The new and improved two-sample T test

Abstract This article considers the problem of comparing two independent groups in terms of some measure of location. It is well known that with Student's two-independent-sample t test, the actual level of significance can be well above or below the nominal level, confidence intervals can have inaccurate probability coverage, and power can be low relative to other methods. A solution to deal with heterogeneity is Welch's (1938) test. Welch's test deals with heteroscedasticity but can have poor power under arbitrarily small departures from normality. Yuen (1974) generalized Welch's test to trimmed means; her method provides improved control over the probability of a Type I error, but problems remain. Transformations for skewness improve matters, but the probability of a Type I error remains unsatisfactory in some situations. We find that a transformation for skewness combined with a bootstrap method improves Type I error control and probability coverage even if sample sizes are small.     

Probability gate statistics

A probability gate presents a fixed but controllable transmission probability p to pulses arriving at its input. This paper describes some statistical properties of the transformation imposed by a gate on the input distribution of interpulse intervals. The output distribution of interpulse intervals is derived in terms of the input density and p. A simple relation is shown to hold between the moment generating functions, and the first four output moments are presented in terms of the input moments and p. Certain restrictions on the kinds of obtainable output distributions are discussed. In particular, a condition is established that specifies when the output density is a “replica” of the input distribution.     

Hydrogen-induced cold cracking in heat-affected zone of low-carbon high-strength steel

The Y-groove cracking test by submerged arc welding was employed to study the susceptibility of a low-carbon high-strength steel to hydrogen-induced cold cracking (HICC). The morphology of hydrogen cracks was observed using an electron probe microscope. The results showed that the heat-affected zone (HAZ) has a higher susceptibility to HICC than the weld metal and that increasing heat input can improve the HICC resistance of the weldment. The intergranular microcracking is the main HICC mode at the lowest heat input condition, accompanied with some transgranular microcracks attached to complex inclusions. In combination with phase transformation behaviour in sub-zones, the effect of the phase transformation sequence is proposed to try to illustrate the fact that the fine-grained HAZ has higher probability of hydrogen cracking than the coarse-grained HAZ owing to the occurrence of hydrogen enrichment in the fine-grained HAZ after the transformation.     

Subjective probability and delay.

Human subjects indicated their preference between a hypothetical $1,000 reward available with various probabilities or delays and a certain reward of variable amount available immediately. The function relating the amount of the certain-immediate reward subjectively equivalent to the delayed $1,000 reward had the same general shape (hyperbolic) as the function found by Mazur (1987) to describe pigeons' delay discounting. The function relating the certain-immediate amount of money subjectively equivalent to the probabilistic $1,000 reward was also hyperbolic, provided that the stated probability was transformed to odds against winning. In a second experiment, when human subjects chose between a delayed $1,000 reward and a probabilistic $1,000 reward, delay was proportional to the same odds-against transformation of the probability to which it was subjectively equivalent.     

A note on keats' generalization of the rasch model

An alternative derivation is given for a simple test model which incorporates anability parameter for the subject anddifficulty level andguessing parameters for the problem. The probability of a correct response to the problem is a projective transformation of the problem difficulty. The ability and difficulty parameters separate into additive components.     

Deriving the loudness exponent from categorical judgments

The power function exponent for loudness is traditionally determined by means of a process of magnitude estimation. It is demonstrated in this paper that the exponent can also be obtained by using the procedure of absolute identification of sound intensity. It has been shown that subjects' responses to tones of a given intensity are distributed in a normal distribution whose variance depends on the range, R, over which the tones are distributed. By means of a standard statistical transformation, the normal density in log space is converted to the corresponding probability density in linear space. The power function exponent can then be obtained directly from the linear probability density. We also suggest that there is a direct relationship between the information calculated from experiments on absolute identification of sound intensity and the neurophysiological, poststimulus histogram measured in a nerve fiber in the auditory nerve.     

The distribution of interresponse times in the pigeon during variable-interval reinforcement

Three pigeons' pecks were reinforced on 1- and 2-min variable-interval schedules, and frequency distributions of their interresponse times (IRTs) were recorded. The conditional probability that a response would fall into any IRT category was estimated by the interresponse-times-per-opportunity transformation (IRTs/op). The resulting functions were notable chiefly for the relatively low probability of IRTs in the 0.2- to 0.3-sec range; in other respects they varied within and between subjects. The overall level of the curves generally rose over the course of 32 experimental hours, but their shapes changed unsystematically. The shape of the IRT distribution was much the same for VI 1-min and VI 2-min. The variability of these distributions supports the notion that the VI schedule only loosely controls response rate, permitting wide latitude to adventitious effects. There was no systematic evidence that curves changed over sessions to conform to the distribution of reinforcements by IRT.     

A Representation Theorem for a Decision Theory With Conditionals

This paper investigates the role of conditionals in hypothetical reasoning and rational decision making. Its main result is a proof of a representation theorem for preferences defined on sets of sentences (and, in particular, conditional sentences), where an agent’s preference for one sentence over another is understood to be a preference for receiving the news conveyed by the former. The theorem shows that a rational preference ordering of conditional sentences determines probability and desirability representations of the agent’s degrees of belief and desire that satisfy, in the case of non-conditional sentences, the axioms of Jeffrey’s decision theory and, in the case of conditional sentences, Adams’ expression for the probabilities of conditionals. Furthermore, the probability representation is shown to be unique and the desirability representation unique up to positive linear transformation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Alberta's suicide prevention training programs: a retrospective comparison with Rothman's developmental research model

Developmental research is used to transform existing knowledge into applied programs. A social research and development (R&D) method, developed by Rothman for the human service professions, is compared with the method used to develop the Suicide Prevention Training Programs in Alberta, Canada. A retrospective review compares the development of these programs, which have since been implemented both nationally and internationally, with Rothman's model; discusses social R&D as a transformation methodology; and draws conclusions about the probable success of using developmental research methods in the human services. The review finds that the Alberta method closely paralleled the phases of Rothman's model. Results show that the transformation of knowledge about suicide into widely disseminated suicide prevention training programs can be attributed to the application of social R&D principles. The study concludes that the use of developmental research methods to establish human service programs should increase the probability of their success.     

Reference frames in mental rotation

Four experiments are reported that investigate whether images or reference frames are transformed during a mental rotation task. In all experiments a display of four identical letters (P1) was presented at either +90 degrees or -90 degrees from upright, and subjects had to decide whether the letters were normal or mirror-image reflections. A single letter (P2) was then presented 100 ms later in a variable orientation with the same task instructions. Reaction times to P2 were assessed to determine whether an image of P2 was rotated to upright or whether an internal reference frame was rotated into congruence with P2 from the orientation of P1. The results as a whole suggest that transformations of P2 can be initiated either relative to upright or relative to the orientation of P1. They further indicate that the probability of using each reference orientation can be changed by procedural variations. The findings are most parsimoniously interpreted as suggesting that mental rotation involves the transformation of reference frames rather than the transformation of template-like representations.     

Multidimensional Fechnerian Scaling: Pairwise Comparisons, Regular Minimality, and Nonconstant Self-Similarity

Stimuli presented pairwise for same-different judgments belong to two distinct observation areas (different time intervals and/or locations). To reflect this fact the underlying assumptions of multidimensional Fechnerian scaling (MDFS) have to be modified, the most important modification being the inclusion of the requirement that the discrimination probability functions possess regular minima. This means that the probability with which a fixed stimulus in one observation area (a reference) is discriminated from stimuli belonging to another observation area reaches its minimum when the two stimuli are identical (following, if necessary, an appropriate transformation of the stimulus measurements in one of the two observation areas). The remaining modifications of the underlying assumptions are rather straightforward, their main outcome being that each of the two observation areas has its own Fechnerian metric induced by a metric function obtained in accordance with the regular variation version of MDFS. It turns out that the regular minimality requirement, when combined with the empirical fact of nonconstant self-similarity (which means that the minimum level of the discrimination probability function for a fixed reference stimulus is generally different for different reference stimuli), imposes rigid constraints on the interdependence between discrimination probabilities and metric functions within each of the observation areas and on the interdependence between Fechnerian metrics and metric functions belonging to different observation areas. In particular, it turns out that the psychometric order of the stimulus space cannot exceed 1.     

Psychophysics without physics: extension of Fechnerian scaling from continuous to discrete and discrete-continuous stimulus spaces

The computation of subjective (Fechnerian) distances from discrimination probabilities involves cumulation of appropriately transformed psychometric increments along smooth arcs (in continuous stimulus spaces) or chains of stimuli (in discrete spaces). In a space where any two stimuli that are each other's points of subjective equality are given identical physical labels, psychometric increments are positive differences ψ(x,y)-ψ(x,x) and ψ(y,x)-ψ(x,x), where x≠y and ψ is the probability of judging two stimuli different. In continuous stimulus spaces the appropriate monotone transformation of these increments (called overall psychometric transformation) is determined uniquely in the vicinity of zero, and its extension to larger values of its argument is immaterial. In discrete stimulus spaces, however, Fechnerian distances critically depend on this extension. We show that if overall psychometric transformation is assumed (A) to be the same for a sufficiently rich class of discrete stimulus spaces, (B) to ensure the validity of the Second Main Theorem of Fechnerian Scaling in this class of spaces, and (C) to agree in the vicinity of zero with one of the possible transformations in continuous spaces, then this transformation can only be identity. This result is generalized to the broad class of “discrete-continuous” stimulus spaces, of which continuous and discrete spaces are proper subclasses.     

Moving Beyond the Individual: Examining the Effects of Domestic Violence Policies on Social Norms

To be effective, criminal justice policies should affect the underlying social norms for which the policies were enacted. This study sought to determine whether public perceptions of criminal justice policies on domestic violence affected social norms. Two waves of data were collected via a telephone survey where a random probability sample of 973 residents was drawn from 4 communities. A structural equation model was tested and confirmed. Results provided strong support for the hypothesis that perceptions of criminal justice policies have direct effects on attitudes toward criminal justice response, and indirect effects on victim-blaming attitudes, both underlying social norms related to domestic violence. The enactment of criminal justice policies, therefore, may have an impact beyond victims and perpetrators and lead to a transformation of the community through the emergence of new social norms. Public awareness campaigns designed to disseminate criminal justice policies may be instrumental in provoking social change.     

Random‐effects meta‐analysis of correlations: Monte Carlo evaluation of mean estimators

Several authors have cautioned against using Fisher's z‐transformation in random‐effects meta‐analysis of correlations, which seems to perform poorly in some situations, especially with substantial inter‐study heterogeneity. Attributing this performance largely to the direct z‐to‐r transformation (DZRT) of Fisher z results (e.g. point estimate of mean correlation), in a previous paper Hafdahl (2009) proposed point and interval estimators of the mean Pearson r correlation that instead use an integral z‐to‐r transformation (IZRT). The present Monte Carlo study of these IZRT Fisher z estimators includes comparisons with their DZRT counterparts and with estimators based on Pearson r correlations. The IZRT point estimator was usually more accurate and efficient than its DZRT counterpart and comparable to the two Pearson r point estimators – better in some conditions but worse in others. Coverage probability for the IZRT confidence intervals (CIs) was often near nominal, much better than for the DZRT CIs, and comparable to coverage for the Pearson r CIs; every approach's CI fell markedly below nominal in some conditions. The IZRT estimators contradict warnings about Fisher z estimators' poor performance. Recommendations for practising research synthesists are offered, and an Appendix provides computing code to implement the IZRT as in the real‐data example.     

The induced asynchrony effect: Its role in visual judgments of temporal order and its relation to other dynamic perceptual phenomena

A context-induced “illusion” in visual judgments of temporal order, termed the induced asynchrony effect (IAE), is reported. It consists of an apparent ordering in time of two simultaneous light onsets, produced by the preceding, asynchronous offsets of two other lights. The joint effect of a real stimulus onset asynchrony and a preceding stimulus offset asynchrony bn judgments of onset order appears to be additive, given a Gaussian transformation of response probability. This result is shown to be consistent with a simple statistical decision model, which provides a conceptual framework for drawing inferences from temporal order judgment data. However, it is emphasized that certain interpretations of such models are not empirically testable on the basis of temporal order data alone. An attempt is made to relate the IAE to three other dynamic perceptual phenomena; all four effects may reflect a tendency of observers to perceive the velocity of apparent motion as being constant. Questions raised by the demonstration of the IAE are discussed, and directions for further research are suggested.     

Combining probability values from independent permutation tests: a discrete analog of Fisher's classical method

Permutation tests are based on all possible arrangements of observed data sets. Consequently, such tests yield exact probability values obtained from discrete probability distributions. An exact nondirectional method to combine independent probability values that obey discrete probability distributions is introduced. The exact method is the discrete analog to Fisher's classical method for combining probability values from independent continuous probability distributions. If the combination of probability values includes even one probability value that obeys a sparse discrete probability distribution, then Fisher's classical method may be grossly inadequate.     

Response force in RT tasks: Isolating effects of stimulus probability and response probability

Mattes, UIrich, and Miller (1997) found that as response probability decreases in a simple reaction time (RT) task, participants produce more forceful responses as well as longer RTs, suggesting a direct influence of preparatory processes on the motor system. In this previous study, however, response probability was confounded with stimulus probability, leaving open the possibility that response force was sensitive to stimulus- rather than response-related preparation. The present study was conducted to unravel the effects of stimulus and response probability. Experiment 1 manipulated stimulus probability and revealed that responses to a more probable stimulus are less forceful than responses to a less probable stimulus even when both stimuli require the same response. Experiment 2 demonstrated that this stimulus probability effect does not depend on the overall level of response probability. Experiment 3 showed an analogous effect for response probability when stimulus probability is kept constant. The complete pattern of results suggests that both stimulus probability and response probability affect the forcefulness of a response. It is argued that response probability exerts adirect influence on the motor system, whereas stimulus probability influences the motor system indirectly via premotoric adjustments.     

用文字概率衡量不确定性：特征和问题

文字概率是衡量不确定性的方式之一, 即人们使用诸如“也许”、“未必”的词汇来描述特定事件发生的可能性。文字概率不同于数字概率, 主要体现在文字概率的模糊性、非概率运算性和语义特性上。这使得相对于数字概率, 用文字概率衡量不确定性既有优势也有问题, 进而对人们的不确定信息沟通和风险决策造成影响。虽然文字概率与数字概率存在特征上的差异并且人们在日常交流中偏爱文字概率, 但是大部分风险领域的研究却仅局限于数字概率, 今后有必要研究使用文字概率测量的风险决策。在已有文字概率特征研究的基础上, 还可以进一步探究其不同于数字概率的其它特征(文字/数字概率与双系统模型的联系、文字概率的跨文化差异等)及其对风险决策的影响。     

Multidimensional Fechnerian Scaling: Probability-Distance Hypothesis

The probability-distance hypothesis states that the probability with which one stimulus is discriminated from another is a function of some subjective distance between these stimuli. The analysis of this hypothesis within the framework of multidimensional Fechnerian scaling yields the following results. If the hypothetical subjective metric is internal (which means, roughly, that the distance between two stimuli equals the infimum of the lengths of all paths connecting them), then the underlying assumptions of Fechnerian scaling are satisfied and the metric in question coincides with the Fechnerian metric. Under the probability-distance hypothesis, the Fechnerian metric exists (i.e., the underlying assumptions of Fechnerian scaling are satisfied) if and only if the hypothetical subjective metric is internalizable, which means, roughly, that by a certain transformation it can be made to coincide in the small with an internal metric; and then this internal metric is the Fechnerian metric. The specialization of these results to unidimensional stimulus continua is closely related to the so-called Fechner problem proposed in 1960's as a substitute for Fechner's original theory.     

Developmental change in subjective probability during adolescence

The developmental change in subjective probability during adolescence, an important period for establishing the probability concept, was investigated. 75 Japanese adolescents, from 12 to 23 yr. of age, were asked to make probability judgments for a lottery under 15 conditions. Analysis showed that with increase in age their subjective probability came closer to the objective probability. Discussion of these results took into consideration recent studies on the development of the concept of probability.