Parameter estimation was obtained through maximum likelihood estimation with integral approximation via adaptive Gaussian quadrature with 15 integration points.
The proposed Bayesian algorithm uses prior knowledge about parameters and builds on the Laplace Approximation Maximum Likelihood Estimation (LAMLE) algorithm (Karimi and McAuley, 2014).
Further, we compared the Bayesian MCMC (Markov Chain Monte Carlo) scheme and the MLE (Maximum Likelihood Estimation) with asymptotic normal approximation for parameter estimation to confirm the advantage of the Bayesian MCMC with respect to uncertainty analysis.
The analyses were conducted with the statistical software SAS 9.2 (SAS Institute Inc., Cary, NC, USA), using the GLIMMIX procedure for the multilevel analyses with a maximum likelihood estimation based on Laplace approximation.
Based on simulation evidence, the two-step maximum likelihood estimation provides a close approximation of the underlying construct, particularly in large samples where cell frequencies are high [ 18, 19].
CGM data was analyzed to find optimal values of parameters by using ordinary least squares fitting or maximum likelihood estimation using a kernel-density approximation.
To investigate morphological traits influencing the total number of offspring produced by the subject males, we used generalized linear mixed models (GLMMs) with a logarithm link function and Poisson distribution, fitted using the Laplace approximation to restricted maximum likelihood estimation (lmer procedure in the lme4 R package [ 47]).
Parameters were estimated by computing the maximum likelihood estimation of the parameters without any approximation of the model (that is, no linearization) using the stochastic approximation expectation maximization algorithm combined with a Markov chain Monte Carlo procedure.
The maximum likelihood estimation method in lme4 utilizes a Laplace approximation.
Data of each simulated trial were analyzed using MONOLIX version 4.2 (http://www.lixoft.eu/monolix/product-monolix-overview/) [ 21], a software devoted to maximum likelihood estimation of parameters in NLMEM using an extension of the stochastic approximation expectation-approximation (SAEM) algorithm [ 22, 23].
