Chapter 6 demonstrates one way that RIO can be used for exploratory data analysis: identifying statistically significant interaction terms. We show how exploring the relationships among cases offers important insights into the relationships between variables.

In this chapter, we discuss the use of simulations for clinical trials. Simulation in statistics generally refers to repeated analyses of randomly generated datasets with known properties. Clinical trial simulation is required to explore, compare, and characterise operating characteristics and statistical properties of adaptive and other innovative trials with complex designs. Clinical trial simulation is an important tool that allows for comparison of different design choices during the planning stage to enhance the quality and feasibility of the trial. While simulations are most frequently used in adaptive and other complex trial designs, they can be applied to fixed trial designs.

Quantifying the multiscale hydraulic heterogeneity in aquifers and their effects on solute transport is the task of this chapter. Using spatial statistics, we explain how to quantify spatial variability of hydraulic properties or parameters in the aquifer using the stochastic or random field concept. In particular, we discuss spatial covariance, variogram, statistical homogeneity, heterogeneity, isotropy, and anisotropy concepts. Field examples complement the discussion. We then present a highly parameterized heterogeneous media (HPHM) approach for simulating flow and solute transport in aquifers with spatially varying hydraulic properties to meet our interest and observation scale. However, our limited ability to collect the needed information for this approach promotes alternatives such as Monte Carlo simulation, zonation, and equivalent homogeneous media (EHM) approaches with macrodispersion approaches. This chapter details the EHM with the macordispersion concept.

This short chapter recapitulates the substantive advances made by the individual chapters in this volume before closing remarks on the difference between using probability to represent epistemic uncertainty and modelling variability, two exercises that are easily confused, and on the use of models to answer historical questions.

An intense debate has arisen among scholars concerning the financial sustainability of the grain funds that Greek and Roman cities used to cope with the instabilities of the grain market. In this paper, we apply a Monte Carlo simulation to model their financial dynamics. Due to the uncertainties pertaining to the scope of such funds (targeting urban dwellers only or including rural residents), our model takes into account two scenarios: ‘optimistic’ (urban only) and ‘pessimistic’ (both urban and rural). The analysis reaches several important findings: (1) For both scenarios, we witness a considerable rate of funds collapsing in their first 10 years of operation. After 10 years, however, the probability of failure displays very little change, as if there was a threshold over which the funds had accumulated enough capital to withstand shortages. (2) As expected, the survival rates are significantly higher for the optimistic scenario. (3) The withdrawals seem to have the most dramatic impact on the dynamic of the fund. Overall, while the grain funds do not appear to be sustainable in the urban-rural scenario, they show clear signs of sustainability in the urban-only scenario. The results invite reconsideration of the widespread view that grain funds were an inefficient and precarious response to food crisis.

This chapter introduces the concepts and methods used by the other chapters in the volume, using the long-standing problem of estimating the land carrying capacity of classical Attica to illustrate the benefits of probabilistic modelling. We begin by surveying the development of techniques for managing uncertainty in ancient history (1.1) and past work on the specific problem of Attica’s land carrying capacity (1.2). The chapter then turns to theoretical questions about the nature of uncertainty and probability (1.3), introducing the ‘subjectivist’ conception of probability as degree of belief, a theoretical framework that makes probability a powerful tool for historians. We go on to discuss the procedure of using probability distributions to represent uncertainty about the actual value of a quantity such as average barley yield in ancient Attica (among other variables relevant to the problem of land carrying capacity) (1.4), the need to be aware of cognitive biases that distort our probability judgements (1.5), the use of Monte Carlo simulation to combine uncertainties (1.6), the potential problem of epistemic interdependence (1.7), the interpretation of the outputs of a Monte Carlo simulation (1.8), and the use of sensitivity analysis to identify the most important sources of uncertainty in a simulation. The appendix illustrates model code in R.

This chapter presents a new estimate of the value of coinage in circulation in the mid-second century CE Roman empire. More than 25 years ago, Richard Duncan-Jones revolutionized ancient economic history by offering a first projection through numismatic and statistical methods. At more than 20 bn sesterces, his estimate implied an anomalously high monetization ratio given past and current estimates of Roman GDP, an issue that economic historians have had to deal with ever since. In the first half, a review of the numismatic evidence points to a much smaller role for gold than posited by Duncan-Jones. The second half presents a new model of the money supply. It uses Monte Carlo simulation to estimate the value of centrally-minted precious metal coins produced annually under Hadrian, and then the total coinage in circulation ca 160 CE. Allowance for various uncertainties and other, minor components of the coinage suggests a money supply of around 16 bn sesterces, with less gold and more silver than expected. However, this is not quite low enough to explain away the monetization ratio, implying a higher GDP and more trade-oriented economy than currently thought. Two appendices contain lengthy but essential technical discussions of the assumptions in the estimate.

This chapter argues that wealth was probably not the primary barrier for Pompeiians to enter the Roman senate. One oddity about the historical record of Pompeii is that it reveals not a single certain senator in the imperial period. A previous reconstruction of the local distribution of income offers a possible explanation: it predicts that there were no Pompeian households with a senatorial income, suggesting that a lack of wealth kept the Pompeiians outside the senate. However, a new reconstruction of the top part of the Pompeian wealth distribution suggests the opposite. This reconstruction is based on combining the archaeological remains of the intramural housing stock with an econometric model which assumes that the distribution of elite wealth follows a distinct mathematical function – a power law. Even though this type of cliometric modelling is pervaded by uncertainties, with the help of probabilistic calculations it is possible to conclude that at least several Pompeian households held enough wealth to satisfy the senatorial census qualification, implying that wealth may not have been the primary barrier preventing Pompeiians from embarking on a senatorial career.