Occam's rattle: children's use of simplicity and probability to constrain inference

A growing literature suggests that generating and evaluating explanations is a key mechanism for learning and inference, but little is known about how children generate and select competing explanations. This study investigates whether young children prefer explanations that are simple, where simplicity is quantified as the number of causes invoked in an explanation, and how this preference is reconciled with probability information. Both preschool-aged children and adults were asked to explain an event that could be generated by 1 or 2 causes, where the probabilities of the causes varied across conditions. In 2 experiments, it was found that children preferred explanations involving 1 cause over 2 but were also sensitive to the probability of competing explanations. Adults, in contrast, responded on the basis of probability alone. These data suggest that children employ a principle of parsimony like Occam's razor as an inductive constraint and that this constraint is employed when more reliable bases for inference are unavailable.     

UNDERSTANDING THE IMPACT OF TEST VALIDITY AND BIAS ON SELECTION ERRORS AND ADVERSE IMPACT IN HUMAN RESOURCE SELECTION

We propose an integrative framework for understanding the relationship among 4 closely related issues in human resource (HR) selection: test validity, test bias, selection errors, and adverse impact. One byproduct of our integrative approach is the concept of a previously undocumented source of selection errors we call bias‐based selection errors (i.e., errors that arise from using a biased test as if it were unbiased). Our integrative framework provides researchers and practitioners with a unique tool that generates numerical answers to questions such as the following: What are the anticipated consequences for bias‐based selection errors of various degrees of test validity and test bias? What are the anticipated consequences for adverse impact of various degrees of test validity and test bias? From a theory point of view, our framework provides a more complete picture of the selection process by integrating 4 key concepts that have not been examined simultaneously thus far. From a practical point of view, our framework provides test developers, employers, and policy makers a broader perspective and new insights regarding practical consequences associated with various selection systems that vary on their degree of validity and bias. We present a computer program available online to perform all needed calculations.     

Polytomous multilevel testlet models for testlet‐based assessments with complex sampling designs

A pplications of standard item response theory models assume local independence of items and persons. This paper presents polytomous multilevel testlet models for dual dependence due to item and person clustering in testlet‐based assessments with clustered samples. Simulation and survey data were analysed with a multilevel partial credit testlet model. This model was compared with three alternative models – a testlet partial credit model (PCM), multilevel PCM, and PCM – in terms of model parameter estimation. The results indicated that the deviance information criterion was the fit index that always correctly identified the true multilevel testlet model based on the quantified evidence in model selection, while the Akaike and Bayesian information criteria could not identify the true model. In general, the estimation model and the magnitude of item and person clustering impacted the estimation accuracy of ability parameters, while only the estimation model and the magnitude of item clustering affected the item parameter estimation accuracy. Furthermore, ignoring item clustering effects produced higher total errors in item parameter estimates but did not have much impact on the accuracy of ability parameter estimates, while ignoring person clustering effects yielded higher total errors in ability parameter estimates but did not have much effect on the accuracy of item parameter estimates. When both clustering effects were ignored in the PCM, item and ability parameter estimation accuracy was reduced.     

当结构假设和分布假设不满足时的验证性因子分析：稳健极大似然法估计和贝叶斯估计的比较研究

结构方程模型已被广泛应用于心理学、教育学、以及社会科学领域的统计分析中。结构方程模型分析中最常用的估计方法是基于正 态分布的估计量,比如极大似然估计法。这些方法需要满足两个假设。第一, 理论模型必须正确地反映变量与变量之间的关系,称为结构假 设。第二,数据必须符合多元正态分布,称为分布假设。如果这些假设不满足,基于正态分布的估计量就有可能导致不正确的卡方指数、不 正确的拟合度、以及有偏差的参数估计和参数估计的标准误。在实际应用中,几乎所有的理论模型都不能准确地解释变量与变量之间的关系, 数据也常常呈非多元正态分布。为此,一些新的估计方法得以发展。这些方法要么在理论上不要求数据呈多元正态分布,要么对因数据呈非 正态分布而导致的不正确结果进行纠正。当前较为流行的两种方法是稳健极大似然估计和贝叶斯估计。稳健极大似然估计是应用 Satorra and Bentler (1994) 的方法对不正确的卡方指数和参数估计的标准误进行调整,而参数估计和用极大似然方法得出的完全等同。贝叶斯估计方法则是 基于贝叶斯定理,其要点是：参数的后验分布是由参数的先验分布和数据似然值相乘而得来。后验分布常用马尔科夫蒙特卡洛算法来进行模拟。 对于稳健极大似然估计和贝叶斯估计这两种方法之间的优劣比较,先前的研究只局限于理论模型是正确的情境。而本研究则着重于理论模型 是错误的情境,同时也考虑到数据呈非正态分布的情境。本研究所采用的模型是验证性因子模型,数据全部由计算机模拟而来。数据的生成 取决于三个因素：8 类因子结构,3 种变量分布,和3 组样本量。这三个因素产生72 个模拟条件（72=8x3x3）。每个模拟条件下生成2000 个 数据组,每个数据组都拟合两个模型,一个是正确模型、一个是错误模型。每个模型都用两种估计方法来拟合：稳健极大似然估计法和贝叶 斯估计方法。贝叶斯估计方法中所使用的先验分布是无信息先验分布。结果分析主要着重于模型拒绝率、拟合度、参数估计、和参数估计的 标准误。研究的结果表明：在样本量充足的情况下,两种方法得出的参数估计非常相似。当数据呈非正态分布时,贝叶斯估计法比稳健极大 似然估计法更好地拒绝错误模型。但是,当样本量不足且数据呈正态分布时,贝叶斯估计在拒绝错误模型和参数估计上几乎没有优势,甚至 在一些条件下,比稳健极大似然法要差。     

Limited information estimation and testing of discretized multivariate normal structural models

Discretized multivariate normal structural models are often estimated using multistage estimation procedures. The asymptotic properties of parameter estimates, standard errors, and tests of structural restrictions on thresholds and polychoric correlations are well known. It was not clear how to assess the overall discrepancy between the contingency table and the model for these estimators. It is shown that the overall discrepancy can be decomposed into a distributional discrepancy and a structural discrepancy. A test of the overall model specification is proposed, as well as a test of the distributional specification (i.e., discretized multivariate normality). Also, the small sample performance of overall, distributional, and structural tests, as well as of parameter estimates and standard errors is investigated under conditions of correct model specification and also under mild structural and/or distributional misspecification. It is found that relatively small samples are needed for parameter estimates, standard errors, and structural tests. Larger samples are needed for the distributional and overall tests. Furthermore, parameter estimates, standard errors, and structural tests are surprisingly robust to distributional misspecification. This research was supported by the Department of Universities, Research and Information Society (DURSI) of the Catalan Government, and by grants BSO2000-0661 and BSO2003-08507 of the Spanish Ministry of Science and Technology.     

Measuring, estimating, and understanding the psychometric function: a commentary.

The psychometric function, relating the subject's response to the physical stimulus, is fundamental to psychophysics. This paper examines various psychometric function topics, many inspired by this special symposium issue of Perception & Psychophysics: What are the relative merits of objective yes/no versus forced choice tasks (including threshold variance)? What are the relative merits of adaptive versus constant stimuli methods? What are the relative merits of likelihood versus up-down staircase adaptive methods? Is 2AFC free of substantial bias? Is there no efficient adaptive method for objective yes/no tasks? Should adaptive methods aim for 90% correct? Can adding more responses to forced choice and objective yes/no tasks reduce the threshold variance? What is the best way to deal with lapses? How is the Weibull function intimately related to the d' function? What causes bias in the likelihood goodness-of-fit? What causes bias in slope estimates from adaptive methods? How good are nonparametric methods for estimating psychometric function parameters? Of what value is the psychometric function slope? How are various psychometric functions related to each other? The resolution of many of these issues is surprising.     

Measuring, estimating, and understanding the psychometric function: A commentary

The psychometric function, relating the subject’s response to the physical stimulus, is fundamental to psychophysics. This paper examines various psychometric function topics, many inspired by this special symposium issue ofPerception & Psychophysics: What are the relative merits of objective yes/no versus forced choice tasks (including threshold variance)? What are the relative merits of adaptive versus constant stimuli methods? What are the relative merits of likelihood versus up-down staircase adaptive methods? Is 2AFC free of substantial bias? Is there no efficient adaptive method for objective yes/no tasks? Should adaptive methods aim for 90% correct? Can adding more responses to forced choice and objective yes/no tasks reduce the threshold variance? What is the best way to deal with lapses? How is the Weibull function intimately related to thed’ function? What causes bias in the likelihood goodness-of-fit? What causes bias in slope estimates from adaptive methods? How good are nonparametric methods for estimating psychometric function parameters? Of what value is the psychometric function slope? How are various psychometric functions related to each other? The resolution of many of these issues is surprising.     

Meta-analytic methods of pooling correlation matrices for structural equation modeling under different patterns of missing data

Three methods of synthesizing correlations for meta-analytic structural equation modeling (SEM) under different degrees and mechanisms of missingness were compared for the estimation of correlation and SEM parameters and goodness-of-fit indices by using Monte Carlo simulation techniques. A revised generalized least squares (GLS) method for synthesizing correlations, weighted-covariance GLS (W-COV GLS), was compared with univariate weighting with untransformed correlations (univariate r) and univariate weighting with Fisher's z-transformed correlations (univariate z). These 3 methods were crossed with listwise and pairwise deletion. Univariate z and W-COV GLS performed similarly, with W-COV GLS providing slightly better estimation of parameters and more correct model rejection rates. Missing not at random data produced high levels of relative bias in correlation and model parameter estimates and higher incorrect SEM model rejection rates. Pairwise deletion resulted in inflated standard errors for all synthesis methods and higher incorrect rejection rates for the SEM model with univariate weighting procedures.     

Standard Errors of Kernel Equating: Accounting for Bandwidth Estimation

In standardized testing, equating is used to ensure comparability of test scores across multiple test administrations. One equipercentile observed-score equating method is kernel equating, where an essential step is to obtain continuous approximations to the discrete score distributions by applying a kernel with a smoothing bandwidth parameter. When estimating the bandwidth, additional variability is introduced which is currently not accounted for when calculating the standard errors of equating. This poses a threat to the accuracy of the standard errors of equating. In this study, the asymptotic variance of the bandwidth parameter estimator is derived and a modified method for calculating the standard error of equating that accounts for the bandwidth estimation variability is introduced for the equivalent groups design. A simulation study is used to verify the derivations and confirm the accuracy of the modified method across several sample sizes and test lengths as compared to the existing method and the Monte Carlo standard error of equating estimates. The results show that the modified standard errors of equating are accurate under the considered conditions. Furthermore, the modified and the existing methods produce similar results which suggest that the bandwidth variability impact on the standard error of equating is minimal.     

基于增长模型的非随机缺失数据处理:选择模型和极大似然方法

对含有非随机缺失数据的潜变量增长模型,为了考察基于不同假设的缺失数据处理方法:极大似然(ML)方法与DiggleKenward选择模型的优劣,通过Monte Carlo模拟研究,比较两种方法对模型中增长参数估计精度及其标准误估计的差异,并考虑样本量、非随机缺失比例和随机缺失比例的影响。结果表明,符合前提假设的Diggle-Kenward选择模型的参数估计精度普遍高于ML方法;对于标准误估计值,ML方法存在一定程度的低估,得到的置信区间覆盖比率也明显低于Diggle-Kenward选择模型。     

Robust Methods for Moderation Analysis with a Two-Level Regression Model

Moderation analysis has many applications in social sciences. Most widely used estimation methods for moderation analysis assume that errors are normally distributed and homoscedastic. When these assumptions are not met, the results from a classical moderation analysis can be misleading. For more reliable moderation analysis, this article proposes two robust methods with a two-level regression model when the predictors do not contain measurement error. One method is based on maximum likelihood with Student's t distribution and the other is based on M-estimators with Huber-type weights. An algorithm for obtaining the robust estimators is developed. Consistent estimates of standard errors of the robust estimators are provided. The robust approaches are compared against normal-distribution-based maximum likelihood (NML) with respect to power and accuracy of parameter estimates through a simulation study. Results show that the robust approaches outperform NML under various distributional conditions. Application of the robust methods is illustrated through a real data example. An R program is developed and documented to facilitate the application of the robust methods.     

Some standard errors in item response theory

The mathematics required to calculate the asymptotic standard errors of the parameters of three commonly used logistic item response models is described and used to generate values for some common situations. It is shown that the maximum likelihood estimation of a lower asymptote can wreak havoc with the accuracy of estimation of a location parameter, indicating that if one needs to have accurate estimates of location parameters (say for purposes of test linking/equating or computerized adaptive testing) the sample sizes required for acceptable accuracy may be unattainable in most applications. It is suggested that other estimation methods be used if the three parameter model is applied in these situations.The research reported here was supported, in part, by contract #F41689-81-6-0012 from the Air Force Human Resources Laboratory to McFann-Gray & Associates, Benjamin A. Fairbank, Jr., Principal Investigator. Further support of Wainer's effort was supplied by the Educational Testing Service, Program Statistics Research Project.     

WHEN IS PARSIMONY A VIRTUE?

Parsimony is a virtue of empirical theories. Is it also a virtue of philosophical theories? I review four contemporary accounts of the virtue of parsimony in empirical theorizing, and consider how each might apply to two prominent appeals to parsimony in the philosophical literature, those made on behalf of physicalism and on behalf of nominalism. None of the accounts of the virtue of parsimony extends naturally to either of these philosophical cases. This suggests that in typical philosophical contexts, ontological simplicity has no evidential value.     

LGM模型中缺失数据处理方法的比较:ML方法与Diggle-Kenward选择模型

追踪研究中缺失数据十分常见。本文通过Monte Carlo模拟研究,考察基于不同前提假设的Diggle-Kenward选择模型和ML方法对增长参数估计精度的差异,并考虑样本量、缺失比例、目标变量分布形态以及不同缺失机制的影响。结果表明:(1)缺失机制对基于MAR的ML方法有较大的影响,在MNAR缺失机制下,基于MAR的ML方法对LGM模型中截距均值和斜率均值的估计不具有稳健性。(2)DiggleKenward选择模型更容易受到目标变量分布偏态程度的影响,样本量与偏态程度存在交互作用,样本量较大时,偏态程度的影响会减弱。而ML方法仅在MNAR机制下轻微受到偏态程度的影响。     

Nonparametric Estimation of Standard Errors in Covariance Analysis Using the Infinitesimal Jackknife

The infinitesimal jackknife provides a simple general method for estimating standard errors in covariance structure analysis. Beyond its simplicity and generality what makes the infinitesimal jackknife method attractive is that essentially no assumptions are required to produce consistent standard error estimates, not even the requirement that the population sampled has the covariance structure assumed. Commonly used covariance structure analysis software uses parametric methods for estimating parameters and standard errors. When the population sampled has the covariance structure assumed, but fails to have the distributional form assumed, the parameter estimates usually remain consistent, but the standard error estimates do not. This has motivated the introduction of a variety of nonparametric standard error estimates that are consistent when the population sampled fails to have the distributional form assumed. The only distributional assumption these require is that the covariance structure be correctly specified. As noted, even this assumption is not required for the infinitesimal jackknife. The relation between the infinitesimal jackknife and other nonparametric standard error estimators is discussed. An advantage of the infinitesimal jackknife over the jackknife and the bootstrap is that it requires only one analysis to produce standard error estimates rather than one for every jackknife or bootstrap sample.     

Estimating standard errors in feature network models

Feature network models are graphical structures that represent proximity data in a discrete space while using the same formalism that is the basis of least squares methods employed in multidimensional scaling. Existing methods to derive a network model from empirical data only give the best‐fitting network and yield no standard errors for the parameter estimates. The additivity properties of networks make it possible to consider the model as a univariate (multiple) linear regression problem with positivity restrictions on the parameters. In the present study, both theoretical and empirical standard errors are obtained for the constrained regression parameters of a network model with known features. The performance of both types of standard error is evaluated using Monte Carlo techniques.     

Assessing the informational value of parameter estimates in cognitive models.

Mathematical models of cognition often contain unknown parameters whose values are estimated from the data. A question that generally receives little attention is how informative such estimates are. In a maximum likelihood framework, standard errors provide a measure of informativeness. Here, a standard error is interpreted as the standard deviation of the distribution of parameter estimates over multiple samples. A drawback to this interpretation is that the assumptions that are required for the maximum likelihood framework are very difficult to test and are not always met. However, at least in the cognitive science community, it appears to be not well known that standard error calculation also yields interpretable intervals outside the typical maximum likelihood framework. We describe and motivate this procedure and, in combination with graphical methods, apply it to two recent models of categorization: ALCOVE (Kruschke, 1992) and the exemplar-based random walk model (Nosofsky & Palmeri, 1997). The applications reveal aspects of these models that were not hitherto known and bring a mix of bad and good news concerning estimation of these models.     

Issues in the study and explanation of language behavior

A number of major issues raised in the study of language are examined from the standpoint of the requirements of an empirical science of language. In an effort to critically evaluate some of the basic assumptions and explanatory concepts of nativism in the study of language, the following five issues are examined: (1) What is nativism? (2) Can the creative aspect of language be explained in terms of nativism? (3) What is the explanatory status of nativist concepts? (4) What is the nature of nativist evidence? and (5) Do evaluative statements challenge alternative positions? The discussion points out various objections to and inconsistencies of nativism as well as the inadequacy of the explanatory status of nativist concepts.     

Assessing the informational value of parameter estimates in cognitive models

Mathematical models of cognition often contain unknown parameters whose values are estimated from the data. A question that generally receives little attention is how informative such estimates are. In a maximum likelihood framework, standard errors provide a measure of informativeness. Here, a standard error is interpreted as the standard deviation of the distribution of parameter estimates over multiple samples. A drawback to this interpretation is that the assumptions that are required for the maximum likelihood framework are very difficult to test and are not always met. However, at least in the cognitive science community, it appears to be not well known that standard error calculation also yields interpretable intervals outside the typical maximum likelihood framework. We describe and motivate this procedure and, in combination with graphical methods, apply it to two recent models of categorization: ALCOVE (Kruschke, 1992) and the exemplar-based random walk model (Nosofsky & Palmeri, 1997). The applications reveal aspects of these models that were not hitherto known and bring a mix of bad and good news concerning estimation of these models.     

Estimation Methods for Mixed Logistic Models with Few Clusters

For mixed models generally, it is well known that modeling data with few clusters will result in biased estimates, particularly of the variance components and fixed effect standard errors. In linear mixed models, small sample bias is typically addressed through restricted maximum likelihood estimation (REML) and a Kenward-Roger correction. Yet with binary outcomes, there is no direct analog of either procedure. With a larger number of clusters, estimation methods for binary outcomes that approximate the likelihood to circumvent the lack of a closed form solution such as adaptive Gaussian quadrature and the Laplace approximation have been shown to yield less-biased estimates than linearization estimation methods that instead linearly approximate the model. However, adaptive Gaussian quadrature and the Laplace approximation are approximating the full likelihood rather than the restricted likelihood; the full likelihood is known to yield biased estimates with few clusters. On the other hand, linearization methods linearly approximate the model, which allows for restricted maximum likelihood and the Kenward-Roger correction to be applied. Thus, the following question arises: Which is preferable, a better approximation of a biased function or a worse approximation of an unbiased function? We address this question with a simulation and an illustrative empirical analysis.