While the joint modeling of item responses and response times (RTs) has received considerable attention, most existing approaches remain limited to dichotomous items and are not applicable to assessments involving polytomous or mixed-format items. To address this limitation, this article proposes a novel joint modeling framework for graded item responses and RTs. Specifically, we develop a conditional RT model given item responses and integrate it with a marginal response model based on Samejima’s graded response model, yielding a conditional joint model for graded item responses and RTs. The model is then embedded within a two-level hierarchical framework to account for the relationship between ability and speed at the population level. A key methodological contribution is the development of a stochastic approximation EM (SAEM) algorithm for estimating the proposed model, which efficiently computes its marginal maximum likelihood estimates. Simulation studies demonstrate the accurate parameter recovery of the SAEM algorithm and indicate that the proposed model outperforms the hierarchical model assuming conditional independence across various testing conditions. Finally, an empirical analysis using data from the 2022 Programme for International Student Assessment illustrates the effectiveness of the graded response–response time model in large-scale assessments.

The determination of livestock growth patterns is important for meat or milk production systems, and nonlinear models are used to summarize and interpret the information. The aim of this study was to more accurately estimate growth curve parameters in buffalo cows by evaluating and selecting nonlinear mixed models that employ different types of residuals and include or not contemporary groups (CG) as a covariate. Weight records from 720 animals obtained over a period of 60 months were used. The growth curves were fit using nonlinear mixed-effects models. The Bertalanffy, Gompertz and Logistic models were evaluated. Modelling residuals using four structures (constant, combined, exponential and proportional) and the inclusion or not of CG in the models were also evaluated. The Akaike information criterion (AIC) and Bayesian information criterion (BIC) were used to select the model. In addition to estimating the parameters of the nonlinear growth models and their correlations, the instantaneous growth rate and inflection point were obtained. The Bertalanffy model with a combined residual structure and CG exhibited the lowest AIC and BIC values. Asymptotic weight (A) estimates ranged from 621.8 to 742.1 kg, and the maturity rate (k) ranged from 0.068 to 0.115 kg/month. The correlation between A and k ranged from −0.32 to −0.82 among the models evaluated. The selection criteria indicated that the Bertalanffy model was the most suitable for growth curve analysis in buffaloes.

The objective of the current paper was to apply mixed models to adjust the growth curve of quail lines for meat and laying hens and present the rates of instantaneous, relative and absolute growth. A database was used with birth weight records up to the 148th day of female quail of the lines for meat and posture. The models evaluated were Brody, Von Bertalanffy, Logistic and Gompertz and the types of residues were constant, combined, proportional and exponential. The Gompertz model with the combined residue presented the best fit. Both strains present a high correlation between the parameters asymptotic weight (A) and average growth rate (k). The two strains presented a different growth profile. However, growth rates allow greater discernment of growth profiles. The meat line presented a higher growth rate (6.95 g/day) than the lineage for laying (3.65 g/day). The relative growth rate was higher for lineage for laying (0.15%) in relation to the lineage for meat (0.13%). The inflection point of both lines is on the first third of the growth curve (up to 15 days). All results suggest that changes in management or nutrition could optimize quail production.

Parameter estimation in non linear mixed effects models requires a large number of evaluations of the model to study. For ordinary differential equations, the overall computation time remains reasonable. However when the model itself is complex (for instance when it is a set of partial differential equations) it may be time consuming to evaluate it for a single set of parameters. The procedures of population parametrization (for instance using SAEM algorithms) are then very long and in some cases impossible to do within a reasonable time. We propose here a very simple methodology which may accelerate population parametrization of complex models, including partial differential equations models. We illustrate our method on the classical KPP equation.

Non-linear mixed models defined by stochastic differential equations (SDEs) are considered: the parameters of the diffusion process are random variables and vary among the individuals. A maximum likelihood estimation method based on the Stochastic Approximation EM algorithm, is proposed. This estimation method uses the Euler-Maruyama approximation of the diffusion, achieved using latent auxiliary data introduced to complete the diffusion process between each pair of measurement instants.A tuned hybrid Gibbs algorithm based on conditional Brownian bridges simulations of the unobserved process paths is included in this algorithm. The convergence is proved and the error induced on the likelihood by the Euler-Maruyama approximation is bounded as a function of the step size of the approximation. Results of a pharmacokinetic simulation study illustrate the accuracy of this estimation method. The analysis of the Theophyllin real dataset illustrates the relevance of the SDE approach relative to the deterministic approach.

The stochastic approximation version of EM (SAEM) proposed by Delyon et al. (1999) isa powerful alternative to EM when the E-step is intractable. Convergence ofSAEM toward a maximum of the observed likelihood is established whenthe unobserved data are simulated at each iteration under the conditionaldistribution. We show that this very restrictive assumption can be weakened. Indeed, the results of Benveniste et al. for stochastic approximationwith Markovian perturbations are used to establish the convergenceof SAEM when it is coupled with a Markov chain Monte-Carloprocedure. This result is very useful for many practical applications. Applications to the convolution model and the change-points model are presented to illustrate the proposed method.