Simulating multivariate nonnormal distributions

A method for generating multivariate nonnormal distributions with specified intercorrelations and marginal means, variances, skews, and kurtoses is proposed. As an example, the method is applied to the generation of simulated scores on three psychological tests administered to a single group of individuals.     

Simulating correlated multivariate nonnormal distributions: Extending the fleishman power method

A procedure for generating multivariate nonnormal distributions is proposed. Our procedure generates average values of intercorrelations much closer to population parameters than competing procedures for skewed and/or heavy tailed distributions and for small sample sizes. Also, it eliminates the necessity of conducting a factorization procedure on the population correlation matrix that underlies the random deviates, and it is simpler to code in a programming language (e.g., FORTRAN). Numerical examples demonstrating the procedures are given. Monte Carlo results indicate our procedure yields excellent agreement between population parameters and average values of intercorrelation, skew, and kurtosis.     

Comparing means from nonnormal distributions: The bisquare-weighted analysis of variance

A new procedure analogous to the analysis of variance (ANOVA), called the bisquare-weighted ANOVA (bANOVA), is described. When a traditional ANOVA is calculated, using samples from a distribution with heavy tails, the Type I error rates remain in check, but the Type II error rates increase, relative to those across samples from a normal distribution. The bANOVA is robust with respect to deviations from a normal distribution, maintaining high power with normal and heavy-tailed distributions alike. The more popular rank ANOVA (rANOVA) is also described briefly. However, the rANOVA is not as robust to large deviations from normality as is the bANOVA, and it generates high Type I error rates when applied to three-way designs.     

Using the linear mixed model to analyze nonnormal data distributions in longitudinal designs

Using a Monte Carlo simulation and the Kenward–Roger (KR) correction for degrees of freedom, in this article we analyzed the application of the linear mixed model (LMM) to a mixed repeated measures design. The LMM was first used to select the covariance structure with three types of data distribution: normal, exponential, and log-normal. This showed that, with homogeneous between-groups covariance and when the distribution was normal, the covariance structure with the best fit was the unstructured population matrix. However, with heterogeneous between-groups covariance and when the pairing between covariance matrices and group sizes was null, the best fit was shown by the between-subjects heterogeneous unstructured population matrix, which was the case for all of the distributions analyzed. By contrast, with positive or negative pairings, the within-subjects and between-subjects heterogeneous first-order autoregressive structure produced the best fit. In the second stage of the study, the robustness of the LMM was tested. This showed that the KR method provided adequate control of Type I error rates for the time effect with normally distributed data. However, as skewness increased—as occurs, for example, in the log-normal distribution—the robustness of KR was null, especially when the assumption of sphericity was violated. As regards the influence of kurtosis, the analysis showed that the degree of robustness increased in line with the amount of kurtosis.     

Generating multiple samples of multivariate data with arbitrary population parameters

A method of generating any number of score and correlation matrices with arbitrary population parameters is described. EitherZ scores or stanines are sampled from a normal population to represent factor scores by an IBM 1620 program. These are converted to variates from a population with an a priori factor structure. The effectiveness of the method is illustrated from research data. Some further modifications and uses of the method are discussed.     

Visualization of multivariate data: Human-factors considerations

Recent software provides new tools for visualizing multivariate data that facilitate data analysis. We focus on (1) the learnability and use of visualization systems, and (2) the perceptual and cognitive processes involved in viewing visualizations. Effective visualization systems support a broad range of user tasks and abilities, are easy to learn, and provide powerful and flexible output formatting. Effective visualizations incorporate Gestalt and other perceptual and cognitive principles that encourage more rapid, automatic processing, and less slow, controlled processing.     

Evaluating the robustness of repeated measures analyses: The case of small sample sizes and nonnormal data

Repeated measures analyses of variance are the method of choice in many studies from experimental psychology and the neurosciences. Data from these fields are often characterized by small sample sizes, high numbers of factor levels of the within-subjects factor(s), and nonnormally distributed response variables such as response times. For a design with a single within-subjects factor, we investigated Type I error control in univariate tests with corrected degrees of freedom, the multivariate approach, and a mixed-model (multilevel) approach (SAS PROC MIXED) with Kenward–Roger’s adjusted degrees of freedom. We simulated multivariate normal and nonnormal distributions with varied population variance–covariance structures (spherical and nonspherical), sample sizes (N), and numbers of factor levels (K). For normally distributed data, as expected, the univariate approach with Huynh–Feldt correction controlled the Type I error rate with only very few exceptions, even if samples sizes as low as three were combined with high numbers of factor levels. The multivariate approach also controlled the Type I error rate, but it requires N ≥ K. PROC MIXED often showed acceptable control of the Type I error rate for normal data, but it also produced several liberal or conservative results. For nonnormal data, all of the procedures showed clear deviations from the nominal Type I error rate in many conditions, even for sample sizes greater than 50. Thus, none of these approaches can be considered robust if the response variable is nonnormally distributed. The results indicate that both the variance heterogeneity and covariance heterogeneity of the population covariance matrices affect the error rates.     

Describing the elephant: Structure and function in multivariate data

There is a unity underlying the diversity of models for the analysis of multivariate data. Essentially, they constitute a family models, most generally nonlinear, for structural/functional relations between variables drawn from a behavior domain.     

A unified approach to exploratory factor analysis with missing data,nonnormal data,and in the presence of outliers

Factor analysis is regularly used for analyzing survey data. Missing data, data with outliers and consequently nonnormal data are very common for data obtained through questionnaires. Based on covariance matrix estimates for such nonstandard samples, a unified approach for factor analysis is developed. By generalizing the approach of maximum likelihood under constraints, statistical properties of the estimates for factor loadings and error variances are obtained. A rescaled Bartlett-corrected statistic is proposed for evaluating the number of factors. Equivariance and invariance of parameter estimates and their standard errors for canonical, varimax, and normalized varimax rotations are discussed. Numerical results illustrate the sensitivity of classical methods and advantages of the proposed procedures.This project was supported by a University of North Texas Faculty Research Grant, Grant #R49/CCR610528 for Disease Control and Prevention from the National Center for Injury Prevention and Control, and Grant DA01070 from the National Institute on Drug Abuse. The results do not necessarily represent the official view of the funding agencies. The authors are grateful to three reviewers for suggestions that improved the presentation of this paper.     

The influence of temporal distributions on causal induction from tabular data

Two experiments investigated the joint influence of statistical and temporal information on causal inference from tabular data. Participants were presented with unambiguous data sets containing information about relative effect frequencies in cause-present and cause-absent situations. In addition to contingency information, the stimuli also revealed information about the temporal distribution of effects. The participants took this information into account when making causal judgments, so that the mere advancing or postponing of the effect in time was attached with causal significance, even when the cause did not increase the overall probability of the effect. These results cannot be reconciled with standard contingency accounts of causal induction.     

Testing the significance of a correlation with nonnormal data: Comparison of Pearson, Spearman, transformation, and resampling approaches

It is well known that when data are nonnormally distributed, a test of the significance of Pearson's r may inflate Type I error rates and reduce power. Statistics textbooks and the simulation literature provide several alternatives to Pearson's correlation. However, the relative performance of these alternatives has been unclear. Two simulation studies were conducted to compare 12 methods, including Pearson, Spearman's rank-order, transformation, and resampling approaches. With most sample sizes (n ≥ 20), Type I and Type II error rates were minimized by transforming the data to a normal shape prior to assessing the Pearson correlation. Among transformation approaches, a general purpose rank-based inverse normal transformation (i.e., transformation to rankit scores) was most beneficial. However, when samples were both small (n ≤ 10) and extremely nonnormal, the permutation test often outperformed other alternatives, including various bootstrap tests. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

PLS model building: A multivariate approach to personality test data

The aim of this study was to demonstrate how personality test data can be plotted with a multivariate method known as Partial Least Squares of Latent Structures (PLS). The basic methodology behind PLS modeling is presented and the example demonstrates how a PLS model of personality test data can be used for diagnostic prediction. Principles for validating the models are also presented. The conclusion is that PLS modeling appears to be a powerful method for extracting clinically relevant information from complex personality test data matrixes. It could be used as a complement to more hard modeling methods in the process of examining a new area of interest.     

Political socialization: A multivariate analysis of Swedish political attitude and preference data

Studied the socio-political attitudes and political party preferences of 532 Swedish high school students as a function of seven background variables: (I) the mother's political party preference, (2) the father's political party preference, (3) the mother's education, (4) the father's education, (5) the mother's income, (6) the father's income and (7) social class identification. Multiple classification analysis and multivariate nominal analysis were used to uncover the most important possible determinants of political socialization of the youth in both bivariate and multivariate aspects. The results showed that, of the seven predictor or background variables studied, only three had any substantial relationship with socio-political attitudes and political party preferences of the youth: (a) the mother's political party preference, (b) class identification and (c) the father's political party preference in that general order of importance. Furthermore, the superiority of the mother's political party preference over the father's political party preference was especially marked for girls. Among other things, the results also disclosed that ‘left-wing’ youth tended to be more loyal to parental political beliefs than ‘moderate’ and ‘right-wing’ youth. Several alternative explanations were proposed for these findings.     

Analysis of multivariate mixed longitudinal data: A flexible latent process approach

Multivariate ordinal and quantitative longitudinal data measuring the same latent construct are frequently collected in psychology. We propose an approach to describe change over time of the latent process underlying multiple longitudinal outcomes of different types (binary, ordinal, quantitative). By relying on random‐effect models, this approach handles individually varying and outcome‐specific measurement times. A linear mixed model describes the latent process trajectory while equations of observation combine outcome‐specific threshold models for binary or ordinal outcomes and models based on flexible parameterized non‐linear families of transformations for Gaussian and non‐Gaussian quantitative outcomes. As models assuming continuous distributions may be also used with discrete outcomes, we propose likelihood and information criteria for discrete data to compare the goodness of fit of models assuming either a continuous or a discrete distribution for discrete data. Two analyses of the repeated measures of the Mini‐Mental State Examination, a 20‐item psychometric test, illustrate the method. First, we highlight the usefulness of parameterized non‐linear transformations by comparing different flexible families of transformation for modelling the test as a sum score. Then, change over time of the latent construct underlying directly the 20 items is described using two‐parameter longitudinal item response models that are specific cases of the approach.     

Lexical access revisited: Evidence from picture-pseudoword interference

In the present picture–word interference experiments, auditory pseudoword distractors were constructed to be semantically, phonologically, or doubly (semantically and phonologically) similar to the target. The degree and the location of segmental overlap with the target and/or the semantic alternative were varied systematically. The results reveal that different degrees of overlap with the target and/or the semantic alternative induce a fine-grained pattern of interference effects in picture naming, while the location of overlap is less influential. Implications of the present findings for models of lexical retrieval are discussed.