BootES: An R package for bootstrap confidence intervals on effect sizes

Bootstrap Effect Sizes (bootES; Gerlanc & Kirby, 2012) is a free, open-source software package for R (R Development Core Team, 2012), which is a language and environment for statistical computing. BootES computes both unstandardized and standardized effect sizes (such as Cohen’s d, Hedges’s g, and Pearson’s r) and makes easily available for the first time the computation of their bootstrap confidence intervals (CIs). In this article, we illustrate how to use bootES to find effect sizes for contrasts in between-subjects, within-subjects, and mixed factorial designs and to find bootstrap CIs for correlations and differences between correlations. An appendix gives a brief introduction to R that will allow readers to use bootES without having prior knowledge of R.     

Estimating true score in the compound binomial error model

Several procedures have been proposed in the statistical literature for estimating simultaneously the mean of each ofk binomial populations. In terms of mental test theory, however, it is not clear that these procedures should be used when an item sampling model applies since the binomial error model is usually viewed as an oversimplification of the true situation. In this study we compare empirically several of these estimation techniques. Particular attention is given to situations where observations are generated according to a two-term approximation to the compound binomial distribution.The author would like to thank Shelley Niwa for writing the computer programs used in this study.The work upon which this publication is based was performed pursuant to Grant # NIE-G-76-0083 with the National Institute of Education, Department of Health, Education and Welfare. Points of view or opinions stated do not necessarily represent official NIE position or policy.     

Estimating confidence intervals for principal component loadings: A comparison between the bootstrap and asymptotic results

Confidence intervals (CIs) in principal component analysis (PCA) can be based on asymptotic standard errors and on the bootstrap methodology. The present paper offers an overview of possible strategies for bootstrapping in PCA. A motivating example shows that CI estimates for the component loadings using different methods may diverge. We explain that this results from both differences in quality and in perspective on the rotational freedom of the population loadings. A comparative simulation study examines the quality of various estimated component loading CIs. The bootstrap approach is more flexible and generally yields better CIs than the asymptotic approach. However, in the case of a clear simple structure of varimax rotated loadings, one can be confident that the asymptotic estimates are reasonable as well.     

Extensions of Rasch's multiplicative poisson model

Consideration will be given to a model developed by Rasch that assumes scores observed on some types of attainment tests can be regarded as realizations of a Poisson process. The parameter of the Poisson distribution is assumed to be a product of two other parameters, one pertaining to the ability of the subject and a second pertaining to the difficulty of the test. Rasch's model is expanded by assuming a prior distribution, with fixed but unknown parameters, for the subject parameters. The test parameters are considered fixed. Secondly, it will be shown how additional between- and within-subjects factors can be incorporated. Methods for testing the fit and estimating the parameters of the model will be discussed, and illustrated by empirical examples.     

Constructing bootstrap confidence intervals for principal component loadings in the presence of missing data: a multiple-imputation approach

Earlier research has shown that bootstrap confidence intervals from principal component loadings give a good coverage of the population loadings. However, this only applies to complete data. When data are incomplete, missing data have to be handled before analysing the data. Multiple imputation may be used for this purpose. The question is how bootstrap confidence intervals for principal component loadings should be corrected for multiply imputed data. In this paper, several solutions are proposed. Simulations show that the proposed corrections for multiply imputed data give a good coverage of the population loadings in various situations.     

Empirical bayes estimation of coefficients in the general linear model from data of deficient rank

Empirical Bayes methods are shown to provide a practical alternative to standard least squares methods in fitting high dimensional models to sparse data. An example concerning prediction bias in educational testing is presented as an illustration.The authors would like to thank the referees for several useful comments.The analysis of the data discussed in this report was part of a study funded jointly by the Graduate Management Admission Council and Educational Testing Service.     

Exact and best confidence intervals for the ability parameter of the Rasch model

A commonly used method to evaluate the accuracy of a measurement is to provide a confidence interval that contains the parameter of interest with a given high probability. Smallest exact confidence intervals for the ability parameter of the Rasch model are derived and compared to the traditional, asymptotically valid intervals based on the Fisher information. Tables of the exact confidence intervals, termed Clopper-Pearson intervals, can be routinely drawn up by applying a computer program designed by and obtainable from the author. These tables are particularly useful for tests of only moderate lengths where the asymptotic method does not provide valid confidence intervals.     

An introduction to Bayesian model selection for evaluating informative hypotheses

Most researchers have specific expectations concerning their research questions. These may be derived from theory, empirical evidence, or both. Yet despite these expectations, most investigators still use null hypothesis testing to evaluate their data, that is, when analysing their data they ignore the expectations they have. In the present article, Bayesian model selection is presented as a means to evaluate the expectations researchers have, that is, to evaluate so called informative hypotheses. Although the methodology to do this has been described in previous articles, these are rather technical and havemainly been published in statistical journals. The main objective of thepresent article is to provide a basic introduction to the evaluation of informative hypotheses using Bayesian model selection. Moreover, what is new in comparison to previous publications on this topic is that we provide guidelines on how to interpret the results. Bayesian evaluation of informative hypotheses is illustrated using an example concerning psychosocial functioning and the interplay between personality and support from family.     

On approximate confidence intervals for measures of concordance

The use ofU-statistics based on rank correlation coefficients in estimating the strength of concordance among a group of rankers is examined for cases where the null hypothesis of random rankings is not tenable. The studentizedU-statistics is asymptotically distribution-free, and the Student-t approximation is used for small and moderate sized samples. An approximate confidence interval is constructed for the strength of concordance. Monte Carlo results indicate that the Student-t approximation can be improved by estimating the degrees of freedom.Research partially supported on ONR Contract N00014-82-K-0207.     

Marginal maximum likelihood estimation for a psychometric model of discontinuous development

Item response theory models posit latent variables to account for regularities in students' performances on test items. Wilson's “Saltus” model extends the ideas of IRT to development that occurs in stages, where expected changes can be discontinuous, show different patterns for different types of items, or even exhibit reversals in probabilities of success on certain tasks. Examples include Piagetian stages of psychological development and Siegler's rule-based learning. This paper derives marginal maximum likelihood (MML) estimation equations for the structural parameters of the Saltus model and suggests a computing approximation based on the EM algorithm. For individual examinees, empirical Bayes probabilities of learning-stage are given, along with proficiency parameter estimates conditional on stage membership. The MML solution is illustrated with simulated data and an example from the domain of mixed number subtraction. The authors' names appear in alphabetical order. We would like to thank Karen Draney for computer programming, Kikumi Tatsuoka for allowing us to use the mixed-number subtraction data, and Eric Bradlow, Chan Dayton, Kikumi Tatsuoka, and four anonymous referees for helpful suggestions. The first author's work was supported by Contract No. N00014-88-K-0304, R&T 4421552, from the Cognitive Sciences Program, Cognitive and Neural Sciences Division, Office of Naval Research, and by the Program Research Planning Council of Educational Testing Service. The second author's work was supported by a National Academy of Education Spencer Fellowship and by a Junior Faculty Research Grant from the Committee on Research, University of California at Berkeley. A copy of the Saltus computer program can be obtained from the second author.     

Application of the bootstrap methods in factor analysis

A Monte Carlo experiment is conducted to investigate the performance of the bootstrap methods in normal theory maximum likelihood factor analysis both when the distributional assumption is satisfied and unsatisfied. The parameters and their functions of interest include unrotated loadings, analytically rotated loadings, and unique variances. The results reveal that (a) bootstrap bias estimation performs sometimes poorly for factor loadings and nonstandardized unique variances; (b) bootstrap variance estimation performs well even when the distributional assumption is violated; (c) bootstrap confidence intervals based on the Studentized statistics are recommended; (d) if structural hypothesis about the population covariance matrix is taken into account then the bootstrap distribution of the normal theory likelihood ratio test statistic is close to the corresponding sampling distribution with slightly heavier right tail.This study was carried out in part under the ISM cooperative research program (91-ISM · CRP-85, 92-ISM · CRP-102). The authors would like to thank the editor and three reviewers for their helpful comments and suggestions which improved the quality of this paper considerably.     

三类多层中介效应分析方法比较

比较了贝叶斯法、Monte Carlo法和参数Bootstrap法在2-1-1多层中介分析中的表现。结果发现：1)有先验信息的贝叶斯法的中介效应点估计和区间估计都最准确;2)无先验信息的贝叶斯法、Monte Carlo法、偏差校正和未校正的参数Bootstrap法的中介效应点估计和区间估计表现相当,但Monte Carlo法在第Ⅰ类错误率和区间宽度指标上表现略优于其他三种方法,偏差校正的Bootstrap法在统计检验力上表现略优于其他三种方法,但在第Ⅰ类错误率上表现最差;结果表明,当有先验信息时,推荐使用贝叶斯法;当先验信息不可得时,推荐使用Monte Carlo法。     

A comparison of three simple test theory models

In very simple test theory models such as the Rasch model, a single parameter is used to represent the ability of any examinee or the difficulty of any item. Simple models such as these provide very important points of departure for more detailed modeling when a substantial amount of data are available, and are themselves of real practical value for small or even medium samples. They can also serve a normative role in test design.As an alternative to the Rasch model, or the Rasch model with a correction for guessing, a simple model is introduced which characterizes strength of response in terms of the ratio of ability and difficulty parameters rather than their difference. This model provides a natural account of guessing, and has other useful things to contribute as well. It also offers an alternative to the Rasch model with the usual correction for guessing. The three models are compared in terms of statistical properties and fits to actual data. The goal of the paper is to widen the range of minimal models available to test analysts.This research was supported by grant AP320 from the Natural Sciences and Engineering Research Council of Canada. The author is grateful for discussions with M. Abrahamowicz, I. Molenaar, D. Thissen, and H. Wainer.     

Hierarchical models of simple mechanisms underlying confidence in decision making

Choice confidence is a central measure in psychological decision research, often being reported on a probabilistic scale. Simple mechanisms that describe the psychological processes underlying choice confidence, including those based on error and confirmation biases, have typically received support via fits to data averaged over subjects. While averaged data ease model development, they can also destroy important aspects of the confidence data distribution. In this paper, we develop a hierarchical model of raw confidence judgments using the beta distribution, and we implement two simple confidence mechanisms within it. We use Bayesian methods to fit the hierarchical model to data from a two-alternative confidence experiment, and we use a variety of Bayesian tools to diagnose shortcomings of the simple mechanisms that are overlooked when applied to averaged data. Bugs code for estimating the models is also supplied.     

Bootstrap standard error and confidence intervals for the correlations corrected for indirect range restriction

The standard Pearson correlation coefficient, r, is a biased estimator of the population correlation coefficient, ρ(XY) , when predictor X and criterion Y are indirectly range-restricted by a third variable Z (or S). Two correction algorithms, Thorndike's (1949) Case III, and Schmidt, Oh, and Le's (2006) Case IV, have been proposed to correct for the bias. However, to our knowledge, the two algorithms did not provide a procedure to estimate the associated standard error and confidence intervals. This paper suggests using the bootstrap procedure as an alternative. Two Monte Carlo simulations were conducted to systematically evaluate the empirical performance of the proposed bootstrap procedure. The results indicated that the bootstrap standard error and confidence intervals were generally accurate across simulation conditions (e.g., selection ratio, sample size). The proposed bootstrap procedure can provide a useful alternative for the estimation of the standard error and confidence intervals for the correlation corrected for indirect range restriction.     

Bayesian estimation in the three-parameter logistic model

A joint Bayesian estimation procedure for the estimation of parameters in the three-parameter logistic model is developed in this paper. Procedures for specifying prior beliefs for the parameters are given. It is shown through simulation studies that the Bayesian procedure (i) ensures that the estimates stay in the parameter space, and (ii) produces better estimates than the joint maximum likelihood procedure as judged by such criteria as mean squared differences between estimates and true values. The research reported here was performed pursuant to Grant No. N0014-79-C-0039 with the Office of Naval Research. A related article by Robert J. Mislevy (1986) appeared when the present paper was in the printing stage.     

Parameter estimation in latent trait models

Latent trait models for binary responses to a set of test items are considered from the point of view of estimating latent trait parameters=( ₁, , _n ) and item parameters=( ₁, , _k ), where _j may be vector valued. With considered a random sample from a prior distribution with parameter, the estimation of (, ) is studied under the theory of the EM algorithm. An example and computational details are presented for the Rasch model.This work was supported by Contract No. N00014-81-K-0265, Modification No. P00002, from Personnel and Training Research Programs, Psychological Sciences Division, Office of Naval Research. The authors wish to thank an anonymous reviewer for several valuable suggestions.     

General estimators for the reliability of qualitative data

We study a proportional reduction in loss (PRL) measure for the reliability of categorical data and consider the general case in which each ofN judges assigns a subject to one ofK categories. This measure has been shown to be equivalent to a measure proposed by Perreault and Leigh for a special case when there are two equally competent judges, and the correct category has a uniform prior distribution. We consider a general framework where the correct category is assumed to have an arbitrary prior distribution, and where classification probabilities vary by correct category, judge, and category of classification. In this setting, we consider PRL reliability measures based on two estimators of the correct category—the empirical Bayes estimator and an estimator based on the judges' consensus choice. We also discuss four important special cases of the general model and study several types of lower bounds for PRL reliability.Bruce Cooil is Associate Professor of Statistics, and Roland T. Rust is Professor and area head for Marketing, Owen Graduate School of Management, Vanderbilt University. The authors thank three anonymous reviewers and an Associate Editor for their helpful comments and suggestions. This work was supported in part by the Dean's Fund for Faculty Research of the Owen Graduate School of Management, Vanderbilt University.     

A probabilistic model of cross-categorization

Most natural domains can be represented in multiple ways: we can categorize foods in terms of their nutritional content or social role, animals in terms of their taxonomic groupings or their ecological niches, and musical instruments in terms of their taxonomic categories or social uses. Previous approaches to modeling human categorization have largely ignored the problem of cross-categorization, focusing on learning just a single system of categories that explains all of the features. Cross-categorization presents a difficult problem: how can we infer categories without first knowing which features the categories are meant to explain? We present a novel model that suggests that human cross-categorization is a result of joint inference about multiple systems of categories and the features that they explain. We also formalize two commonly proposed alternative explanations for cross-categorization behavior: a features-first and an objects-first approach. The features-first approach suggests that cross-categorization is a consequence of attentional processes, where features are selected by an attentional mechanism first and categories are derived second. The objects-first approach suggests that cross-categorization is a consequence of repeated, sequential attempts to explain features, where categories are derived first, then features that are poorly explained are recategorized. We present two sets of simulations and experiments testing the models’ predictions about human categorization. We find that an approach based on joint inference provides the best fit to human categorization behavior, and we suggest that a full account of human category learning will need to incorporate something akin to these capabilities.     

Bootstrap standard error and confidence intervals for the correlation corrected for range restriction: a simulation study

The standard Pearson correlation coefficient is a biased estimator of the true population correlation, rho, when the predictor and the criterion are range restricted. To correct the bias, the correlation corrected for range restriction, rc, has been recommended, and a standard formula based on asymptotic results for estimating its standard error is also available. In the present study, the bootstrap standard-error estimate is proposed as an alternative. Monte Carlo simulation studies involving both normal and nonnormal data were conducted to examine the empirical performance of the proposed procedure under different levels of rho, selection ratio, sample size, and truncation types. Results indicated that, with normal data, the bootstrap standard-error estimate is more accurate than the traditional estimate, particularly with small sample size. With nonnormal data, performance of both estimates depends critically on the distribution type. Furthermore, the bootstrap bias-corrected and accelerated interval consistently provided the most accurate coverage probability for rho.