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This chapter describes the conceptual underpinnings of the Cox regression model and demonstrates how to fit it to data. Section 14.1 begins by developing the Cox model specification itself, demonstrating why it is a sensible representation. Section 14.2 describes how the model is fit. Section 14.3 examines the results of model fitting, showing how to interpret parameters, test hypotheses, evaluate goodness-of-fit, and summarize effects. Section 14.4 concludes by presenting strategies for displaying results graphically.

Comparison between the real and simulated settlement distributions allows us to assess the goodness-of-fit between them. Since our agents optimize their locations, this is also a measure of the extent to which the real households optimized their locations, which we call “settlement efficiency.” Settlement efficiency tends to peak at the end of each population cycle, as households were leaving the area. We also develop methods for determining which combinations of parameters maximize the goodness-of-fit as that changes through time. This reveals two distinct settlement regimes, separated at AD 1060.

This chapter addresses the conceptual challenge posed by Europeanisation by reflecting on the utility of the existing goodness-of-fit model for exploring domestic adaptation aimed at uploading national policy preferences onto the European Union (EU) arena. It argues that the goodness-of-fit model is ill-suited to conceptualising strategic adaptation to EU membership: that is, the reform of national policy-making processes for the purpose of enhancing the coordination and projection of national EU policy. As an alternative to the conventional goodness-of-fit model, the chapter proposes an innovative strategic-projection model that sought to delineate between four modes of Europeanisation: the effective obligation of membership, differential empowerment and strategic adaptation within government, administrative transfer through intergovernmental learning, and the desire to maximise the compatibility of domestic and EU structures.

This chapter examines the effects of ageing on cognitive processes in two choice tasks. It fits the diffusion model to the response time and accuracy data for each task, and interprets the effects of ageing in terms of the components of processing identified by the model. The chapter addresses whether the interpretations are specific to the diffusion model. To address this question, it fits two other models, the accumulator model and the leaky competing accumulator model of Usher and McClelland, to the data from young and older subjects for six experiments. The chapter finds that, although the diffusion model fits the data better than the other models for most of the experiments, the models' explanations of how ageing affects components of processing do not differ significantly.

This chapter covers basic statistical concepts. Most statistics relate to hypothesis testing, and others to variable selection and model fitting. The name is because an exact match between a theoretical and empirical distribution is as rare as a unicorn. (1) Dispersion—measures of random variations—variance and its inflation factor, covariance and correlations {Pearson’s product-moment, Spearman’s rank order}, and the Mahalanobis distance. (2) Goodness-of-fit—do observations match expectations? This applies to both continuous dependent variables {R-squared and adjusted R2} and categorical {Pearson’s chi-square, Hosmer–Lemeshow statistic}. (3) Likelihood—assesses estimates’ goodness-of-fit to binary dependent variables {log-likelihood, deviance}, plus the Akaike and Bayesian information criteria used to penalize complexity. (4) The Holy Trinity of Statistics—i) Neyman–Pearson’s ‘likelihood ratio’—the basis for model comparisons; ii) Wald’s chi-square—for potential variable removal; iii) Rao’s score chi-square—for potential variable inclusion. These are all used in Logistic Regression.

There are several additional statistical procedures that can be conducted after a habitat analysis. The statistical model produced by a habitat analysis can be assessed for fit to the data. Model fit describes how well the predictor variables explain the variance in the response variable, typically species presence–absence or abundance. When more than one statistical model has been produced by the habitat analysis, these can be compared by a formal procedure called model comparison. This usually involves identifying the model with the lowest Akaike information criterion (AIC) value. If the statistical model is considered a predictive tool then its predictive accuracy needs to be assessed. There are many metrics for assessing the predictive performance of a model and quantifying rates of correct and incorrect classification; the latter are error rates. Many of these metrics are based on the numbers of true positive, true negative, false positive, and false negative observations in an independent dataset. “True” and “false” refer to whether species presence–absence was correctly predicted or not. Predictive performance can also be assessed by constructing a receiver operating characteristic (ROC) curve and calculating area under the curve (AUC) values. High AUC values approaching 1 indicate good predictive performance, whereas a value near 0.5 indicates a poor model that predicts species presence–absence no better than a random guess.

Parametric bootstrapping (BS) provides an attractive alternative, both theoretically and numerically, to asymptotic theory for estimating sampling distributions. This chapter summarizes its use not only for calculating confidence intervals for estimated parameters and functions of parameters, but also to obtain log-likelihood-based confidence regions from which confidence bands for cumulative distribution and regression functions can be obtained. All such BS calculations are very easy to implement. Details are also given for calculating critical values of EDF statistics used in goodness-of-fit (GoF) tests, such as the Anderson-Darling A² statistic whose null distribution is otherwise difficult to obtain, as it varies with different null hypotheses. A simple proof is given showing that the parametric BS is probabilistically exact for location-scale models. A formal regression lack-of-fit test employing parametric BS is given that can be used even when the regression data has no replications. Two real data examples are given.

This chapter illustrates use of (i) the score statistic and (ii) a goodness-of-fit statistic to test if an embedded model provides an adequate fit, in the latter case with critical values calculated by bootstrapping. Also illustrated is (iii) calculation of parameter confidence intervals and CDF confidence bands using both asymptotic theory and bootstrapping, and (iv) use of profile log-likelihood plots to display the form of the maximized log-likelihood and scatterplots for checking convergence to normality of estimated parameter distributions. Two different data sets are analysed. In the first, the generalized extreme value (GEVMin) distribution and its embedded model the simple extreme value (EVMin) are fitted to Kevlar-fibre breaking strength data. In the second sample, the four-parameter Burr XII distribution, its three-parameter embedded models, the GEVMin, Type II generalized logistic and Pareto and two-parameter embedded models, the EVMin and shifted exponential, are fitted to carbon-fibre strength data and compared.

This chapter evaluates the role of political economy and domestic institutional arrangements in the materialisation of hegemonic discourses. It examines whether the nature of political economy is sufficient to account for the materialisation process of hegemonic discourses and evaluates how useful is the ‘goodness-of-fit’ hypothesis found in Europeanisation studies. This chapter also analyses whether the combination of the nature of political economy and the nature of interest representation can account for the materialisation process of hegemonic discourses.

This chapter describes quantifying events in America and their historical context. The cotton gin is invented and has tremendous impact on the country, bringing sentiments of taxation and slavery to the fore, for state’s rights. Events leading to the American Civil War are described, as are other circumstances leading to the Industrial Revolution, first in England and then moving to America. Karl Pearson is introduced with description of his The Grammar of Science, as well as his approach to scholarship as first defining a philosophy of science, which has dominated much of scientific research from the time of the book’s publication to today. Pearson’s invention of the coefficient of correlation is described, and his other contributions to statistics are mentioned: standard deviation, skewness, kurtosis, and goodness of fit, as well as his formal introduction of the contingency table.

In this chapter, we focus on the use of Structural Equation Modeling (SEM) to compare path models across two or more cultural groups. SEM can be used to test the goodness of fit of a causal model, as well as to test equivalence of causal relationships among variables of interest across various cultural groups. We will demonstrate SEM through the use of Stata for these purposes. We begin with a rationale for path mode analysis, move on to provide context for the construction of theories in path models using SEM, and provide examples of SEM models for various cultural groups for comparison. We provide examples of Stata commands for examining differences in direct and indirect effects along with goodness of fit statistics across various cultural groups using SEM.

The chapter “Model Selection” provides a brief overview of alternative ways of thinking about what is (are) the best model(s) and shows the core motivation behind the Akaike information criterion as a measure of the predictive ability of a model fitted via maximum likelihood. It then gives stepwise practical guidance for using information theory as a basis of model selection, including nested versus non-nested models, goodness of fit, and overdispersion. Advanced topics cover some of the philosophy of information theory, other types of information theory criteria, and other ways of evaluating the predictive ability of models. As an example, the chapter examines the case of the Western monarch butterfly (Danaus plexippus plexippus), which, over the past two decades, has experienced a 97% decline from its historical average abundance, declining by 86% from 2017 to 2018 alone. Undoubtedly, there is more than one cause—indeed, overwintering habitat loss and pesticide use are both believed to be important contributors to the decline. Adopting a hypothesis-evaluation framework makes it possible to consider multiple alternative hypotheses simultaneously and measure degrees of support for alternative hypotheses.

The second chapter introduces the reader to some core principles of temperament and how these dimensions often combine to yield five different temperamental types (mellow, moderate, anxious, vigorous, and agitated). Instructions and tools will be provided to help the reader identify their child’s temperament traits and type as well as their own. The well-known goodness-of-fit theory will be described to support the idea for why it is so important to incorporate temperament when making good parenting choices. This chapter will also briefly cover topics related to sex differences and the causes of temperament. The basic knowledge about temperament acquired in this chapter will be applied to future chapters concerning specific parenting debates.

For the most part, the book up until this point has focused on comparative constructions which, together, may be understood as the core, “regular” comparatives. Yet a variety of other types have been purported to exist. The goal of this chapter is to re-examine whether that purported variety should bear on the analysis of comparative morphology proposed in the book. Ultimately, the chapter concludes that there is at least a bifurcation in the data, but shows how the relevant distinctions can be accounted for based on an interaction between abstract syntax and “what is measured” in the two cases. In particular, the chapter suggests that while “regular” comparatives involve the measure and comparison of entities that are, for the most part, idiosyncratic to the lexical category targeted, “categorizing” comparatives involve comparing the extent to which a given predication is accurate.

This chapter begins with a very successful demonstration that illustrates many of the general principles of statistical inference, including estimation, bias, and the concept of the sampling distribution. Students each take a “random” sample of different size candies, weigh them, and estimate the total weight of all candies. Then various demonstrations and examples are presented that take the students on the transition from probability to hypothesis testing, confidence intervals, and more advanced concepts such as statistical power and multiple comparisons. These activities include use an inflatable globe, short-term memory test, first digits of street addresses, and simulated student IQs.