概化理论方差分量估计的跨分布分析

方差分量估计是进行概化理论分析的关键。采用MonteCarlo模拟技术,探讨心理与教育测量数据分布对概化理论各种方法估计方差分量的影响。数据分布包括正态、二项和多项分布,估计方法包括Traditional、Jackknife、Bootstrap和MCMC方法。结果表明:(1)Traditional方法估计正态分布和多项分布数据的方差分量相对较好,估计二项分布数据需要校正,Jackknife方法准确地估计了三种分布数据的方差分量,校正的Bootstrap方法和有先验信息的MCMC方法(MCMCinf)估计三种分布数据的方差分量结果较好;(2)心理与教育测量数据分布对四种方法估计概化理论方差分量有影响,数据分布制约着各种方差分量估计方法性能的发挥,需要加以区分地使用。     

Bootstrapping the estimated latent distribution of the two‐parameter latent trait model

This paper focuses on the two‐parameter latent trait model for binary data. Although the prior distribution of the latent variable is usually assumed to be a standard normal distribution, that prior distribution can be estimated from the data as a discrete distribution using a combination of EM algorithms and other optimization methods. We assess with what precision we can estimate the prior from the data, using simulations and bootstrapping. A novel calibration method is given to check that near optimality is achieved for the bootstrap estimates. We find that there is sufficient information on the prior distribution to be informative, and that the bootstrap method is reliable. We illustrate the bootstrap method for two sets of real data.     

Cooperation in Asymmetric Social Dilemmas When Equality Is Perceived as Unfair1

Three experiments examined the hypothesis that in an asymmetric social dilemma, perceived fairness of the distribution promotes cooperation. In support of the hypothesis, Studies 1 and 2 showed that willingness to voluntarily and anonymously pay for child care was related to ratings of perceived fairness of equal and equitable distributions of the quality of child care when it was a public good (provided by the municipality) or a market good (provided by a private business). Study 1 also showed that an equal distribution of the public good was perceived to be fairer than an equal distribution of the market good, whereas the reverse was true for an equitable distribution. Study 2 showed that when quality differences were explicit, an equitable distribution was perceived as fairer than an equal distribution. Study 3 showed that an equitable distribution of quantity of child care was perceived to be fairer and increased willingness to pay as compared to an equal distribution.     

Contextual effects in social judgment

Judges were asked to evaluate the overall performance of hypothetical students, given their scores on two examinations. The distribution of total scores was manipulated in order to investigate the loci of contextual effects. The interaction between the two exams was reversed by manipulation of the distribution. When the distribution of total scores was positively skewed, judgments showed a convergent interaction as a function of the two exams; when the distribution was negatively skewed, the interaction was divergent. The data were consistent with the hypothesis that the distribution of total scores affects only the transformation from integrated impressions to overt responses. This transformation (judgment function) was well-described by an extension of range-frequency theory. The finding that the interaction can be manipulated by changing the stimulus distribution has methodological implications for the popular interpretation of interactions or lack thereof. A good model may be improperly rejected or a bad one improperly retained through lack of attention to contextual effects.     

Dimension in latent variable models

Psychologists would like to say that a probability distribution on {0,1}ⁿ is d-dimensional if (1) the distribution can be represented by some smooth d-dimensional latent variable model and (2) the distribution cannot be represented by any smooth d−1 dimensional model. This does not work out because for d>1, every distribution that can be represented by a smooth d-dimensional model can also be represented by a smooth one-dimensional model. A proof and discussion of some implications of this mathematical result follow.     

Item Response Theory with Estimation of the Latent Population Distribution Using Spline-Based Densities

The purpose of this paper is to introduce a new method for fitting item response theory models with the latent population distribution estimated from the data using splines. A spline-based density estimation system provides a flexible alternative to existing procedures that use a normal distribution, or a different functional form, for the population distribution. A simulation study shows that the new procedure is feasible in practice, and that when the latent distribution is not well approximated as normal, two-parameter logistic (2PL) item parameter estimates and expected a posteriori scores (EAPs) can be improved over what they would be with the normal model. An example with real data compares the new method and the extant empirical histogram approach.     

Equality ratio equity,general linear equity and framing effects in judgments of allocation divisions

It is argued here that there is not necessarily a contradiction between the general linear and equal ratio approaches to equity, and the two may be integrated to make more precise predictions. It was hypothesized that, (a) equity is best represented by a limited equal ratio rule; (b) the more a linear distribution systematically deviates from equal input/outcome ratios, the more unfair it will be judged, but (c) subjects will prefer a distribution that accords with a general linear formula in certain cases where an equal ratio distribution of available outcomes would actually infringe a preeminent or established equal ratio distribution. Four studies are presented, the results of which seem broadly to support these ideas. However, the results also suggest that equity judgments can be strongly influenced by framing effects.     

On the Bayesian Nonparametric Generalization of IRT-Type Models

We study the identification and consistency of Bayesian semiparametric IRT-type models, where the uncertainty on the abilities’ distribution is modeled using a prior distribution on the space of probability measures. We show that for the semiparametric Rasch Poisson counts model, simple restrictions ensure the identification of a general distribution generating the abilities, even for a finite number of probes. For the semiparametric Rasch model, only a finite number of properties of the general abilities’ distribution can be identified by a finite number of items, which are completely characterized. The full identification of the semiparametric Rasch model can be only achieved when an infinite number of items is available. The results are illustrated using simulated data.     

On the sampling interpretation of confidence intervals and hypothesis tests in the context of conditional maximum likelihood estimation

In the context ofconditional maximum likelihood (CML) estimation, confidence intervals can be interpreted in three different ways, depending on the sampling distribution under which these confidence intervals contain the true parameter value with a certain probability. These sampling distributions are (a) the distribution of the data given theincidental parameters, (b) the marginal distribution of the data (i.e., with the incidental parameters integrated out), and (c) the conditional distribution of the data given the sufficient statistics for the incidental parameters. Results on the asymptotic distribution of CML estimates under sampling scheme (c) can be used to construct asymptotic confidence intervals using only the CML estimates. This is not possible for the results on the asymptotic distribution under sampling schemes (a) and (b). However, it is shown that theconditional asymptotic confidence intervals are also valid under the other two sampling schemes. I am indebted to Theo Eggen, Norman Verhelst and one of Psychometrika's reviewers for their helpful comments.     

Generalized Fiducial Inference for Binary Logistic Item Response Models

Generalized fiducial inference (GFI) has been proposed as an alternative to likelihood-based and Bayesian inference in mainstream statistics. Confidence intervals (CIs) can be constructed from a fiducial distribution on the parameter space in a fashion similar to those used with a Bayesian posterior distribution. However, no prior distribution needs to be specified, which renders GFI more suitable when no a priori information about model parameters is available. In the current paper, we apply GFI to a family of binary logistic item response theory models, which includes the two-parameter logistic (2PL), bifactor and exploratory item factor models as special cases. Asymptotic properties of the resulting fiducial distribution are discussed. Random draws from the fiducial distribution can be obtained by the proposed Markov chain Monte Carlo sampling algorithm. We investigate the finite-sample performance of our fiducial percentile CI and two commonly used Wald-type CIs associated with maximum likelihood (ML) estimation via Monte Carlo simulation. The use of GFI in high-dimensional exploratory item factor analysis was illustrated by the analysis of a set of the Eysenck Personality Questionnaire data.     

Calculate or wait: Is man an eager or a lazy intuitive statistician?

Research on people's ability to act as intuitive statisticians has mainly focused on the accuracy of estimates of central tendency and variability. In this paper, we investigate two hypothesised cognitive processes by which people make judgements of distribution shape. The first claims that people spontaneously induce abstract representations of distribution properties from experience, including about distribution shape. The second process claims that people construct beliefs about distribution properties post hoc by retrieval from long-term memory of small samples from the distribution, implying format dependence with accuracy that differs depending on judgement format. Results from two experiments confirm the predicted format dependence, suggesting that people are often constrained by the post hoc assessment of distribution properties by sampling from long-term memory. The results, however, also suggest that, although post hoc sampling from memory seems to be the default process, under certain predictable circumstances people do induce abstract representations of distribution shape.     

Estimating response time hazard functions: An exposition and extension

The hazard function of a response time distribution describes the temporal dimension of a response process and completely characterizes the distribution. The function can be estimated by a quadratic spline which is constrained to have one of a number of simple functional forms. The estimator can also be employed when the response time is an unobserved component in a parallel or serial pair of processes.     

竞争和合作对学龄前儿童分配模式的影响:心理理论和抑制控制的作用

研究考察在双方贡献不相等时,3~6岁儿童分配模式的发展趋势及其原因。98名儿童在竞争或合作情境下和玩偶进行钓鱼游戏,然后对6枚冰箱贴进行分配。结果发现5~6岁儿童比3~4岁儿童进行平均分配的比例更高,一级心理理论和抑制控制能力完全中介了年龄的效应。和竞争情境相比,合作情境促进儿童进行平均分配,同时情境调节了抑制控制对分配模式的影响。研究表明一级心理理论和抑制控制能力的发展能够解释学龄前儿童分配模式的变化。     

Examining mixed unknown distributions (mud)

A function, written in R, for testing whether the distribution of responses in one condition can be considered a combination of the distributions from two other conditions is described. The important aspect of this function is that it does not make any assumptions about the shape of the distributions. It is based o nthe Kolmogorov-Smirnov D statistic. The function also allows the user to test more specific and, hence, more statistically powerful hypotheses. One hypothesis, that the mixture does not capture the middle third of the distribution, is included as a built-in option, and code is provided so that other alternatives can easily be run. A power analysis reveals that the function is most likely to detect a difference between the combined conditions' distribution and the other distribution when the center of the other distribution is near the midpoint of the two original distributions. Critical p values are estimated for each set of distributions, using bootstrap methods. An example from human memory research, exploring the blending hypothesis of the misinformation effect, is used for illustrative purposes.     

The Normal Distribution Derived from Qualitative Conditions

The normal distribution is characterized in a measurement theoretic framework. The qualitative conditions guarantee that representations can be regarded as random variables. Additional axioms, also qualitative in the measurement sense, yield the normal. One characterization draws on a limit theorem. The main result derives the normal distribution from conjoint measurement axioms. This approach consists of formulating properties of a linear model as a component structure with error as one component. The normal distribution of errors is shown to be a consequence of the measurement theoretic assumptions. The possible impact of these results on statistical models is discussed. Copyright 2001 Academic Press.     

Shapes of reaction-time distributions and shapes of learning curves: a test of the instance theory of automaticity.

The instance theory assumes that automatic performance is based on single-step direct-access retrieval from memory of prior solutions to present problems. The theory predicts that the shape of the learning curve depends on the shape of the distribution of retrieval times. One can deduce from the fundamental assumptions of the theory that (1) the entire distribution of reaction times, not just the mean, will decrease as a power function of practice; (2) asymptotically, the retrieval-time distribution must be a Weibull distribution; and (3) the exponent of the Weibull, which is the parameter that determines its shape, must be the reciprocal of the exponent of the power function. These predictions were tested and mostly confirmed in 12 data sets from 2 experiments. The ability of the instance theory to predict the power law is contrasted with the ability of other theories to account for it.     

Operant Variability: A Conceptual Analysis

Some researchers claim that variability is an operant dimension of behavior. The present paper reviews the concept of operant behavior and emphasizes that differentiation is the behavioral process that demonstrates an operant relation. Differentiation is conceived as change in the overlap between two probability distributions: the distribution of reinforcement probability as a function of some response property (S distribution) and the probability distribution of the response property itself (R distribution). This concept implies that the differentiation process can be measured only if S distribution and R distribution are both established on the same response property. To determine whether the differentially reinforced behavioral variability fits the proposed concept of operant behavior, I examine the main procedures (lag n and threshold procedures) and the main dependent variable (U value) employed in the studies of operant variability. Because lag n and threshold procedures establish their S distributions on properties distinct from U value, differentiation cannot be measured over the change in U value. I conclude that studies of operant variability have failed to provide a direct demonstration that variability is an operant dimension of behavior. Hence, studies in which measures of variability provide a basis to measure differentiation can better support the claim that variability is an operant dimension of behavior.     

校正的Bootstrap方法对概化理论方差分量及其变异量估计的改善

Bootstrap方法是一种有放回的再抽样方法, 可用于概化理论的方差分量及其变异量估计。用Monte Carlo技术模拟四种分布数据, 分别是正态分布、二项分布、多项分布和偏态分布数据。基于p×i设计, 探讨校正的Bootstrap方法相对于未校正的Bootstrap方法, 是否改善了概化理论估计四种模拟分布数据的方差分量及其变异量。结果表明：跨越四种分布数据, 从整体到局部, 不论是“点估计”还是“变异量”估计, 校正的Bootstrap方法都要优于未校正的Bootstrap方法, 校正的Bootstrap方法改善了概化理论方差分量及其变异量估计。     

Bayesian estimation and testing of structural equation models

The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those based on the maximum likelihood solution, for example, output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters underidentified models, as we illustrate on a simple errors-in-variables model.We thank David Spiegelhalter for suggesting applying the Gibbs sampler to structural equation models to the first author at a 1994 workshop in Wiesbaden. We thank Ulf Böckenholt, Chris Meek, Marijtje van Duijn, Clark Glymour, Ivo Molenaar, Steve Klepper, Thomas Richardson, Teddy Seidenfeld, and Tom Snijders for helpful discussions, mathematical advice, and critiques of earlier drafts of this paper.     

A semi-parametric within-subject mixture approach to the analyses of responses and response times

In item response theory, modelling the item response times in addition to the item responses may improve the detection of possible between- and within-subject differences in the process that resulted in the responses. For instance, if respondents rely on rapid guessing on some items but not on all, the joint distribution of the responses and response times will be a multivariate within-subject mixture distribution. Suitable parametric methods to detect these within-subject differences have been proposed. In these approaches, a distribution needs to be assumed for the within-class response times. In this paper, it is demonstrated that these parametric within-subject approaches may produce false positives and biased parameter estimates if the assumption concerning the response time distribution is violated. A semi-parametric approach is proposed which resorts to categorized response times. This approach is shown to hardly produce false positives and parameter bias. In addition, the semi-parametric approach results in approximately the same power as the parametric approach.