Direction of Effects in Multiple Linear Regression Models

Previous studies analyzed asymmetric properties of the Pearson correlation coefficient using higher than second order moments. These asymmetric properties can be used to determine the direction of dependence in a linear regression setting (i.e., establish which of two variables is more likely to be on the outcome side) within the framework of cross-sectional observational data. Extant approaches are restricted to the bivariate regression case. The present contribution extends the direction of dependence methodology to a multiple linear regression setting by analyzing distributional properties of residuals of competing multiple regression models. It is shown that, under certain conditions, the third central moments of estimated regression residuals can be used to decide upon direction of effects. In addition, three different approaches for statistical inference are discussed: a combined D’Agostino normality test, a skewness difference test, and a bootstrap difference test. Type I error and power of the procedures are assessed using Monte Carlo simulations, and an empirical example is provided for illustrative purposes. In the discussion, issues concerning the quality of psychological data, possible extensions of the proposed methods to the fourth central moment of regression residuals, and potential applications are addressed.     

Direction dependence analysis: A framework to test the direction of effects in linear models with an implementation in SPSS

In nonexperimental data, at least three possible explanations exist for the association of two variables x and y: (1) x is the cause of y, (2) y is the cause of x, or (3) an unmeasured confounder is present. Statistical tests that identify which of the three explanatory models fits best would be a useful adjunct to the use of theory alone. The present article introduces one such statistical method, direction dependence analysis (DDA), which assesses the relative plausibility of the three explanatory models on the basis of higher-moment information about the variables (i.e., skewness and kurtosis). DDA involves the evaluation of three properties of the data: (1) the observed distributions of the variables, (2) the residual distributions of the competing models, and (3) the independence properties of the predictors and residuals of the competing models. When the observed variables are nonnormally distributed, we show that DDA components can be used to uniquely identify each explanatory model. Statistical inference methods for model selection are presented, and macros to implement DDA in SPSS are provided. An empirical example is given to illustrate the approach. Conceptual and empirical considerations are discussed for best-practice applications in psychological data, and sample size recommendations based on previous simulation studies are provided.     

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.     

Emotional attention: effects of emotion and gaze direction on overt orienting of visual attention

In the present study we considered the two factors that have been advocated for playing a role in emotional attention: perception of gaze direction and facial expression of emotions. Participants performed an oculomotor task in which they had to make a saccade towards one of the two lateral targets, depending on the colour of the fixation dot which appeared at the centre of the computer screen. At different time intervals (stimulus onset asynchronies, SOAs: 50,100,150 ms) following the onset of the dot, a picture of a human face (gazing either to the right or to the left) was presented at the centre of the screen. The gaze direction of the face could be congruent or incongruent with respect to the location of the target, and the expression could be neutral or angry. In Experiment 1 the facial expressions were presented randomly in a single block, whereas in Experiment 2 they were shown in separate blocks. Latencies for correct saccades and percentage of errors (saccade direction errors) were considered in the analyses. Results showed that incongruent trials determined a significantly higher percentage of saccade direction errors with respect to congruent trials, thus confirming that gaze direction, even when task-irrelevant, interferes with the accuracy of the observer’s oculomotor behaviour. The angry expression was found to hold attention for a longer time with respect to the neutral one, producing delayed saccade latencies. This was particularly evident at 100 ms SOA and for incongruent trials. Emotional faces may then exert a modulatory effect on overt attention mechanisms.     

Statistical Significance Levels of Nonparametric Tests Biased by Heterogeneous Variances of Treatment Groups

The statistical significance levels of the Wilcoxon-Mann-Whitney test and the Kruskal-Wallis test are substantially biased by heterogeneous variances of treatment groups—even when sample sizes are equal. Under these conditions, the Type I error probabilities of the nonparametric tests, performed at the .01, .05, and .10 significance levels, increase by as much as 40%-50% in many cases and sometimes as much as 300%. The bias increases systematically as the ratio of standard deviations of treatment groups increases and remains fairly constant for various sample sizes. There is no indication that Type I error probabilities approach the significance level asymptotically as sample size increases.     

《老子》美学意蕴的现象学解读

借助现象学存在论阐释，《老子》的美学意蕴展现为：“观”的审美态度，“有无相生”的生存体验，以语言悖论为特征的美学话语。它们奠定了中国美学尤其道家美学的基本形态。     

Modelling monotonic effects of ordinal predictors in Bayesian regression models

Ordinal predictors are commonly used in regression models. They are often incorrectly treated as either nominal or metric, thus under- or overestimating the information contained. Such practices may lead to worse inference and predictions compared to methods which are specifically designed for this purpose. We propose a new method for modelling ordinal predictors that applies in situations in which it is reasonable to assume their effects to be monotonic. The parameterization of such monotonic effects is realized in terms of a scale parameter b representing the direction and size of the effect and a simplex parameter modelling the normalized differences between categories. This ensures that predictions increase or decrease monotonically, while changes between adjacent categories may vary across categories. This formulation generalizes to interaction terms as well as multilevel structures. Monotonic effects may be applied not only to ordinal predictors, but also to other discrete variables for which a monotonic relationship is plausible. In simulation studies we show that the model is well calibrated and, if there is monotonicity present, exhibits predictive performance similar to or even better than other approaches designed to handle ordinal predictors. Using Stan, we developed a Bayesian estimation method for monotonic effects which allows us to incorporate prior information and to check the assumption of monotonicity. We have implemented this method in the R package brms, so that fitting monotonic effects in a fully Bayesian framework is now straightforward.     

Order effects in sequentially judged options due to the direction of comparison

Many real-world judgment tasks present options in sequence. Typically, these judgments are made step-by-step, immediately after considering each option, or end-of-sequence, after all have been seen. We report similar order effects in both procedures, due to direction of comparison. It appears that judges form an impression of each new option by comparing it to those that preceded it. Using that option’s features as a “checklist,” more weight is given to unique ones than to ones shared with previous options. This unidirectional comparison process produces increasing ratings in options with unique positive features, and decreasing ratings when options have unique negative features. By manipulating the barriers to making multiple comparisons, we show that the direction-of-comparison effect is not limited to judgment tasks with sequential presentation. Even simultaneously presented options may show order effects, if they are judged one at a time—in sequence. Possible explanations and implications are discussed.     

A test of the hypothesis that Cronbach's alpha reliability coefficient is the same for two tests administered to the same sample

In measurement studies the researcher may wish to test the hypothesis that Cronbach's alpha reliability coefficient is the same for two measurement procedures. A statistical test exists for independent samples of subjects. In this paper three procedures are developed for the situation in which the coefficients are determined from the same sample. All three procedures are computationally simple and give tight control of Type I error when the sample size is 50 or greater.The author is indebted to Jerry S. Gilmer for development of the computer programs used in this study.     

Comparing the predictive powers of alternative multiple regression models

Mean squared error of prediction is used as the criterion for determining which of two multiple regression models (not necessarily nested) is more predictive. We show that an unrestricted (or true) model witht parameters should be chosen over a restricted (or misspecified) model withm parameters if (P _t ² ?P _m ² )>(1?P _t ² )(t?m)/n, whereP _t ² andP _m ² are the population coefficients of determination of the unrestricted and restricted models, respectively, andn is the sample size. The left-hand side of the above inequality represents the squared bias in prediction by using the restricted model, and the right-hand side gives the reduction in variance of prediction error by using the restricted model. Thus, model choice amounts to the classical statistical tradeoff of bias against variance. In practical applications, we recommend thatP ² be estimated by adjustedR ² . Our recommendation is equivalent to performing theF-test for model comparison, and using a critical value of 2?(m/n); that is, ifF>2?(m/n), the unrestricted model is recommended; otherwise, the restricted model is recommended.     

Selecting the number of classes under latent class regression: a factor analytic analogue

Recently, the regression extension of latent class analysis (RLCA) model has received much attention in the field of medical research. The basic RLCA model summarizes shared features of measured multiple indicators as an underlying categorical variable and incorporates the covariate information in modeling both latent class membership and multiple indicators themselves. To reduce complexity and enhance interpretability, one usually fixes the number of classes in a given RLCA. Often, goodness of fit methods comparing various estimated models are used as a criterion to select the number of classes. In this paper, we propose a new method that is based on an analogous method used in factor analysis and does not require repeated fitting. Two ideas with application to many settings other than ours are synthesized in deriving the method: a connection between latent class models and factor analysis, and techniques of covariate marginalization and elimination. A Monte Carlo simulation study is presented to evaluate the behavior of the selection procedure and compare to alternative approaches. Data from a study of how measured visual impairments affect older persons’ functioning are used for illustration.This work was supported by National Institute on Aging (NIA) Program Project P01-AG-10184-03. The author wishes to thank Dr. Karen Bandeen-Roche for her stimulating comments and helpful discussions, and Drs. Gary Rubin and Sheila West for kindly making the Salisbury Eye Evaluation data available.     

Using information criteria to determine the number of regimes in threshold autoregressive models

Threshold autoregressive models can be used to study processes that are characterized by recurrent switches between two or more regimes, where switching is triggered by a manifest threshold variable. In this paper the performance of diverse information criteria for selecting the number of regimes in small to moderate sample sizes (i.e., n=50,100,200) is investigated. In addition it is investigated whether these information criteria can be used to determine whether the residual variances are identical across the regimes. It is concluded that for small sample sizes should be preferred, while for larger sample sizes either BIC or should be considered: The latter is the only information criterion that includes a penalty for the unknown threshold parameters.     

A PRP-study to determine the locus of target priming effects

Visual stimuli that are made invisible by a following mask can nonetheless affect motor responses. To localize the origin of these target priming effects we used the psychological refractory period paradigm. Participants classified tones as high or low, and responded to the position of a visual target that was preceded by a prime. The stimulus onset asynchrony (SOA) between both tasks varied. In Experiment 1 the tone task was followed by the position task and SOA dependent target priming effects were observed. When the visual position task preceded the tone task in Experiment 2, with short SOA the priming effect propagated entirely to the tone task yielding faster responses to tones on visually congruent trials and delayed responses to tones on visually incongruent trials. Together, results suggest that target priming effects arise from processing before and at the level of the central bottleneck such as sensory analysis and response selection.     

Can we disregard the whole model? Omnibus non-inferiority testing for R2 in multi-variable linear regression and in ANOVA

Determining a lack of association between an outcome variable and a number of different explanatory variables is frequently necessary in order to disregard a proposed model (i.e., to confirm the lack of a meaningful association between an outcome and predictors). Despite this, the literature rarely offers information about, or technical recommendations concerning, the appropriate statistical methodology to be used to accomplish this task. This paper introduces non-inferiority tests for ANOVA and linear regression analyses, which correspond to the standard widely used F test for and R², respectively. A simulation study is conducted to examine the Type I error rates and statistical power of the tests, and a comparison is made with an alternative Bayesian testing approach. The results indicate that the proposed non-inferiority test is a potentially useful tool for ‘testing the null’.     

Fisher与Neyman–Pearson的分歧与心理统计中的假设检验争议

假设检验思想的提出者Fisher与Neyman–Pearson在统计模型的方法论基础、两类错误的性质、显著性水平的理解、以及假设检验的功能等方面存在诸多分歧, 使得心理统计中最常用的原假设显著性检验模式呈现出隐含的各种矛盾, 从而引发了应用上的争议。心理统计不仅需要检讨现有检验模型的模糊之处和提出其他补充性的统计推论方式,更应注重反思心理统计的教育传统, 以建立更加开放和多元的统计应用视野, 使心理统计为更好地心理学研究服务。     

Context effects in a temporal discrimination task" further tests of the Scalar Expectancy Theory and Learning-to-Time models

Pigeons were trained on two temporal bisection tasks, which alternated every two sessions. In the first task, they learned to choose a red key after a 1-s signal and a green key after a 4-s signal; in the second task, they learned to choose a blue key after a 4-s signal and a yellow key after a 16-s signal. Then the pigeons were exposed to a series of test trials in order to contrast two timing models, Learning-to-Time (LeT) and Scalar Expectancy Theory (SET). The models made substantially different predictions particularly for the test trials in which the sample duration ranged from 1 s to 16 s and the choice keys were Green and Blue, the keys associated with the same 4-s samples: LeT predicted that preference for Green should increase with sample duration, a context effect, but SET predicted that preference for Green should not vary with sample duration. The results were consistent with LeT. The present study adds to the literature the finding that the context effect occurs even when the two basic discriminations are never combined in the same session.     

Evaluating performance in continuous experiments with feedback to subjects

Consider an experiment in which a subject guesses repeatedly at a randomly chosen target on a continuum. To guarantee a positive probability of success, the continuum is partitioned into a finite but large number of segments. The subject is given directional feedback. General guessing strategies are characterized, and an optimal strategy is identified. The hypothesis that the subject's performance can be explained by chance alone is of interest in such experiments. A test is developed based on comparing the subject's performance to expected performance using the optimal strategy. A skill-scoring procedure is developed for assessing a subject's performance in light of the strategy used, and a test based on skill-scoring is advanced.Research by the first author is supported in part by the Air Force Office of Scientific Research under grant AFOSR 77-3180.The authors wish to thank the referees for remarks that led to improvements in both content and clarity.     

Separating within-person from between-person effects in the longitudinal co-occurrence of depression and different anxiety syndromes in youth

Cross-lagged panel models (CLPM) are often used to study anxiety-depression co-occurrence. However, the CLPM aggregates within- and between-person variance, which can lead to incorrect estimates. The latent curve model with structured residuals (LCM-SR) parses these sources of variance. We utilized the LCM-SR to examine prospective associations between anxiety (physical, social, separation) and depression. Youth (N = 680; M_age = 11.8; 55% female) completed measures of depression and anxiety every 3 months for 3 years (13 timepoints). The LCM-SR describing anxiety-depression co-occurrence fit well (RMSEA = 0.05, SRMR = 0.06). Depression predicted within-person change in social (b = 0.09), physical (b = 0.04), and separation anxiety (b = 0.06) over 13 timepoints. Separation anxiety predicted within-person change in depression (b = 0.08); social and physical anxiety did not. Findings advance knowledge of within-person development of anxiety-depression co-occurrence.     

On the use of heterogeneous thresholds ordinal regression models to account for individual differences in response style

This paper proposes a general approach to accounting for individual differences in the extreme response style in statistical models for ordered response categories. This approach uses a hierarchical ordinal regression modeling framework with heterogeneous thresholds structures to account for individual differences in the response style. Markov chain Monte Carlo algorithms for Bayesian inference for models with heterogeneous thresholds structures are discussed in detail. A simulation and two examples based on ordinal probit models are given to illustrate the proposed methodology. The simulation and examples also demonstrate that failing to account for individual differences in the extreme response style can have adverse consequences for statistical inferences.The author is grateful to Ulf Böckenholt, an associate editor, and three anonymous reviewers for helpful comments, and Kristine Kuhn and Kshiti Joshi for providing the data.     

Transfer tests of stimulus value in concurrent chains

We report two experiments that use transfer tests to investigate whether in concurrent chains the value of a terminal-link stimulus is affected by the alternate terminal link. In Experiment 1, two groups of pigeons were trained on multiple concurrent-chains schedules in which switching between the schedules was via pecking a changeover key. For one group, the terminal links were fixed-interval 8 s versus fixed-interval 16 s in one component and fixed-interval 16 s versus fixed-interval 32 s in the other component. For a second group, the terminal links were variable-interval 10 s versus variable-interval 20 s in one component and variable-interval 20 s versus variable-interval 40 s in the other. After sufficient baseline training had been given so that performances had stabilized, transfer tests were conducted in which the two chains with equal terminal-link schedules were presented together as a new concurrent pair. For 6 of the 7 subjects, initial-link responding changed fairly rapidly during the test in the manner predicted if the values of the terminal links were equal. In Experiment 2, pigeons were trained on multiple concurrent chains using a two-key procedure, and the terminal links were the same variable-interval schedules as in Experiment 1. After baseline training, transfer tests were conducted that assessed (a) the relative reinforcing strength of the terminal-link stimuli in a novel initial-link situation and (b) the relative ability of those stimuli to evoke responding. The data from the reinforcing strength test were consistent with those from Experiment 1, but those from the evocation strength test were not. Although this discrepancy shows that responding in transfer tests is not solely a function of stimulus value, the results from both experiments suggest, overall, that value is determined by the stimulus—reinforcer relation independently of the alternative terminal link.