Showing papers in "British Journal of Mathematical and Statistical Psychology in 2001"
TL;DR: A method of composite reliability estimation using covariance structure analysis with nonlinear constraints is outlined, demonstrating that in many cases the widely used coefficient alpha is an unsatisfactory index of scale reliability already at the population level.
Abstract: A method of composite reliability estimation using covariance structure analysis with nonlinear constraints is outlined. To motivate the developments, initially a short overview of research is presented, demonstrating that in many cases the widely used coefficient alpha is an unsatisfactory index of scale reliability already at the population level. As an alternative, the proposed covariance structure analysis procedure is based on the theoretical formula of the scale reliability coefficient in terms of parameters pertaining to a given set of congeneric components. The described approach is illustrated with several numerical examples and its performance compared with that of coefficient alpha.
357 citations
TL;DR: It is argued that, by only reporting probability values and referring to statistical analyses as repeated measures ANOVA, authors convey neither the type of analysis that was used nor the validity of the reported probability value.
Abstract: Repeated measures ANOVA can refer to many different types of analysis. Specifically, this vague term can refer to conventional tests of significance, one of three univariate solutions with adjusted degrees of freedom, two different types of multivariate statistic, or approaches that combine univariate and multivariate tests. Accordingly, it is argued that, by only reporting probability values and referring to statistical analyses as repeated measures ANOVA, authors convey neither the type of analysis that was used nor the validity of the reported probability value, since each of these approaches has its own strengths and weaknesses. The various approaches are presented with a discussion of their strengths and weaknesses, and recommendations are made regarding the 'best' choice of analysis. Additional topics discussed include analyses for missing data and tests of linear contrasts.
251 citations
TL;DR: This work looks at the quantitative effect of outliers on estimators and test statistics based on normal theory maximum likelihood and the asymptotically distribution-free procedures.
Abstract: A small proportion of outliers can distort the results based on classical procedures in covariance structure analysis. We look at the quantitative effect of outliers on estimators and test statistics based on normal theory maximum likelihood and the asymptotically distribution-free procedures. Even if a proposed structure is correct for the majority of the data in a sample, a small proportion of outliers leads to biased estimators and significant test statistics. An especially unfortunate consequence is that the power to reject a model can be made arbitrarily--but misleadingly--large by inclusion of outliers in an analysis.
93 citations
TL;DR: A Bayesian approach for the multisample factor analysis model with continuous and polytomous variables is developed and it is shown that the conditional distributions involved in the implementation are the familiar uniform, gamma, normal, univariate truncated normal and Wishart distributions.
Abstract: The main purpose of this paper is to develop a Bayesian approach for the multisample factor analysis model with continuous and polytomous variables. Joint Bayesian estimates of the thresholds, the factor scores and the structural parameters subjected to some simple constraints across groups are obtained simultaneously. The Gibbs sampler is used to produce the joint Bayesian estimates. It is shown that the conditional distributions involved in the implementation are the familiar uniform, gamma, normal, univariate truncated normal and Wishart distributions. The Bayes factor is introduced to test hypotheses involving constraints among the structural parameters of the factor analysis models across groups. Two procedures for computing the test statistics are developed, one based on the Schwarz criterion (or Bayesian information criterion), while the other computes the posterior densities and likelihood ratios by means of draws from the appropriate conditional distributions via the Gibbs sampler. The empirical performance of the proposed Bayesian procedure and its sensitivity to prior distributions are illustrated by some simulation results and two real-life examples.
69 citations
TL;DR: In this article, the assumption of measurement invariance across occasions yields three-mode models that are suited for the analysis of multivariate longitudinal data, including autoregressive models and latent curve models as special cases.
Abstract: Multivariate longitudinal data are characterized by three modes: variables, occasions and subjects. Three-mode models are described as special cases of a linear latent variable model. The assumption of measurement invariance across occasions yields three-mode models that are suited for the analysis of multivariate longitudinal data. These so-called longitudinal three-mode models include autoregressive models and latent curve models as special cases. Empirical data from the field of industrial psychology are used in an example of how to test substantive hypotheses with the longitudinal, autoregressive and latent curve three-mode models.
53 citations
TL;DR: A new method is proposed for comparing all predictors in a multiple regression model, which generates a measure of predictor criticality, which is distinct from and has several advantages over traditional indices of predictor importance.
Abstract: A new method is proposed for comparing all predictors in a multiple regression model. This method generates a measure of predictor criticality, which is distinct from and has several advantages over traditional indices of predictor importance. Using the bootstrapping (resampling with replacement) procedure, a large number of samples are obtained from a given data set which contains one response variable and p predictors. For each sample, all 2 p 2 1 subset regression models are ® tted and the best subset model is selected. Thus, the (multinomial) distribution ±
39 citations
Journal Article•
32 citations
TL;DR: This note compares two heteroscedastic methods for testing H0 and finds that in terms of Type I errors, the nested bootstrap performed best in simulations when using rho, and generally an adjusted percentile bootstrap, used in conjunction with r, performed better than the nestedbootstrap.
Abstract: Let (Yb Xi), i = 1, ..., n, be a random sample from some bivariate distribution, and let rho be the (Pearson) population correlation between X and Y. The usual Student's t test of H0: rho = 0 is valid when X and Y are independent, so in particular the conditional variance of Y, given X, does not vary with X. But when the conditional variance does vary with X, Student's t uses an incorrect estimate of the standard error. In effect, when rejecting H0, this might be due to rho not equal to 0, but perhaps the main reason for rejecting is that there is heteroscedasticity. This note compares two heteroscedastic methods for testing H0 and finds that in terms of Type I errors, the nested bootstrap performed best in simulations when using rho. When using one of two robust analogues of rho (Spearman's rho and the percentage bend correlation), little or no advantage was found, in terms of Type I error probabilities, when using a nested bootstrap versus the basic percentile method. As for power, generally an adjusted percentile bootstrap, used in conjunction with r, performed better than the nested bootstrap, even in situations where, for the null case, the estimated probability of a Type I error was lower when using the adjusted percentile method. As for computing a confidence interval when correlations are positive, situations are found where all methods perform in an unsatisfactory manner.
27 citations
TL;DR: The simulated results show that Johnson's transformation with trimmed mean and the approximate test is valid in terms of Type I error rate control, and that the magnitude of the statistical power for non-normal distributions is better than that of conventional methods.
Abstract: The present study proposes a procedure that combines Johnson's transformation and the trimmed means method to deal with the problem of non-normality. An approximate test such as the Alexander-Govern test or Welch-James type test is then employed to deal with the heterogeneity of cell variance in the non-orthogonal two-way fixed effects completely randomized design. Both unweighted and weighted means analyses are considered. The empirical Type I error rates and the statistical power for comparing population means are investigated by Monte Carlo simulation. The simulated results show that Johnson's transformation with trimmed mean and the approximate test is valid in terms of Type I error rate control, and that the magnitude of the statistical power for non-normal distributions is better than that of conventional methods.
22 citations
TL;DR: In this article, the Monte Carlo EM algorithm was used for the maximum likelihood estimation of Thurstonian paired comparison models, even when the number of items is large, and a paired comparison study was presented in detail to illustrate the estimation approach.
Abstract: Thurstonian models provide a flexible framework for the analysis of multiple paired comparison judgments because they allow a wide range of hypotheses about the judgments' mean and covariance structures to be tested. However, applications have been limited to a large extent by the computational intractability involved in fitting this class of models. This paper demonstrates that the Monte Carlo EM algorithm facilitates maximum likelihood estimation of Thurstonian paired comparison models even when the number of items is large. A paired comparison study is presented in detail to illustrate the estimation approach.
13 citations
TL;DR: A class of Rasch model tests is proposed, all of them based on the Mantel-Haenszel chi-squared statistic, and three applications of the general procedure are presented, two on unidimensionality and one on item dependence in educational testing.
Abstract: A class of Rasch model tests is proposed, all of them based on the Mantel± Haenszel chi-squared statistic. All tests make use of thesuf® cient statistics' property the Rasch model possesses. One element of our general class, the test for item bias developed by Holland and Thayer, has been discussed extensively in the psycho- metric literature. Three applications of the general procedure are presented, two on unidimensionality and one on item dependence in educational testing. In each case, simulation results are reported. Our procedure is also applied to real data.
TL;DR: This paper presents a unidimensional item response model intended for personality and attitude items that use a continuous response format that takes into account the bounded nature of the item responses and assumes that their conditional distributions for a fixed trait level are truncated normal.
Abstract: This paper presents a unidimensional item response model intended for personality and attitude items that use a continuous response format. The model's starting point is the linear congeneric model for item scores, but it takes into account the bounded nature of the item responses and assumes that their conditional distributions for a fixed trait level are truncated normal. This assumption leads to nonlinear item-trait regressions and considerably modifies some aspects of the linear model. The linear model is considered as an approximation to the modified version, and an interval is defined in which the approximation is satisfactory. Procedures for estimating the item and subject parameters are described. The applicability of the model is illustrated using real data.
TL;DR: A new measure is constructed to define and detect conditional local influences and use the linear regression model for illustration to demonstrate that new information can be revealed by this proposed measure.
Abstract: The local influence approach proposed by Cook (1986) makes use of the normal curvature and the direction achieving the maximum curvature to assess the local influence of minor perturbation of statistical models. When the approach is applied to the linear regression model, the result provides information concerning the data structure different from that contributed by Cook's distance. One of the main advantages of the local influence approach is its ability to handle the simultaneous effect of several cases, namely, the ability to address the problem of ‘masking’. However, Lawrance (1995) points out that there are two notions of ‘masking’ effects, the joint influence and the conditional influence, which are distinct in nature. The normal curvature and the direction of maximum curvature are capable of addressing effects under the category of joint influences but not conditional influences. We construct a new measure to define and detect conditional local influences and use the linear regression model for illustration. Several reported data sets are used to demonstrate that new information can be revealed by this proposed measure.
TL;DR: An evaluation of an integer linear programming method for asymmetric seriation using five moderately sized matrices from the psychological literature, as well as 80 synthetic matrices, which found the solution to the linear programming relaxation of the integer-programming model was integer-optimal.
Abstract: Integer linear programming approaches for seriation of asymmetric n × n proximity matrices have generally been perceived as intractable for problems of practical size. However, to date, no computational evidence has been provided regarding the effectiveness (or ineffectiveness) of such approaches. This paper presents an evaluation of an integer linear programming method for asymmetric seriation using five moderately sized matrices (15 ≤ n ≤ 36) from the psychological literature, as well as 80 synthetic matrices (20 ≤ n ≤ 30). The solution to the linear programming relaxation of the integer-programming model was integer-optimal for each of the five empirical matrices and many of the synthetic matrices. In such instances, branch-and-bound integer programming was not required and optimal orderings were obtained within a few seconds. In all cases where the solution to the linear programming relaxation was not integer-optimal, branch-and-bound integer programming was able to find an optimal seriation in 18 minutes or less. A pragmatic solution strategy for larger matrices (n > 30) is also presented. This approach exploits the fact that, in many instances, only a modest percentage of all possible transitivity constraints are required to obtain an optimal solution.
TL;DR: An approach to generalizing an (ultrametric) representation is proposed in which the nested character of the partition sequence is relaxed and replaced by the weaker requirement that the classes within each partition contain objects consecutive with respect to a fixed ordering of the objects.
Abstract: Methods for the hierarchical clustering of an object set produce a sequence of nested partitions such that object classes within each successive partition are constructed from the union of object classes present at the previous level. Any such sequence of nested partitions can in turn be characterized by an ultrametric. An approach to generalizing an (ultrametric) representation is proposed in which the nested character of the partition sequence is relaxed and replaced by the weaker requirement that the classes within each partition contain objects consecutive with respect to a fixed ordering of the objects. A method for fitting such a structure to a given proximity matrix is discussed, along with several alternative strategies for graphical representation. Using this same ultrametric extension, additive tree representations can also be generalized by replacing the ultrametric component in the decomposition of an additive tree (into an ultrametric and a centroid metric). A common numerical illustration is developed and maintained throughout the paper.
TL;DR: A computationally efficient algorithm for computing the asymptotic standard errors for the promax factor solution and the reasons why it is more efficient than the augmented information approach are discussed.
Abstract: A computationally efficient algorithm for computing the asymptotic standard errors for the promax factor solution is proposed. The algorithm covers promax rotation with or without row normalization of the pre-rotated factor matrix. It also covers situations with either even- or odd-powered promax targets. With some modifications, the algorithm applies to Procrustean rotations with fixed or random independent targets. Simulation results show that the algorithm provides reasonable approximate standard errors for the promax solution with N = 200. In a real data example, the numerical results of the standard error computation using the proposed algorithm match those of an existing method based largely on the augmented information approach. The reasons why the proposed algorithm is more efficient than the augmented information approach are discussed.
TL;DR: The present study proposes Hall's or Johnson's transformation in conjunction with the trimmed mean to deal with the one-sample t test and results indicate that the proposed methods can control Type I error well in very extreme conditions and are more powerful than the conventional methods.
Abstract: If the assumption of normality is not satisfied, there is no simple solution to this problem for the one-sample t test. The present study proposes Hall's or Johnson's transformation in conjunction with the trimmed mean to deal with the problem. Computer simulation is carried out to evaluate the small-sample behaviour of the proposed methods in terms of Type I error rate and statistical power. The proposed methods are compared with the conventional Student t, Yuen's trimmed t, Johnson's transformation untrimmed t, and Hall's transformation untrimmed t statistics for one-sided and two-sided tests. The simulation results indicate that the proposed methods can control Type I error well in very extreme conditions and are more powerful than the conventional methods.
TL;DR: The minimax sequential strategy is compared for the anorexia nervosa example with other procedures that exist for similar classification decision problems in the literature in terms of average number of patients to be tested, classification accuracy and average loss.
Abstract: The purpose of this paper is to derive optimal rules for sequential testing problems in psychodiagnostics. In sequential psychodiagnostic testing, each time a patient is exposed to a new treatment, the decision then is to declare this new treatment effective, ineffective, or to continue testing and exposing the new treatment to another random patient suffering from the same mental health problem. The framework of minimax sequential decision theory is proposed for solving such testing problems; that is, optimal rules are obtained by minimizing the maximum expected losses associated with all possible decision rules at each stage of testing. The main advantage of this approach is that costs of testing can be explicitly taken into account. The sequential testing procedure is applied to an empirical example for determining the effectiveness of a cognitive-analytic therapy for patients suffering from anorexia nervosa. For a given maximum number of patients to be tested, the appropriate action is indicated at each stage of testing for different numbers of positive reactions to the cognitive-analytic therapy. The paper concludes with a simulation study, in which the minimax sequential strategy is compared for the anorexia nervosa example with other procedures that exist for similar classification decision problems in the literature in terms of average number of patients to be tested, classification accuracy and average loss.
TL;DR: Alternative approaches are outlined, which are straightforward and simpler than the reparameterization recently proposed by Alanen, Leskinen and Kuusinen.
Abstract: Covariance structure analysis methods for testing invariance in reliability and stability coefficients in multi-wave, multi-indicator models are discussed. Alternative approaches are outlined, which are straightforward and simpler than the reparameterization recently proposed by Alanen, Leskinen and Kuusinen.
TL;DR: A simulation study designed to evaluate the pseudo-R2T proposed in an earlier paper by Spiess and Keller suggests that this measure represents the goodness of fit not only of the systematic part, but also of the assumed correlation structure in binary panel probit models.
Abstract: A simulation study designed to evaluate the pseudo-R2T proposed in an earlier paper by Spiess and Keller suggests that, for the models considered, this measure represents the goodness of fit not only of the systematic part, but also of the assumed correlation structure in binary panel probit models.
TL;DR: It is shown that both over- and underdispersion may arise in latent class models of which the other three are special cases, so the score distribution is not always indicative of the lack of fit of the mixture binomial when in fact the latent class model is true.
Abstract: Four scenarios of homogeneity/heterogeneity with respect to the performance of the subjects and the task difficulties are considered: first, the unconstrained latent class model providing for heterogeneity with respect to both; second, the mixture binomial assuming constant task difficulty within each mixing component, but different levels of performance of the subjects; third, the model of independence which is equivalent to the one-class latent class model allowing for different task difficulties but no variability of the subjects; and fourth, the binomial with success probability constant across tasks and subjects. It is shown that both over- and underdispersion may arise in latent class models of which the other three are special cases. As a consequence, the latent class model and the mixture binomial may generate nearly indistinguishable score distributions where overdispersion is present. So the score distribution is not always indicative of the lack of fit of the mixture binomial when in fact the latent class model is true. It may, therefore, be misleading to accept mixture binomials as well-fitting models without having additionally assessed the fit of latent class models. This, however, is often the case in empirical research. A long series of investigations on Piaget's water-level tasks serves as a good example.
TL;DR: Three common factor models are proposed for the analysis of k x k ordinal data arising from test validity or reliability situations, which represent an extension of the polychoric correlation model and item response theory.
Abstract: Three common factor models are proposed for the analysis of k x k ordinal data arising from test validity or reliability situations. These models represent an extension of the polychoric correlation model and item response theory. Identification is complete in the most usual reliability situation, where data from only two indicators (raters) are available. Full maximum likelihood estimation is available together with associated informative deviance tests and goodness-of-fit tests, examples of which are provided.
TL;DR: This paper considers the direct products of relational systems of the same type and scales defined thereon, as well as the invariance properties of functions on such product structures, and applies the results to a theory of multidimensional scale types.
Abstract: This paper considers the direct products of relational systems of the same type and scales defined thereon, as well as the invariance properties of functions on such product structures. The results will be applied to a theory of multidimensional scale types.