Coefficient of variation calculated from the range for skewed distributions

In this research a coefficient of variation (CVS(high.low)) is developed that is calculated from the highest and lowest values in a set of data for samples from skewed distributions. A correction factor is determined such that CVS(high-low) is a dose estimate of the population coefficient of variation when sampling from three skewed chi-squared distributions and three skewed empirical distributions. The empirical distributions are from "real-world" data sets in psychology and education.     

On the relation between the mean and the variance of a diffusion model response time distribution

Almost every empirical psychological study finds that the variance of a response time (RT) distribution increases with the mean. Here we present a theoretical analysis of the nature of the relationship between RT mean and RT variance, based on the assumption that a diffusion model (e.g., Ratcliff (1978) Psychological Review, 85, 59-108; Ratcliff (2002). Psychonomic Bulletin & Review, 9, 278-291), adequately captures the shape of empirical RT distributions. We first derive closed-form analytic solutions for the mean and variance of a diffusion model RT distribution. Next, we study how systematic differences in two important diffusion model parameters simultaneously affect the mean and the variance of the diffusion model RT distribution. Within the range of plausible values for the drift rate parameter, the relation between RT mean and RT standard deviation is approximately linear. Manipulation of the boundary separation parameter also leads to an approximately linear relation between RT mean and RT standard deviation, but only for low values of the drift rate parameter.     

Intrapersonal and social comparison determinants of happiness: a range-frequency analysis

Examined whether intrapersonal comparisons and social comparisons operate in similar ways to determine ratings of happiness. Events were varied to create positively and negatively skewed distributions. The events in each distribution were ascribed to either a single person or a group of people; Ss rated how happy they would feel if they experienced specific events within the distribution. Ratings for both intrapersonal and social comparisons were fit well by Parducci's (1984) range-frequency theory. Individual events received higher ratings when presented within the positively skewed context. Overall happiness, as measured by both the mean of the happiness ratings as well as direct ratings, was highest for the negatively skewed distributions. The effects of skewing were more pronounced for intrapersonal comparisons, but ratings were more closely defined by the range of experimental stimuli for social comparisons.     

Interpersonal and intrapersonal variance of exponent of power function observed in grip strength task.

For grip strength there is a power function with an exponent of 1.7 between the subjective magnitude and the actual force exerted by a subject, but large variabilities among and within individuals were found. We focused on these variabilities and investigated the relationship between them by conducting a ratio production procedure requiring trials of maximum effort and half of maximum effort. For 30 adults we conducted four measurement trials, two on the same day, and the remaining two trials on a day or two later. The mean value of the exponent, the standard deviation, and the coefficient of variation of the four trials for each subject were calculated. The mean value of the exponent of the power function for all subjects was 1.6. This value approximated the value of 1.7 reported by Stevens and Mack. The values ranged from .50 to 5.39. The correlation between subjects' mean exponent value and standard deviation was .90, and the correlation between the mean value of the exponent and the coefficient of variation was .50. There was a close relationship between interpersonal and intrapersonal variance.     

A closer look at the effect of preliminary goodness-of-fit testing for normality for the one-sample t-test

Student's one-sample t-test is a commonly used method when inference about the population mean is made. As advocated in textbooks and articles, the assumption of normality is often checked by a preliminary goodness-of-fit (GOF) test. In a paper recently published by Schucany and Ng it was shown that, for the uniform distribution, screening of samples by a pretest for normality leads to a more conservative conditional Type I error rate than application of the one-sample t-test without preliminary GOF test. In contrast, for the exponential distribution, the conditional level is even more elevated than the Type I error rate of the t-test without pretest. We examine the reasons behind these characteristics. In a simulation study, samples drawn from the exponential, lognormal, uniform, Student's t-distribution with 2 degrees of freedom (t(2) ) and the standard normal distribution that had passed normality screening, as well as the ingredients of the test statistics calculated from these samples, are investigated. For non-normal distributions, we found that preliminary testing for normality may change the distribution of means and standard deviations of the selected samples as well as the correlation between them (if the underlying distribution is non-symmetric), thus leading to altered distributions of the resulting test statistics. It is shown that for skewed distributions the excess in Type I error rate may be even more pronounced when testing one-sided hypotheses.     

Short report: Reaction time analysis with outlier exclusion: Bias varies with sample size

To remove the influence of spuriously long response times, many investigators compute “restricted means”, obtained by throwing out any response time more than 2.0, 2.5, or 3.0 standard deviations from the overall sample average. Because reaction time distributions are skewed, however, the computation of restricted means introduces a bias: the restricted mean underestimates the true average of the population of response times. This problem may be very serious when investigators compare restricted means across conditions with different numbers of observations, because the bias increases with sample size. Simulations show that there is substantial differential bias when comparing conditions with fewer than 10 observations against conditions with more than 20. With strongly skewed distributions and a cutoff of 3.0 standard deviations, differential bias can influence comparisons of conditions with even more observations.     

Influence of vision and static stretch of the calf muscles on postural sway during quiet standing

The purpose of this experimental study was to evaluate the effects of vision and stretching of the calf muscles on postural sway during quiet standing. Under pre-stretch conditions, participants stood on a force plate for 30s and the sway of the ground reaction force center of pressure was recorded. The following postural sway variables were calculated off-line: sweep speed, sway speed, standard deviation, maximal mediolateral range, maximal anteroposterior range, mean mediolateral position and mean anteroposterior position. For post-stretch conditions, participants stood quietly on a device that was utilized to impose a static 3 min ankle joint dorsiflexion stretch. Immediately thereafter, participants moved onto the force platform where postural sway parameters were again recorded. Randomized eyes-open and eyes-closed conditions were tested in both cases. Results showed that postural sway significantly increased due to stretch (sweep speed, sway speed, standard deviation, maximal anteroposterior range, mean anteroposterior position), as well as eye closure (sweep speed, sway speed, standard deviation, maximal mediolateral range, maximal anteroposterior range). The interaction between stretch and eye closure was also significant (sweep speed, sway speed, standard deviation, maximal mediolateral range), suggesting that there were only minor increases in postural sway after stretch under the eyes-open condition. It was suggested that stretching of the calf muscles has the effect of increasing postural sway, although this effect can be greatly compensated for when vision is included.     

Neuromotor profiles: what are they and what can we learn from them?

Eye movements, alternating movements, rapid pointing movements, and various tremors were measured on patients with Parkinson's disease (n = 21), on Cree subjects exposed to methylmercury (n = 36), and on healthy control subjects (n = 30). Neuromotor profiles were created according to thirty characteristics extracted from test results of four subgroups matched for age and composed of six subjects each. Z scores were calculated with respect to the mean and standard deviation of the control group for each of the 30 characteristics. The subgroup with the lower methylmercury blood level had larger z scores than the control subgroup and with a few positive values above one standard deviation. The subgroup with the higher methylmercury blood level had several z scores above two standard deviations. Interestingly, the abnormal values for the subgroup with Parkinson's disease were mostly limited to static tremor recorded with no visual feedback and reached up to 5 standard deviations. These results indicate that neuromotor profiles can be used to summarize information extracted from different neuromotor tests and to differentiate neurological conditions.     

Measures of attitudinal polarization

Abstract.— Criteria for measures of attitudinal polarization, i. e., degree of opposition among people on a specific issue, are proposed and some formulas, including the standard deviation, are evaluated in relation to the criteria. The formulas were also tested on empirical data with respect to level, dispersion and agreement of received values. The measures on the whole showed a high degree of agreement. There clearly exist instances where the standard deviation is not an adequate measure of attitudinal polarization. Some guidelines are given for the choice of constant values in one of the formulas.     

On robusiness of the normal-theory based asymptotic distributions of three reliability coefficient estimates

This paper studies the asymptotic distributions of three reliability coefficient estimates: Sample coefficient alpha, the reliability estimate of a composite score following a factor analysis, and the estimate of the maximal reliability of a linear combination of item scores following a factor analysis. Results indicate that the asymptotic distribution for each of the coefficient estimates, obtained based on a normal sampling distribution, is still valid within a large class of nonnormal distributions. Therefore, a formula for calculating the standard error of the sample coefficient alpha, recently obtained by van Zyl, Neudecker and Nel, applies to other reliability coefficients and can still be used even with skewed and kurtotic data such as are typical in the social and behavioral sciences.This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of North Texas faculty research grant. We would like to thank the Associate Editor and two reviewers for suggestions that helped to improve the paper.     

The differentiation of response numerosities in the pigeon

Two experiments examined how pigeons differentiate response patterns along the dimension of number. In Experiment 1, 5 pigeons received food after pecking the left key at least N times and then switching to the right key (Mechner's Fixed Consecutive Number schedule). Parameter N varied across conditions from 4 to 32. Results showed that run length on the left key followed a normal distribution whose mean and standard deviation increased linearly with N; the coefficient of variation approached a constant value (the scalar property). In Experiment 2, 4 pigeons received food with probability p for pecking the left key exactly four times and then switching. If that did not happen, the pigeons still could receive food by returning to the left key and pecking it for a total of at least 16 times and then switching. Parameter p varied across conditions from 1.0 to .25. Results showed that when p= 1.0 or p=.5, pigeons learned two response numerosities within the same condition. When p=.25, each pigeon adapted to the schedule differently. Two of them emitted first runs well described by a mixture of two normal distributions, one with mean close to 4 and the other with mean close to 16 pecks. A mathematical model for the differentiation of response numerosity in Fixed Consecutive Number schedules is proposed.     

A Theoretical Model of Psychometric Effects of Faking on Assessment Procedures: Empirical findings and implications for personality at work

This article proposes a theoretical model for explaining the psychometric effects of faking on assessment procedures (e.g., biodata, interviews, assessment center, personality inventories, and self‐reported measures). The model hypothesizes that faking is a phenomenon of homogenization of scores, consisting of a double mechanism that increases the mean, on one hand, and decreases the standard deviation of distributions of scores, on the other. Subsequently, this affects the covariance, reliability, and validity of assessment procedures negatively. The model predicts that a mean ratio (faking ratio) greater than 1 and a coefficient of homogeneity u smaller than 1 characterizes faking. Meta‐analysis was used to test several predictions of the theoretical model in the case of the personality measures. A database of 46 independent studies was created. All the studies used the NEO‐PI‐R for assessing the Big Five and their facets. The general pattern of data fully supported the model predictions. Implications for personnel assessment and, particularly, for personality assessment are discussed.     

An investigation of the sampling distributions of item discrimination indices

The sampling properties of four item discrimination indices (biserialr, Cook's indexB, theU–L 27 per cent index, and DeltaP) were investigated in order to ascertain their sampling properties when small samples drawn from actual test data rather than constructed data were employed. The empirical results indicated that the mean index values approximated the population values and that values of the standard deviations computed from large sample formulas were good approximations to the standard deviations of the observed distributions based on samples of size 120 or less. Goodness of fit tests comparing the observed distributions with the corresponding distribution of the product-moment correlation coefficient based upon a bivariate normal population indicated that this correlational model was inappropriate for the data. The lack of adequate mathematical models for the sampling distributions of item discrimination indices suggests that one should avoid indices whose only real reason for existence was the simplification of computational procedures.This research reported herein was performed pursuant to a contract (OE-2-10-071) with the United States Office of Education, Department of Health, Education and Welfare.     

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.