146 research outputs found
Randomization test of mean is compuationally inaccessible when the number of groups exceeds two
With the advent of fast computers, the randomization test of mean (also called the permutation test) received some attention in the recent years. Here we show that the randomization test is possible only for two-group design; comparing three groups requires a number of permutations so vast that even three groups of ten participants is beyond the current capabilities of modern computers. Further, we show that the rate of increase in the number of permutation is so large that simply adding one more participant per group to the data results in a computation time increased by at least one order of magnitude (in the three-group design) or more. Hence, the exhaustive randomization test may never be a viable alternative to ANOVAs
Using Mathematica within E-Prime
When programming complex experiments (for example, involving the generation of stimuli online), the traditional experiment programming software are not well equipped. One solution is to give up entirely the use of such software in favor of a low-level programming language. Here we show how E-Prime can be connected to Mathematica so that the easiness and reliability of this software can be preserved while at the same time granting it the full computational power of a high-level programming language. As an example, we show how to generate noisy images with noise proportional to the rate of success of the participants with as few as 12 lines of codes in E-Prime
A semantic method to prove strong normalization from weak normalization
Deduction modulo is an extension of first-order predicate logic where axioms are replaced by rewrite rules and where many theories, such as arithmetic, simple type theory and some variants of set theory, can be expressed. An important question in deduction modulo is to find a condition of the theories that have the strong normalization property. In a previous paper we proposed a refinement of the notion of model for theories expressed in deduction modulo, in a way allowing not only to prove soundness, but also completeness: a theory has the strong normalization property if and only if it has a model of this form. In this paper, we present how we can use these techniques to prove that all weakly normalizing theories expressed in minimal deduction modulo, are strongly normalizing
TLA+ Proofs
TLA+ is a specification language based on standard set theory and temporal
logic that has constructs for hierarchical proofs. We describe how to write
TLA+ proofs and check them with TLAPS, the TLA+ Proof System. We use Peterson's
mutual exclusion algorithm as a simple example to describe the features of
TLAPS and show how it and the Toolbox (an IDE for TLA+) help users to manage
large, complex proofs.Comment: A shorter version of this article appeared in the proceedings of the
conference Formal Methods 2012 (FM 2012, Paris, France, Springer LNCS 7436,
pp. 147-154
Embedding Pure Type Systems in the lambda-Pi-calculus modulo
The lambda-Pi-calculus allows to express proofs of minimal predicate logic.
It can be extended, in a very simple way, by adding computation rules. This
leads to the lambda-Pi-calculus modulo. We show in this paper that this simple
extension is surprisingly expressive and, in particular, that all functional
Pure Type Systems, such as the system F, or the Calculus of Constructions, can
be embedded in it. And, moreover, that this embedding is conservative under
termination hypothesis
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Boundaries of Creativity: Thick or Thin Organization?
Semantic organization of knowledge has a long history in theories of creativity. Flexibility of thinking and distant connec-tions are indispensable elements of a creative network. Simultaneously, convergence of thoughts and evaluation of ideas areessential at many stages of the creative process. The current study evaluates these complementary aspects through the lensof an exploratory concept known as mental boundaries. Correlation analyses are used to compare flexible and rigid ten-dencies of organizing the world, the concepts of intellect, schizotypy, perfectionism, divergent thinking and self-perceivedcreativity. Results (n = 316) reveal an interesting contrasting pattern where divergent thinking is significantly related toflexible internal and external organizations, whereas self-perceived creativity is significantly related to rigid external andnon-significantly related to rigid internal organizations. The present findings have implications for the measurement ofcreativity and the identification of factors that facilitate the creative process
Analysis of frequency data: The ANOFA framework
Analyses of frequencies are commonly done using a chi-square test. This test, derived from a normal approximation, is deemed generally efficient (controlling type-I error rates fairly well and having good statistical power). However, in the case of factorial designs, it is difficult to decompose a total test statistic into additive interaction effects and main effects. Herein, we present an alternative test based on the statistic. The test has similar type-I error rates and power as the former one. However, it is based on a total statistic that is naturally decomposed additively into interaction effects, main effects, simple effects, contrast effects, etc., mimicking precisely the logic of ANOVAs. We call this set of tools ANOFA (Analysis of Frequency data) to highlight its similarities with ANOVA. We also examine how to render plots of frequencies along with confidence intervals. Finally, quantifying effect sizes and planning statistical power are described under this framework. The ANOFA is a tool that assesses the significance of effects instead of the significance of parameters; as such, it is more intuitive to most researchers than alternative approaches based on generalized linear models
A review of effect sizes and their confidence intervals, Part {I}: The Cohen's d family
Effect sizes and confidence intervals are important statistics to assess the magnitude and the precision of an effect. The various standardized effect sizes can be grouped in three categories depending on the experimental design: measures of the difference between two means (the family), measures of strength of association (e.., , , , ), and risk estimates (e.g., odds ratio, relative risk, phi; Kirk, 1996). Part I of this study reviews the family, with a special focus on Cohen's and Hedges' for two-independent groups and two-repeated measures (or paired samples) designs. The present paper answers questions concerning the family via Monte Carlo simulations. First, four different denominators are often proposed to standardize the mean difference in a repeated measures design. Which one should be used? Second, the literature proposes several approximations to estimate the standard error. Which one most closely estimates the true standard deviation of the distribution? Lastly, central and noncentral methods have been proposed to construct a confidence interval around . Which method leads to more precise coverage, and how to calculate it? Results suggest that the best way to standardize the effect in both designs is by using the pooled standard deviation in conjunction with a correction factor to unbias . Likewise, the best standard error approximation is given by substituting the gamma function from the true formula by its approximation. Lastly, results from the confidence interval simulations show that, under the normality assumption, the noncentral method is always superior, especially with small sample sizes. However, the central method is equivalent to the noncentral method when is greater than 20 in each group for a between-group design and when is greater than 24 pairs of observations for a repeated measures design. A practical guide to apply the findings of this study can be found after the general discussion
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