How to create a (total) Ghrelin calibration curve?

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== Computing Ghrelin calibration curve using R ==
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===This article describes creating calibration curve, and measuring the total-ghrelin concentration ===
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The calibration curve can be created using R, and libracy "drc". Optionally, one can use library "sfmisc" for formatting of the labels on plot axis.
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We will assume a 4-parameter log-logistic model:
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x = {\left( {\frac{{{e^b}\left( {y - d} \right)}}{{c - y}}} \right)^{\frac{1}{b}}}
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[[Image(LL4.png, 300px)]]
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The R code is attached below.
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The example assumes the data to be available in file "ghrelin_conc std_a std_b avg.csv"
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The measured data:
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|| '''Ghrelin (ng/ml)''' || '''Standard a''' || '''Standard b''' ||
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|| 1000000 ||-0.040596823||-0.052699697 ||
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|| 100000 ||0.136105144||0.119766263 ||
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|| 10000 ||0.61356354||0.606906959 ||
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|| 1000 ||0.846543873||0.839887292 ||
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|| 100 ||0.887693646||0.88345764 ||
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||0||0.896770802||0.896165658 ||
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{{{
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##### Install libraries
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install.packages("drc")
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install.packages("sfsmisc")
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require(drc)
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library(sfsmisc)
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##### Read the data
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hormone.data <- read.csv("ghrelin_conc std_a std_b avg.csv")
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hormone.data <- hormone.data[,1:3]
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colnames(hormone.data)[1:3] <- c("Concentration","Response_1", "Response_2")
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##### Reorganize the data
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hormone.data <- reshape(hormone.data, varying=c("Response_1","Response_2"), direction="long", v.names=c("Response"))
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hormone.data <- hormone.data[,c("Concentration", "Response")]
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##### Fitting the model (4-parameter log-logistic function)
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hormone.data.model <- drm(Response ~ Concentration, data = hormone.data, fct = LL.4())
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summary(hormone.data.model)
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}}}
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The resultant parameters of a log-logistic equation are:
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{{{
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Model fitted: Log-logistic (ED50 as parameter) (4 parms)
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Parameter estimates:
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Estimate Std. Error t-value p-value
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b:(Intercept) 9.5057e-01 2.2294e-02 4.2638e+01 0
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c:(Intercept) -7.6010e-02 6.9075e-03 -1.1004e+01 0
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d:(Intercept) 8.9163e-01 3.3216e-03 2.6843e+02 0
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e:(Intercept) 2.5221e+04 7.7727e+02 3.2448e+01 0
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}}}
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The calibration curve can be plotted using the commands below:
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{{{
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##### Plotting a nice plot
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par(pty="s", mar=c(5,5,1,1))
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plot(hormone.data.model, type="confidence", cex.lab=2, axes=F, xlim=c(-10,10^6))
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axis(side=1, at=hormone.data[1:6,1], labels=pretty10exp(hormone.data[1:6,1]), cex.axis=1.2)
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axis(side=2, at=seq(0,1,0.2), labels=seq(0,1,0.2))
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plot(hormone.data.model, type="all", add=T, pch=21, col="red", lwd=1, cex=2, bg="green")
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}}}
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[[Image(Ghrelin.png)]]
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The parameters of the eqution can be plugged into the formula below (an inverse of the model), and used in Excel, or other spreadsheet program.
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[[Image(LL4-inv.png, 230px)]]
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However, the concentration can be also easily estimated in R using "ED" function of the "drc" library. The code below demonstrates the concentration estimated from the response of 0.1, assuming alpha=0.05. The code returns the estimation, the error, and the condfidence interval.
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{{{
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##### Computing the concentration from the response, for instance for a response=0.1, and alpha=1-0.95
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ED(hormone.data.model, respLev=0.1, interval="delta", type="absolute", level=0.95)
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}}}