Wolf Schwarz

The analysis of covariance (Ancova) is a widely used statistical technique for the comparison of groups with respect to a quantitative dependent variable in such a way that the comparison takes into account concomitant differences in a quantitative covariate. Despite its widespread use, some of the main features of this technique have remained elusive, contentious, or misconceived in applied settings. For example, some authors claim that the validity of an Ancova depends on the assumption that the expected value of the covariate is the same for all participants, or that the adjusted mean difference evaluated in an Ancova has a useful interpretation as the difference between the mean change scores of each group, whereas these claims are disputed by other authors. I suggest that these issues are best addressed and settled in the context of the underlying exact sampling distribution theory since significance statements, effect size estimates, and statistical power all derive directly from the statistical sampling distribution theory implied by the Ancova model. The distributional approach also clarifies the central distinction between conditional and marginal means, and the way in which various study designs (controlled, randomized, observational) affect and modify conclusions derived from an Ancova. The tutorial provides an explicit distributional account of the standard Ancova model to compare two independent groups; it clarifies the assumptions underlying the Ancova model, the nature and limitations of the conclusions it provides, and corrects some common misconceptions associated with its applications.

Le Her is a strategic two-person prize game the classic version of which uses a deck of cards. Despite its simple rules, the analysis of the game is surprisingly complex. I describe a modern variant of Le Her that is amenable to elementary analysis, yet retains and highlights all central aspects of the classical game. This elementary analysis combines naturally central concepts of calculus (partial derivatives, multidimensional optimization, double integrals, fixed and saddle points) with major probabilistic concepts (random variable, event, independence) and basic notions in decision theory (Nash equilibrium). Due to its stimulating context as a strategic prize game Le Her offers an attractive illustration and an appealing application of learning goals and curricular contents which are otherwise often perceived as remote and

In many applied single-point Yes/No signal-detection studies, the main interest is to evaluate the observer’s sensitivity, based on the observed rates of hits and false alarms. For example, Kostopoulou, Nurek, Cantarella et al. (2019, Medical Decision Making, 39, 21–31) presented general practitioners (GPs) with clinical vignettes of patients showing various cancer-related symptoms, and asked them to decide if urgent referral was required; the standard discrimination index d′ was calculated for each GP. An alternative conditional approach to statistical inference emphasizes explicitly the conditional nature of the inferences drawn, and argues on the basis of the response marginal (the number of “yes” responses) that was actually observed. It is closely related to, for example, Fisher’s exact test or the Rasch model in item response theory which have long been valuable and prominent in psychology. The conditional framework applied to single-point Yes/No detection studies is based on the noncentral hypergeometric sampling distribution and permits, for samples of any size, exact inference because it eliminates nuisance (i.e., bias) parameters by conditioning. We describe in detail how the conditional approach leads to conditional maximum likelihood sample estimates of sensitivity, and to exact confidence intervals for the underlying (log) odds ratio. We relate the conditional approach to classical (logistic) detection models also leading to analyses of the odds ratio, compare its statistical power to that of the unconditional approach, and conclude by discussing some of its pros and cons.

We describe a mathematically simple yet precise model of activation suppression that can explain the negative-going delta plots often observed in standard

We propose to interpret tasks evoking the classical Müller-Lyer illusion as one form of a conflict paradigm involving relevant (line length) and irrelevant (arrow orientation) stimulus attributes. Eight practiced observers compared the lengths of two line–arrow combinations; the length of the lines and the orientation of their arrows was varied unpredictably across trials so as to obtain psychometric and chronometric functions for congruent and incongruent line–arrow combinations. To account for decision speed and accuracy in this parametric data set, we present a diffusion model based on two assumptions: inward (outward)-pointing arrows added to a line (i) add (subtract) a separate, task-irrelevant drift component, and (ii) they reduce (increase) the distance to the barrier associated with the response identifying this line as being longer. The model was fitted to the data of each observer separately, and accounted in considerable quantitative detail for many aspects of the data obtained, including the fact that arrow-congruent responses were most prominent in the earliest RT quartile-bin. Our model gives a specific, process-related meaning to traditional static interpretations of the Müller-Lyer illusion, and combines within a single coherent framework structural and strategic mechanisms contributing to the illusion. Its central assumptions correspond to the general interpretation of geometrical-optical illusions as a manifestation of the resolution of a perceptual conflict (Day & Smith, 1989; Westheimer, 2008).

Following the classical work of Moyer and Landauer (1967), experimental studies investigating the way in which humans process and compare symbolic numerical information regularly used one of two...

When objects are manually lifted to compare their weight, then smaller objects are judged to be heavier than larger objects of the same physical weights: the classical size–weight illusion (Gregory, 2004 ). It is also well established that increasing numerical magnitude is strongly associated with increasing physical size: the number–size congruency effect e.g., (Besner & Coltheart Neuropsychologia, 17, 467–472 1979 ); Henik & Tzelgov Memory & Cognition, 10, 389–395 1982 ). The present study investigates the question suggested by combining these two classical effects: if smaller numbers are associated with smaller size, and objects of smaller size appear heavier, then are numbered objects (balls) of equal weight and size also judged as heavier when they carry smaller numbers? We present two experiments testing this hypothesis for weight comparisons of numbered (1 to 9) balls of equal size and weight, and report results which largely conform to an interpretation in terms of a new “number–weight illusion”.

Using a standard repeated measures model with arbitrary true score distribution and normal error variables, we present some fundamental closed‐form results which explicitly indicate the conditions under which regression effects towards ( RTM ) and away from the mean are expected. Specifically, we show that for skewed and bimodal distributions many or even most cases will show a regression effect that is in expectation away from the mean, or that is not just towards but actually beyond the mean. We illustrate our results in quantitative detail with typical examples from experimental and biometric applications, which exhibit a clear regression away from the mean (‘egression from the mean’) signature. We aim not to repeal cautionary advice against potential RTM effects, but to present a balanced view of regression effects, based on a clear identification of the conditions governing the form that regression effects take in repeated measures designs.

Physical size modulates the efficiency of digit comparison, depending on whether the relation of numerical magnitude and physical size is congruent or inco

We present two experiments in which participants classify stimuli having two potentially conflicting attributes, one of which is response-relevant whereas