Variation of Isometric Response Force in the Rat

Hungry, unrestrained rats (N = 7) were rewarded for pressing a response beam in excess of 11 different force requirements. Changes in peak force production as a function of peak force requirement were examined by analyses of the first four moments of distributions of peak response forces: constant error, the within-subject standard deviation, skewness, and kurtosis. Results were similar to those previously obtained with human subjects: Constant error was positive at low and negative at high force requirements, the within-subject standard deviation increased as a negatively accelerating function of force requirement, and skewness and kurtosis were positive at low force requirements and decreased to negative values at the highest increments. Additional analyses of response kinetics indicated that rats, like humans, meet increasing force requirements by altering the rate of rise of force. The performance similarities suggest that common processes are engaged by the human and rat motor control systems to solve the problem of generating forces that are appropriate to the prevailing environmental constraints.     

Effects of skewness and kurtosis on normal-theory based maximum likelihood test statistic in multilevel structural equation modeling

A simulation study investigated the effects of skewness and kurtosis on level-specific maximum likelihood (ML) test statistics based on normal theory in multilevel structural equation models. The levels of skewness and kurtosis at each level were manipulated in multilevel data, and the effects of skewness and kurtosis on level-specific ML test statistics were examined. When the assumption of multivariate normality was violated, the level-specific ML test statistics were inflated, resulting in Type I error rates that were higher than the nominal level for the correctly specified model. Q-Q plots of the test statistics against a theoretical chi-square distribution showed that skewness led to a thicker upper tail and kurtosis led to a longer upper tail of the observed distribution of the level-specific ML test statistic for the correctly specified model.     

The effect of skewness and kurtosis on the robustness of linear mixed models

This study analyzes the robustness of the linear mixed model (LMM) with the Kenward–Roger (KR) procedure to violations of normality and sphericity when used in split-plot designs with small sample sizes. Specifically, it explores the independent effect of skewness and kurtosis on KR robustness for the values of skewness and kurtosis coefficients that are most frequently found in psychological and educational research data. To this end, a Monte Carlo simulation study was designed, considering a split-plot design with three levels of the between-subjects grouping factor and four levels of the within-subjects factor. Robustness is assessed in terms of the probability of type I error. The results showed that (1) the robustness of the KR procedure does not differ as a function of the violation or satisfaction of the sphericity assumption when small samples are used; (2) the LMM with KR can be a good option for analyzing total sample sizes of 45 or larger when their distributions are normal, slightly or moderately skewed, and with different degrees of kurtosis violation; (3) the effect of skewness on the robustness of the LMM with KR is greater than the corresponding effect of kurtosis for common values; and (4) when data are not normal and the total sample size is 30, the procedure is not robust. Alternative analyses should be performed when the total sample size is 30.     

Effects of differing response-force requirements on food-maintained responding in CD-1 mice

The effect of force requirements on response effort was examined using outbred (CD-1) mice trained to press a disk with their snout. Lateral peak forces greater than 2 g were defined as threshold responses (i.e., all measured responses). Different force requirements were used to define criterion responses (a subclass of threshold responses) that exceeded the requirement. The reinforcer was sweetened, condensed milk, and it was delivered upon response termination. All mice were exposed to two ascending series of criterion force requirements (2, 4, 8, 16, and 32 g). Increasing the force requirement decreased criterion response rates, but increased threshold response rates. The time-integral of force (area under the force-time curve for individual responses, which is proportional to energy expenditure for each response) increased with the increase in the force requirement. These results conflict with the hypothesis that higher force requirements have aversive qualities and suggest that increased force requirements are more analogous to intermittent schedules of reinforcement. These data suggest that estimations of effort or energy expenditure should be measured independently of the force requirement. Individual differences in responding were found for the CD-1 outbred stock.     

Problematic standard errors and confidence intervals for skewness and kurtosis

Many statistics packages print skewness and kurtosis statistics with estimates of their standard errors. The function most often used for the standard errors (e.g., in SPSS) assumes that the data are drawn from a normal distribution, an unlikely situation. Some textbooks suggest that if the statistic is more than about 2 standard errors from the hypothesized value (i.e., an approximate value for the critical value from the t distribution for moderate or large sample sizes when α = 5%), the hypothesized value can be rejected. This is an inappropriate practice unless the standard error estimate is accurate and the sampling distribution is approximately normal. We show distributions where the traditional standard errors provided by the function underestimate the actual values, often being 5 times too small, and distributions where the function overestimates the true values. Bootstrap standard errors and confidence intervals are more accurate than the traditional approach, although still imperfect. The reasons for this are discussed. We recommend that if you are using skewness and kurtosis statistics based on the 3^rd and 4^th moments, bootstrapping should be used to calculate standard errors and confidence intervals, rather than using the traditional standard. Software in the freeware R for this article provides these estimates.     

Test skewness and kurtosis as functions of item parameters

Indexes of skewness and kurtosis for a test-score distribution are expressed in terms of item parameters. Both are shown to depend, in part, on item means, variances, and covariances. The index of skewness depends also on trivariances. A trivariance is a product moment involving first powers of deviation scores for three items. The index of kurtosis depends on quadrivariances, as well as trivariances. A quadrivariance is a product moment involving first powers of deviation scores for four items. Empirical data are presented for responses of groups of subjects to 25 triads and 25 tetrads of items from five tests.Certain parts of this article represent the results of doctoral research conducted by Hundleby and Goldstein under the direction of Ray in the Department of Psychology at Pennsylvania State University. The authors are indebted to Professor Lester Guest and Professor William Lepley for their supervisory assistance in the final stages of the two dissertations during the absence of the senior author.     

Confirmation of linear system theory prediction: Rate of change of Herrnstein's kappa as a function of response-force requirement

Four human subjects worked on all combinations of five variable-interval schedules and five reinforcer magnitudes (¢/reinforcer) in each of two phases of the experiment. In one phase the force requirement on the operandum was low (1 or 11 N) and in the other it was high (25 or 146 N). Estimates of Herrnstein's κ were obtained at each reinforcer magnitude. The results were: (1) response rate was more sensitive to changes in reinforcement rate at the high than at the low force requirement, (2) κ increased from the beginning to the end of the magnitude range for all subjects at both force requirements, (3) the reciprocal of κ was a linear function of the reciprocal of reinforcer magnitude for seven of the eight data sets, and (4) the rate of change of κ was greater at the high than at the low force requirement by an order of magnitude or more. The second and third findings confirm predictions made by linear system theory, and replicate the results of an earlier experiment (McDowell & Wood, 1984). The fourth finding confirms a further prediction of the theory and supports the theory's interpretation of conflicting data on the constancy of Herrnstein's κ.     

How to Generate Non-normal Data for Simulation of Structural Equation Models

A procedure for generating non-normal data for simulation of structural equation models is proposed. A simple transformation of univariate random variables is used for the generation of data on latent and error variables under some restrictions for the elements of the covariance matrices for these variables. Data on the observed variables is then computed from latent and error variables according to the model. It is shown that by controlling univariate skewness and kurtosis on pre-specified random latent and error variables, observed variables can be made to have a relatively wide range of univariate skewness and kurtosis characteristics according to the pre-specified model. Univariate distributions are used for the generation of data which enables a user to choose from a large number of different distributions. The use of the proposed procedure is illustrated for two different structural equation models and it is shown how PRELIS can be used to generate the data.     

Univariate and multivariate skewness and kurtosis for measuring nonnormality: Prevalence,influence and estimation

Nonnormality of univariate data has been extensively examined previously (Blanca et al., Methodology: European Journal of Research Methods for the Behavioral and Social Sciences, 9(2), 78–84, 2013; Miceeri, Psychological Bulletin, 105(1), 156, 1989). However, less is known of the potential nonnormality of multivariate data although multivariate analysis is commonly used in psychological and educational research. Using univariate and multivariate skewness and kurtosis as measures of nonnormality, this study examined 1,567 univariate distriubtions and 254 multivariate distributions collected from authors of articles published in Psychological Science and the American Education Research Journal. We found that 74 % of univariate distributions and 68 % multivariate distributions deviated from normal distributions. In a simulation study using typical values of skewness and kurtosis that we collected, we found that the resulting type I error rates were 17 % in a t-test and 30 % in a factor analysis under some conditions. Hence, we argue that it is time to routinely report skewness and kurtosis along with other summary statistics such as means and variances. To facilitate future report of skewness and kurtosis, we provide a tutorial on how to compute univariate and multivariate skewness and kurtosis by SAS, SPSS, R and a newly developed Web application.     

A device for facilitating the computation of the first four moments about the mean

A mechanical device is described which materially reduces the time, labor, and probability of errors which are involved in the computation of certain moments about the mean. The use of the device is illustrated with the computation of the mean, standard deviation,g ₁ as a measure of skewness, andg ₂ as a measure of kurtosis of data from a given study.Grateful acknowledgment is made to the Holter Research Foundation of Helena, Montana for the grant-in-aid which made possible the development of this device.     

Effect of drugs on response-duration differentiation VII: response-force requirements

Rats were trained to press a lever for at least 1 s but for less than 1.3 s. The force required to press the lever was then increased or decreased by 10, 15, or 20 g. Increases in the force requirements for lever pressing decreased timing accuracy, but decreases in the force requirement had the opposite effect. Accuracy decreases at increasing force requirements were characterized by an increase in the relative frequency of responses that were too short to meet the reinforcement criterion. In contrast, increases in accuracy when the force requirements were decreased were characterized by increases in response durations that met the reinforcement criterion and decreases in the relative frequency of responses that were too short to produce the reinforcer. Phencyclidine (PCP) and methamphetamine produced dose-dependent decreases in accuracy that were associated primarily with increases in the relative frequency of short response durations, although methamphetamine also produced increases in long response durations at some doses. When the effects of PCP were determined with the force requirement increased by 10 g or decreased by 15 g, the cumulative response-duration distribution shifted toward even shorter response durations. When the effects of methamphetamine were determined with the force requirement on the lever increased by 10 g, the cumulative frequency distribution was shifted toward shorter response durations to about the same extent as it had been before force requirements increased; however, when the force required to press the lever was decreased by 15 g, these shifts toward shorter response durations almost completely disappeared. These results show that increases and decreases in the force requirements for lever pressing have different effects on the accuracy of temporal response differentiation.     

Force and rate relations in responding during variable-interval reinforcement

Four rats responded on one-minute variable-interval schedules with several variations in peak-force of response required for food reinforcement. Measures of peak force and rate were taken for the responses, which were the downward exertions of force against a static force-transducing operandum. The analysis distinguished responses, a generic class of measured behavior, from criterion responses, an operationally specified subclass required for reinforcement. Absolute rate of response showed no systematic change, but the rate of responses meeting a newly required criterion of peak-force invariably increased through changes in the absolute rate of response, the relative-frequency distributions of peak force, or some combination of both. The relative frequency of responses meeting an elevated force criterion during variable-interval reinforcement exceeded that maintained with the same criterion with continuous reinforcement. The requirement of more effortful responding for reinforcement does not necessarily reduce response rate. Conformity of the behavior to the requirement for reinforcement is the salient effect.     

The role of response force on the persistence and structure of behavior during extinction

Behavior Momentum Theory has emerged as a prominent account of resistance to change in both basic and applied research. Although laboratory studies often define precise, repeatable responses, application research often deals with response classes that may vary widely along a number of dimensions. In general, Behavior Momentum Theory has not addressed how response dimensions impact resistance to change, providing an opportunity to expand the model in new directions. Four rats pressed a force transducer under a multiple variable interval (VI) 60‐s VI 60‐s schedule of reinforcement. In one component, responses satisfied the schedule only if the response force fell within a “low” force band requirement; responses in the other schedule were required to satisfy a “high” force band. Once responding stabilized, extinction was programmed for three sessions. Then, the procedures were replicated. The results showed that response force came under discriminative control, but force requirements had no impact on resistance to extinction. In a follow‐up condition, the schedule was changed to a multiple VI 30‐s VI 120‐s schedule and the low‐force band operated in both components. The results showed that behavior maintained by the VI 30‐s schedule was generally more resistant to extinction. A secondary analysis showed that force distributions created under baseline maintained during extinction. Overall, the results suggest that differential response force requirements prevailing in steady state do not affect the course of extinction.     

Effects of effort on response rate

Pigeons were trained to peck for food reinforcement, and the minimum effective force for a peck was varied. In a single-key situation, the response rate was unaffected up to a certain force value, beyond which increments caused a proportional decrement in response rate. Transient enhancement and suppression effects were observed following a change in the force requirement. A change from a high force requirement to a low force requirement resulted in a temporary enhancement of response rate above the stable performance level, whereas a change from a low to a high requirement produced a temporary suppression of the rate below the stable rate. In a two-key situation, the response rate on a key with a given force remained unaffected by the force requirement on the other key. When the rate of response was plotted against the required force, the resultant function was remarkably similar to that obtained from the single-key experiment. When the rates of reinforcement on the two keys differed, the decrement in response rate produced by an increase in the force requirement was proportionally greater on the key with the lower rate of reinforcement.     

Asymptotic confidence intervals for the Pearson correlation via skewness and kurtosis

When bivariate normality is violated, the default confidence interval of the Pearson correlation can be inaccurate. Two new methods were developed based on the asymptotic sampling distribution of Fisher's z′ under the general case where bivariate normality need not be assumed. In Monte Carlo simulations, the most successful of these methods relied on the (Vale & Maurelli, 1983, Psychometrika, 48, 465) family to approximate a distribution via the marginal skewness and kurtosis of the sample data. In Simulation 1, this method provided more accurate confidence intervals of the correlation in non-normal data, at least as compared to no adjustment of the Fisher z′ interval, or to adjustment via the sample joint moments. In Simulation 2, this approximate distribution method performed favourably relative to common non-parametric bootstrap methods, but its performance was mixed relative to an observed imposed bootstrap and two other robust methods (PM1 and HC4). No method was completely satisfactory. An advantage of the approximate distribution method, though, is that it can be implemented even without access to raw data if sample skewness and kurtosis are reported, making the method particularly useful for meta-analysis. Supporting information includes R code.     

When item recognition and visual search functions are similar

The effect of prolonged practice upon item recognition performance was investigated under conditions of nested positive sets and complete response consistency. Nesting is defined by each positive set containing all the items contained in smaller positive sets. Response consistency is defined by each item in the stimulus set consistently requiring only a positive or only a negative response. A low error level was maintained. Twelve Ss worked with three positive set sizes in each of 36 sessions. Half the Ss worked with digit stimuli and half with pictures. The item recognition function (that function relating response latency and positive set size) was found to be negatively accelerated throughout the course of practice. The effect of positive set size decreased significantly (p < .001) with practice, and set size effects were significantly (p < .03) greater for positive response trials than for negative response trials. Kind of item had no effect on the set size effect. A theoretical framework consistent with these results is suggested. Results from the present study are compared with findings obtained previously from visual search studies. It is concluded that when the procedures in both tasks include response consistency, nested positive sets, and low error levels, the effects of prolonged practice upon the set size from item recognition and visual search are qualitatively very similar.     

Direction of dependence in measurement error models

Methods to determine the direction of a regression line, that is, to determine the direction of dependence in reversible linear regression models (e.g., x→y vs. y→x), have experienced rapid development within the last decade. However, previous research largely rested on the assumption that the true predictor is measured without measurement error. The present paper extends the direction dependence principle to measurement error models. First, we discuss asymmetric representations of the reliability coefficient in terms of higher moments of variables and the attenuation of skewness and excess kurtosis due to measurement error. Second, we identify conditions where direction dependence decisions are biased due to measurement error and suggest method of moments (MOM) estimation as a remedy. Third, we address data situations in which the true outcome exhibits both regression and measurement error, and propose a sensitivity analysis approach to determining the robustness of direction dependence decisions against unreliably measured outcomes. Monte Carlo simulations were performed to assess the performance of MOM-based direction dependence measures and their robustness to violated measurement error assumptions (i.e., non-independence and non-normality). An empirical example from subjective well-being research is presented. The plausibility of model assumptions and links to modern causal inference methods for observational data are discussed.     

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.