A proof for Rhiel's range estimator of the coefficient of variation for skewed distributions

In this research study is proof that the coefficient of variation (CV(high-low)) calculated from the highest and lowest values in a set of data is applicable to specific skewed distributions with varying means and standard deviations. Earlier Rhiel provided values for d(n), the standardized mean range, and a(n), an adjustment for bias in the range estimator of micro. These values are used in estimating the coefficient of variation from the range for skewed distributions. The d(n) and an values were specified for specific skewed distributions with a fixed mean and standard deviation. In this proof it is shown that the d(n) and an values are applicable for the specific skewed distributions when the mean and standard deviation can take on differing values. This will give the researcher confidence in using this statistic for skewed distributions regardless of the mean and standard deviation.     

The true distributions of the range of rank totals in the two-way classification

This paper presents the results from an investigation of the true probability distributions of the range of rank totals. A procedure for generating an approximation to the true distributions is also given. A comparison of the results of this approximation with an extensive criterion of generated true and sample distributions, and with other approximations is indicated. Accurate estimates of the critical ranges necessary to reach significance at three commonly used alpha levels, where the number of judges and items is less than or equal to sixteen, are presented in tabular form.     

Implementing continuous non-normal skewed distributions in latent growth mixture modeling: An assessment of specification errors and class enumeration

Abstract

Recent advances have allowed for modeling mixture components within latent growth modeling using robust, skewed mixture distributions rather than normal distributions. This feature adds flexibility in handling non-normality in longitudinal data, through manifest or latent variables, by directly modeling skewed or heavy-tailed latent classes rather than assuming a mixture of normal distributions. The aim of this study was to assess through simulation the potential under- or over-extraction of latent classes in a growth mixture model when underlying data follow either normal, skewed-normal, or skewed-t distributions. In order to assess this, we implement skewed-t, skewed-normal, and conventional normal (i.e., not skewed) forms of the growth mixture model. The skewed-t and skewed-normal versions of this model have only recently been implemented, and relatively little is known about their performance. Model comparison, fit, and classification of correctly specified and mis-specified models were assessed through various indices. Findings suggest that the accuracy of model comparison and fit measures are dependent on the type of (mis)specification, as well as the amount of class separation between the latent classes. A secondary simulation exposed computation and accuracy difficulties under some skewed modeling contexts. Implications of findings, recommendations for applied researchers, and future directions are discussed; a motivating example is presented using education data.     

Determination of joint distributions from marginal distributions in dichotomous systems

In many areas of research in which the experimenter deals with a set of dichotomous dimensions it is often desirable to specify them in terms of marginal distributions and some conditional distributions. This paper develops a set of systematic procedures for determination of joint distributions with an arbitrary number of dimensions. Most of these procedures require only addition and/or subtraction. An extension of the procedures permits easy determination of covariance and correlation matrices.This research was supported in part by National Institutes of Health grant No. MH-12541. Part of this work was done at Stanford University during work sessions of the 1967 Summer Conference on Mathematical Models in Perception and Learning. The author is indebted to Richard S. Bogartz for many helpful comments.     

Estimating averages from distributions of tone durations

We examined whether estimating average duration was influenced by the distribution peak location. We presented participants with samples of various tone durations and then presented comparison tone durations. Participants judged whether each comparison duration was longer than the average sample duration. Estimates of the averages were inferred from the psychophysical functions. The durations were sampled from three distributions: one positively skewed, one symmetric, and one negatively skewed. In Experiment 1, every participant was presented with every distribution. Estimates of the averages were unbiased for the symmetric distribution but were biased toward the long tail of each skewed distribution. This would occur if participants combined the sample to be judged with the previous, irrelevant samples, or with the comparison durations. In Experiment 2, each participant was presented with samples from only one of the distributions. Estimates of the averages were still biased toward the long tails of the skewed distributions. This would occur if participants combined the sample to be judged with the comparison durations, which were the same for the three distributions. In Experiment 3, each participant was presented with only one distribution, and each distribution was tested with its own comparison durations, selected as percentiles from the distribution. The estimates were accurate for the smallest population mean (positively skewed distribution) but underestimated the larger means. These results could be explained by subjective shortening of the durations in memory, with a simple equation from scalar timing theory. This equation correctly predicted two results: The estimated averages were a linear function of the stimulus means, and the variances were a linear function of the squared stimulus means. Neither prediction was dependent on the skewness of the stimulus durations.     

An abac for the sample range

An abac is computed which gives the probability that at leastq% of the whole population will be included in the range of a random sample of given size. Applications are suggested for testing homogeneity of a sampling from a given population.     

Constraints on the range of context-independent priming from ambiguous words

The semantic priming technique was used to explore the semantic information activated by the contextually irrelevant meanings of ambiguous words in sentence contexts. The nature of the relationship between prime and target was systematically manipulated. Evidence of priming was obtained when the target was an approximate synonym of the prime (e.g., coach-bus), and when the target was a category coordinate of low similarity, but high category typicality (e.g., raincoat jumper, in Italian). No priming was obtained when the target was event related to the prime (coach-driver), nor when it was a highly similar category co-ordinate (raincoat overcoat, in Italian). However, in these latter cases, robust priming effects were obtained when the prime was presented out of context. The results are discussed in terms of the different kinds of semantic-retrieval operations that may be performed, depending upon whether or not the subjects attend to the meaning of the prime.     

On the asymptotic distributions of two statistics for two-level covariance structure models within the class of elliptical distributions

Since data in social and behavioral sciences are often hierarchically organized, special statistical procedures for covariance structure models have been developed to reflect such hierarchical structures. Most of these developments are based on a multivariate normality distribution assumption, which may not be realistic for practical data. It is of interest to know whether normal theory-based inference can still be valid with violations of the distribution condition. Various interesting results have been obtained for conventional covariance structure analysis based on the class of elliptical distributions. This paper shows that similar results still hold for 2-level covariance structure models. Specifically, when both the level-1 (within cluster) and level-2 (between cluster) random components follow the same elliptical distribution, the rescaled statistic recently developed by Yuan and Bentler asymptotically follows a chi-square distribution. When level-1 and level-2 have different elliptical distributions, an additional rescaled statistic can be constructed that also asymptotically follows a chi-square distribution. Our results provide a rationale for applying these rescaled statistics to general non-normal distributions, and also provide insight into issues related to level-1 and level-2 sample sizes. The authors thank an associate editor and three referees for their constructive comments, which led to an improved version of the paper. This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of Notre Dame faculty research grant.     

Marginal distributions for the estimation of proportions inm groups

A Bayesian Model II approach to the estimation of proportions inm groups (discussed by Novick, Lewis, and Jackson) is extended to obtain posterior marginal distributions for the proportions. It is anticipated that these will be useful in applications (such as Individually Prescribed Instruction) where decisions are to be made separately for each proportion, rather than jointly for the set of proportions. In addition, the approach is extended to allow greater use of prior information than previously and the specification of this prior information is discussed.We are grateful to a reviewer for suggestions that made possible a more concise and complete presentation of our work.     

Assessing the evidence for response time mixture distributions

I describe a technique for comparing two simple accounts of a distribution of response times: A mixture model and a generalized-shift model. In the mixture model, a target distribution is assumed to be a mixture of response times from two other (reference) distributions. In the generalized-shift model, the target distribution is assumed to be a quantile average of the reference distributions. In order to distinguish these two possibilities, quantiles for the target distribution are estimated from the quantiles of the reference distributions assuming either a shift or a mixture, and the predicted quantiles are used to calculate the multinomial likelihood of the obtained data. Monte Carlo simulations reported here demonstrate that the index is relatively unbiased, is effective with moderate sample sizes and modest spreads between the reference distributions, is relatively unaffected by changes in the number of bins or by data trimming, can be used with data aggregated across subjects, and is relatively insensitive to a range of subject variations in distribution shape and in mixture or shift proportion. As an illustration, the index is applied to the interpretation of three effects from distinct paradigms: residual switch costs in the task-switching paradigm, the psychological refractory period effect, and sequential effects in the Simon task. I conclude that the multinomial likelihood index provides a useful and easily applied tool for the interpretation of effects on response time distributions.     

Sample size planning for the coefficient of variation from the accuracy in parameter estimation approach

The accuracy in parameter estimation approach to sample size planning is developed for the coefficient of variation, where the goal of the method is to obtain an accurate parameter estimate by achieving a sufficiently narrow confidence interval. The first method allows researchers to plan sample size so that the expected width of the confidence interval for the population coefficient of variation is sufficiently narrow. A modification allows a desired degree of assurance to be incorporated into the method, so that the obtained confidence interval will be sufficiently narrow with some specified probability (e.g., 85% assurance that the 95 confidence interval width will be no wider than to units). Tables of necessary sample size are provided for a variety of scenarios that may help researchers planning a study where the coefficient of variation is of interest plan an appropriate sample size in order to have a sufficiently narrow confidence interval, optionally with somespecified assurance of the confidence interval being sufficiently narrow. Freely available computer routines have been developed that allow researchers to easily implement all of the methods discussed in the article.     

随机截距潜在转变分析(RI-LTA)——个案自我转变与个案间差异的分离

传统的潜在转变分析属于单水平分析, 但其同样也可以看作二水平分析。Muthén和Asparouhov就以二水平分析的视角在单水平分析框架内提出了随机截距潜在转变分析(RI-LTA), 其中跨时间点产生的自我转变可以看作在水平1进行分析, 跨时间点不变的个案间差异可以看作在水平2进行分析, 使个案的自我转变和个案间的初始差异分离, 避免了高估保留在初始类别的概率。某研究型大学2016级本科生的追踪调查数据被用于演示使用随机截距潜在转变分析的过程。该方法的最大优势是通过引入随机截距避免了高估保留在本类别的转变概率。未来研究可以运用蒙特卡洛模拟研究探究随机截距潜在转变分析模型的适用性, 也可以用多水平分析的思路为灵感, 探究多水平随机截距潜在转变分析在统计软件中的实现。     

A Profile-Based Framework for Factorial Similarity and the Congruence Coefficient

We present a novel profile-based framework for understanding factorial similarity in the context of exploratory factor analysis in general, and for understanding the congruence coefficient (a commonly used index of factor similarity) specifically. First, we introduce the profile-based framework articulating factorial similarity in terms of 3 intuitive components: general saturation similarity, differential saturation similarity, and configural similarity. We then articulate the congruence coefficient in terms of these components, along with 2 additional profile-based components, and we explain how these components resolve ambiguities that can be—and are—found when using the congruence coefficient. Finally, we present secondary analyses revealing that profile-based components of factorial are indeed linked to experts' actual evaluations of factorial similarity. Overall, the profile-based approach we present offers new insights into the ways in which researchers can examine factor similarity and holds the potential to enhance researchers' ability to understand the congruence coefficient.     

A Reliability Generalization Study of Coefficient Alpha for the Life Orientation Test

The Pd, K and L scales of the MMPI and four of Kolberg's Moral Judgment Situations were administered to three criterion groups; adolescent delinquents, college students, and individuals belonging to a therapy collective which provides free community service. Greater than 50% of both the delinquent and therapy collective subjects were classified as psychopathic, (Pd + K T score > 70). The three subject groups obtained significantly different moral maturity scores with the therapy collective subjects receiving the highest moral maturity mean score. The results of two stepwise regression analyses found no relationship between Pd scores and moral maturity scores.     

A Reliability Generalization Study of Coefficient Alpha for the UCLA Loneliness Scale

The goal of this investigation was the development of an Inconsistency scale (ICN–SF) for the Personality Assessment Inventory–Short Form (PAI–SF). In Study 1, 503 inpatient profiles were randomly assigned to a derivation or cross-validation sample. Ten correlated item pairs were identified using the derivation sample and placed on the ICN–SF. Psychometric properties of the ICN–SF total scores were comparable in the derivation and cross-validation samples. Total ICN–SF scores in both samples were significantly lower than scores obtained from computer-generated random samples. Diagnostic efficiency statistics are reported using multiple cut-off scores at various base rate estimates. ICN–SF scores greater than 8 reasonably balanced sensitivity and specificity rates. This cutoff correctly classified 92% of the random protocols and inaccurately classified 9% of the patient protocols in Study 1. In Study 2, PAI–SF scores from 627 forensic and civil inpatients produced similar results, effectively identifying cases with elevated scores on the full-form Inconsistency scale. Overall the results of both studies suggest that the ICN–SF can aid examiners in assessing for inconsistent responding.     

A mixture model for distributions of correlation coefficients

An old problem in personnel psychology is to characterize distributions of test validity correlation coefficients. The proposed model views histograms of correlation coefficients as observations from a mixture distribution which, for a fixed sample sizen, is a conditional mixture distributionh(r|n) = _{j j} h(r; _j ,n), whereR is the correlation coefficient, _j are population correlation coefficients and _j are the mixing weights. The associated marginal distribution ofR is regarded as the parent distribution underlying histograms of empirical correlation coefficients. Maximum likelihood estimates of the parameters _j and _j can be obtained with an EM algorithm solution and tests for the number of componentst are achieved after the (one-component) density ofR is replaced with a tractable modeling densityh(r; _j ,n). Two illustrative examples are provided.     

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.