Comparing the regression slopes of independent groups

In this study, we explore the effects of non‐normality and heteroscedasticity when testing the hypothesis that regression lines associated with two independent groups have the same slopes. Our results indicate that some recently proposed methods that allow heteroscedasticity and perform well in extant simulation studies do not perform well for the situation at hand. Two of the methods studied here are recommended for general use.     

Multilevel SEM with random slopes in discrete data using the pairwise maximum likelihood

Pairwise maximum likelihood (PML) estimation is a promising method for multilevel models with discrete responses. Multilevel models take into account that units within a cluster tend to be more alike than units from different clusters. The pairwise likelihood is then obtained as the product of bivariate likelihoods for all within-cluster pairs of units and items. In this study, we investigate the PML estimation method with computationally intensive multilevel random intercept and random slope structural equation models (SEM) in discrete data. In pursuing this, we first reconsidered the general ‘wide format’ (WF) approach for SEM models and then extend the WF approach with random slopes. In a small simulation study we the determine accuracy and efficiency of the PML estimation method by varying the sample size (250, 500, 1000, 2000), response scales (two-point, four-point), and data-generating model (mediation model with three random slopes, factor model with one and two random slopes). Overall, results show that the PML estimation method is capable of estimating computationally intensive random intercept and random slopes multilevel models in the SEM framework with discrete data and many (six or more) latent variables with satisfactory accuracy and efficiency. However, the condition with 250 clusters combined with a two-point response scale shows more bias.     

基于多元回归的调节效应分析

在心理学和其他社科研究领域,大量实证研究建立调节模型,以分析自变量对因变量关系的影响机制,但在基于多元回归的调节效应分析实践中仍存在不足。我们回顾了均值中心化在基于多元回归的调节效应分析中的作用,均值中心化不影响乘积项(即调节效应)的检验,仅对一阶项(即主效应)的检验有影响。讨论了简单斜率的检验方法,建议在调节变量为连续变量时,使用Johnson-Neyman法进行简单斜率检验;在调节变量为类别变量或研究者对某个调节变量值感兴趣时,使用选点法。并用一个实际例子演示如何进行调节效应分析。随后展望了调节效应检验的拓展方向。     

Using the Criterion-Predictor Factor Model to Compute the Probability of Detecting Prediction Bias with Ordinary Least Squares Regression

The study of prediction bias is important and the last five decades include research studies that examined whether test scores differentially predict academic or employment performance. Previous studies used ordinary least squares (OLS) to assess whether groups differ in intercepts and slopes. This study shows that OLS yields inaccurate inferences for prediction bias hypotheses. This paper builds upon the criterion-predictor factor model by demonstrating the effect of selection, measurement error, and measurement bias on prediction bias studies that use OLS. The range restricted, criterion-predictor factor model is used to compute Type I error and power rates associated with using regression to assess prediction bias hypotheses. In short, OLS is not capable of testing hypotheses about group differences in latent intercepts and slopes. Additionally, a theorem is presented which shows that researchers should not employ hierarchical regression to assess intercept differences with selected samples.     

Robust inference with binary data

In this paper robustness properties of the maximum likelihood estimator (MLE) and several robust estimators for the logistic regression model when the responses are binary are analysed. It is found that the MLE and the classical Rao's score test can be misleading in the presence of model misspecification which in the context of logistic regression means either misclassification's errors in the responses, or extreme data points in the design space. A general framework for robust estimation and testing is presented and a robust estimator as well as a robust testing procedure are presented. It is shown that they are less influenced by model misspecifications than their classical counterparts. They are finally applied to the analysis of binary data from a study on breastfeeding.The author is partially supported by the Swiss National Science Foundation. She would like to thank Rand Wilcox, Eva Cantoni and Elvezio Ronchetti for their helpful comments on earlier versions of the paper, as well as Stephane Heritier for providing the routine to compute the OBRE.     

Associative symmetry in the pigeon after successive matching-to-sample training

If an organism is explicitly taught an A→B association, then might it also spontaneously learn the symmetrical B→A association? Little evidence attests to such “associative symmetry” in nonhuman animals. We report for the first time a clear case of associative symmetry in the pigeon. Experiment 1 used a successive go/no go matching‐to‐sample procedure, which showed all of the training and testing stimuli in one location and intermixed arbitrary and identity matching trials. We found symmetrical responding that was as robust during testing (B→A) as during training (A→B). In Experiment 2, we trained different pigeons using only arbitrary matching trials before symmetry testing. No symmetrical responding was found. In Experiment 3, we trained other pigeons with only arbitrary matching trials and then tested for symmetry. When these pigeons, too, did not exhibit symmetrical responding, we retrained them with intermixed identity and arbitrary matching trials. Less robust symmetrical responding was obtained here than in Experiment 1. Collectively, these results suggest that identity matching may have to be learned concurrently with arbitrary matching from the outset of training for symmetry to emerge.     

Some results on extensions and modifications of the Theil — Sen regression estimator

Many robust regression estimators have been proposed that have a high, finite‐sample breakdown point, roughly meaning that a large porportion of points must be altered to drive the value of an estimator to infinity. But despite this, many of them can be inordinately influenced by two properly placed outliers. With one predictor, an estimator that appears to correct this problem to a fair degree, and simultaneously maintain good efficiency when standard assumptions are met, consists of checking for outliers using a projection‐type method, removing any that are found, and applying the Theil — Sen estimator to the data that remain. When dealing with multiple predictors, there are two generalizations of the Theil — Sen estimator that might be used, but nothing is known about how their small‐sample properties compare. Also, there are no results on testing the hypothesis of zero slopes, and there is no information about the effect on efficiency when outliers are removed. In terms of hypothesis testing, using the more obvious percentile bootstrap method in conjunction with a slight modification of Mahalanobis distance was found to avoid Type I error probabilities above the nominal level, but in some situations the actual Type I error probabilities can be substantially smaller than intended when the sample size is small. An alternative method is found to be more satisfactory.     

Timing relations in speech and the identification of voice-onset times: a stable perceptual boundary for voicing categories across speaking rates

This study shows that the ratio of voice onset time (VOT) to syllable duration for /t/ and /d/ presents distributions with a stable boundary across speaking rates and that this boundary constitutes a perceptual criterion by which listeners judge the category affiliation of VOT. In Experiment 1, best-fit regression lines for VOT ratios of intervocalic /t/ and /d/ against speaking rate had zero slopes, and there was an inferable boundary between the distributions. In Experiment 2, listeners' identifications of syllable-initial stops conformed to this boundary ratio. In Experiment 3, VOT was held constant, while VOT ratios were altered by modifying the duration of the following vowel. As VOT ratios exceeded the boundary estimated from the data of Experiment 1, listeners' identifications shifted from /d/ to /t/. Timing relations in speech production can determine the identification of voicing categories across speaking rates.     

The Traditional Regression Model for Measuring Test Bias is Incorrect and Biased Against Minorities

For 25 years psychologists have measured systematic measurement bias in terms of regression lines. According to this traditional approach a test is an unbiased predictor of a criterion for all subgroups if all subgroups have identical Y regression lines (i.e., identical slopes and identical Y intercepts). This paper shows that the traditional model is fundamentally incorrect and identical Y regression lines are not expected to occur with an unbiased test in a testing situation in which one group score lower than another group on both the test and criterion. This is the case even if the test is perfectly reliable. The traditional model for measuring bias actually results in a consistent error or bias against groups which score lower than average on both the test and criterion. In practice this bias operates against minority groups. Tests now thought to be unbiased or even biased in favor of minority groups may in fact be biased against minority groups. A new model of test bias, which is based solely on measurement principles, is briefly introduced. In this model unbiased tests produce groups with identical test-criterion common-factor axes having a slope of S _YC/S _XC and with each axis intersecting the group centroids.     

纳入式分类分析法在潜在剖面模型的后续多元回归中的应用

纳入式分类分析法能克服传统的分类分析法对后续一元回归模型参数的低估,发挥潜在类别模型的后续分析简化变量间交互作用的功能。本文进一步将纳入式分类分析法拓展至潜在剖面模型后续的多元统计分析中。通过蒙特卡洛模拟实验,比较各种纳入变量的方法思路与后续分析模型在四种常见的多元回归模型中参数估计的表现。结果发现,纳入式分类分析法所需纳入的变量取决于后续分析中与因变量、潜类别变量的关系,且后续分析使用含交互作用的模型更为稳健。     

Testing equivalence with repeated measures: tests of the difference model of two-alternative forced-choice performance

Solving theoretical or empirical issues sometimes involves establishing the equality of two variables with repeated measures. This defies the logic of null hypothesis significance testing, which aims at assessing evidence against the null hypothesis of equality, not for it. In some contexts, equivalence is assessed through regression analysis by testing for zero intercept and unit slope (or simply for unit slope in case that regression is forced through the origin). This paper shows that this approach renders highly inflated Type I error rates under the most common sampling models implied in studies of equivalence. We propose an alternative approach based on omnibus tests of equality of means and variances and in subject-by-subject analyses (where applicable), and we show that these tests have adequate Type I error rates and power. The approach is illustrated with a re-analysis of published data from a signal detection theory experiment with which several hypotheses of equivalence had been tested using only regression analysis. Some further errors and inadequacies of the original analyses are described, and further scrutiny of the data contradict the conclusions raised through inadequate application of regression analyses.     

Computing and comparing correlation and regression coefficients using a pocket calculator

Programs suitable for pocket calculators using reverse Polish notation are described. Program 1 computes regression coefficients, correlation coefficient, and standard error of estimate for paired data. Program 2 performs at test to compare the slopes of two regression lines. Program 3 computes F ratios to test the departure of a regression slope from zero and to test linearity of the regression. Programs 4 and 5 test the significance of (differences between independent and correlated correlation coefficients, respectively.     

Estimating the Reliability of Aggregated and Within-Person Centered Scores in Ecological Momentary Assessment

Consider two independent groups with K measures for each subject. For the j^th group and k^th measure, let μ^tjk be the population trimmed mean, j = 1, 2; k = 1, ..., K. This article compares several methods for testing H 0 : u1k = t2k such that the probability of at least one Type I error is, and simultaneous probability coverage is 1 - α when computing confidence intervals for μ^t1k - μ^t2k . The emphasis is on K = 4 and α = .05. For zero trimming the problem reduces to comparing means, but it is well known that when comparing means, arbitrarily small departures from normality can result in extremely low power relative to using say 20% trimming. Moreover, when skewed distributions are being compared, conventional methods for comparing means can be biased for reasons reviewed in the article. A consequence is that in some realistic situations, the probability of rejecting can be higher when the null hypothesis is true versus a situation where the means differ by a half standard deviation. Switching to robust measures of location is known to reduce this problem, and combining robust measures of location with some type of bootstrap method reduces the problem even more. Published articles suggest that for the problem at hand, the percentile t bootstrap, combined with a 20% trimmed mean, will perform relatively well, but there are known situations where it does not eliminate all problems. In this article we consider an extension of the percentile bootstrap approach that is found to give better results.     

The homogeneity assumption in differential prediction analysis: does it really matter?

In simulation studies, the F test for differences in regression slopes has tended to distort nominal Type I and II error rates when the 2 subgroup error variances exceeded a 1.50:1 ratio. This study examines the frequency and extent that this ratio is violated within data sets relevant to applied psychology. The General Aptitude Test Battery (GATB) validity study database contained ability data and overall job performance ratings. The Project A military database contained both ability and personality data, along with job performance factor scores and an overall job performance rating. Results suggest that subgroup (White-Black, male-female) error variances are often homogeneous enough to support F test results from past empirical work. Enough heterogeneity was found, however, to urge applied psychologists investigating differential prediction to explore their data and consider the possibility of alternative statistical tests.     

Robust Bayesian growth curve modelling using conditional medians

Growth curve models have been widely used to analyse longitudinal data in social and behavioural sciences. Although growth curve models with normality assumptions are relatively easy to estimate, practical data are rarely normal. Failing to account for non-normal data may lead to unreliable model estimation and misleading statistical inference. In this work, we propose a robust approach for growth curve modelling using conditional medians that are less sensitive to outlying observations. Bayesian methods are applied for model estimation and inference. Based on the existing work on Bayesian quantile regression using asymmetric Laplace distributions, we use asymmetric Laplace distributions to convert the problem of estimating a median growth curve model into a problem of obtaining the maximum likelihood estimator for a transformed model. Monte Carlo simulation studies have been conducted to evaluate the numerical performance of the proposed approach with data containing outliers or leverage observations. The results show that the proposed approach yields more accurate and efficient parameter estimates than traditional growth curve modelling. We illustrate the application of our robust approach using conditional medians based on a real data set from the Virginia Cognitive Aging Project.     

On the Usefulness of a Multilevel Logistic Regression Approach to Person-Fit Analysis

The logistic person response function (PRF) models the probability of a correct response as a function of the item locations. Reise (2000) Reise, S. P. 2000. Using multilevel logistic regression to evaluate person-fit in IRT models. Multivariate Behavioral Research, 35: 543–568. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] proposed to use the slope parameter of the logistic PRF as a person-fit measure. He reformulated the logistic PRF model as a multilevel logistic regression model and estimated the PRF parameters from this multilevel framework. An advantage of the multilevel framework is that it allows relating person fit to explanatory variables for person misfit/fit. We critically discuss Reise's approach. First, we argue that often the interpretation of the PRF slope as an indicator of person misfit is incorrect. Second, we show that the multilevel logistic regression model and the logistic PRF model are incompatible, resulting in a multilevel person-fit framework, which grossly violates the bivariate normality assumption for residuals in the multilevel model. Third, we use a Monte Carlo study to show that in the multilevel logistic regression framework estimates of distribution parameters of PRF intercepts and slopes are biased. Finally, we discuss the implications of these results and suggest an alternative multilevel regression approach to explanatory person-fit analysis. We illustrate the alternative approach using empirical data on repeated anxiety measurements of cardiac arrhythmia patients who had a cardioverter-defibrillator implanted.     

Doubly Robust-Type Estimation for Covariate Adjustment in Latent Variable Modeling

Due to the difficulty in achieving a random assignment, a quasi-experimental or observational study design is frequently used in the behavioral and social sciences. If a nonrandom assignment depends on the covariates, multiple group structural equation modeling, that includes the regression function of the dependent variables on the covariates that determine the assignment, can provide reasonable estimates under the condition of correct specification of the regression function. However, it is usually difficult to specify the correct regression function because the dimensions of the dependent variables and covariates are typically large. Therefore, the propensity score adjustment methods have been proposed, since they do not require the specification of the regression function and have been applied to several applied studies. However, these methods produce biased estimates if the assignment mechanism is incorrectly specified. In order to make a more robust inference, it would be more useful to develop an estimation method that integrates the regression approach with the propensity score methodology. In this study we propose a doubly robust-type estimation method for marginal multiple group structural equation modeling. This method provides a consistent estimator if either the regression function or the assignment mechanism is correctly specified. A simulation study indicates that the proposed estimation method is more robust than the existing methods. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists (B), 187-30406.     

A comparison of two‐stage procedures for testing least‐squares coefficients under heteroscedasticity

This study explores the performance of several two‐stage procedures for testing ordinary least‐squares (OLS) coefficients under heteroscedasticity. A test of the usual homoscedasticity assumption is carried out in the first stage of the procedure. Subsequently, a test of the regression coefficients is chosen and performed in the second stage. Three recently developed methods for detecting heteroscedasticity are examined. In addition, three heteroscedastic robust tests of OLS coefficients are considered. A major finding is that performing a test of heteroscedasticity prior to applying a heteroscedastic robust test can lead to poor control over Type I errors.     

Should metacognition be measured by logistic regression?

Are logistic regression slopes suitable to quantify metacognitive sensitivity, i.e. the efficiency with which subjective reports differentiate between correct and incorrect task responses? We analytically show that logistic regression slopes are independent from rating criteria in one specific model of metacognition, which assumes (i) that rating decisions are based on sensory evidence generated independently of the sensory evidence used for primary task responses and (ii) that the distributions of evidence are logistic. Given a hierarchical model of metacognition, logistic regression slopes depend on rating criteria. According to all considered models, regression slopes depend on the primary task criterion. A reanalysis of previous data revealed that massive numbers of trials are required to distinguish between hierarchical and independent models with tolerable accuracy. It is argued that researchers who wish to use logistic regression as measure of metacognitive sensitivity need to control the primary task criterion and rating criteria.     

Robust regression: Testing global hypotheses about the slopes when there is multicollinearity or heteroscedasticity

A well-known concern regarding the usual linear regression model is multicollinearity. As the strength of the association among the independent variables increases, the squared standard error of regression estimators tends to increase, which can seriously impact power. This paper examines heteroscedastic methods for dealing with this issue when testing the hypothesis that all of the slope parameters are equal to zero via a robust ridge estimator that guards against outliers among the dependent variable. Included are results related to leverage points, meaning outliers among the independent variables. In various situations, the proposed method increases power substantially.