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 \sigma^2(v) denote the variance function evaluated at v, the two-component variance function is defined as \sigma^2(v) = a^2 + b^2 * v^{2}, 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

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

# Generate some synthetic data using the two-component error model and use
# different variance functions, also with fixed sigma in order to avoid
# overparameterisation in the case of a constant term in the variance function
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)

# The error model parameters 3 and 0.07 are approximately recovered
intervals(f_varConstProp_sf)