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Using a hierarchical construction, we develop methods for a wide and flexible class of models by taking a fully parametric approach to generalized linear mixed models with complex covariance dependence. The Laplace approximation is used to marginally estimate covariance parameters while integrating out all fixed and latent random effects. The Laplace approximation relies on Newton-Raphson updates, which also leads to predictions for the latent random effects. We develop methodology for complete marginal inference, from estimating covariance parameters and fixed effects to making predictions for unobserved data, for any patterned covariance matrix in the hierarchical generalized linear mixed models framework. The marginal likelihood is developed for six distributions that are often used for binary, count, and positive continuous data, and our framework is easily extended to other distributions. The methods are illustrated with simulations from stochastic processes with known parameters, and their efficacy in terms of bias and interval coverage is shown through simulation experiments. Examples with binary and proportional data on election results, count data for marine mammals, and positive-continuous data on heavy metal concentration in the environment are used to illustrate all six distributions with a variety of patterned covariance structures that include spatial models (e.g., geostatistical and areal models), time series models (e.g., first-order autoregressive models), and mixtures with typical random intercepts based on grouping.

Quality metrics in health care refer to a variety of measures used mainly to characterize what should have been done for a patient or the health consequences of what was done. When estimating quality of health care, often many metrics are measured and then combined to provide an overall estimate either at the patient level or at higher levels of accountability, such as the provider organization, insurer, or even geographic area. Racial/ethnic disparities are defined as the mean difference in overall quality between minorities and Whites not justified by underlying health conditions or patient preferences. However, several statistical features of health care quality data have frequently been ignored: quality is a theoretical construct that is not directly observed; the quality metrics are measured on different scales or, if measured on the same scale, have different baseline rates; the structure of the construct is likely multidimensional; and metrics are correlated within-patients. We address these features and utilize multi-dimensional item response theory models to estimate racial/ethnic quality disparities. Quality metrics measured on 93,000 adults with schizophrenia residing in 5 U.S. states illustrate approaches.

Advanced science and technology provide a wealth of big data from different sources for extreme value analysis.Classic extreme value theory was extended to obtain an accelerated max-stable distribution family for modelling competing risk-based extreme data in Cao and Zhang (2021). In this paper, we establish probability models for power normalized maxima and minima from competing risks. The limit distributions consist of an extensional new accelerated max-stable and min-stable distribution family (termed as the accelerated p-max/p-min stable distribution), and its left-truncated version. The limit types of distributions are determined principally by the sample generating process and the interplay among the competing risks, which are illustrated by common examples. Further, the statistical inference concerning the maximum likelihood estimation and model diagnosis of this model was investigated. Numerical studies show first the efficient approximation of all limit scenarios as well as its comparable convergence rate in contrast with those under linear normalization, and then present the maximum likelihood estimation and diagnosis of accelerated p-max/p-min stable models for simulated data sets. Finally, two real datasets concerning annual maximum of ground level ozone and survival times of Stanford heart plant demonstrate the performance of our accelerated p-max and accelerated p-min stable models.

We propose a sparse algebra for samplet compressed kernel matrices, to enable efficient scattered data analysis. We show the compression of kernel matrices by means of samplets produces optimally sparse matrices in a certain S-format. It can be performed in cost and memory that scale essentially linearly with the matrix size $N$, for kernels of finite differentiability, along with addition and multiplication of S-formatted matrices. We prove and exploit the fact that the inverse of a kernel matrix (if it exists) is compressible in the S-format as well. Selected inversion allows to directly compute the entries in the corresponding sparsity pattern. The S-formatted matrix operations enable the efficient, approximate computation of more complicated matrix functions such as ${\bm A}^\alpha$ or $\exp({\bm A})$. The matrix algebra is justified mathematically by pseudo differential calculus. As an application, efficient Gaussian process learning algorithms for spatial statistics is considered. Numerical results are presented to illustrate and quantify our findings.

An old problem in multivariate statistics is that linear Gaussian models are often unidentifiable, i.e. some parameters cannot be uniquely estimated. In factor (component) analysis, an orthogonal rotation of the factors is unidentifiable, while in linear regression, the direction of effect cannot be identified. For such linear models, non-Gaussianity of the (latent) variables has been shown to provide identifiability. In the case of factor analysis, this leads to independent component analysis, while in the case of the direction of effect, non-Gaussian versions of structural equation modelling solve the problem. More recently, we have shown how even general nonparametric nonlinear versions of such models can be estimated. Non-Gaussianity is not enough in this case, but assuming we have time series, or that the distributions are suitably modulated by some observed auxiliary variables, the models are identifiable. This paper reviews the identifiability theory for the linear and nonlinear cases, considering both factor analytic models and structural equation models.

Across research disciplines, cluster randomized trials (CRTs) are commonly implemented to evaluate interventions delivered to groups of participants, such as communities and clinics. Despite advances in the design and analysis of CRTs, several challenges remain. First, there are many possible ways to specify the causal effect of interest (e.g., at the individual-level or at the cluster-level). Second, the theoretical and practical performance of common methods for CRT analysis remain poorly understood. Here, we present a general framework to formally define an array of causal effects in terms of summary measures of counterfactual outcomes. Next, we provide a comprehensive overview of CRT estimators, including the t-test, generalized estimating equations (GEE), augmented-GEE, and targeted maximum likelihood estimation (TMLE). Using finite sample simulations, we illustrate the practical performance of these estimators for different causal effects and when, as commonly occurs, there are limited numbers of clusters of different sizes. Finally, our application to data from the Preterm Birth Initiative (PTBi) study demonstrates the real-world impact of varying cluster sizes and targeting effects at the cluster-level or at the individual-level. Specifically, the relative effect of the PTBI intervention was 0.81 at the cluster-level, corresponding to a 19% reduction in outcome incidence, and was 0.66 at the individual-level, corresponding to a 34% reduction in outcome risk. Given its flexibility to estimate a variety of user-specified effects and ability to adaptively adjust for covariates for precision gains while maintaining Type-I error control, we conclude TMLE is a promising tool for CRT analysis.

In many fields, including environmental epidemiology, researchers strive to understand the joint impact of a mixture of exposures. This involves analyzing a vector of exposures rather than a single exposure, with the most significant exposure sets being unknown. Examining every possible interaction or effect modification in a high-dimensional vector of candidates can be challenging or even impossible. To address this challenge, we propose a method for the automatic identification and estimation of exposure sets in a mixture with explanatory power, baseline covariates that modify the impact of an exposure and sets of exposures that have synergistic non-additive relationships. We define these parameters in a realistic nonparametric statistical model and use machine learning methods to identify variables sets and estimate nuisance parameters for our target parameters to avoid model misspecification. We establish a prespecified target parameter applied to variable sets when identified and use cross-validation to train efficient estimators employing targeted maximum likelihood estimation for our target parameter. Our approach applies a shift intervention targeting individual variable importance, interaction, and effect modification based on the data-adaptively determined sets of variables. Our methodology is implemented in the open-source SuperNOVA package in R. We demonstrate the utility of our method through simulations, showing that our estimator is efficient and asymptotically linear under conditions requiring fast convergence of certain regression functions. We apply our method to the National Institute of Environmental Health Science mixtures workshop data, revealing correct identification of antagonistic and agonistic interactions built into the data. Additionally, we investigate the association between exposure to persistent organic pollutants and longer leukocyte telomere length.

This work studies an experimental design problem where {the values of a predictor variable, denoted by $x$}, are to be determined with the goal of estimating a function $m(x)$, which is observed with noise. A linear model is fitted to $m(x)$ but it is not assumed that the model is correctly specified. It follows that the quantity of interest is the best linear approximation of $m(x)$, which is denoted by $\ell(x)$. It is shown that in this framework the ordinary least squares estimator typically leads to an inconsistent estimation of $\ell(x)$, and rather weighted least squares should be considered. An asymptotic minimax criterion is formulated for this estimator, and a design that minimizes the criterion is constructed. An important feature of this problem is that the $x$'s should be random, rather than fixed. Otherwise, the minimax risk is infinite. It is shown that the optimal random minimax design is different from its deterministic counterpart, which was studied previously, and a simulation study indicates that it generally performs better when $m(x)$ is a quadratic or a cubic function. Another finding is that when the variance of the noise goes to infinity, the random and deterministic minimax designs coincide. The results are illustrated for polynomial regression models and the general case is also discussed.

Learning on big data brings success for artificial intelligence (AI), but the annotation and training costs are expensive. In future, learning on small data is one of the ultimate purposes of AI, which requires machines to recognize objectives and scenarios relying on small data as humans. A series of machine learning models is going on this way such as active learning, few-shot learning, deep clustering. However, there are few theoretical guarantees for their generalization performance. Moreover, most of their settings are passive, that is, the label distribution is explicitly controlled by one specified sampling scenario. This survey follows the agnostic active sampling under a PAC (Probably Approximately Correct) framework to analyze the generalization error and label complexity of learning on small data using a supervised and unsupervised fashion. With these theoretical analyses, we categorize the small data learning models from two geometric perspectives: the Euclidean and non-Euclidean (hyperbolic) mean representation, where their optimization solutions are also presented and discussed. Later, some potential learning scenarios that may benefit from small data learning are then summarized, and their potential learning scenarios are also analyzed. Finally, some challenging applications such as computer vision, natural language processing that may benefit from learning on small data are also surveyed.

Recent years have witnessed the enormous success of low-dimensional vector space representations of knowledge graphs to predict missing facts or find erroneous ones. Currently, however, it is not yet well-understood how ontological knowledge, e.g. given as a set of (existential) rules, can be embedded in a principled way. To address this shortcoming, in this paper we introduce a framework based on convex regions, which can faithfully incorporate ontological knowledge into the vector space embedding. Our technical contribution is two-fold. First, we show that some of the most popular existing embedding approaches are not capable of modelling even very simple types of rules. Second, we show that our framework can represent ontologies that are expressed using so-called quasi-chained existential rules in an exact way, such that any set of facts which is induced using that vector space embedding is logically consistent and deductively closed with respect to the input ontology.