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In this chapter, we examine how to quantify uncertainty about model parameters, highlighting two main approaches: frequentist and Bayesian. We start by modelling a data-generating mechanism with a parametric family, where different parameter values correspond to different models. Assuming our model family can describe the mechanism, we use data to infer plausible parameters and quantify uncertainty. In frequentist inference, we build parameter estimators and study their sampling distributions across repeated data collection. Here, parameters are fixed unknown constants, and only estimators are treated probabilistically. In Bayesian inference, parameters are latent random variables. We express uncertainty through probability, combining prior beliefs about parameter values with observed data using Bayes’ rule to obtain a posterior distribution. The posterior and the frequentist sampling distribution often play similar roles and can resemble each other in practice. Computational tools like bootstrapping and Markov chain Monte Carlo help estimate sampling and posterior distributions, respectively.
From social networks to biological systems, networks are a fundamental part of modern life. Network analysis is increasingly popular across the mathematical, physical, life and social sciences, offering insights into a range of phenomena, from developing new drugs based on intracellular interactions, to understanding the influence of social interactions on behaviour patterns. This book provides a toolkit for analyzing random networks, together with theoretical justification of the methods proposed. It combines methods from both probability and statistics, teaching how to build and analyze plausible models for random networks, and how to validate such models, to detect unusual features in the data, and to make predictions. Theoretical results are motivated by applications across a range of fields, and classical data sets are used for illustration throughout the book. This book offers a comprehensive introduction to the field for graduate students and researchers.
Many journals ask authors to report confidence intervals (to quantify estimation precision or uncertainty) and measures of effect size (to quantify a factor’s explanatory power). Arguments for such practices focus on benefits to interpreting and applying scientific findings that go beyond merely detecting effects, thereby implying that effect sizes and confidence intervals should be reported and discussed. Accordingly, we examined 150 recent articles from 6 journals that publish research on Judgment and Decision Making (JDM) to survey current practices for reporting and discussing results. We recorded which of those articles report p-values, standardized effect sizes, and confidence/credibility intervals in their Results sections. We examined the articles’ narrative sections (Abstract, Discussion/Conclusion) for explicit reference to the presence/absence of an effect, an effect’s size, and the precision or range associated with an estimate. Ninety-one percent of articles reported p-values, CI0.95 [85%, 95%], and all discussed the presence or absence of effects. Most articles gave effect size information, with 73%, CI0.95 [65%, 79%], reporting standardized effect sizes, and 63%, CI0.95 [55%, 71%], reporting confidence/credibility intervals or graphical SE bars. However, an estimation perspective was less apparent in the articles’ Discussion sections, wherein 59%, CI0.95 [51%, 66%], discussed effect size information—though often with limited detail—and only 3%, CI0.95 [1%, 6%], discussed interval estimates. Mostly, it seems, JDM researchers follow guidelines for reporting effect size and the uncertainty and precision for effect estimates. Yet, one might ask whether this impacts researchers’ interpretation and communication of those effects as it should.
The political consequences of climate change have been topics at numerous political science conferences. Contrary to the plurality of discussions at these meetings, it is striking that there is no systematic account of the carbon footprint of political science conferences themselves. Applying a GIS-based approach I estimate the travel induced greenhouse gas emissions of the last six ECPR General Conferences (2013–18). The results show that for the five conferences that took part in Europe the average emissions per attendee were between 0.5–1.3 tons CO2-equivalents. At the 2015 conference in Montreal it were even 1.9–3.4 tons. Compared to estimations based on the latest IPCC reports which call for a reduction of per capita emissions to 2.5 tons by 2030 and even 0.7 tons by 2050 in order to keep on track with the 1.5-degree goal, the travel induced GHG-emissions of ECPR conferences are very high. Yet, further estimations demonstrate that significant emission reductions are possible: by choosing more central conference venues, promoting low-emission landbound means of transportation and introducing online participation for researchers from far away, the carbon footprint could be reduced by 75–90 per cent. The article also gives concrete recommendations how the carbon footprint of conferences could be reduced.
We introduce a family of parsimonious network models that are intended to generalize the configuration model to temporal settings. We present consistent estimators for the model parameters and perform numerical simulations to illustrate the properties of the estimators on finite samples. We also derive analytical solutions for the basic and effective reproduction numbers for the early stage of the discrete-time SIR spreading process for our temporal configuration model (TCM). We apply three distinct TCMs to empirical student proximity networks and compare their performance.
This chapter discusses techniques to build predictive models from data and to quantify the uncertainty of the model parameters and of the model predictions. The chapter discusses important concepts of linear and nonlinear regression and focuses on a couple of major paradigms used for estimation: maximum likelihood and Bayesian estimation. The chapter also discusses how to incorporate prior knowledge in the estimation process.
This chapter discusses techniques that help us estimate parameters and summarizing statistics for random variables from data. The chapter discusses techniques such as the method of moments, least-squares, and maximum likelihood. The chapter also touches on concepts of Monte Carlo simulation, which is a technique that can be used to approximate the summarizing statistics of random variables from random samples or from data. The chapter also highlights how one can characterize the quality of such approximations using the central limit theorem and the law of large numbers.
Maritime transport plays a vital role in global logistics and trade; however, its environmental impact, particularly CO₂ emissions, has become a growing concern. Current estimation methodologies are divided into top-down and bottom-up approaches. Top-down methods rely on macro-statistical data but often lack specificity regarding individual ship characteristics, leading to high uncertainty. Bottom-up methods, increasingly prevalent due to advancements in ship equipment and big data technology, estimate CO₂ emissions based on detailed ship activity trajectories, offering greater precision. This study integrates data from multiple vessel-position transmitting devices — AIS, V-Pass, and LTE-Maritime — to estimate CO₂ emissions from maritime activities in the coastal regions of South Korea. By combining these data sources, the study develops a comprehensive and accurate emissions assessment, improving reliability and supporting more informed decision-making in maritime environmental management and policy development.
The glomerular filtration rate (GFR), estimated from serum creatinine (SCr), is widely used in clinical practice for kidney function assessment, but SCr-based equations are limited by non-GFR determinants and may introduce inaccuracies across racial groups. Few studies have evaluated whether advanced modeling techniques enhance their performance.
Methods:
Using multivariable fractional polynomials (MFP), generalized additive models (GAM), random forests (RF), and gradient boosted machines (GBM), we developed four SCr-based GFR-estimating equations in a pooled data set from four cohorts (n = 4665). Their performance was compared to that of the refitted linear regression-based 2021 CKD-EPI SCr equation using bias (median difference between measured GFR [mGFR] and estimated GFR [eGFR]), precision, and accuracy metrics (e.g., P10 and P30, percentage of eGFR within 10% and 30% of mGFR, respectively) in a pooled validation data set from three additional cohorts (n = 2215).
Results:
In the validation data set, the greatest bias and lowest accuracy, were observed in Black individuals for all equations across subgroups defined by race, sex, age, and eGFR. The MFP and GAM equations performed similarly to the refitted CKD-EPI SCr equation, with slight improvements in P10 and P30 in subgroups including Black individuals and females. The GBM and RF equations demonstrated smaller biases, but lower accuracy compared to other equations. Generally, differences among equations were modest overall and across subgroups.
Conclusions:
Our findings suggest that advanced methods provide limited improvement in SCr-based GFR estimation. Future research should focus on integrating novel biomarkers for GFR estimation and improving the feasibility of GFR measurement.
There is an odd contradiction about much of the empirical (experimental) literature: The data is analysed using statistical tools which presuppose that there is some noise or randomness in the data, but the source and possible nature of the noise are rarely explicitly discussed. This paper argues that the noise should be brought out into the open, and its nature and implications openly discussed. Whether the statistical analysis involves testing or estimation, the analysis inevitably is built upon some assumed stochastic structure to the noise. Different assumptions justify different analyses, which means that the appropriate type of analysis depends crucially on the stochastic nature of the noise. This paper explores such issues and argues that ignoring the noise can be dangerous.
This chapter reviews alternative methods proposed in the literature for estimating discrete-time stochastic volatility models and illustrates the details of their application. The methods reviewed are classified as either frequentist or Bayesian. The methods in the frequentist class include generalized method of moments, quasi-maximum likelihood, empirical characteristic function, efficient method of moments, and simulated maximum likelihood based on Laplace-based importance sampler. The Bayesian methods include single-move Markov chain Monte Carlo, multimove Markov chain Monte Carlo, and sequential Monte Carlo.
We propose Rényi information generating function (RIGF) and discuss its properties. A connection between the RIGF and the diversity index is proposed for discrete-type random variables. The relation between the RIGF and Shannon entropy of order q > 0 is established and several bounds are obtained. The RIGF of escort distribution is derived. Furthermore, we introduce the Rényi divergence information generating function (RDIGF) and discuss its effect under monotone transformations. We present nonparametric and parametric estimators of the RIGF. A simulation study is carried out and a real data relating to the failure times of electronic components is analyzed. A comparison study between the nonparametric and parametric estimators is made in terms of the standard deviation, absolute bias, and mean square error. We have observed superior performance for the newly proposed estimators. Some applications of the proposed RIGF and RDIGF are provided. For three coherent systems, we calculate the values of the RIGF and other well-established uncertainty measures, and similar behavior of the RIGF is observed. Further, a study regarding the usefulness of the RDIGF and RIGF as model selection criteria is conducted. Finally, three chaotic maps are considered and then used to establish a validation of the proposed information generating function.
Garbarino et al. (J Econ Sci Assoc. https://doi.org/10.1007/s40881-018-0055-4, 2018) describe a new method to calculate the probability distribution of the proportion of lies told in “coin flip” style experiments. I show that their estimates and confidence intervals are flawed. I demonstrate two better ways to estimate the probability distribution of what we really care about—the proportion of liars—and I provide R software to do this.
Fully Bayesian estimation of item response theory models with logistic link functions suffers from low computational efficiency due to posterior density functions that do not have known forms. To improve algorithmic computational efficiency, this paper proposes a Bayesian estimation method by adopting a new data-augmentation strategy in uni- and multidimensional IRT models. The strategy is based on the Pólya–Gamma family of distributions which provides a closed-form posterior distribution for logistic-based models. In this paper, an overview of Pólya–Gamma distributions is described within a logistic regression framework. In addition, we provide details about deriving conditional distributions of IRT, incorporating Pólya–Gamma distributions into the conditional distributions for Bayesian samplers’ construction, and random drawing from the samplers such that a faster convergence can be achieved. Simulation studies and applications to real datasets were conducted to demonstrate the efficiency and utility of the proposed method.
Quantitative psychology is concerned with the development and application of mathematical models in the behavioral sciences. Over time, models have become more complex, a consequence of the increasing complexity of research designs and experimental data, which is also a consequence of the utility of mathematical models in the science. As models have become more elaborate, the problems of estimating them have become increasingly challenging. This paper gives an introduction to a computing tool called automatic differentiation that is useful in calculating derivatives needed to estimate a model. As its name implies, automatic differentiation works in a routine way to produce derivatives accurately and quickly. Because so many features of model development require derivatives, the method has considerable potential in psychometric work. This paper reviews several examples to demonstrate how the methodology can be applied.
Methodological development of the model-implied instrumental variable (MIIV) estimation framework has proved fruitful over the last three decades. Major milestones include Bollen’s (Psychometrika 61(1):109–121, 1996) original development of the MIIV estimator and its robustness properties for continuous endogenous variable SEMs, the extension of the MIIV estimator to ordered categorical endogenous variables (Bollen and Maydeu-Olivares in Psychometrika 72(3):309, 2007), and the introduction of a generalized method of moments estimator (Bollen et al., in Psychometrika 79(1):20–50, 2014). This paper furthers these developments by making several unique contributions not present in the prior literature: (1) we use matrix calculus to derive the analytic derivatives of the PIV estimator, (2) we extend the PIV estimator to apply to any mixture of binary, ordinal, and continuous variables, (3) we generalize the PIV model to include intercepts and means, (4) we devise a method to input known threshold values for ordinal observed variables, and (5) we enable a general parameterization that permits the estimation of means, variances, and covariances of the underlying variables to use as input into a SEM analysis with PIV. An empirical example illustrates a mixture of continuous variables and ordinal variables with fixed thresholds. We also include a simulation study to compare the performance of this novel estimator to WLSMV.
Nonlinear random coefficient models (NRCMs) for continuous longitudinal data are often used for examining individual behaviors that display nonlinear patterns of development (or growth) over time in measured variables. As an extension of this model, this study considers the finite mixture of NRCMs that combine features of NRCMs with the idea of finite mixture (or latent class) models. The efficacy of this model is that it allows the integration of intrinsically nonlinear functions where the data come from a mixture of two or more unobserved subpopulations, thus allowing the simultaneous investigation of intra-individual (within-person) variability, inter-individual (between-person) variability, and subpopulation heterogeneity. Effectiveness of this model to work under real data analytic conditions was examined by executing a Monte Carlo simulation study. The simulation study was carried out using an R routine specifically developed for the purpose of this study. The R routine used maximum likelihood with the expectation–maximization algorithm. The design of the study mimicked the output obtained from running a two-class mixture model on task completion data.
This research concerns a mediation model, where the mediator model is linear and the outcome model is also linear but with a treatment–mediator interaction term and a residual correlated with the residual of the mediator model. Assuming the treatment is randomly assigned, parameters in this mediation model are shown to be partially identifiable. Under the normality assumption on the residual of the mediator and the residual of the outcome, explicit full-information maximum likelihood estimates of model parameters are introduced given the correlation between the residual for the mediator and the residual for the outcome. A consistent variance matrix of these estimates is derived. Currently, the coefficients of this mediation model are estimated using the iterative feasible generalized least squares (IFGLS) method that is originally developed for seemingly unrelated regressions (SURs). We argue that this mediation model is not a system of SURs. While the IFGLS estimates are consistent, their variance matrix is not. Theoretical comparisons of the FIMLE variance matrix and the IFGLS variance matrix are conducted. Our results are demonstrated by simulation studies and an empirical study. The FIMLE method has been implemented in a freely available R package iMediate.
An observer is to make inference statements about a quantity p, called a propensity and bounded between 0 and 1, based on the observation that p does or does not exceed a constant c. The propensity p may have an interpretation as a proportion, as a long-run relative frequency, or as a personal probability held by some subject. Applications in medicine, engineering, political science, and, most especially, human decision making are indicated. Bayes solutions for the observer are obtained based on prior distributions in the mixture of beta distribution family; these are then specialized to power-function prior distributions. Inference about log p and log odds is considered. Multiple-action problems are considered in which the focus of inference shifts to the process generating the propensities p, both in the case of a process parameter π known to the subject and unknown. Empirical Bayes techniques are developed for observer inference about c when π is known to the subject. A Bayes rule, a minimax rule and a beta-minimax rule are constructed for the subject when he is uncertain about π.
This chapter introduces communication and information theoretical aspects of molecular communication, relating molecular communication to existing techniques and results in communication systems. Communication models are discussed, as well as detection and estimation problems. The information theory of molecular communication is introduced, and calculation of the Shannon capacity is discussed.