Juan M. Rodríguez-Díaz

Optimal designs for compartmental models with correlated observations

Abstract The flow of internally deposited radioisotope particles inside the body of people exposed to inhalation, ingestion, injection or other ways is usually evaluated using compartmental models (see Sánchez & López-Fidalgo, (2003, and López-Fidalgo & Sánchez, 2005). The International Commission on Radiological Protection (ICRP, 1994) describes the model of the human respiratory tract, represented by two main regions. One of these, the thoracic region (lungs) is divided into different compartments. The retention in the lungs is given by a large combination of ratios of exponential sums depending on time. The aim of this work is to provide optimal times for making bioassays when there has been an accidental radioactivity intake and there is interest in estimating it. In this paper, a large two–parameter model is studied and a simplified model is proposed in order to obtain optimal designs in a more suitable way. Local c-optimal designs for the main parameters are obtained using the results of López-Fidalgo & Rodríguez-Díaz, 2004). Efficiencies for all the computed designs are provided and compared.

Integral approximations for computing optimum designs in random effects logistic regression models

In the context of nonlinear models, the analytical expression of the Fisher information matrix is essential to compute optimum designs. The Fisher information matrix of the random effects logistic regression model is proved to be equivalent to the information matrix of the linearized model, which depends on some integrals. Some algebraic approximations for these integrals are proposed, which are consistent with numerical integral approximations but much faster to be evaluated. Therefore, these algebraic integral approximations are very useful from a computational point of view. Locally D -, A -, c -optimum designs and the optimum design to estimate a percentile are computed for the univariate logistic regression model with Gaussian random effects. Since locally optimum designs depend on a chosen nominal value for the parameter vector, a Bayesian D -optimum design is also computed. In order to find Bayesian optimum designs it is essential to apply the proposed integral approximations, because the use of numerical approximations makes the computation of these optimum designs very slow.

Characteristic Polynomial Criteria in Optimal Experimental Design

Many optimality criteria have been used in the literature of experimental design. Two of the most common are A-optimality and D-optimality. They are the first and the last coefficients of the characteristic polynomial of the inverse information matrix. In this paper, criteria from the remaining coefficients are considered, and some properties are studied. While A-optimality focuses on the average of the estimate variances and D-optimality focuses on all of the covariances, these criteria take into consideration the covariances considered in groups of two, three, four,…

Optimizing the test power for a radiation retention model in the human body

The model that describes the retention in lungs of radioisotope particles is studied in this paper, considering the situation of an accident in facilities that handle radioactive materials. Optimal times to make the bioassays are computed for D- and c-optimality, and efficiencies for the computed designs are provided and compared. Moreover, the test power is checked by means of simulations and replications. After that the inverse of the Fisher information matrix is compared to an estimation of the covariance matrix of the parameters. Finally, a study taking into consideration the randomness of the designs space is performed.

Computation of c-optimal designs for models with correlated observations

In the optimal design of experiments setup, different optimality criteria can be considered depending on the objectives of the practitioner. One of the most used is c-optimality, which for a given model looks for the design that minimizes the variance of the linear combination of the parameters’ estimators given by vector c. c-optimal designs are needed when dealing with standardized criteria, and are specially useful when c is taken to be each one of the Euclidean vectors since in that case they provide the best designs for estimating the individual parameters. The well known procedure proposed by Elfving for independent observations is the origin of the procedure that can be used in the correlation framework. Some analytical results are shown for the model with constant covariance, but even in this case the computational task can become quite hard. For this reason, an algorithmic procedure is proposed; it can be used when dealing with a general model and some covariance structures.

Optimal Designs for Random Blocks Model Using Corrected Criteria

Many industrial experiments involve random factors. The random blocks model defines a covariance structure in the data, thus generalized least square estimators of the parameters are used, and their covariance matrix is usually computed using the inverse of the generalized least square estimators information matrix. Many optimality criteria are based on this approximation of the covariance matrix. However, this approach underestimates the true covariance matrix of the parameters, and thus, the optimality criteria should be corrected in order to pay attention to the actual covariance. The bias in the estimation of the covariance matrix is negligible (or even null) for many models, and for this reason in those cases, it has no sense to deal with the corrected criteria because of the complexity of the calculations involved. But for some models, the correction does have importance, and thus, the modified criteria should be considered when designing; otherwise, the practitioner may risk to deal with poor designs. Some analytical results are presented for simpler models, and optimal designs taking into account the corrected variance will be computed and compared with those using the traditional approach for more complex models, showing that the loss in efficiency may be very important when the correction for the covariance matrix is ignored. Copyright

Discrimination Between Random and Fixed Effect Logistic Regression Models

D s- and KL-optimum designs are computed for discriminating between univariate logistic regression models with or without random effects. Both these competing optimum designs are constructed numerically. The main problem in finding them is the computation of some integrals at each step of the numerical procedure. In order to improve the convergence speed of this numerical procedure some integral approximations are suggested.