Fit a Calibration Line or Curve
Fit a calibration line or curve based on linear regression.
calibrate(formula, data, test.higher.orders = TRUE, max.order = 4, p.crit = 0.05,
F.test = "partial", weights, subset, na.action, method = "qr", model = FALSE,
x = FALSE, y = FALSE, contrasts = NULL, warn = TRUE, ...)formula |
a |
data |
an optional data frame, list or environment (or object coercible by |
test.higher.orders |
logical scalar indicating whether to start with a model that contains a single predictor
variable and test the fit of higher order polynomials to consider for the calibration
curve ( |
max.order |
integer indicating the maximum order of the polynomial to consider for the
calibration curve. The default value is |
p.crit |
numeric scaler between 0 and 1 indicating the p-value to use for the stepwise regression
when determining which polynomial model to use. The default value is |
F.test |
character string indicating whether to perform the stepwise regression using the
standard partial F-test ( |
weights |
optional vector of observation weights; if supplied, the algorithm fits to minimize the sum of the
weights multiplied into the squared residuals. The length of weights must be the same as
the number of observations. The weights must be nonnegative and it is strongly recommended
that they be strictly positive, since zero weights are ambiguous, compared to use of the
|
subset |
optional expression saying which subset of the rows of the data should be used in the fit. This can be a logical vector (which is replicated to have length equal to the number of observations), or a numeric vector indicating which observation numbers are to be included, or a character vector of the row names to be included. All observations are included by default. |
na.action |
optional function which indicates what should happen when the data contain |
method |
optional method to be used; for fitting, currently only |
model, x, y, qr |
optional logicals. If |
contrasts |
an optional list. See the argument |
warn |
logical scalar indicating whether to issue a warning ( |
... |
additional arguments to be passed to the low level regression fitting functions
(see |
A simple and frequently used calibration model is a straight line where the response variable S denotes the signal of the machine and the predictor variable C denotes the true concentration in the physical sample. The error term is assumed to follow a normal distribution with mean 0. Note that the average value of the signal for a blank (C = 0) is the intercept. Other possible calibration models include higher order polynomial models such as a quadratic or cubic model.
In a typical setup, a small number of samples (e.g., n = 6) with known concentrations are measured and the signal is recorded. A sample with no chemical in it, called a blank, is also measured. (You have to be careful to define exactly what you mean by a “blank.” A blank could mean a container from the lab that has nothing in it but is prepared in a similar fashion to containers with actual samples in them. Or it could mean a field blank: the container was taken out to the field and subjected to the same process that all other containers were subjected to, except a physical sample of soil or water was not placed in the container.) Usually, replicate measures at the same known concentrations are taken. (The term “replicate” must be well defined to distinguish between for example the same physical samples that are measured more than once vs. two different physical samples of the same known concentration.)
The function calibrate initially fits a linear calibration model. If the argument
max.order is greater than 1, calibrate then performs forward stepwise linear
regression to determine the “best” polynomial model.
In the case where replicates are not availble, calibrate uses standard stepwise
ANOVA to compare models (Draper and Smith, 1998, p.335). In this case, if the p-value
for the partial F-test to compare models is greater than or equal to p.crit, then
the model with fewer terms is used as the final model.
In the case where replicates are available, if F.test="lof", then for each model
calibrate computes the p-value of the ANOVA for lack-of-fit vs. pure error
(Draper and Smith, 1998, Chapters 2; see anovaPE). If the p-value is
greater than or equal to p.crit, then this is the final model; otherwise the next
higher-order term is added to the polynomial and the model is re-fit. If, during the
stepwise procedure, the degrees of freedom associated with the residual sums of squares
of a model to be tested is less than or equal to the number of observations minus the
number of unique observations, calibrate uses the partial F-test instead of the
lack-of-fit F-test.
The stepwise algorithm terminates when either the p-value is greater than or equal to
p.crit, or the currently selected model in the algorithm is of order
max.order. The algorithm will terminate earlier than this if the next model to be
fit includes singularities so that not all coefficients can be estimted.
Almost always the process of determining the concentration of a chemical in a soil,
water, or air sample involves using some kind of machine that produces a signal, and
this signal is related to the concentration of the chemical in the physical sample.
The process of relating the machine signal to the concentration of the chemical is
called calibration. Once calibration has been performed, estimated
concentrations in physical samples with unknown concentrations are computed using
inverse regression (see inversePredictCalibrate). The uncertainty
in the process used to estimate the concentration may be quantified with decision,
detection, and quantitation limits.
Steven P. Millard (EnvStats@ProbStatInfo.com)
Draper, N., and H. Smith. (1998). Applied Regression Analysis. Third Edition. John Wiley and Sons, New York, Chapter 3 and p.335.
Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring. Second Edition. John Wiley & Sons, Hoboken. Chapter 6, p. 111.
Helsel, D.R. (2012). Statistics for Censored Environmental Data Using Minitab and R, Second Edition. John Wiley \& Sons, Hoboken, New Jersey. Chapter 3, p. 22.
Millard, S.P., and N.K. Neerchal. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton, FL, pp.562-575.
# The data frame EPA.97.cadmium.111.df contains calibration data for
# cadmium at mass 111 (ng/L) that appeared in Gibbons et al. (1997b)
# and were provided to them by the U.S. EPA.
# Display a plot of these data along with the fitted calibration line
# and 99% non-simultaneous prediction limits. See
# Millard and Neerchal (2001, pp.566-569) for more details on this
# example.
Cadmium <- EPA.97.cadmium.111.df$Cadmium
Spike <- EPA.97.cadmium.111.df$Spike
calibrate.list <- calibrate(Cadmium ~ Spike, data = EPA.97.cadmium.111.df)
newdata <- data.frame(Spike = seq(min(Spike), max(Spike), len = 100))
pred.list <- predict(calibrate.list, newdata = newdata, se.fit = TRUE)
pointwise.list <- pointwise(pred.list, coverage = 0.99, individual = TRUE)
dev.new()
plot(Spike, Cadmium, ylim = c(min(pointwise.list$lower),
max(pointwise.list$upper)), xlab = "True Concentration (ng/L)",
ylab = "Observed Concentration (ng/L)")
abline(calibrate.list, lwd = 2)
lines(newdata$Spike, pointwise.list$lower, lty = 8, lwd = 2)
lines(newdata$Spike, pointwise.list$upper, lty = 8, lwd = 2)
title(paste("Calibration Line and 99% Prediction Limits",
"for US EPA Cadmium 111 Data", sep = "\n"))
#----------
# Clean up
#---------
rm(Cadmium, Spike, newdata, calibrate.list, pred.list, pointwise.list)
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