Constant Plus Proportion Variance Function
Description
This function is a constructor for the varConstProp class,
representing a variance function structure corresponding to
a two-component error model (additive and proportional error). Letting
v denote the variance covariate and σ2(v)
denote the variance function evaluated at v, the two-component variance
function is defined as
σ2(v)=a2+b2∗v2, where a is
the additive component and b is the relative error component. In order
to avoid overparameterisation of the model, it is recommended to use
the possibility to fix sigma, preferably to a value of 1 (see examples).
Usage
varConstProp(const, prop, form, fixed)
Arguments
const, prop
|
optional numeric vectors, or lists of numeric
values, with, respectively, the coefficients for the constant
and the proportional error terms. Both arguments must have length one,
unless a grouping factor is specified in form. If either argument
has length greater than one, it must have names which identify its
elements to the levels of the grouping factor defined in
form. If a grouping factor is present in
form and the argument has length one, its value will be
assigned to all grouping levels. Only positive values are allowed
for const. Default is 0.1 for both const and
prop.
|
form |
an optional one-sided formula of the form ~ v, or
~ v | g, specifying a variance covariate v and,
optionally, a grouping factor g for the coefficients. The
variance covariate must evaluate to a numeric vector and may involve
expressions using ".", representing a fitted model object
from which fitted values (fitted(.)) and residuals
(resid(.)) can be extracted (this allows the variance
covariate to be updated during the optimization of an object
function). When a grouping factor is present in form,
a different coefficient value is used for each of its
levels. Several grouping variables may be
simultaneously specified, separated by the * operator, as
in ~ v | g1 * g2 * g3. In this case, the levels of each
grouping variable are pasted together and the resulting factor is
used to group the observations. Defaults to ~ fitted(.)
representing a variance covariate given by the fitted values of a
fitted model object and no grouping factor.
|
fixed |
an optional list with components const and/or
power, consisting of numeric vectors, or lists of numeric
values, specifying the values at which some or all of the
coefficients in the variance function should be fixed. If a grouping
factor is specified in form, the components of fixed
must have names identifying which coefficients are to be
fixed. Coefficients included in fixed are not allowed to vary
during the optimization of an objective function. Defaults to
NULL, corresponding to no fixed coefficients.
|
Value
a varConstProp object representing a constant plus proportion variance
function structure, also inheriting from class varFunc.
Note
The error model underlying this variance function structure can be understood
to result from two uncorrelated sequences of standardized random variables
(Lavielle(2015), p. 55) and has been proposed for use in analytical chemistry
(Werner et al. (1978), Wilson et al. (2004)) and chemical degradation
kinetics (Ranke and Meinecke (2019)). Note that the two-component error
model proposed by Rocke and Lorenzato (1995) assumed a log-normal
distribution of residuals at high absolute values, which is not
compatible with the varFunc structures in package nlme.
Author(s)
José Pinheiro and Douglas Bates (for varConstPower) and
Johannes Ranke (adaptation to varConstProp()).
References
Lavielle, M. (2015)
Mixed Effects Models for the Population Approach: Models, Tasks,
Methods and Tools, Chapman and Hall/CRC.
doi:10.1201/b17203
Pinheiro, J.C., and Bates, D.M. (2000)
Mixed-Effects Models in S and S-PLUS, Springer.
doi:10.1007/b98882
Ranke, J., and Meinecke, S. (2019)
Error Models for the Kinetic Evaluation of Chemical Degradation Data.
Environments 6(12), 124
doi:10.3390/environments6120124
Rocke, David M. and Lorenzato, Stefan (1995)
A Two-Component Model for Measurement Error in Analytical Chemistry.
Technometrics 37(2), 176–184.
doi:10.1080/00401706.1995.10484302
Werner, Mario, Brooks, Samuel H., and Knott, Lancaster B. (1978)
Additive, Multiplicative, and Mixed Analytical Errors.
Clinical Chemistry 24(11), 1895–1898.
doi:10.1093/clinchem/24.11.1895
Wilson, M.D., Rocke, D.M., Durbin, B. and Kahn, H.D (2004)
Detection Limits and Goodness-of-Fit Measures for the Two-Component Model
of Chemical Analytical Error.
Analytica Chimica Acta 2004, 509, 197–208
doi:10.1016/j.aca.2003.12.047
See Also
varClasses,
varWeights.varFunc,
coef.varFunc
Examples
times <- c(0, 1, 3, 7, 14, 28, 56, 120)
pred <- 100 * exp(- 0.03 * times)
sd_pred <- sqrt(3^2 + 0.07^2 * pred^2)
n_replicates <- 2
set.seed(123456)
syn_data <- data.frame(
time = rep(times, each = n_replicates),
value = rnorm(length(times) * n_replicates,
rep(pred, each = n_replicates),
rep(sd_pred, each = n_replicates)))
syn_data$value <- ifelse(syn_data$value < 0, NA, syn_data$value)
f_const <- gnls(value ~ SSasymp(time, 0, parent_0, lrc),
data = syn_data, na.action = na.omit,
start = list(parent_0 = 100, lrc = -3))
f_varPower <- gnls(value ~ SSasymp(time, 0, parent_0, lrc),
data = syn_data, na.action = na.omit,
start = list(parent_0 = 100, lrc = -3),
weights = varPower())
f_varConstPower <- gnls(value ~ SSasymp(time, 0, parent_0, lrc),
data = syn_data, na.action = na.omit,
start = list(parent_0 = 100, lrc = -3),
weights = varConstPower())
f_varConstPower_sf <- gnls(value ~ SSasymp(time, 0, parent_0, lrc),
data = syn_data, na.action = na.omit,
control = list(sigma = 1),
start = list(parent_0 = 100, lrc = -3),
weights = varConstPower())
f_varConstProp <- gnls(value ~ SSasymp(time, 0, parent_0, lrc),
data = syn_data, na.action = na.omit,
start = list(parent_0 = 100, lrc = -3),
weights = varConstProp())
f_varConstProp_sf <- gnls(value ~ SSasymp(time, 0, parent_0, lrc),
data = syn_data, na.action = na.omit,
start = list(parent_0 = 100, lrc = -3),
control = list(sigma = 1),
weights = varConstProp())
AIC(f_const, f_varPower, f_varConstPower, f_varConstPower_sf,
f_varConstProp, f_varConstProp_sf)
intervals(f_varConstProp_sf)