Study on the interior equilibrium point of a special class of 2 × 2 × 2 asymmetric evolutionary games.

Many behavioural interactions in real life involve three individuals. When each individual has two alternative strategies, they can be abstracted into mathematical models by means of 2 × 2 × 2 asymmetric games. In this paper, we explore a special class of 2 × 2 × 2 asymmetric games satisfying fixed conditions. Firstly, we analyse two solitary interior equilibrium points and provide the judgement condition for their instability based on the Jacobi matrix local stability analysis method. Secondly, we analyse the interior equilibrium points that are continuously distributed within a line and probe into their stability conditions based on generalized Hamiltonian systems theory. Under the circumstances, the stable interior equilibrium point is surrounded by closed orbits in phase space, which presents an observable stable state where two strategies coexist and fluctuate in each of the three game populations. This work enriches the study of 2 × 2 × 2 asymmetric games' evolutionary dynamics.

Likelihood Ratio Test and the Evidential Approach for 2 × 2 Tables.

Categorical data analysis of 2 × 2 contingency tables is extremely common, not least because they provide risk difference, risk ratio, odds ratio, and log odds statistics in medical research. A χ2 test analysis is most often used, although some researchers use likelihood ratio test (LRT) analysis. Does it matter which test is used? A review of the literature, examination of the theoretical foundations, and analyses of simulations and empirical data are used by this paper to argue that only the LRT should be used when we are interested in testing whether the binomial proportions are equal. This so-called test of independence is by far the most popular, meaning the χ2 test is widely misused. By contrast, the χ2 test should be reserved for where the data appear to match too closely a particular hypothesis (e.g., the null hypothesis), where the variance is of interest, and is less than expected. Low variance can be of interest in various scenarios, particularly in investigations of data integrity. Finally, it is argued that the evidential approach provides a consistent and coherent method that avoids the difficulties posed by significance testing. The approach facilitates the calculation of appropriate log likelihood ratios to suit our research aims, whether this is to test the proportions or to test the variance. The conclusions from this paper apply to larger contingency tables, including multi-way tables.

Design of randomized clinical trials with a binary endpoint: Conditional versus unconditional analyses of a two-by-two table.

When designing a randomized clinical trial to compare two treatments, the sample size required to have desired power with a specified type 1 error depends on the hypothesis testing procedure. With a binary endpoint (e.g., response), the trial results can be displayed in a 2 × 2 table. If one does the analysis conditional on the number of positive responses, then using Fisher's exact test has an actual type 1 error less than or equal to the specified nominal type 1 error. Alternatively, one can use one of many unconditional "exact" tests that also preserve the type 1 error and are less conservative than Fisher's exact test. In particular, the unconditional test of Boschloo is always at least as powerful as Fisher's exact test, leading to smaller required sample sizes for clinical trials. However, many statisticians have argued over the years that the conditional analysis with Fisher's exact test is the only appropriate procedure. Since having smaller clinical trials is an extremely important consideration, we review the general arguments given for the conditional analysis of a 2 × 2 table in the context of a randomized clinical trial. We find the arguments not relevant in this context, or, if relevant, not completely convincing, suggesting the sample-size advantage of the unconditional tests should lead to their recommended use. We also briefly suggest that since designers of clinical trials practically always have target null and alternative response rates, there is the possibility of using this information to improve the power of the unconditional tests.

Illustrations of interactions needed when investigating sleep using a type of AM-PM PM-AM design.

Sleep has long been thought of and promoted to be beneficial for memory. Some claims that sleep aids memory have been made in the absence of a critical interaction. This condition is necessary when using a commonly-used experimental design (a type of AM-PM PM-AM design). We propose that a sleep effect exists only if there is an interaction between groups (experimental and time-of-day controls) and the time of test or study (morning and evening). We show different patterns of results that would and would not support a sleep effect with empirical and model-generated data from recognition memory experiments and hypothetical data. While we use these data to make our point, our suggestions apply to any memory and non-memory-related investigation (e.g., emotional memory, false memory susceptibility, language learning, problem-solving). Testing for and finding the proper interaction will add to the evidence that sleep boosts performance.

Estimands for factorial trials.

Factorial trials offer an efficient method to evaluate multiple interventions in a single trial, however the use of additional treatments can obscure research objectives, leading to inappropriate analytical methods and interpretation of results. We define a set of estimands for factorial trials, and describe a framework for applying these estimands, with the aim of clarifying trial objectives and ensuring appropriate primary and sensitivity analyses are chosen. This framework is intended for use in factorial trials where the intent is to conduct "two-trials-in-one" (ie, to separately evaluate the effects of treatments A and B), and is comprised of four steps: (i) specifying how additional treatment(s) (eg, treatment B) will be handled in the estimand, and how intercurrent events affecting the additional treatment(s) will be handled; (ii) designating the appropriate factorial estimator as the primary analysis strategy; (iii) evaluating the interaction to assess the plausibility of the assumptions underpinning the factorial estimator; and (iv) performing a sensitivity analysis using an appropriate multiarm estimator to evaluate to what extent departures from the underlying assumption of no interaction may affect results. We show that adjustment for other factors is necessary for noncollapsible effect measures (such as odds ratio), and through a trial re-analysis we find that failure to consider the estimand could lead to inappropriate interpretation of results. We conclude that careful use of the estimands framework clarifies research objectives and reduces the risk of misinterpretation of trial results, and should become a standard part of both the protocol and reporting of factorial trials.

Large-Dynamic-Range and High-Stability Phase Demodulation Technology for Fiber-Optic Michelson Interferometric Sensors.

A large-dynamic-range and high-stability phase demodulation technology for fiber-optic Michelson interferometric sensors is proposed. This technology utilizes two output signals from a 2 × 2 fiber-optic coupler, the interferometric phase difference of which is π. A linear-fitting trigonometric-identity-transformation differential cross-multiplication (LF-TIT-DCM) algorithm is used to interrogate the phase signal from the two output signals from the coupler. The interferometric phase differences from the two output signals from the 2 × 2 fiber-optic couplers with different coupling ratios are all equal to π, which ensures that the LF-TIT-DCM algorithm can be applied perfectly. A fiber-optic Michelson interferometric acoustic sensor is fabricated, and an acoustic signal testing system is built to prove the proposed phase demodulation technology. Experimental results show that excellent linearity is observed from 0.033 rad to 3.2 rad. Moreover, the influence of laser wavelength and optical power is researched, and variation below 0.47 dB is observed at different sound pressure levels (SPLs). Long-term stability over thirty minutes is tested, and fluctuation is less than 0.36 dB. The proposed phase demodulation technology obtains large dynamic range and high stability at rather low cost.

The Effect of Information Formats and Incidental Affect on Prior and Posterior Probability Judgments.

Background. Interpreting medical test results involves judging probabilities, including making Bayesian inferences such as judging the positive and negative predictive values. Although prior work has shown that experience formats (e.g., slide shows of representative patient cases) produce more accurate Bayesian inferences than description formats (e.g., verbal statistical summaries), there are disadvantages of using the experience format for real-world medical decision making that may be solved by presenting relevant information in a 2 × 2 table format. Furthermore, medical decisions are often made in stressful contexts, yet little is known about the influence of acute stress on the accuracy of Bayesian inferences. This study aimed to a) replicate the description-experience format effect on probabilistic judgments; b) examine judgment accuracy across description, experience, and a new 2 × 2 table format; and c) assess the effect of acute stress on probability judgments. Method. The study employed a 2 (stress condition) × 3 (format) factorial between-subjects design. Participants (N = 165) completed a Bayesian inference task in which information about a medical screening test was presented in 1 of 3 formats (description, experience, 2 × 2 table), following a laboratory stress induction or a no-stress control condition. Results. Overall, the 2 × 2 table format produced the most accurate probability judgments, including Bayesian inferences, compared with the description and experience formats. Stress had no effect on judgment accuracy. Discussion. Given its accuracy and practicality, a 2 × 2 table may be better suited than description or experience formats for communicating probabilistic information in medical contexts.

De-escalated radiotherapy for indolent primary cutaneous B-cell lymphoma.

OBJECTIVE: Radiotherapy (RT) has an established role in the curative treatment of indolent primary cutaneous B­cell lymphoma (PCBCL). With the role of low-dose regimens such as 2 × 2 Gy being uncertain, we compared conventional-dose RT to a low-dose approach and investigated outcome and toxicities. MATERIALS AND METHODS: We retrospectively reviewed the medical records of 26 patients with 44 cutaneous lesions treated at our institution between 2007 and 2017, comprising 22 marginal zone lymphoma (PCMZL) lesions and 22 follicle center lymphoma (PCFCL) lesions. Seven lesions (16%) were treated with low-dose RT (LDRT) (4 Gy) and 37 (84%) with conventional-dose RT (≥24 Gy, median 40 Gy). Median follow-up duration was 76 months. RESULTS: The overall response rate (ORR) was 91% (complete response rate [CRR]: 75%). The 5­year local control rate (LCR) was 88% and the 10-year LCR was 84%. The response rates were significantly higher following conventional-dose RT (ORR: 92% vs. 86%; CRR: 84% vs. 29%; P = 0.007). In terms of radiation dose, the rate of infield relapses (14% vs. 11%, P = 0.4) and the 5­year LCR (86% vs. 90%, P = 0.4) were comparable in the LDRT and conventional-dose RT groups. During RT courses, about two-thirds of patients experienced mild toxicities, with grade I and II acute toxicity rates of 61% and 9%, respectively, with lower incidences of grade I (14% vs. 70%) and grade II (0% vs. 8%, P = 0.004) toxicities following LDRT. CONCLUSION: This long-term analysis confirms the excellent outcome of RT in the management of PCBCL. The LDRT concept with 4 Gy was associated with a comparable LCR and reduced rates of acute toxicity. However, the response rates were significantly lower for this group and LDRT may therefore not be recommended as standard treatment.

Testing the 2 × 2 model of perfectionism in Ecuadorian adolescent population.

This study aimed to test the 2 × 2 model of dispositional perfectionism in an Ecuadorian sample of 1562 students aged between 12 and 17 years (M = 14.83 years, standard deviation = 1.86 years). The Child and Adolescent Perfectionism Scale and the brief 21-item version of the Depression Anxiety Stress Scales were used. Cluster analysis revealed four profiles: Pure Self-Oriented Perfectionism, Pure Socially Prescribed Perfectionism, Mixed Perfectionism, and Non-perfectionism. Mixed Perfectionism and Non-perfectionism obtained, respectively, the highest and lowest scores in anxiety, depression, and stress. The results are discussed in light of cultural differences in the interpretation and consequences of perfectionism.

Two-tailed significance tests for 2 × 2 contingency tables: What is the alternative?

Two-tailed significance testing for 2 × 2 contingency tables has remained controversial. Within the medical literature, different tests are used in different papers and that choice may decide whether findings are adjudged to be significant or nonsignificant; a state of affairs that is clearly undesirable. In this paper, it is argued that a part of the controversy is due to a failure to recognise that there are two possible alternative hypotheses to the Null. It is further argued that, while one alternative hypothesis can lead to tests with greater power, the other choice is more applicable in medical research. That leads to the recommendation that, within medical research, 2 × 2 tables should be tested using double the one-tailed exact probability from Fisher's exact test or, as an approximation, the chi-squared test with Yates' correction for continuity.

An Eye-Tracking Study of Statistical Reasoning With Tree Diagrams and 2 × 2 Tables.

Changing the information format from probabilities into frequencies as well as employing appropriate visualizations such as tree diagrams or 2 × 2 tables are important tools that can facilitate people's statistical reasoning. Previous studies have shown that despite their widespread use in statistical textbooks, both of those visualization types are only of restricted help when they are provided with probabilities, but that they can foster insight when presented with frequencies instead. In the present study, we attempt to replicate this effect and also examine, by the method of eye tracking, why probabilistic 2 × 2 tables and tree diagrams do not facilitate reasoning with regard to Bayesian inferences (i.e., determining what errors occur and whether they can be explained by scan paths), and why the same visualizations are of great help to an individual when they are combined with frequencies. All ten inferences of N = 24 participants were based solely on tree diagrams or 2 × 2 tables that presented either the famous "mammography context" or an "economics context" (without additional textual wording). We first asked participants for marginal, conjoint, and (non-inverted) conditional probabilities (or frequencies), followed by related Bayesian tasks. While solution rates were higher for natural frequency questions as compared to probability versions, eye-tracking analyses indeed yielded noticeable differences regarding eye movements between correct and incorrect solutions. For instance, heat maps (aggregated scan paths) of distinct results differed remarkably, thereby making correct and faulty strategies visible in the line of theoretical classifications. Moreover, the inherent structure of 2 × 2 tables seems to help participants avoid certain Bayesian mistakes (e.g., "Fisherian" error) while tree diagrams seem to help steer them away from others (e.g., "joint occurrence"). We will discuss resulting educational consequences at the end of the paper.

A simulations approach for meta-analysis of genetic association studies based on additive genetic model.

Meta-analysis of genetic association studies is being increasingly used to assess phenotypic differences between genotype groups. When the underlying genetic model is assumed to be dominant or recessive, assessing the phenotype differences based on summary statistics, reported for individual studies in a meta-analysis, is a valid strategy. However, when the genetic model is additive, a similar strategy based on summary statistics will lead to biased results. This fact about the additive model is one of the things that we establish in this paper, using simulations. The main goal of this paper is to present an alternate strategy for the additive model based on simulating data for the individual studies. We show that the alternate strategy is far superior to the strategy based on summary statistics.

Efficient treatment allocation in 2 × 2 multicenter trials when costs and variances are heterogeneous.

At the design stage of a study, it is crucial to compute the sample size needed for treatment effect estimation with maximum precision and power. The optimal design depends on the costs, which may be known at the design stage, and on the outcome variances, which are unknown. A balanced design, optimal for homogeneous costs and variances, is typically used. An alternative to the balanced design is a design optimal for the known and possibly heterogeneous costs, and homogeneous variances, called costs considering design. Both designs suffer from loss of efficiency, compared with optimal designs for heterogeneous costs and variances. For 2 × 2 multicenter trials, we compute the relative efficiency of the balanced and the costs considering designs, relative to the optimal designs. We consider 2 heterogeneous costs and variance scenarios (in 1 scenario, 2 treatment conditions have small and 2 have large costs and variances; in the other scenario, 1 treatment condition has small, 2 have intermediate, and 1 has large costs and variances). Within these scenarios, we examine the relative efficiency of the balanced design and of the costs considering design as a function of the extents of heterogeneity of the costs and of the variances and of their congruence (congruent when the cheapest treatment has the smallest variance, incongruent when the cheapest treatment has the largest variance). We find that the costs considering design is generally more efficient than the balanced design, and we illustrate this theory on a 2 × 2 multicenter trial on lifestyle improvement of patients in general practices.

Profiles of Perfectionism and School Anxiety: A Review of the 2 × 2 Model of Dispositional Perfectionism in Child Population.

The 2 × 2 model of dispositional perfectionism has been very well received by researchers of the topic, leading to the creation of new studies that have analyzed the way in which the four proposed subtypes are distinctly associated with measures of adaptation and maladjustment. The goal of this study was to determine the possible existence of four profiles of child perfectionism that are congruent with the subtypes proposed by the 2 × 2 model, and whether these subtypes are associated with school anxiety, in accordance with the hypotheses established by the model. The sample was composed of 2157 students from Spanish Primary Education aged between 8 and 11 years (M = 9.60, SD = 1.24). The Child and Adolescent Perfectionism Scale was used to assess Socially Prescribed Perfectionism and Self-Oriented Perfectionism, and the School Anxiety Inventory for Primary Education was used to measure school anxiety. The results of cluster analysis identified four differential groups of perfectionists similar to the subtypes defined by the 2 × 2 model: Non-Perfectionism, Pure Personal Standards Perfectionism (Pure PSP), Pure Evaluative Concerns Perfectionism (Pure ECP), and Mixed Perfectionism. The four groups presented a differentiable pattern of association with school anxiety, with the exception of Pure PSP and Pure ECP, which showed no significant differences. Participants classified as Non-perfectionists presented the most adaptive outcomes, whereas subjects included in the Mixed Perfectionism group scored significantly higher on school anxiety than the three remaining groups. To conclude, the results partially supported the hypotheses of the 2 × 2 model, questioning the consideration of Self-Oriented Perfectionism as a positive manifestation of perfectionism and showing that it is the combination of high scores in both perfectionist dimensions, Self-Oriented Perfectionism and Socially Prescribed Perfectionism that implies higher levels of school anxiety. These findings should be taken into account when generalizing the 2 × 2 model to child population.

Non-factorial analyses of two-by-two factorial trial designs.

BACKGROUND/AIMS: Factorial analyses of 2 × 2 trial designs are known to be problematic unless one can be sure that there is no interaction between the treatments (A and B). Instead, we consider non-factorial analyses of a factorial trial design that addresses clinically relevant questions of interest without any assumptions on the interaction. Primary questions of interest are as follows: (1) is A better than the control treatment C, (2) is B better than C, (3) is the combination of A and B (AB) better than C, and (4) is AB better than A, B, and C. METHODS: A simple three-step procedure that tests the first three primary questions of interest using a Bonferroni adjustment at the first step is proposed. A Hochberg procedure on the four primary questions is also considered. The two procedures are evaluated and compared in limited simulations. Published results from three completed trials with factorial designs are re-evaluated using the two procedures. RESULTS: Both suggested procedures (that answer multiple questions) require a 50%-60% increase in per arm sample size over a two-arm design asking a single question. The simulations suggest a slight advantage to the three-step procedure in terms of power (for the primary and secondary questions). The proposed procedures would have formally addressed the questions arising in the highlighted published trials arguably more simply than the pre-specified factorial analyses used. CONCLUSION: Factorial trial designs are an efficient way to evaluate two treatments, alone and in combination. In situations where a statistical interaction between the treatment effects cannot be assumed to be 0, simple non-factorial analyses are possible that directly assess the questions of interest without the zero interaction assumption.

Efficient treatment allocation in 2 × 2 cluster randomized trials, when costs and variances are heterogeneous.

Typically, clusters and individuals in cluster randomized trials are allocated across treatment conditions in a balanced fashion. This is optimal under homogeneous costs and outcome variances. However, both the costs and the variances may be heterogeneous. Then, an unbalanced allocation is more efficient but impractical as the outcome variance is unknown in the design stage of a study. A practical alternative to the balanced design could be a design optimal for known and possibly heterogeneous costs and homogeneous variances. However, when costs and variances are heterogeneous, both designs suffer from loss of efficiency, compared with the optimal design. Focusing on cluster randomized trials with a 2 × 2 design, the relative efficiency of the balanced design and of the design optimal for heterogeneous costs and homogeneous variances is evaluated, relative to the optimal design. We consider two heterogeneous scenarios (two treatment arms with small, and two with large, costs or variances, or one small, two intermediate, and one large costs or variances) at each design level (cluster, individual, and both). Within these scenarios, we compute the relative efficiency of the two designs as a function of the extents of heterogeneity of the costs and variances, and the congruence (the cheapest treatment has the smallest variance) and incongruence (the cheapest treatment has the largest variance) between costs and variances. We find that the design optimal for heterogeneous costs and homogeneous variances is generally more efficient than the balanced design and we illustrate this theory on a trial that examines methods to reduce radiological referrals from general practices. Copyright © 2016 John Wiley & Sons, Ltd.

An Effective Approach to Improving Cadmium Telluride (111)A Surface by Molecular-Beam-Epitaxy Growth of Tellurium Monolayer.

The surface cleansing treatment of non-natural cleavage planes of semiconductors is usually performed in vacuum using ion sputtering and subsequent annealing. In this Research Article, we report on the evolution of surface atomic structure caused by different ways of surface treatment as monitored by in situ core-level photoemission measurements of Cd-4d and Te-4d atomic levels and reflection high-energy electron diffraction (RHEED). Sputtering of surface increases the density of the dangling bonds by 50%. This feature and the less than ideal ordering can be detrimental to device applications. An effective approach is employed to improve the quality of this surface. One monolayer (ML) of Te grown by the method of molecular beam epitaxy (MBE) on the target surface with heating at 300 °C effectively improves the surface quality as evidenced by the improved sharpness of RHEED pattern and a reduced diffuse background in the spectra measured by high-resolution ultraviolet photoemission spectroscopy (HRUPS). Calculations have been performed for various atomic geometries by employing first-principles geometry optimization. In conjunction with an analysis of the core level component intensities in terms the layer-attenuation model, we propose a "vacancy site" model of the modified 1 ML-Te/CdTe(111)A (2 × 2) surface.

Effects of visualizing statistical information - an empirical study on tree diagrams and 2 × 2 tables.

In their research articles, scholars often use 2 × 2 tables or tree diagrams including natural frequencies in order to illustrate Bayesian reasoning situations to their peers. Interestingly, the effect of these visualizations on participants' performance has not been tested empirically so far (apart from explicit training studies). In the present article, we report on an empirical study (3 × 2 × 2 design) in which we systematically vary visualization (no visualization vs. 2 × 2 table vs. tree diagram) and information format (probabilities vs. natural frequencies) for two contexts (medical vs. economical context; not a factor of interest). Each of N = 259 participants (students of age 16-18) had to solve two typical Bayesian reasoning tasks ("mammography problem" and "economics problem"). The hypothesis is that 2 × 2 tables and tree diagrams - especially when natural frequencies are included - can foster insight into the notoriously difficult structure of Bayesian reasoning situations. In contrast to many other visualizations (e.g., icon arrays, Euler diagrams), 2 × 2 tables and tree diagrams have the advantage that they can be constructed easily. The implications of our findings for teaching Bayesian reasoning will be discussed.

The effect of heterogeneous variance on efficiency and power of cluster randomized trials with a balanced 2 × 2 factorial design.

Sample size calculation for cluster randomized trials (CRTs) with a [Formula: see text] factorial design is complicated due to the combination of nesting (of individuals within clusters) with crossing (of two treatments). Typically, clusters and individuals are allocated across treatment conditions in a balanced fashion, which is optimal under homogeneity of variance. However, the variance is likely to be heterogeneous if there is a treatment effect. An unbalanced allocation is then more efficient, but impractical because the optimal allocation depends on the unknown variances. Focusing on CRTs with a [Formula: see text] design, this paper addresses two questions: How much efficiency is lost by having a balanced design when the outcome variance is heterogeneous? How large must the sample size be for a balanced allocation to have sufficient power under heterogeneity of variance? We consider different scenarios of heterogeneous variance. Within each scenario, we determine the relative efficiency of a balanced design, as a function of the level (cluster, individual, both) and amount of heterogeneity of the variance. We then provide a simple correction of the sample size for the loss of power due to heterogeneity of variance when a balanced allocation is used. The theory is illustrated with an example of a published 2 x2 CRT.

Recommended confidence intervals for two independent binomial proportions.

The relationship between two independent binomial proportions is commonly estimated and presented using the difference between proportions, the number needed to treat, the ratio of proportions or the odds ratio. Several different confidence intervals are available, but they can produce markedly different results. Some of the traditional approaches, such as the Wald interval for the difference between proportions and the Katz log interval for the ratio of proportions, do not perform well unless the sample size is large. Better intervals are available. This article describes and compares approximate and exact confidence intervals that are - with one exception - easy to calculate or available in common software packages. We illustrate the performances of the intervals and make recommendations for both small and moderate-to-large sample sizes.