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Error calculation

Source: vignettes/error-calculation.Rmd
error-calculation.Rmd

Introduction

In the parameter identification (PI) process, measuring the discrepancy between simulated results and observed data is crucial. The ospsuite.parameteridentification package offers two primary error models to quantify this difference: the least squares error (lsq) and the M3 method (m3) for handling data below the quantification limit (BQL).

Error Models

LSQ - Least Squares Error

The lsq method is a traditional approach that minimizes the sum of the squared differences between observed and simulated values. This method is widely used in pharmacokinetic modeling, offering a straightforward and intuitive error calculation. By default, selecting this method will transform BQL data by replacing it with LLOQ/2.

M3 - Extended Least Squares Error

The m3 method extends the least squares error by incorporating the likelihood of observed data being below the quantification limit. This method treats BQL data as censored, applying maximum likelihood estimation to include all data in the model fit, thus offering a more accurate and statistically robust approach to handling BQL observations.

Configuring Error Models in PI

To apply an error model, set the targetFunctionType attribute in objectFunctionOptions of the PIConfiguration object:

piConfiguration <- PIConfiguration$new()
piConfiguration$objectiveFunctionOptions$objectiveFunctionType <- "lsq" # or "m3"

The error calculation between simulated results and observer datasets within each PIOutputMapping directly affects the optimization process, aggregating errors across all mappings.

Advanced Error Model Customization

Residual Weighting

PIConfiguration offers further customization for error calculation through the residualWeightingMethod. This option specifies how residuals are weighted. These methods are advantageous when observed data comes from diverse populations, when datasets comprise different observation counts, or when error of dependent variable is measured. Available options include:

  • none: (default): No residual weighting.
  • std: Useful when variability in observed data points is high, as it normalizes residuals by the standard deviation, mitigating the impact of outliers.
  • mean: Beneficial for datasets with significant differences in observation magnitudes, scaling residuals by the mean to ensure equal contribution across data points.
  • error: Ideal when individual data points come with their variance estimates, allowing for weighting by the inverse of these variances, which can provide a more tailored fit given the know error.

To apply residual weighting, set the residualWeightingMethod attribute in objectFunctionOptions of the PIConfiguration object:

piConfiguration$objectiveFunctionOptions$residualWeightingMethod <- "std" # or "mean" or "error"

Robust Residual Calculation

For enhanced flexibility in handling outliers, ´PIConfiguration´ allows specifying a robust method for residual adjustment. Implementing a robustMethod for residuals directly addresses outlier influence, refining model robustness and ensuring more reliable parameter identification outcomes. Available options include:

  • none (default): No robust adjustments, treating all residuals equally.
  • huber: Minimizes the impact of outliers by combining squared and absolute values, making it less sensitive to extreme deviations.
  • bisquare: Further reduces the influence of outliers by applying a weighting scheme that diminishes the role of residuals as they move away from the median.

To apply robust residual calculation, set the robustMethod attribute in objectFunctionOptions of the PIConfiguration object:

piConfiguration$objectiveFunctionOptions$robustMethod <- "huber" # or "bisquare"

Scaling Impact

The PIOutputMapping$scaling attribute, adjustable to log or lin, crucially shapes the error model’s effectiveness and residual calculations. Logarithmic scaling is particularly advantageous for datasets with broad ranges, ensuring proportionate errors and uniform residuals, thereby reducing skewness in error distribution and enhancing model convergence stability.

To change the default ´linscaling, set thescalingattribute inPIOutputMapping`:

outputMapping <- PIOutputMapping$new(quantity = testQuantity)
outputMapping$scaling <- "log"

References

  • Beal SL. (2001). Ways to fit a PK model with some data below the quantification limit. Journal of Pharmacokinetics and Pharmacodynamics, 28(5), 481-504. DOI: 10.1023/a:1012299115260.

  • Fox, J. & Weisberg, S. (2013). Robust Regression. College of Liberal Arts - Statistics, Minneapolis.