3.22 Facultad de Matemáticas

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Browsing 3.22 Facultad de Matemáticas by Subject "519.5"

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    Modelling predictive validity problems : a partial identification approach
    (2021) Alarcón Bustamante, Eduardo Sebastián; San Martín, Ernesto; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
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    New contributions to joint models of longitudinal and survival outcomes : two-stage approaches
    (2021) Leiva Yamaguchi, Valeria; Silva, Danilo Alvares da; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
    Joint models of longitudinal and survival outcomes have gained much popularity over the last three decades. This type of modeling consists of two submodels, one longitudinal and one survival, which are connected by some common term. Unsurprisingly, sharing information makes the inferential process highly time-consuming. This problem can be overcome by estimating the parameters of each submodel separately, leading to a natural reduction in the complexity of joint models, but often producing biased estimates. Hence, we propose different two-stage strategies that first fits the longitudinal submodel and then plug the shared information into the survival submodel. Our proposals are developed for both the frequentist and Bayesian paradigms. Specifically, our frequentist two-stage approach is based on the simulation-extrapolation algorithm. On the other hand, we propose two Bayesian approaches, one inspired by frailty models and another that uses maximum a posteriori estimations and longitudinal likelihood to calculate posterior distributions of random effects and survival parameters. Based on simulation studies and real applications, we empirically compare our two-stage approaches with their main competitors. The results show that our methodologies are very promising, since they reduce the estimation bias compared to other two-stage methods and require less processing time than joint specification approaches.
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    Spatiotemporal modeling of count data
    (2021) Morales Navarrete, Diego Fabián; Castro Cepero, Luis Mauricio; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
    Modeling spatial and spatio-temporal data is a challenging task in statistics. In many applications, the observed data can be modeled using Gaussian, skew-Gaussian or even restricted random field models. However, in several fields, such as population genetics, epidemiology, aquaculture, among others, the data of interest are often count data, and therefore the mentioned models are not suitable for the analysis of this type of data. Consequently, there is a need for spatial and spatio-temporal models that are able to properly describe data coming from counting processes. Commonly two approaches are used to model this type of data: generalized linear mixed models (GLMMs) with Gaussian random field (GRF) effects, and copula models. Unfortunately, these approaches do not give an explicit characterization of the count random field such us their q-dimensional distribution or correlation function. It is important to stress that GLMMs models induces a discontinuity in the path. Therefore, the correlation function is not continuous at the origin and samples located nearby are more dissimilar than in the continuous case. Moreover, there are cases in which the copula representation for discrete distributions is not unique, so it is unidentifiable. Hence, to deal with the latter mentioned issues, we propose a novel approach to model spatial and spatio-temporal count data in an efficient and accurate manner. Briefly, starting from independent copies of a “parent” GRF, a set of transformations can be applied, and the result is a non-Gaussian random field. This approach is based on the characterization of count random fields that inherit some of the well-known geometric properties from GRFs. For instance, if one chooses an isotropic correlation function defined in the parent GFR, then the count random fields have an isotropic correlation function. Firstly, we define a general class of count random fields. Then, three particular count random fields are studied. The first one is a Poisson random field, the second one is a count random field that considers excess zeros and the last one is a count random field that considers over-dispersion. Additionally, a simulation study will be developed to assess the performance of the proposed models. In that way, we are going to evaluate them through several simulation scenarios, making variations in the parameters. The results show accurate estimations of the parameters for different scenarios. Additionally, we assess the performance of the optimal linear prediction of the proposed models and it is compared with GLMMs and copula models. The results show that the proposed models have a better performance than GLMMs models and a quite similar performance with copula models. Finally, we analyze two real data applications. The first one considers a zero inflated version of the proposed Poisson random field to deal with excess zeros and the second one considers an over-dispersed count random field.
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    Statistical methods for the analysis of Polytomous response data in non-cognitive tests
    (2021) Calderón Maldonado, Francisca Loreto; González Burgos, Jorge Andrés; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
    In psychology, education, and other social science disciplines, questionnaires and surveys are useful instruments to measure latent variables such as behaviors, ability, or perceptions about specific constructs. Measuring latent traits, abilities, and in general, any type of nonobservable variables is much more complicated than measuring observable features. Latent variables cannot be measured directly but only indirectly through multiple observed variables called indicators (i.e., observed variables of either polytomous or dichotomous type). The scores on items in the questionnaires can be considered indicators of latent variables and are thus used to measure the unobserved constructs of interest. The main theme of this dissertation is the study and implementation of statistical models and methods for the analysis of polytomous response data in non-cognitive tests. Polytomous data arise when items are scored in more than two categories (e.g., strongly disagree, disagree, agree, strongly agree), as in surveys and questionnaires. We have adapted and extended existing statistical models and methods to meet the requirements of various approaches based on polytomous data. The empirical data sets used for the applications of the models are meant as exemplars of a broader category and a more extensive range of domains.