I consider the error-statistical account as both a theory of evidence and as a theory of inference. I seek to show how inferences regarding the truth of hypotheses can be upheld by avoiding a certain kind of alternative hypothesis problem. In addition to the testing of assumptions behind the experimental model, I discuss the role of judgments of implausibility. A benefit of my analysis is that it reveals a continuity in the application of error-statistical assessment to low-level empirical (...) hypotheses and highly general theoretical principles. This last point is illustrated with a brief sketch of the issues involved in the parametric framework analysis of tests of physical theories such as General Relativity and of fundamental physical principles such as the Einstein Equivalence Principle. (shrink)

The error statistical account of testing uses statistical considerations, not to provide a measure of probability of hypotheses, but to model patterns of irregularity that are useful for controlling, distinguishing, and learning from errors. The aim of this paper is (1) to explain the main points of contrast between the error statistical and the subjective Bayesian approach and (2) to elucidate the key errors that underlie the central objection raised by Colin Howson at our PSA 96 Symposium.

In empirical modeling, an important desiderata for deeming theoretical entities and processes as real is that they can be reproducible in a statistical sense. Current day crises regarding replicability in science intertwines with the question of how statistical methods link data to statistical and substantive theories and models. Different answers to this question have important methodological consequences for inference, which are intertwined with a contrast between the ontological commitments of the two types of models. The key to untangling them is (...) the realization that behind every substantive model there is a statistical model that pertains exclusively to the probabilistic assumptions imposed on the data. It is not that the methodology determines whether to be a realist about entities and processes in a substantive field. It is rather that the substantive and statistical models refer to different entities and processes, and therefore call for different criteria of adequacy. (shrink)

Error is protean, ubiquitous and crucial in scientific process. In this paper it is argued that understanding scientific process requires what is currently absent: an adaptable, context-sensitive functional role for error in science that naturally harnesses error identification and avoidance to positive, success-driven, science. This paper develops a new account of scientific process of this sort, error and success driving Self-Directed Anticipative Learning (SDAL) cycling, using a recent re-analysis of ape-language research as test example. The example (...) shows the limitations of other accounts of error, in particular Mayo’s (Error and the growth of experimental knowledge, 1996) error-statistical approach, and SDAL cycling shows how they can be fruitfully contextualised. (shrink)

No one has a well developed solution to Duhem's problem, the problem of how experimental evidence warrants revision of our theories. Deborah Mayo proposes a solution to Duhem's problem in route to her more ambitious program of providing a philosophical account of inductive inference and experimental knowledge. This paper is a response to Mayo's Error Statistics (ES) program, paying particular attention to her response to Duhem's problem. It turns out that Mayo's purported solution to Duhem's problem is very (...) significant to her project, for the epistemic license claimed by ES and the philosophical underpinnings to her account of experimental knowledge depend on this solution. By introducing the partition problem, I argue that ES fails to solve Duhem's problem and therefore fails to provide an adequate account of experimental knowledge. (shrink)

We argue for a naturalistic account for appraising scientific methods that carries non-trivial normative force. We develop our approach by comparison with Laudan’s (American Philosophical Quarterly 24:19–31, 1987, Philosophy of Science 57:20–33, 1990) “normative naturalism” based on correlating means (various scientific methods) with ends (e.g., reliability). We argue that such a meta-methodology based on means–ends correlations is unreliable and cannot achieve its normative goals. We suggest another approach for meta-methodology based on a conglomeration of tools and strategies (from statistical modeling, (...) experimental design, and related fields) that affords forward looking procedures for learning from error and for controlling error. The resulting “error statistical” appraisal is empirical—methods are appraised by examining their capacities to control error. At the same time, this account is normative, in that the strategies that pass muster are claims about how actually to proceed in given contexts to reach reliable inferences from limited data. (shrink)

This paper analyzes Deborah Mayo's error-statistical (ES) account of scientific evidence in order to clarify the kinds of "material postulates" it requires and to explain how those assumptions function. A secondary aim is to explain and illustrate the importance of the security of an inference. After finding that, on the most straightforward reading of the ES account, it does not succeed in its stated aims, two remedies are considered: either relativize evidence claims or introduce stronger assumptions. The choice between (...) these approaches turns on the value attached to two aims of inquiry that are in tension: drawing strong, informative conclusions and reasoning securely. (shrink)

Many biologists, especially in ecology and evolution, analyze their data by estimating fits to a set of candidate models and selecting the best model according to the Akaike Information Criterion (AIC) or the Bayesian Information Criteria (BIC). When the candidate models represent alternative hypotheses, biologists may want to limit the chance of a false positive to a specified level. Existing model selection methodology, however, allows for only indirect control over error rates by setting a threshold for the difference in (...) AIC scores. We present a novel theoretical framework for parametric Neyman-Pearson (NP) model selection using information criteria that does not require a pre-data null and applies to three or more non-nested models simultaneously. We apply the theoretical framework to the Error Control for Information Criteria (ECIC) procedure introduced by Cullan et al. (J Appl Stat 47: 2565–2581, 2019), and we show it shares many of the desirable properties of AIC-type methods, including false positive and negative rates that converge to zero asymptotically. We discuss implications for the compatibility of evidentialist and severity-based approach to evidence in philosophy of science. (shrink)

After showing how Deborah Mayo’s error-statistical philosophy of science might be applied to address important questions about the evidential status of computer simulation results, I argue that an error-statistical perspective offers an interesting new way of thinking about computer simulation models and has the potential to significantly improve the practice of simulation model evaluation. Though intended primarily as a contribution to the epistemology of simulation, the analysis also serves to fill in details of Mayo’s epistemology of experiment.

In seeking general accounts of evidence, confirmation, or inference, philosophers have looked to logical relationships between evidence and hypotheses. Such logics of evidential relationship, whether hypothetico-deductive, Bayesian, or instantiationist fail to capture or be relevant to scientific practice. They require information that scientists do not generally have (e.g., an exhaustive set of hypotheses), while lacking slots within which to include considerations to which scientists regularly appeal (e.g., error probabilities). Building on my co-symposiasts contributions, I suggest some directions in which (...) a new and more adequate philosophy of evidence can move. (shrink)

Error statistics is an important methodological view in philosophy of statistics and philosophy of science that can be applied to scientific experiments such as clinical trials. In this paper, I raise two potential issues for ES when it comes to guiding, and explaining early stopping of randomized controlled trials : ES provides limited guidance in cases of early unfavorable trends due to the possibility of trend reversal; ES is silent on how to prospectively control error rates (...) in experiments requiring multiple interim analyses. The method of conditional power, together with a rationing principle for RCTs, can assist ES in addressing such issues. (shrink)

I argue that the Bayesian Way of reconstructing Duhem's problem fails to advance a solution to the problem of which of a group of hypotheses ought to be rejected or "blamed" when experiment disagrees with prediction. But scientists do regularly tackle and often enough solve Duhemian problems. When they do, they employ a logic and methodology which may be called error statistics. I discuss the key properties of this approach which enable it to split off the task of (...) testing auxiliary hypotheses from that of appraising a primary hypothesis. By discriminating patterns of error, this approach can at least block, if not also severely test, attempted explanations of an anomaly. I illustrate how this approach directs progress with Duhemian problems and explains how scientists actually grapple with them. (shrink)

The main aim of this paper is to revisit the curve fitting problem using the reliability of inductive inference as a primary criterion for the ‘fittest' curve. Viewed from this perspective, it is argued that a crucial concern with the current framework for addressing the curve fitting problem is, on the one hand, the undue influence of the mathematical approximation perspective, and on the other, the insufficient attention paid to the statistical modeling aspects of the problem. Using goodness-of-fit as the (...) primary criterion for ‘best', the mathematical approximation perspective undermines the reliability of inference objective by giving rise to selection rules which pay insufficient attention to ‘accounting for the regularities in the data'. A more appropriate framework is offered by the error-statistical approach, where (i) statistical adequacy provides the criterion for assessing when a curve captures the regularities in the data adequately, and (ii) the relevant error probabilities can be used to assess the reliability of inductive inference. Broadly speaking, the fittest curve (statistically adequate) is not determined by the smallness if its residuals, tempered by simplicity or other pragmatic criteria, but by the nonsystematic (e.g. white noise) nature of its residuals. The advocated error-statistical arguments are illustrated by comparing the Kepler and Ptolemaic models on empirical grounds. ‡I am grateful to Deborah Mayo and Clark Glymour for many valuable suggestions and comments on an earlier draft of the paper; estimating the Ptolemaic model was the result of Glymour's prompting and encouragement. †To contact the author, please write to: Department of Economics, Virginia Tech 3019 Pamplin Hall (0316), Blacksburg, VA 24061; e-mail: [email protected]. (shrink)

Experimenters sometimes insist that it is unwise to examine data before determining how to analyze them, as it creates the potential for biased results. I explore the rationale behind this methodological guideline from the standpoint of an error statistical theory of evidence, and I discuss a method of evaluating evidence in some contexts when this predesignation rule has been violated. I illustrate the problem of potential bias, and the method by which it may be addressed, with an example from (...) the search for the top quark. A point in favor of the error statistical theory is its ability, demonstrated here, to explicate such methodological problems and suggest solutions, within the framework of an objective theory of evidence. (shrink)

This chapter seeks to recover an approach to consciousness from a general theory of brain function, namely the prediction error minimization theory. The way this theory applies to mental and developmental disorder demonstrates its relevance to consciousness. The resulting view is discussed in relation to a contemporary theory of consciousness, namely the idea that conscious perception depends on Bayesian metacognition; this theory is also supported by considerations of psychopathology. This Bayesian theory is first disconnected from the higher-order thought theory, (...) and then, via a prediction error conception of action, connected instead to the global neuronal workspace theory. Considerations of mental and developmental disorder therefore show that a very general theory of brain function is relevant to explaining the structure of conscious perception; furthermore, this theory can subsume and unify two contemporary approaches to consciousness, in a move that seeks to elucidate the fundamental mechanism for selection of representational content into consciousness. (shrink)

In the United States, the racial and ethnic statistics published by the National Center for Health Statistics (NCHS) assume that each member of the U.S. population has a race and ethnicity and that if a member is black or white with respect to his risk of one disease, he is the same race with respect to his risk of another. Such an assumption is mistaken. Race and ethnicity are taken by the NCHS to be an intrinsic property of (...) members of a population, when they should be taken to depend on interest. The actual or underlying race or ethnicity of members of a population depends on the risk whose variation within the population we wish to describe or explain. (shrink)

The AI ethics of statistical fairness is an error, the approach should be abandoned, and the accumulated academic work deleted. The argument proceeds by identifying four recurring mistakes within statistical fairness. One conflates fairness with equality, which confines thinking to similars being treated similarly. The second and third errors derive from a perspectival ethical view which functions by negating others and their viewpoints. The final mistake constrains fairness to work within predefined social groups instead of allowing unconstrained fairness to (...) subsequently define group composition. From the nature of these misconceptions, the larger argument follows. Because the errors are integral to how statistical fairness works, attempting to resolve the difficulties only deepens them. Consequently, the errors cannot be corrected without undermining the larger project, and statistical fairness collapses from within. While the collapse ends a failure in ethics, it also provokes distinct possibilities for fairness, data, and algorithms. Quickly indicating some of these directions is a secondary aim of the paper, and one that aligns with what fairness has consistently meant and done since Aristotle. (shrink)

This text provides a critique of the subjective Bayesian view of statistical inference, and proposes the author's own error-statistical approach as an alternative framework for the epistemology of experiment. It seeks to address the needs of researchers who work with statistical analysis.

Three new versions of Wason’s 2-4-6 rule discovery task incorporating error rates or feedback of uncertainty reduction, inspired by the error-statistical account in philosophy of science, were employed. In experiments 1 and 2, participants were instructed that some experimenter feedback would be erroneous (control was original 2-4-6 without error). The results showed that performance was impaired when there was probabilistic error. In experiment 3, participants were given uncertainty reduction feedback as they generated different number triples and (...) the negative effects of probabilistic error were not observed. These findings are informative not only about rule discovery tasks in general but also about contexts of inference under uncertainty. (shrink)

When Kangas suggested in 1986 that wildlife reserve designs could be much smaller than previously thought, community ecologists attacked his views on methodological grounds (island biogeographical theory is beset with uncertainties) and on conservation grounds (Kangas seemed to encourage deforestation and extinction). Kangas' defenders, like Simberloff, argued that in a situation of biological uncertainty (the degree/type of deforestation-induced extinction), scientists ought to follow the epistemologically conservative course and risk type-II error (the risk of not rejecting a null hypothesis that (...) is false), rather than type-I error. (This is the risk of rejecting a null hypothesis that is true). Kangas' opponents, like Noss, argued that, in a situation of scientific uncertainty, scientists ought to risk type-I, rather than type-II, error. This essay argues that there are different types of rationality appropriate to science and applied science and, therefore, in cases of applied science (like conservation biology), the more conservative course of action is for scientists to risk type-I error. The essay argues further that, on grounds of scientific rationality, Kangas, Simberloff, and others were correct in risking type-II error, but that, on grounds of decision-theoretic rationality, Noss, Waide, and others were correct in risking type-I error. (shrink)

One commits the availability error when one pays attention to, or is compelled by, the readily available evidence – the evidence that is obvious, memorable, or psychologically compelling – instead of taking into account all the evidence or the reliable evidence. This chapter focuses on one of the common fallacies in Western philosophy called availability error. The availability error contributes to confirmation bias, the tendency to only pay attention to the evidence that confirms what we believe and (...) ignore the evidence that doesn't. The availability error also encourages other fallacies, such as hasty generalization – where you generalize about an entire group based on a statistically insignificant portion of that group. In fact, memory is especially susceptible to the availability error because we are apt to remember only what is memorable, and that is often only what is psychologically compelling. (shrink)

A main message from the causal modelling literature in the last several decades is that under some plausible assumptions, there can be statistically consistent procedures for inferring (features of) the causal structure of a set of random variables from observational data. But whether we can control the error probabilities with a finite sample size depends on the kind of consistency the procedures can achieve. It has been shown that in general, under the standard causal Markov and Faithfulness assumptions, the (...) procedures can only be pointwise but not uniformly consistent without substantial background knowledge. This implies the impossibility of choosing a finite sample size to control the worst case error probabilities. In this paper, I consider the simpler task of inferring causal directions when the skeleton of the causal structure is known, and establish a similarly negative result concerning the possibility of controlling error probabilities. Although the result is negative in form, it has an interesting positive implication for causal discovery methods. (shrink)

An important theme to have emerged from the new experimentalist movement is that much of actual scientific practice deals not with appraising full-blown theories but with the manifold local tasks required to arrive at data, distinguish fact from artifact, and estimate backgrounds. Still, no program for working out a philosophy of experiment based on this recognition has been demarcated. I suggest why the new experimentalism has come up short, and propose a remedy appealing to the practice of standard error (...) statistics. I illustrate a portion of my proposal using Galison's experimental narrative on neutral currents. (shrink)

Recently, the practice of deciding legal cases on purely statistical evidence has been widely criticised. Many feel uncomfortable with finding someone guilty on the basis of bare probabilities, even though the chance of error might be stupendously small. This is an important issue: with the rise of DNA profiling, courts are increasingly faced with purely statistical evidence. A prominent line of argument—endorsed by Blome-Tillmann 2017; Smith 2018; and Littlejohn 2018—rejects the use of such evidence by appealing to epistemic norms (...) that apply to individual inquirers. My aim in this paper is to rehabilitate purely statistical evidence by arguing that, given the broader aims of legal systems, there are scenarios in which relying on such evidence is appropriate. Along the way I explain why popular arguments appealing to individual epistemic norms to reject legal reliance on bare statistics are unconvincing, by showing that courts and individuals face different epistemic predicaments (in short, individuals can hedge when confronted with statistical evidence, whilst legal tribunals cannot). I also correct some misconceptions about legal practice that have found their way into the recent literature. (shrink)

Statistical Default Logic (SDL) is an expansion of classical (i.e., Reiter) default logic that allows us to model common inference patterns found in standard inferential statistics, e.g., hypothesis testing and the estimation of a population‘s mean, variance and proportions. This paper presents an embedding of an important subset of SDL theories, called literal statistical default theories, into stable model semantics. The embedding is designed to compute the signature set of literals that uniquely distinguishes each extension on a statistical default (...) theory at a pre-assigned error-bound probability. (shrink)

A question, long discussed by legal scholars, has recently provoked a considerable amount of philosophical attention: ‘Is it ever appropriate to base a legal verdict on statistical evidence alone?’ Many philosophers who have considered this question reject legal reliance on bare statistics, even when the odds of error are extremely low. This paper develops a puzzle for the dominant theories concerning why we should eschew bare statistics. Namely, there seem to be compelling scenarios in which there are (...) multiple sources of incriminating statistical evidence. As we conjoin together different types of statistical evidence, it becomes increasingly incredible to suppose that a positive verdict would be impermissible. I suggest that none of the dominant views in the literature can easily accommodate such cases, and close by offering a diagnosis of my own. (shrink)

Over the last twenty-five years, climate scientists working on the attribution of climate change to humans have developed increasingly sophisticated statistical models in a process that can be understood as a kind of calibration: the gradual changes to the statistical models employed in attribution studies served as iterative revisions to a measurement(-like) procedure motivated primarily by the aim of neutralizing particularly troublesome sources of error or uncertainty. This practice is in keeping with recent work on the evaluation of models (...) more generally that views models as tools for particular tasks: what drives the process is the desire for models that provide more reliable grounds for inference rather than accuracy to the underlying mechanisms of data-generation. (shrink)

This paper is concerned with the role of rational belief change theory in the philosophical understanding of experimental error. Today, philosophers seek insight about error in the investigation of specific experiments, rather than in general theories. Nevertheless, rational belief change theory adds to our understanding of just such cases: R. A. Fisher’s criticism of Mendel’s experiments being a case in point. After an historical introduction, the main part of this paper investigates Fisher’s paper from the point of view (...) of rational belief change theory: what changes of belief about Mendel’s experiment does Fisher go through and with what justification. It leads to surprising insights about what Fisher had done right and wrong, and, more generally, about the limits of statistical methods in detecting error. (shrink)

Psychological research has shown that cognitive illusions, of which visual illusions are just a special case, are systematic and pervasive, raising epistemological questions about how error in all forms of research can be identified and eliminated. The quantitative sciences make use of statistical techniques for this purpose, but it is not clear what the qualitative equivalent is, particularly in view of widespread scepticism about validity and objectivity. I argue that, in the light of cognitive psychology, the ‘error question’ (...) cannot be dismissed as a positivist obsession, and that the concepts of truth and objectivity are unavoidable. However, they constitute only a ‘minimal realism’, which does not necessarily bring a commitment to ‘absolute’ truth, certainty, correspondence, causation, reductionism, or universal laws in its wake. The assumption that it does reflects a misreading of positivism and, ironically, precipitates a ‘crisis of legitimation and representation’, as described by constructivist authors. (shrink)

Three new versions of Wason’s 2-4-6 rule discovery task incorporating error rates or feedback of uncertainty reduction, inspired by the error-statistical account in philosophy of science, were employed. In experiments 1 and 2, participants were instructed that some experimenter feedback would be erroneous. The results showed that performance was impaired when there was probabilistic error. In experiment 3, participants were given uncertainty reduction feedback as they generated different number triples and the negative effects of probabilistic error (...) were not observed. These findings are informative not only about rule discovery tasks in general but also about contexts of inference under uncertainty. (shrink)

Statistical model specification and validation raise crucial foundational problems whose pertinent resolution holds the key to learning from data by securing the reliability of frequentist inference. The paper questions the judiciousness of several current practices, including the theory-driven approach, and the Akaike-type model selection procedures, arguing that they often lead to unreliable inferences. This is primarily due to the fact that goodness-of-fit/prediction measures and other substantive and pragmatic criteria are of questionable value when the estimated model is statistically misspecified. Foisting (...) one's favorite model on the data often yields estimated models which are both statistically and substantively misspecified, but one has no way to delineate between the two sources of error and apportion blame. The paper argues that the error statistical approach can address this Duhemian ambiguity by distinguishing between statistical and substantive premises and viewing empirical modeling in a piecemeal way with a view to delineate the various issues more effectively. It is also argued that Hendry's general to specific procedures does a much better job in model selection than the theory-driven and the Akaike-type procedures primary because of its error statistical underpinnings. (shrink)

Although both philosophers and scientists are interested in how to obtain reliable knowledge in the face of error, there is a gap between their perspectives that has been an obstacle to progress. By means of a series of exchanges between the editors and leaders from the philosophy of science, statistics and economics, this volume offers a cumulative introduction connecting problems of traditional philosophy of science to problems of inference in statistical and empirical modelling practice. Philosophers of science and (...) scientific practitioners are challenged to reevaluate the assumptions of their own theories - philosophical or methodological. Practitioners may better appreciate the foundational issues around which their questions revolve and thereby become better 'applied philosophers'. Conversely, new avenues emerge for finally solving recalcitrant philosophical problems of induction, explanation and theory testing. (shrink)

Conflicting results undermine making inferences from the empirical literature. So far, the replication crisis is mainly seen as resulting from honest errors and questionable research practices such as p-hacking or the base-rate fallacy. I discuss the malleability (researcher degrees of freedom) of quantitative research and argue that conflicting results can emerge from two studies using different but plausible designs (e.g., eligibility criteria, operationalization of concepts, outcome measures) and statistical methods. I also explore how the choices regarding study design and statistical (...) techniques bias results in a way that makes them more or less relevant for a given policy or clinical question. (shrink)

: Bayesians and error statisticians have relied heavily upon examples from physics in developing their accounts of scientific inference. The present essay demonstrates it is possible to analyze H.B.D. Kettlewell's classic study of natural selection from Deborah Mayo's error statistical point of view (Mayo 1996). A comparison with a previous analysis of this episode from a Bayesian perspective (Rudge 1998) reveals that the error statistical account makes better sense of investigations such as Kettlewell's because it clarifies how (...) core elements in the design of experiments are used to minimize erroneous inferences rather than dwelling on whether the strategies used are reasonable. (shrink)

The null-hypothesis significance-test procedure (NHSTP) is defended in the context of the theory-corroboration experiment, as well as the following contrasts: (a) substantive hypotheses versus statistical hypotheses, (b) theory corroboration versus statistical hypothesis testing, (c) theoretical inference versus statistical decision, (d) experiments versus nonexperimental studies, and (e) theory corroboration versus treatment assessment. The null hypothesis can be true because it is the hypothesis that errors are randomly distributed in data. Moreover, the null hypothesis is never used as a categorical proposition. Statistical (...) significance means only that chance influences can be excluded as an explanation of data; it does not identify the nonchance factor responsible. The experimental conclusion is drawn with the inductive principle underlying the experimental design. A chain of deductive arguments gives rise to the theoretical conclusion via the experimental conclusion. The anomalous relationship between statistical significance and the effect size often used to criticize NHSTP is more apparent than real. The absolute size of the effect is not an index of evidential support for the substantive hypothesis. Nor is the effect size, by itself, informative as to the practical importance of the research result. Being a conditional probability, statistical power cannot be the apriori probability of statistical significance. The validity of statistical power is debatable because statistical significance is determined with a single sampling distribution of the test statistic based on H0, whereas it takes two distributions to represent statistical power or effect size. Sample size should not be determined in the mechanical manner envisaged in power analysis. It is inappropriate to criticize NHSTP for nonstatistical reasons. At the same time, neither effect size, nor confidence interval estimate, nor posterior probability can be used to exclude chance as an explanation of data. Neither can any of them fulfill the nonstatistical functions expected of them by critics. (shrink)

The replication crisis has prompted many to call for statistical reform within the psychological sciences. Here we examine issues within Frequentist statistics that may have led to the replication crisis, and we examine the alternative—Bayesian statistics—that many have suggested as a replacement. The Frequentist approach and the Bayesian approach offer radically different perspectives on evidence and inference with the Frequentist approach prioritising error control and the Bayesian approach offering a formal method for quantifying the relative strength of (...) evidence for hypotheses. We suggest that rather than mere statistical reform, what is needed is a better understanding of the different modes of statistical inference and a better understanding of how statistical inference relates to scientific inference. (shrink)

This paper is an extended review of the book Error and Inference, edited by Deborah Mayo and Aris Spanos, about their frequentist and philosophical perspective on testing of hypothesis and on the criticisms of alternatives like the Bayesian approach.

This paper recommends how authors of statistical studies can communicate to general audiences fully, clearly, and comfortably. The studies may use statistical methods to explore issues in science, engineering, and society or they may address issues in statistics specifically. In either case, readers without explicit statistical training should have no problem understanding the issues, the methods, or the results at a non-technical level. The arguments for those results should be clear, logical, and persuasive. This paper also provides advice for (...) editors of general journals on selecting high quality statistical articles without the need for exceptional work or expense. Finally, readers are also advised to watch out for some common errors or misuses of statistics that can be detected without a technical statistical background. (shrink)