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.