "SQ" } #Rename SQ column if it is mispelled but can be identified and there is only 1.
if(length(A)!=1) { write("There is a problem with the 'SQ' column.\n\n",file="Analysis Log.txt",append=T) } #Add error message to analysis log.
if(length(A)>1) { cat("ERROR: multiple 'SQ' columns detected.",colnames(DAT)[A],sep="\n") }
if(length(A)==0) { print("ERROR: cannot detect 'SQ' column.") }
}
## Ensure data is in the proper format:
DAT$Target <- as.factor(DAT$Target)
DAT$Lab <- as.factor(DAT$Lab) #ML
DAT$Cq <- suppressWarnings(as.numeric(as.character(DAT$Cq))) #Non-numerical values (i.e. negative wells) will be converted to NAs
DAT$SQ <- suppressWarnings(as.numeric(as.character(DAT$SQ))) #Non-numerical values (i.e. NTC) will be converted to NAs
## ML Remove positive controls
dim(DAT)
DAT <- DAT[DAT$Content!='Pos Ctrl',]
dim(DAT)
## ML assume SQs == NA are zero - negative controls
DAT$SQ[is.na(DAT$SQ)] <- 0
dim(DAT)
summary(DAT)
DAT.df <- data.frame(DAT)
## ML Chris omitted negative controls from data set
## ML negative controls will be added to CERC, Monroe, UVIC below
```
# Process/Summarize samples by Lab; Compute the Poisson estimates of SQ
Hindson et al "High-Throughput Droplet Digital PCR System for Absolute
Quantitation of DNA Copy Number", Anal. Chem., 2011, 83 (22), pp 8604–8610
use a Poisson approximation for quantitation.
Before that, Dube et al. 2008, "Mathematical analysis of copy number
variation in a DNA sample
using digital PCR on a nanofluidic device", PloS One, Vol 3, Issue 8, e2876,
model the number of molecules in each chamber as a Poisson process, giving the
relationship between $p$ and $\lambda$.
```{r detect, echo=FALSE}
## Summarize data by (Target, SQ), detect=#detections, n=#tech reps
DAT.Tar.SQ <- DAT.df %>%
group_by(Target, SQ) %>%
summarise(detect=sum(!is.na(Cq)), n=n(), Cqmean=mean(Cq, na.rm=TRUE),
Lab=Lab[1] )
DAT.Tar.SQ <- droplevels(data.frame(DAT.Tar.SQ))
dim(DAT.Tar.SQ); cat('dim, before negative controls added')
summary(DAT.Tar.SQ)
uLabs <- unique(DAT.Tar.SQ$Lab) #unique labs
## ML check the data for each Lab
# if(sink.indicator){
# sink(file='gBlock-SummariesRaw.txt', split=TRUE)}
# for(i in uLabs){
# print(knitr::kable(DAT.Tar.SQ[DAT.Tar.SQ$Lab==i,], format="pandoc",
# digits=3, caption=i), results="asis")
# }
# if(sink.indicator){ sink()}
## All labs had negative controls, some of which were omitted in gBlock file
## If lab has no technical replicates with SQ=0,
## Add in Negative Control (ntc) zeroes (24/(48 for Monroe) technical replicates)
ntc.rows <- tibble(Target=factor(0), SQ=0, detect=0, n=0, Cqmean=0, Lab=factor(0))
for (i in unique(DAT.Tar.SQ$Target)){
nn=24
if ((DAT.Tar.SQ$Lab[DAT.Tar.SQ$Target==i])[1]=="Monroe") nn=48
if(min(DAT.Tar.SQ$SQ[DAT.Tar.SQ$Target==i])!=0){
ntc.rows <- ntc.rows %>% add_row(Target=i,
SQ=0, detect=0, n=nn, Cqmean=NA,
Lab=(DAT.Tar.SQ$Lab[DAT.Tar.SQ$Target==i])[1])
}
}
ntc.rows <- data.frame(ntc.rows[-1,]) #remove the first row
# DAT.Tar.SQ <- DAT.Tar.SQ %>% add_row(ntc.rows) #no longer works
suppressWarnings(DAT.Tar.SQ <- DAT.Tar.SQ %>% bind_rows(ntc.rows))
DAT.Tar.SQ$Target <- as.factor(DAT.Tar.SQ$Target)
DAT.Tar.SQ$Lab <- as.factor(DAT.Tar.SQ$Lab)
DAT.Tar.SQ <- arrange(DAT.Tar.SQ, Lab, Target, SQ) #sort data by SQ in Target in Lab
#write.csv(DAT.Tar.SQ, "DAT.Tar.SQ.csv") #write the data to file
## Add variables to data set: L10.SQ, phat, ...
DAT.Tar.SQ <- within(DAT.Tar.SQ, {
L10.SQ <- log10(SQ)
phat <- detect/n #sample proportion detect
vphat <- phat*(1-phat)/n #var of phat
lamhat <- -log(1-phat)
vlamhat <- phat/n/(1-phat) #var of lamhat using the delta method
sdlamhat <- sqrt(vlamhat) #sd of lamhat using the delta method
MElamhat <- 1.96*sdlamhat #margin of error for lambda hat using delta method
}
)
dim(DAT.Tar.SQ); cat('dim, Added 24/48 negative controls for Targets with zero ntc')
# head(DAT.Tar.SQ)
## All Targets and Labs **DO NOT DUPLICATE Target names over Labs!!
uLabs <- unique(DAT.Tar.SQ$Lab)
uTargets <- unique(DAT.Tar.SQ$Target)
nTargets <- length(uTargets)
uLabsTargets <- unique(DAT.Tar.SQ[,c('Lab','Target')])
uLabsTargets$Lab <- as.character(uLabsTargets$Lab)
#ensure ulabsTargets in same order as uTargets
uLabsTargets <- uLabsTargets[match(uLabsTargets$Target, uTargets),]
uLabsTargets.names <- apply(uLabsTargets, 1, paste, collapse=', ')
## Print summary tables
if(sink.indicator){
sink(file='gBlock-Summaries.txt', split=TRUE)}
for(i in uLabs){
print(knitr::kable(DAT.Tar.SQ[DAT.Tar.SQ$Lab==i, c(1:4, 12)], format="pandoc", digits=3, caption=i),
results="asis")
}
if(sink.indicator){ sink()}
```
```{r ExactTransCI, echo=FALSE}
## Exact 95% Binomial Confidence intervals => backtransform given alpha and beta
## Binomial bound from Julious 2005, Stat in Medicine
DAT.Tar.SQ <- within(DAT.Tar.SQ, {
CIexphat.lower <- 1 - qbeta(.975, n-detect+1, detect) #exact phat bounds
CIexphat.upper <- qbeta(.975, detect+1, n-detect)
## Use transformed exact phat bounds
Lamhatex.Lower <- -log(1 - CIexphat.lower)
Lamhatex.Upper <- -log(1 - CIexphat.upper)
}
)
```
# Plot the Poisson estimates (and CI) of SQ for levels that had non-detects
Only the first levels of SQ that had non-detects are analyzed.
Blue line is least squares linear regression line.
*ML* ??will error if all phats==1
```{r PlotPois, warning=FALSE, echo=FALSE}
#pdf('Outputs\\lamhat.pdf')
#postscript('\\Outputs\\lamhat.eps')
if(nTargets>3) par(mfrow=c(2,2))
if(nTargets==2) par(mfrow=c(1,2))
## Use observations with nonempty data and where phat <1
## Only the SQ's before the first one with phat==1 are used
## Allows for variable numbers of SQ levels per Target
nndetect <- vector("list", nTargets)
nrowTarget <- rep(0, length=nTargets)
for(i in 1:nTargets) {
Target.dat <- subset(DAT.Tar.SQ, Target==uTargets[i])
bSQ <- !is.na(Target.dat$phat)
lastSQ <- as.logical(cumprod(Target.dat$phat!=1 & bSQ))
## removes first observations with SQ with phat=1 and larger SQs
Target.dat <- Target.dat[lastSQ,]
nndetect[i] <- list(Target.dat )
nrowTarget[i] <- nrow(Target.dat)
if(nrow(nndetect[[i]]) < 2) {cat(paste('Too few values for ', uTargets[i])); next}
maxSQ <- max(Target.dat$SQ)
maxlamhat <- max(Target.dat$lamhat)
plot(Target.dat$SQ, Target.dat$lamhat, xlog=TRUE, ylab='Lambda hat',
xlab='Starting copy number',
ylim=c(0, maxlamhat), xlim=c(0, maxSQ), main=uLabsTargets.names[i])
## Transformed Exact CI
arrows(Target.dat$SQ, Target.dat$Lamhatex.Lower, Target.dat$SQ,
Target.dat$Lamhatex.Upper,
length=0.05, angle=90, code=3)
## overlay simple regression line and R-squared
jlm <- lm(lamhat ~ SQ, data=Target.dat)
abline(jlm, col=2)
legend("topleft", paste('lm Rsq=',round(summary(jlm)$r.squared, 2)), bty="n")
cat("\n\n")
}
#dev.off()
par(mfrow=c(1,1))
```
\pagebreak
# Supplemental Material
This file contains the outputs from all 29 data sets from Klymus et al. (2019) whose data are summarized in Table 4 of the manuscript. Results from both no intercept and intercept models are presented.
# Estimate Poisson models - no intercept model
```{r MLEfit0s, warning=FALSE, echo=FALSE, fig.height=4.5, fig.width=6.5}
#pdf('Outputs\\MLfit0.pdf')
#postscript('Outputs\\MLfit.eps')
par(mfrow=c(1,2))
## List of results - 0 in name denotes fits through the origin
Calib.fit.estimates0 <- vector("list", nTargets) #list of fit estimates
names(Calib.fit.estimates0) <- uTargets
Calib.fit.all0 <- vector("list", nTargets) #list of fits
Calib.fit.LLRp0 <- vector("numeric", nTargets)
names(Calib.fit.LLRp0) <- uTargets
Calib.fit.res0 <- matrix(0, nrow=nTargets, ncol=4)
rownames(Calib.fit.res0) <- uTargets
colnames(Calib.fit.res0) <- c("convergence", "LLR", "degf","Pval")
Calib.fit.res0 <- data.frame(Calib.fit.res0)
for(i in 1:nTargets){
if(nrow(nndetect[[i]]) < 3) {Calib.fit.estimates0[[i]] <- NULL
cat(paste('Too few values for ', uTargets[i]), '\n')
next}
Target.dat <- nndetect[[i]]
# Could use following for starting value
# j.glm <- glm(cbind(n-detect, detect)~SQ-1, data=Target.dat, family=binomial(link='log'))
Calib.fit <- optim(par=c(1), fn=CalibOr.LLik, nd=Target.dat$detect,
S=Target.dat$SQ, nn=Target.dat$n,
method="BFGS", control=list(fnscale=-1), gr=CalibOr.dLLik,
hessian=TRUE)
Calib.fit.all0[[i]] <- Calib.fit
## Variance estimates
if(nrow(Target.dat)==2) {
Calib.fit.Var <- matrix(0, 2, 2) #singular hessian for 2 observations
} else {
Calib.fit.Var <- solve(-Calib.fit$hessian)
}
cmat <- cbind(Calib.fit$par, sqrt(diag(Calib.fit.Var)))
cmat <- cbind(cmat, cmat[,1]/cmat[,2])
cmat <- cbind(cmat, 2*pnorm(-cmat[, 3]))
colnames(cmat) <- c("Estimate", "Std.Err", "Z value", "Pr(>z)")
if (nrow(cmat)==1) {rownames(cmat) <- c("beta")
} else rownames(cmat) <- c("alpha","beta")
Calib.fit.estimates0[[i]] <- cmat
## Likelihood ratio statistic and pvalue for goodnes-of-fit of the model
Calib.degf <- nrow(Target.dat) - length(Calib.fit$par)
#SQ=0 do not contribute to the likelihood for no intercept model
bool <- Target.dat$SQ !=0
Calib.LLR <- 2*(Bin.LLik(Target.dat$detect[bool], Target.dat$n[bool])
- Calib.fit$value)
Calib.LLR.pv <- pchisq(Calib.LLR, Calib.degf, lower.tail = FALSE)
Calib.fit.LLRp0[i] <- Calib.LLR.pv
Calib.fit.res0[i,] <- c(Calib.fit$convergence, Calib.LLR, Calib.degf, Calib.LLR.pv)
# if(sink.indicator){
# sink(file(paste("Outputs\\ML",uTargets[i],".txt", sep=""),
# encoding="UTF-8"), split=TRUE)
# }
# cat("\n", as.character(uTargets[i]), "\n")
# cat("Convergence=", Calib.fit$convergence, "\n")
# printCoefmat(cmat, digits=3)
# cat('LLR test stat=', Calib.LLR, ', df= ', Calib.degf, ', p-value=', Calib.LLR.pv, "\n\n")
# if(sink.indicator) sink()
## Compute fitted values for ML model
Calib.fitted <- Calib.fit$par * Target.dat$SQ
# Plot calibration curve on lambda scale
maxSQ <- max(Target.dat$SQ)
maxlamhat <- max(Target.dat$lamhat)
plot(Target.dat$SQ, Target.dat$lamhat, xlog=TRUE,
ylab='Lambda hat', xlab='Starting copy number',
ylim=c(0, maxlamhat), xlim=c(0, maxSQ), las=1,
main=uTargets[i])
abline(0, Calib.fit$par, col=4)
legend("topleft", legend=c('ML fit'), lty=1, col=4, bty="n")
arrows(Target.dat$SQ, Target.dat$Lamhatex.Lower, Target.dat$SQ,
Target.dat$Lamhatex.Upper,
length=0.05, angle=90, code=3)
## Plot calibration curve on phat scale
## Compute minSQ such that phat~=1 (1-phat = .99)
## sqs <- seq(0, maxSQ)
maxSQa <- max( -( log(.01))/Calib.fit$par, maxSQ)
sqs <- seq(0, maxSQa, by=.1)
Calib.phat <- 1 - exp(-( Calib.fit$par * sqs))
plot(sqs, Calib.phat, xlog=TRUE,
ylab='Proportion detect', xlab='Starting copy number', type='l', col=4, las=1,
ylim=c(0, 1), xlim=c(0, maxSQa), main=uTargets[i])
points(Target.dat$SQ, Target.dat$phat)
legend("topleft", legend=c('ML fit'), lty=1, col=4, bty="n")
cat('\n\n\n')
if(sink.indicator){
sink(file(paste("Outputs\\ML",uTargets[i],".txt", sep=""),
encoding="UTF-8"), split=TRUE)
}
cat("\n", as.character(uTargets[i]), "\n")
cat("Convergence=", Calib.fit$convergence, "\n")
printCoefmat(cmat, digits=3)
cat('LLR test stat=', Calib.LLR, ', df= ', Calib.degf, ', p-value=', Calib.LLR.pv, "\n\n")
if(sink.indicator) sink()
knitr::asis_output("\n\\newpage\n")
}
#dev.off()
par(mfrow=c(1,1))
```
\newpage
# Estimate predicted Sq given number detects and technical replicates - no intercept (not shown)
The estimated SQ is easily obtained for given new values of nn0=number of replicates,
nd0=number detected and the estimated slope betaS
as: Shat <- -(log((nn0 - nd0)/nn0)) / betaS[1]
The standard errors are obtained from the Hessian matrix
(or via the function CalibS0Or.ddLLik()) .
```{r MLES0fits0, warning=FALSE, echo=FALSE, eval=FALSE, include=FALSE}
#Calibration estimate of S0 given new nd0=number detected, nn0=number replicates
# includes standard errors and p-values
# Values of (nd0, nn0) such that nd0/nn0 < 1 - exp(-alpha_hat)
CalibS0.fit.estimates0 <- vector("list", nTargets) #list of fits including test value
for(i in 1:nTargets){
nd0 <- 32; nn0 <- 96 #test number detects, number technical replicates
# nd0 <- 3; nn0 <- 8
# nd0 <- 1; nn0 <- 3
# i <- 5 # for manuscript MYPI-6t
if(nrow(nndetect[[i]]) < 3) {CalibS0.fit.estimates0[[i]] <- NULL
cat(paste('Too few values for ', uTargets[i]), '\n')
next}
Target.dat <- nndetect[[i]]
CalibS0.fit <- optim(par=c(1, 3), fn=CalibS0Or.LLik, nd=Target.dat$detect,
S=Target.dat$SQ, nn=Target.dat$n, nd0=nd0, nn0=nn0,
method="BFGS", control=list(fnscale=-1), gr=CalibS0Or.dLLik,
hessian=TRUE)
CalibS0.fit.Var <- solve(-CalibS0.fit$hessian)
betaS <- CalibS0.fit$par
cmat <- cbind(betaS, sqrt(diag(CalibS0.fit.Var)))
cmat <- cbind(cmat, cmat[,1]/cmat[,2])
cmat <- cbind(cmat, 2*pnorm(-cmat[, 3]))
if (nrow(cmat)==2) {rownames(cmat) <- c("beta", "SQ0")
} else rownames(cmat) <- c("alpha","beta", "SQ0")
colnames(cmat) <- c("Estimate", "Std.Err", "Z value", "Pr(>z)")
cat('\n\n','ML estimate of SQ for nd0 number of detects and nn0 replicates', '\n')
if(sink.indicator) sink(file(paste("Outputs\\MLSQ",UTargets[i],".txt", sep=""),
encoding="UTF-8"), split=TRUE)
cat(as.character(uTargets[i]), 'y0=', nd0, 'n0=' , nn0, "\n\n")
printCoefmat(cmat, digits=3)
if(sink.indicator) sink()
CalibS0.fit.estimates0[[i]] <- cmat
}
```
\newpage
# Estimate Poisson models - intercept model
```{r MLEfits, warning=FALSE, echo=FALSE, fig.height=4.5, fig.width=6.5}
#pdf('Outputs\\MLfit.pdf')
#postscript('Outputs\\MLfit.eps')
# if(nTargets>3) par(mfrow=c(2,2))
# if(nTargets==2) par(mfrow=c(1,2))
par(mfrow=c(1,2))
# List of results
Calib.fit.estimates <- vector("list", nTargets) #list of fit estimates
names(Calib.fit.estimates) <- uTargets
Calib.fit.all <- vector("list", nTargets) #list of fits all
Calib.fit.LLRp <- vector("numeric", nTargets)
names(Calib.fit.LLRp) <- uTargets
Calib.fit.res <- matrix(0, nrow=nTargets, ncol=4)
rownames(Calib.fit.res) <- uTargets
colnames(Calib.fit.res) <- c("convergence", "LLR", "degf","Pval")
Calib.fit.res <- data.frame(Calib.fit.res)
for(i in 1:nTargets){
if(nrow(nndetect[[i]]) < 3) {Calib.fit.estimates[[i]] <- NULL
cat(paste('Too few values for ', uTargets[i]), '\n')
next}
Target.dat <- nndetect[[i]]
jlm <- lm(lamhat ~ SQ, data=Target.dat) #starting values for alpha and beta
Calib.fit <- optim(par=pmax(c(0.01, 0.01), coef(jlm)), fn=Calib.LLik,
nd=Target.dat$detect,
S=Target.dat$SQ, nn=Target.dat$n, gr=Calib.dLLik,
method="BFGS", control=list(fnscale=-1), hessian=TRUE)
Calib.fit.all[[i]] <- Calib.fit
if(nrow(Target.dat)==2) {
Calib.fit.Var <- matrix(0, 2, 2) #singular hessian for 2 observations
} else {
Calib.fit.Var <- solve(-Calib.fit$hessian)
}
cmat <- cbind(Calib.fit$par, sqrt(diag(Calib.fit.Var)))
cmat <- cbind(cmat, cmat[,1]/cmat[,2])
cmat <- cbind(cmat, 2*pnorm(-cmat[, 3]))
colnames(cmat) <- c("Estimate", "Std.Err", "Z value", "Pr(>z)")
if (nrow(cmat)==1) {rownames(cmat) <- c("beta")
} else rownames(cmat) <- c("alpha","beta")
Calib.fit.estimates[[i]] <- cmat
#Likelihood ratio statistic and pvalue for goodnes-of-fit of the model
Calib.degf <- nrow(Target.dat) - length(Calib.fit$par)
Calib.LLR <- 2*(Bin.LLik(Target.dat$detect, Target.dat$n)
- Calib.fit$value)
Calib.LLR.pv <- pchisq(Calib.LLR, Calib.degf, lower.tail = FALSE)
Calib.fit.LLRp[i] <- Calib.LLR.pv
Calib.fit.res[i,] <- c(Calib.fit$convergence, Calib.LLR, Calib.degf, Calib.LLR.pv)
# if(sink.indicator){
# sink(file(paste("Outputs\\ML",uTargets[i],".txt", sep=""),
# encoding="UTF-8"), split=TRUE)
# }
# cat("\n", as.character(uTargets[i]), "\n")
# cat("Convergence=", Calib.fit$convergence, "\n")
# printCoefmat(cmat, digits=3)
# cat('LLR test stat=', Calib.LLR, ', df= ', Calib.degf, ', p-value=', Calib.LLR.pv, "\n\n")
# if(sink.indicator) sink()
#Compute fitted values for ML model
Calib.fitted <- Calib.fit$par[1] + Calib.fit$par[2]* Target.dat$SQ
# Plot calibration curve on lambda scale
maxSQ <- max(Target.dat$SQ)
maxlamhat <- max(Target.dat$lamhat)
plot(Target.dat$SQ, Target.dat$lamhat, xlog=TRUE,
ylab='Lambda hat', xlab='Starting copy number',
ylim=c(0, maxlamhat), xlim=c(0, maxSQ), las=1,
main=uTargets[i])
abline(Calib.fit$par[1], Calib.fit$par[2], col=4)
legend("topleft", legend=c('ML fit'), lty=1, col=4, bty="n")
arrows(Target.dat$SQ, Target.dat$Lamhatex.Lower, Target.dat$SQ,
Target.dat$Lamhatex.Upper,
length=0.05, angle=90, code=3)
# Plot calibration curve on phat scale
# Compute minSQ such that phat~=1
maxSQa <-max( -(Calib.fit$par[1] + log(.01))/Calib.fit$par[2], maxSQ)
sqs <- seq(0, maxSQa, by=.1)
Calib.phat <- 1 - exp(-(Calib.fit$par[1] + Calib.fit$par[2] * sqs))
plot(sqs, Calib.phat, xlog=TRUE,
ylab='Proportion detect', xlab='Starting copy number', type='l', col=4, las=1,
ylim=c(0, 1), xlim=c(0, maxSQa), main=uTargets[i])
points(Target.dat$SQ, Target.dat$phat)
legend("topleft", legend=c('ML fit'), lty=1, col=4, bty="n")
cat("\n\n\n")
if(sink.indicator){
sink(file(paste("Outputs\\ML",uTargets[i],".txt", sep=""),
encoding="UTF-8"), split=TRUE)
}
cat("\n", as.character(uTargets[i]), "\n")
cat("Convergence=", Calib.fit$convergence, "\n")
printCoefmat(cmat, digits=3)
cat('LLR test stat=', Calib.LLR, ', df= ', Calib.degf, ', p-value=', Calib.LLR.pv, "\n\n")
if(sink.indicator) sink()
knitr::asis_output("\n\\newpage\n")
}
#dev.off()
par(mfrow=c(1,1))
```
\newpage
# Estimate predicted Sq given number detects and technical replicates - intercept model (not shown)
```{r MLES0fits, warning=FALSE, echo=FALSE, eval=FALSE, include=FALSE}
#Calibration estimate of S0 given new nd0=number detected, nn0=number replicates
# includes standard errors and p-values
# Values of (nd0, nn0) such that nd0/nn0 < 1 - exp(-alpha_hat)
CalibS0.fit.estimates <- vector("list", nTargets) #list of fits including test value
for(i in 1:nTargets){
nd0 <- 32; nn0 <- 96 #test number detects, number technical replicates
if(nrow(nndetect[[i]]) < 3) {CalibS0.fit.estimates[[i]] <- NULL
cat(paste('Too few values for ', uTargets[i]), '\n')
next}
Target.dat <- nndetect[[i]]
jlm <- lm(lamhat ~ SQ, data=Target.dat) #starting values for alpha and beta
jpar <- pmax(c(0.01, 0.01), coef(jlm))
Stilde <- -(log((nn0 - nd0)/nn0) + jpar[1]) / jpar[2]
CalibS0.fit <- optim(par=c(jpar, Stilde),
fn=CalibS0.LLik, nd=Target.dat$detect,
S=Target.dat$SQ, nn=Target.dat$n, nd0=nd0, nn0=nn0,
control=list(fnscale=-1), method="BFGS",
gr=CalibS0.dLLik, hessian=TRUE)
CalibS0.fit.Var <- solve(-CalibS0.fit$hessian)
betaS <- CalibS0.fit$par
hess <- CalibS0.ddLLik(betaS, Target.dat$detect, Target.dat$SQ,
Target.dat$n, nd0, nn0)
cmat <- cbind(betaS, sqrt(diag(CalibS0.fit.Var)))
cmat <- cbind(cmat, cmat[,1]/cmat[,2])
cmat <- cbind(cmat, 2*pnorm(-cmat[, 3]))
if (nrow(cmat)==2) {rownames(cmat) <- c("beta", "SQ0")
} else rownames(cmat) <- c("alpha","beta", "SQ0")
colnames(cmat) <- c("Estimate", "Std.Err", "Z value", "Pr(>z)")
cat('\n\n','ML estimate of SQ for nd0 number of detects and nn0 replicates', '\n')
if(sink.indicator) sink(file(paste("Outputs\\MLSQ",UTargets[i],".txt", sep=""),
encoding="UTF-8"), split=TRUE)
cat(as.character(uTargets[i]), 'y0=', nd0, 'n0=' , nn0, "\n\n")
printCoefmat(cmat, digits=3)
if(sink.indicator) sink()
# Shat <- -(log((nn0 - nd0)/nn0) + betaS[1]) / betaS[2]
# cat('\n', 'Shat=', Shat)
# cat('\n', 'SEs ', sqrt(diag(solve(-hess))))
# cat('\n', 'Marginal ', (1/sqrt(-hess[3,3])))
# cat('\n', 'Gradient ', CalibS0.dLLik(betaS, Target.dat$detect, Target.dat$SQ,
# Target.dat$n, nd0, nn0), '\n')
CalibS0.fit.estimates[[i]] <- cmat
}
```
\newpage
# Determine Lc, Ld, Lq - no intercept model
Follows Lavagnini and Magno 2007, Mass Spectrometry Reviews
*The notation in the paper was changed from the Lavagnini 2007 paper and is shown in brackets here.*
- Lc (*LOB Limit of blank*) = critical level is the assay signal above which a response is reliably attributed to the presence of analyte
- Ld (*Ld = expected number detects out of NN replicates at concentration LOD*) = signal corresponding to an analyte concentration xd (*=LOD Limit of Detection*) level which may be a priori expected to be recognized
- Lq = quantification limit is a signal with a precision which satisfies an expected value ($=\gamma_Q$)
Lc corresponds to a critical response level or a false positive rate,
i.e. critical number of detects given NN replicates,
above which we would reject the null hypothesis that the concentration/copy number is zero at
alpha = alphaLc ($=\gamma_{FP}$). It is the critical response level corresponding to the false positive rate
of alphaLc. Essentially, the test is positive if the Y~Binomial(m, p) > Lc.
The False Positive Rate is P(Y > Lc | S=0).
Ld is computed to correspond to the false negative rate, beta = betaLd ($=\gamma_{FN}$) here. It is computed so
that the probability of observing a new (unknown concentration) response less than or equal to
Lc is less than or equal to betaLd. The probability of observing Lc or less detects if the
concentration is xd (*=LOD Limit of Detection*) or more is less than or equal to betaLd. The values of Lc depend on
the number of replicates, NN, so xd does as well. Ld is the expected number of detects at
values xd and NN replicates. False negative rate Ld computation: P(Y <= Lc | p_xd) <= betaLd,
and solve for xd.
Lq is less well defined. The literature suggests using Lq = beta0 + 10 s.e.(beta0), but this
uses the normality assumption. Other literature suggests using the "analyte concentration xq (*=LOQ Limit of Quantification*)
for which the experimental relative standard deviation of the responses reaches a fixed level ($=\gamma_Q$),
for example, the level 0.1." Lavagnini and Magno 2007. I interpret the term "relative
standard deviation" to mean the coefficient of variation, CV = sd/mean.
In the exercise below, we use the fits from the ML models to estimate the Lc, Ld and Lq, for
various values of NN replicates for a new observation, i.e. a new (unknown concentration) response
number of detects.
```{r LcLdLqNN0, echo=FALSE}
# No intercept model computations - Here Lc==0
# Lc computation: P(Y > Lc | S=0) <= alphaLc
# where Y ~ Bin(m, p=1 - exp(-betas[1]))
# to incorporate estimation uncertaintly in betas[1], use upper limit of conf int
# i.e. Y ~ Bin(m, p=1 - exp(-(betas[1] + 1.96 * s.e.(betas[1])))
# Ld computation: P(Y <= Lc | p_xd) <= betaLd
# use relationship between Binomial and Beta distribution, p 278 Bain
# Lq computation: (Forootan 2017) choose level, Sj, such that CV<=gammaLq
# not clear which scale: response, back-transformed values, ... ?
#
##* Note in R: The quantile is defined as the smallest value x such
## that F(x) ≥ p, where F is the distribution function.
## Set values for alphaLC, betaLd, gammaLq
alphaLc <- betaLd <- .05; gammaLq <- .20
NN <- c(3, 8,16,24,32,48,64,96) #test number of replicates
#set up tables for manuscript output
Lc.all0 <- matrix(0, nrow=nTargets, ncol=length(NN))
row.names(Lc.all0) <- uTargets
colnames(Lc.all0) <- paste(NN)
xdd.all0 <- xd.all0 <- xd_upper.all0 <- xd_lower.all0 <- xq.all0 <- xq_upper.all0 <-
xq_lower.all0 <- Lc.all0
for(i in 1:nTargets){
#use beta estimated from fits above
if(nrow(nndetect[[i]]) < 3) {
cat(paste('Too few values for ', uTargets[i]), '\n')
next}
betas <- (Calib.fit.estimates0[[i]])[,1]
#ML Takes into account uncertainty in new observation and s.e. of betas
betas.upper <- betas + 1.96 * (Calib.fit.estimates0[[i]])[,2]
betas.lower <- betas - 1.96 * (Calib.fit.estimates0[[i]])[,2]
#Want: P(Y > Lc | S=0) <= alphaLc where Y ~ Bin(m, p=1 - exp(-betas[1]))
#Lc at xc=0 values for new observation
# For no intercept model, P(Y = 0 | S=0)=1 and P(i'th tech rep detect| S=0)=0
# Lc==0, and the P(Y > Lc | S=0) <= alphaLc
# since P(Y > 0 | S=0)==0.
# We are saying that sample is negative if Y=0 and positive if Y>0
Lc <- rep(0, length(NN))
#Want xd, P(Y <= Lc | p_xd) <= betaLd
#Ld and xd calculation
pxd <- 1 - qbeta(betaLd, NN-Lc, Lc+1) #proportion detected
Ld <- NN * pxd
xd <- ( - log(1 - pxd)) / betas #concentration
xdlower <- ( - log(1 - pxd)) / betas.upper
xdupper <- ( - log(1 - pxd)) / betas.lower
names(pxd) <- names(xd) <- names(xdlower) <- names(xdupper) <- paste(NN)
#Compute confidence interval for xd
xd.all0[i,] <- xd
xd_upper.all0[i,] <- xdupper
xd_lower.all0[i,] <- xdlower
#Compute model based xq
xq <- -(log(1 - 1/(1 + gammaLq^2*NN)))/betas
names(xq) <- paste(NN)
#Compute model based xq using lower estimates of beta - upper bound for xq
xq_lower <- -(log(1 - 1/(1 + gammaLq^2*NN)))/betas.lower
names(xq_lower) <- paste(NN)
#Compute model based xq using upper estimates of beta
xq_upper <- -(log(1 - 1/(1 + gammaLq^2*NN)))/betas.upper
names(xq_upper) <- paste(NN)
xq.all0[i,] <- xq
xq_lower.all0[i,] <- xq_lower
xq_upper.all0[i,] <- xq_upper
}
#For a given p_hat, one can compute the sample size m required to attain a
# CV <= gammaLq, as m >= (1-phat)/(phat*gammaLq)
```
# Determine Lc, Ld, Lq - intercept model
```{r LcLdLqNN, echo=FALSE}
# Lc computation: P(Y > Lc | S=0) <= alphaLc - intercept model computations
# where Y ~ Bin(m, p=1 - exp(-betas[1]))
# to incorporate estimation uncertaintly in betas[1], use upper limit of conf int
# i.e. Y ~ Bin(m, p=1 - exp(-(betas[1] + 1.96 * s.e.(betas[1])))
# Ld computation: P(Y <= Lc | p_xd) <= betaLd
# use relationship between Binomial and Beta distribution, p 278 Bain
# Lq computation: (Forootan 2017) choose level, Sj, such that CV<=gammaLq
# not clear which scale: response, back-transformed values, ... ?
#
##* Note in R: The quantile is defined as the smallest value x such
## that F(x) ≥ p, where F is the distribution function.
## Set values for alphaLC, betaLd, gammaLq
alphaLc <- betaLd <- .05; gammaLq <- .20
NN <- c(3, 8,16,24,32,48,64,96) #test number of replicates
#set up tables for manuscript output
Lc.all <- matrix(0, nrow=nTargets, ncol=length(NN))
row.names(Lc.all) <- uTargets
colnames(Lc.all) <- paste(NN)
Lc.upper.all <- xd.all <- xd_upper.all <- xd_lower.all <-
xq.all <- xq_lower.all <- xq_upper.all <- Lc.all
for(i in 1:nTargets){
#use beta estimated from fits above
if(nrow(nndetect[[i]]) < 3) {
cat(paste('Too few values for ', uTargets[i]), '\n')
next}
betas <- (Calib.fit.estimates[[i]])[,1]
#ML Takes into account uncertainty in new observation and s.e. of betas
betas.upper <- betas + 1.96 * (Calib.fit.estimates[[i]])[,2]
betas.lower <- pmax(0, betas - 1.96 * (Calib.fit.estimates[[i]])[,2])
#Want: P(Y > Lc | S=0) <= alphaLc where Y ~ Bin(m, p=1 - exp(-betas[1]))
#Lc at xc=0 values for new observation
p.new <- 1 - exp(-betas[1])
Lc.new <- qbinom(1 - alphaLc, size=NN, prob=p.new)
names(Lc.new) <- paste(NN)
Lc.all[i,] <- Lc.new
#Lc.upper at xc=0 values for new observation, incl s.e. of betas
p.upper <- 1 - exp(-betas.upper[1])
Lc.upper <- qbinom(1 - alphaLc, size=NN, prob=p.upper)
names(Lc.upper) <- paste(NN)
Lc.upper.all[i,] <- Lc.upper
#Want xd, P(Y <= Lc | p_xd) <= betaLd
#Ld and xd calculation
pxd <- 1 - qbeta(betaLd, NN-Lc.new, Lc.new+1) #proportion detected
Ld <- NN * pxd
xd <- (-betas[1] - log(1 - pxd)) / betas[2] #concentration
names(pxd) <- names(xd) <- paste(NN)
# pxd_upper <- 1 - qbeta(betaLd, NN-Lc.upper, Lc.upper+1) #proportion detected\
# Ld_upper <- NN * pxd_upper
# xd_upper <- (-betas[1] - log(1 - pxd_upper)) / betas[2] #concentration
xd_lower <- pmax(0, (-betas.upper[1] - log(1 - pxd)) / betas.upper[2])
xd_upper <- (-betas.lower[1] - log(1 - pxd)) / betas.lower[2]
names(xd_upper) <- names(xd_lower) <- paste(NN)
xd.all[i,] <- xd
xd_upper.all[i,] <- xd_upper
xd_lower.all[i,] <- xd_lower
#Compute model based xq
xq <- -(betas[1] + log(1 - 1/(1 + gammaLq^2*NN)))/betas[2]
names(xq) <- paste(NN)
#Compute model based xq using lower estimates of beta
xq_lower <- -(betas.lower[1] + log(1 - 1/(1 + gammaLq^2*NN)))/betas.lower[2]
names(xq_lower) <- paste(NN)
#Compute model based xq using upper estimates of beta
xq_upper <- -(betas.upper[1] + log(1 - 1/(1 + gammaLq^2*NN)))/betas.upper[2]
names(xq_upper) <- paste(NN)
xq.all[i,] <- xq
xq_lower.all[i,] <- xq_lower
xq_upper.all[i,] <- xq_upper
}
#For a given p_hat, one can compute the sample size m required to attain a
# CV <= gammaLq, as m >= (1-phat)/(phat*gammaLq)
```
\pagebreak
# Estimates, Lc, Ld, Lq for a given number of technical reps NN[NNi]
Choose the model (intercept versus no intercept) with the best LLR test fit.
```{r ChooseModel, echo=FALSE}
# Set the index into NN which is defined in chunk LcLdLqNN0 and LcLdLqNN
# NNi <- 2 corresponds to the 2nd entry of NN
NNi <- 2
#cat('Limits intercept model for N=', NN[NNi])
xdxq.all <- cbind(Lc=Lc.all[,NNi],
# LcUp=Lc.upper.all[,NNi],
SdLow=xd_lower.all[,NNi],
Sd=xd.all[,NNi], SdUp=xd_upper.all[,NNi],
SqLow=xq_upper.all[,NNi],
Sq=xq.all[,NNi], SqUp=xq_lower.all[,NNi])
#cat('Limits for no intercept model for N=', NN[NNi])
xdxq.all0 <- cbind(Lc=0, SdLow=xd_lower.all0[,NNi],
Sd=xd.all0[,NNi], SdUp=xd_upper.all0[,NNi],
SqLow=xq_upper.all0[,NNi],
Sq=xq.all0[,NNi], SqUp=xq_lower.all0[,NNi])
#Include alpha and beta estimates in table
alphabeta.se <- alphabeta0.se <- matrix(0, nrow=nTargets, ncol=4)
colnames(alphabeta.se) <- c("alpha","aSE", "beta", "bSE")
colnames(alphabeta0.se) <- c("alpha","aSE", "beta", "bSE")
rownames(alphabeta0.se) <- uTargets
for (i in 1:nTargets){
if(nrowTarget[i]>2){
alphabeta.se[i,1:2] <- Calib.fit.estimates[[i]][1, 1:2]
alphabeta.se[i, 3:4] <- Calib.fit.estimates[[i]][2, 1:2]
alphabeta0.se[i, 3:4] <- Calib.fit.estimates0[[i]][1, 1:2]
}
}
Calib.choice <- Calib.fit.LLRp > Calib.fit.LLRp0
xdxq.choice <- cbind(InterModel=Calib.choice, alphabeta0.se, xdxq.all0)
xdxq.choice[Calib.choice, 6:12] <- xdxq.all[Calib.choice,]
xdxq.choice[Calib.choice, 2:5] <- alphabeta.se[Calib.choice,]
# NNi <- 2 This is set above
cat('Limits for best choice model for N=', NN[NNi], '\n')
print(round(xdxq.choice[nrowTarget>2,], digits=2))
```
\pagebreak
# Tables and Graphs for the manuscript
Include MYPI-6t and eASMO9, eFISH1, eONKI4 in paper.
Index into uTargets (5, 25, 27, 31)
```{r Manuscript, echo=FALSE, eval=TRUE, include=TRUE}
# Include MYPI-6t and eASMO9, eFISH1, eONKI4 in paper. (5, 25, 27, 31)
ManuTargets <- c(5, 25, 27, 31)
Manusink <- FALSE #writes to output files when set to TRUE
#Manusink <- TRUE
if(Manusink){
sink(file(paste("Outputs\\gBlockLimits.txt", sep=""),
encoding="UTF-8"), split=TRUE)
}
xdxq.choice.PR <- xdxq.choice[nrowTarget>2, -1]
xdxq.choice.PR <- xdxq.choice.PR[order(row.names(xdxq.choice.PR)), ]
colnames(xdxq.choice.PR)[6:11] <- c('LODL','LOD','LODU','LOQL','LOQ','LOQU')
Klymus <- read.csv('AssaySummary-ML.csv')
#xdxq.choice.PR <- cbind(Assay=Klymus[,1], xdxq.choice.PR,
xdxq.choice.PR <- cbind(xdxq.choice.PR,
K.LOD8=Klymus[,c(12)], K.LOQ=Klymus[,c(7)])
#write.csv(xdxq.choice.PR, 'Table3.csv')
print(knitr::kable(round(xdxq.choice.PR,
digits=1),
format="pandoc", digits=1,
# caption='Estimates, Critical, Detection, Quantification Limits and CIs',
results="asis"))
if(Manusink){ sink()}
if(Manusink){
sink(file("Outputs\\gBlockData.txt", encoding="UTF-8"),
split=TRUE)
}
for(Targeti in ManuTargets){
jmat <- nndetect[[Targeti]][, c(2:4, 12, 10)]
row.names(jmat) <- NULL
names(jmat) <- c("S", "num.detect", "n", "p.tilde", "lambda.tilde")
print(knitr::kable(jmat,
format="pandoc", digits=3, caption=uTargets[Targeti]),
results="asis")
}
if(Manusink){ sink()}
if(Manusink){
sink(file(paste("Outputs\\gBlockResults.txt", sep=""),
encoding="UTF-8"), split=TRUE)
}
# Print regression outputs
for(Targeti in ManuTargets){
if (Calib.choice[Targeti]){
cat("\n", as.character(uTargets[Targeti]), "\n")