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 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.     

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.     

Changes in pinch force with bidirectional load forces

The forces used to grasp an object were measured while positive (push) and negative (pull) load forces were applied to the hand under varying frictional conditions. Subjects held between the tips of their thumb and index finger a manipulandum composed of two symmetrically mounted disks. The manipulandum was connected to the stage of an electromagnetic linear motor that generated load forces under computer control. In the first experiment, subjects held the position of the manipulandum constant while the motor generated forces in first the positive and then the negative direction. The motor force at which the manipulandum slipped from the fingers was measured in the second experiment. In both experiments, friction was varied by changing the surface (sandpaper, suede, or plastic) of the manipulandum disks. The pinch forces produced by subjects were linearly related to changes in motor force in both the positive and negative directions, with the slope of this relation varying as a function of the surface properties of the manipulandum. The modulation of pinch force with motor force was influenced, however, by the direction of the load force; higher forces were produced in response to negative load forces. Slip forces varied as a function of pinch force and surface texture; higher forces were associated with materials with lower coefficients of friction. These findings suggest that the friction between the skin and an object being grasped changes as a function of the direction of force that the object applies to the skin, possibly due to the anisotropic nature of glabrous skin, and that this mechanical property contributes to variations in pinch force.     

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.     

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.     

Energetic and motor responses to increasing force requirements

The effects of increasing work (force) requirements on energy expenditure and response topography were examined in 7 rats pressing a beam to earn food. For the 1st 16 days, the force requirement was 5.52 x 10(-2) N (5.625 g). This increased by 4.91 x 10(-2) N (5 g) every 7th session until Ss had experienced 10 upward shifts. Following the 54.57 x 10(-2) N (55.625 g) condition, the original criterion was reinstated. During the augmented phase. Ss maintained stable reinforcement rates across conditions by increasing the peak force of beam pressing. These higher forces, occurring within 20 reinforcements of changing the force criterion, were produced primarily by increases in the rate of change of force (delta F/delta T). Also, while the rate of work performed on the beam increased, the overall energy expenditure fell. In contrast to these rapid adjustments, reinstating the original 5.52 x 10(-2) N (5.625 g) criterion resulted in only gradual alterations in motor performance.     

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.     

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.     

项目难度与被试能力分布最优匹配的模拟研究

该文运用蒙特卡罗方法对被测试能力分布与测验项目难度分布的匹配问题进行模拟分析,分析表明当能力分布为正态分布正偏态分布和负偏态分布时分别与测验项目难度分布与为正态分布,正偏态分布和负偏态分布匹配,比别的匹配有更高测验期望信息值,测验最大信息测验 系数,并且测验信息曲线最大值的能力点与能力分布的众数愈相一致,测验项目参数估计值性真实值的相关也更高。     

Independence of response force and reinforcement rate on concurrent variable-interval schedule performance

Five pigeons were trained over 43 experimental conditions on a variety of concurrent variable-interval schedules on which the forces required on the response keys were varied. The results were well described by the generalized matching law with log reinforcement ratios and log force ratios exerting independent (noninteractive) effects on preference. A further analysis using the Akaike criterion, an information-theoretic measure of the efficiency of a model, showed that overall reinforcement rate and overall force requirement did not affect preference. Unlike reinforcement rate changes, force requirement increases did not change the response rate on the alternate key, and an extension of Herrnstein's absolute response rate function for force variation on a single variable-interval schedule is suggested.     

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.     

JEAB: Past,present, and future

The effects of response force on microstructure were evaluated. A strain-gauge operandum permitted the manipulation of the force required to produce reinforcers (criterion responses) independently from the force defining response threshold. Thus, we could detect subcriterion forces that fell short of the force criterion. Eight rats earned food according to variable-interval (VI) 30- and 120-s schedules. The force requirements were set to 5.6 or 32.0 g; the response threshold was fixed at 5.6 g. Interresponse times were computed when subcriterion responses were both included and omitted from the analysis. Log-survivor functions of interresponse times showed that increasing force requirements elevated the mean between-bout interval of the VI 120-s schedule, but only if subcriterion behavior was excluded. Omitting subcriterion responses thus leads to overestimation of intervals separating response bouts. Increasing force requirements also increased the skewness of the between-bout distribution. A subsequent analysis found that subcriterion responses are most plentiful following reinforcer delivery, which helps to explain why their omission might inflate between-bout intervals, as this period is an important transition from reinforcer consumption to engagement in operant activity. The data suggest caution interpreting the effects of force on microstructure when subcriterion behavior is not or cannot be measured.     

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.     

Impulse variability in isometric tasks

An isometric elbow flexion task was used in two experiments that examined the influence of force-production characteristics on impulse variability. Impulse size was held constant while peak force, time to peak force, rate of force, and, hence, the shape of the criterion force-time curve were manipulated. The results indicated that changes in the force-time curve under conditions of equal impulse bring about systematic changes in impulse variability, and this effect is more pronounced for larger impulse conditions. The inability of existing functions to account for the peak force variability findings led to the generation of a new predicted force variability function. The proposed function accounts for changes in the standard deviation and coefficient of variation of peak force, impulse, and rate of force over a range of force-time conditions.     

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.     

Directional Variability of the Isometric Force Vector Produced by the Human Hand in Multijoint Planar Tasks

Numerous studies have examined control of force magnitude, but relatively little research has considered force direction control. The subjects applied isometric forces to a handle and the authors compared within-trial variability when force is produced in different directions. The standard deviation of the force parallel to the prescribed direction of force production increased linearly with the targeted force level, as did the standard deviation of the force perpendicular to the instructed direction. In contrast, the standard deviation of the angle of force production decreased with increased force level. In the 4 (of 8) instructed force directions where the endpoint force was generated due to a joint torque in only 1 joint (either the shoulder or elbow) the principal component axes in force space were well aligned with the prescribed direction of force production. In the other directions, the variance was approximately equal along the 2 force axes. The variance explained by the first principal component was significantly larger in torque space compared to the force space, and mostly corresponded to positive correlation between the joint torques. Such coordinated changes suggest that the torque variability was mainly due to the variability of the common drive to the muscles serving 2 joints, although this statement needs to be supported by direct studies of muscle activation in the future.