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Parameter Estimation

  • parameter values define β€œgoodness” of the model

Objective Function Value

  • represents the goodness of fit [πŸ“–]
  • proportional to minus 2 times the log likelihood (-2LL)
  • extended least squares ELS objective function [πŸ“–]
  • the preference is given to lower OFV [πŸ“–]
    • by iterative β€œhill climbing” procedure to find the lowest OFV, or minima, within a given search space. [πŸ“–]
  • Initial parameter estimates have an important role [πŸ“–]
    • estimation can be β€œtrapped” gradient search in local OFV minima, and β€œmask” the global minimum

Figure was adopted from [πŸ“–].

Algorithms

Gradient-based algorithms

  • Taylor series approximations for numerical solution of the likelihood function

FOCE

  • First-Order Conditional Estimation algorithm
  • linearised by conditioning on the individual etas [πŸ“–] [πŸ“–]

FOCEI

  • First-Order Conditional Estimation algorithm with interaction
  • considering the interaction between Ξ΅ and Ξ· [πŸ“–]

LAPLACE

  • [πŸ“–]
  • second-order approximation
  • only gradient-based estimation method
  • can be used for categorical data
  • can be used to consider observations below LLOQ
  • more unstable than e.g. FOCE

SAEM

  • Stochastic Approximation Expectation Maximisation
  • [πŸ“–]
  • step E: stochastic approximation
  • step M: maximises the expected likelihood
  • includes one burn-in and one accumulation phase [πŸ“–]
    • burn-in: approximation is done on few samples per individual, and maximised and the process is repeated until the estimates have stabilised
    • accumulation: the individual random-effects are sampled and averaged together

IMP

  • IMPortance sampling
  • [πŸ“–]
    • step E: Monte-Carlo integration to assess the conditional mean and variance of \(Ξ·_i\)
    • step M: maximises the expected likelihood
  • objective function is commonly generated by few iterations of IMP for the final parameter estimates.

Note: [πŸ“–]
step E expectation evaluates the expected likelihood with respect to the conditional distribution of \(Ξ·_i\) based on the current parameter estimates and the observed data;
step M maximisation maximises the expected likelihood (from step E) to generate new parameter estimates.