Detection of interactions between a dichotomous moderator and a continuous predictor in moderated multiple regression with heterogeneous error variance

Moderated multiple regression (MMR) has been widely used to investigate the interaction or moderating effects of a categorical moderator across a variety of subdisciplines in the behavioral and social sciences. In view of the frequent violation of the homogeneity of error variance assumption in MMR applications, the weighted least squares (WLS) approach has been proposed as one of the alternatives to the ordinary least squares method for the detection of the interaction effect between a dichotomous moderator and a continuous predictor. Although the existing result is informative in assuring the statistical accuracy and computational ease of the WLS-based method, no explicit algebraic formulation and underlying distributional details are available. This article aims to delineate the fundamental properties of the WLS test in connection with the well-known Welch procedure for regression slope homogeneity under error variance heterogeneity. With elaborately systematic derivation and analytic assessment, it is shown that the notion of WLS is implicitly embedded in the Welch approach. More importantly, extensive simulation study is conducted to demonstrate the conditions in which the Welch test will substantially outperform the WLS method; they may yield different conclusions. Welch’s solution to the Behrens-Fisher problem is so entrenched that the use of its direct extension within the linear regression framework can arguably be recommended. In order to facilitate the application of Welch’s procedure, the SAS and R computing algorithms are presented. The study contributes to the understanding of methodological variants for detecting the effect of a dichotomous moderator in the context of moderated multiple regression. Supplemental materials for this article may be downloaded from brm.psychonomic-journals.org/content/supplemental.     

Effect size,statistical power,and sample size for assessing interactions between categorical and continuous variables

The reporting and interpretation of effect size estimates are widely advocated in many academic journals of psychology and related disciplines. However, such concern has not been adequately addressed for analyses involving interactions between categorical and continuous variables. For the purpose of improving current practice, this article presents fundamental features and theoretical developments for the variance of standardized slopes as a desirable standardized effect size measure for the degree of disparity between several slope coefficients. To estimate the effect size, a consistent and nearly unbiased estimator is described and a simple refinement is emphasized for extreme situations whenever appropriate. The essential problems of power and sample size calculations for testing the equality of slope coefficients are also considered. According to the analytic justification and empirical assessment, the exact approach has a clear advantage over the approximate methods. Both SAS and R computer codes are provided to facilitate practical accessibility of the proposed techniques in interaction studies.     

Effect size and power in assessing moderating effects of categorical variables using multiple regression: a 30-year review

The authors conducted a 30-year review (1969-1998) of the size of moderating effects of categorical variables as assessed using multiple regression. The median observed effect size (f(2)) is only .002, but 72% of the moderator tests reviewed had power of .80 or greater to detect a targeted effect conventionally defined as small. Results suggest the need to minimize the influence of artifacts that produce a downward bias in the observed effect size and put into question the use of conventional definitions of moderating effect sizes. As long as an effect has a meaningful impact, the authors advise researchers to conduct a power analysis and plan future research designs on the basis of smaller and more realistic targeted effect sizes.     

Probing three-way interactions in moderated multiple regression: development and application of a slope difference test

Researchers often use 3-way interactions in moderated multiple regression analysis to test the joint effect of 3 independent variables on a dependent variable. However, further probing of significant interaction terms varies considerably and is sometimes error prone. The authors developed a significance test for slope differences in 3-way interactions and illustrate its importance for testing psychological hypotheses. Monte Carlo simulations revealed that sample size, magnitude of the slope difference, and data reliability affected test power. Application of the test to published data yielded detection of some slope differences that were undetected by alternative probing techniques and led to changes of results and conclusions. The authors conclude by discussing the test's applicability for psychological research.     

A square root method of selecting a minimum set of variables in multiple regression: I. The method

An extension of Dwyer's “square root” method has been made to the problem of selecting a minimum set of variables in a multiple regression problem. The square root method of selection differs from the Wherry-Doolittle method primarily in that (1) the computations required are more compact, (2) anF ratio criterion is used which leads to the selection of fewer variables. The method provides solutions for the problems of test selection, item analysis, analysis of variance with disproportionate frequencies, and other problems requiring the rejection of superfluous variables. In a subsequent article a worked example will be given, and the square root and Wherry-Doolittle methods compared.     

Multivariate multiple regression analyses: a permutation method for linear models

A multivariate extension of a univariate procedure for the analysis of experimental designs is presented. A Euclidean-distance permutation procedure is used to evaluate multivariate residuals obtained from a regression algorithm, also based on Euclidean distances. Applications include various completely randomized and randomized block experimental designs such as one-way, Latin square, factorial, nested, and split-plot designs, with and without covariates. Unlike parametric procedures, the only required assumption is the randomization of subjects to treatments.     

Models with discrete latent variables for analysis of categorical data: A framework and a MATLAB MDLV toolbox

Studies in the social and behavioral sciences often involve categorical data, such as ratings, and define latent constructs underlying the research issues as being discrete. In this article, models with discrete latent variables (MDLV) for the analysis of categorical data are grouped into four families, defined in terms of two dimensions (time and sampling) of the data structure. A MATLAB toolbox (referred to as the “MDLV toolbox”) was developed for applying these models in practical studies. For each family of models, model representations and the statistical assumptions underlying the models are discussed. The functions of the toolbox are demonstrated by fitting these models to empirical data from the European Values Study. The purpose of this article is to offer a framework of discrete latent variable models for data analysis, and to develop the MDLV toolbox for use in estimating each model under this framework. With this accessible tool, the application of data modeling with discrete latent variables becomes feasible for a broad range of empirical studies.     

A square root method of selecting a minimum set of variables in multiple regression: II. A worked example

Thesquare root method of selection has been explained in a previous article. In the present article a worked example is given which illustrates the compactness of the procedure. The square root method is compared with the Wherry-Doolittle method.     

Searching for two things at once: Establishment of multiple attentional control settings on a trial-by-trial basis

Recent work has demonstrated that attention can be configured to multiple potential targets in spatial search. However, this previous work relied on a fixed set of targets across multiple trials, allowing observers to offload attentional control settings to longer-term representations. In the present experiments, we demonstrate multiple attentional control settings that operate independently of space (Experiments 1 and 2). More important, we show that observers can be cued to different control settings on a trial-by-trial basis (Experiment 3). The latter result suggests that observers were capable of maintaining multiple control settings when the demands of the task required an attentional search for specific feature values. Attention can be configured to extract multiple feature values in a goal-directed manner, and this configuration can be can be dynamically engaged on a trial-by-trial basis. These results support recent findings that reveal the high precision, complexity, and flexibility of attentional control settings.     

Parallel processing across neural systems: implications for a multiple memory system hypothesis

A common conceptualization of the organization of memory systems in brain is that different types of memory are mediated by distinct neural systems. Strong support for this view comes from studies that show double (or triple) dissociations between spatial, response, and emotional memories following selective lesions of hippocampus, striatum, and the amygdala. Here, we examine the extent to which hippocampal and striatal neural activity patterns support the multiple memory systems view. A comparison is made between hippocampal and striatal neural correlates with behavior during asymptotic performance of spatial and response maze tasks. Location- (or place), movement, and reward-specific firing patterns were found in both structures regardless of the task demands. Many, but not all, place fields of hippocampal and striatal neurons were similarly affected by changes in the visual and reward context regardless of the cognitive demands. Also, many, but not all, hippocampal and striatal movement-sensitive neurons showed significant changes in their behavioral correlates after a change in visual context, irrespective of cognitive strategy. Similar partial reorganization was observed following manipulations of the reward condition for cells recorded from both structures, again regardless of task. Assuming that representations that persist across context changes reflect learned information, we make the following conclusions. First, the consistent pattern of partial reorganization supports a view that the analysis of spatial, response, and reinforcement information is accomplished via an error-driven, or match-mismatch, algorithm across neural systems. Second, task-relevant processing occurs continuously within hippocampus and striatum regardless of the cognitive demands of the task. Third, given the high degree of parallel processing across allegedly different memory systems, we propose that different neural systems may effectively compete for control of a behavioral expression system. The strength of the influence of any one neural system on behavioral output is likely modulated by factors such as motivation, experience, or hormone status.     

Cross‐magnitude interactions across development: Longitudinal evidence for a general magnitude system

There is general agreement that humans represent numerical, spatial, and temporal magnitudes from early in development. However, there is disagreement about whether different magnitudes converge within a general magnitude system and whether this system supports behavioral demonstrations of cross‐magnitude interactions at different developmental time points. Using a longitudinal design, we found a relation between children's cross‐magnitude interactions assessed at two developmental time points with different behavioral measures. More specifically, stronger cross‐magnitude interactions in infancy (M = 9.3 months) predicted a stronger cross‐magnitude congruity effect at preschool age (M = 44.2 months), even when controlling for performance on measures of inhibitory control, analogical reasoning, and verbal competence at preschool age. The results suggest a common mechanism for cross‐magnitude interactions at different points in development as well as stability of the underlying individual differences. We argue that this mechanism reflects a nonverbal general magnitude system that is operational early in life and that displays continuity from infancy to preschool age.     

Toward a refined view of aggressive fantasy as a risk factor for aggression: interaction effects involving cognitive and situational variables

Over three decades of research have established a positive connection between fantasizing about aggression and enacting aggression. Such findings have provided strong evidence against the catharsis view of aggressive fantasy. However, little attention has been paid to the potentially nuanced nature of the link between fantasy aggression and actual aggression. In the present article, we examined the influence of four variables in the aggressive fantasy–aggressive behavior link: gender, exposure to violence, fantasy absorption, and level of fantasy about harm befalling loved ones and the self (dysphoric fantasy). Using data from a diverse, community‐based sample of 7–14‐year olds and their mothers, we replicated the general finding that aggressive fantasy is positively associated with real‐world aggressive behavior. However, we also found that the interaction of aggressive fantasy and exposure to violence related significantly to aggression, as did the relation between aggressive fantasy and dysphoric fantasy. When exposure to violence was low, even high levels of aggressive fantasizing did not predict aggressive behavior, and, when aggressive fantasizing was low, even high levels of exposure to violence did not predict aggressive behavior. Similarly, when dysphoric fantasy was high, the connection between fantasy aggression and real aggression was markedly attenuated. The implications of these findings for intervention efforts and future research are considered. Aggr. Behav. 35:313–323, 2009. © 2009 Wiley‐Liss, Inc.     

Male-female interactions across the female estrous cycle: a comparison of two species of dwarf hamster (Phodopus campbelli and Phodopus sungorus)

Recent studies in our laboratory have suggested that monogamy may be the preferred mating system in the Djungarian hamster (Phodopus campbelli), whereas the available evidence for the closely related Siberian hamster (P. sungorus) does not show the same pattern. Here we examine the behavior of male-female dyads of both species interacting during 1-hr tests in large, familiar habitats containing defensible nest boxes, food, and water. Levels of aggression within pairs were low, compared with those seen during brief intrasexual encounters, whereas affiliative behaviors, such as sniffs, were high. P. campbelli scent marked more than twice as frequently as P. sungorus. Females of both species scent marked at a constant rate irrespective of their location in the habitat, whereas males scent marked at a higher rate in the female's home area. Two major features of the copulatory pattern differed between the two species: (a) The duration of the ejaculatory lock was five times longer in P. sungorus than in P. campbelli. (b) Both species had approximately the same number of mounts in each ejaculatory series, but the intromission/mount ratio was significantly higher in P. sungorus.     

Weight loss as a primary objective of therapeutic groups for obese women: two preliminary studies

The studies reported here explored whether therapeutic groups for women who eat compulsively can demonstrate weight loss as a primary result as well as the improvements in emotional functioning reported by other investigators. In both studies questionnaire data showed little change in self-esteem or attitudes as measured by the Rosenberg Self-Esteem and the Eating Disorders Examination. However, interview data showed considerable changes in attitudes and emotional functioning. The initial study demonstrated no weight loss but marked changes in participants’ attitudes to themselves and food, likely to lead to weight loss. A second, longer, uncontrolled study again showed changes in emotional functioning plus weight loss of 5% or more of initial body weight at the end of the intervention in 75% of participants who completed the study (n=8) or 50% of those who were initially recruited (n=12). Weight loss has been maintained in 75% of participants at 18-month follow-up. The data are based on a small sample, which can only be suggestive but supports a further study.     

Enuresis in cross-cultural perspective: a comparison of training for elimination control in three Israeli ethnic groups

Enuresis was investigated in the context of different styles of training for bladder control among three Israeli ethnic groups (Jews of Moroccan, Kurdish or Eastern European descent). Semistructured interviews were conducted in ethnically homogeneous agricultural villages with 46 mothers of 248 children between the ages of 3 and 18 years. In contrast to previous British and American studies, no sex differences were found, but there were higher rates of primary enuresis and lower rates of secondary (regressive) enuresis. Enuresis was correlated among siblings in the Kurdish group only and with disorderly sleeping arrangements in the Moroccan group. The Moroccan and Kurdish groups had higher rates of enuresis than the Ashkenazi group. The higher rates appear to be related to differences in the age of onset of training and a lack of age-appropriate changes in the parent-child interaction, which led to chronic enuresis and the inability to seek effective assistance. These results are discussed in terms of a proposed typology for training: an early symbiotic style, a strict toddler style, and a communicative partnership.     

Factor structure and psychometric properties of the Brief Symptom Inventory-18 in women: a MACS approach to testing for invariance across racial/ethnic groups

This study used data from 3 sites to examine the invariance and psychometric characteristics of the Brief Symptom Inventory-18 across Black, Hispanic, and White mothers of 5th graders (N = 4,711; M = 38.07 years of age, SD = 7.16). Internal consistencies were satisfactory for all subscale scores of the instrument regardless of ethnic group membership. Mean and covariance structures analysis indicated that the hypothesized 3-factor structure of the instrument was not robust across ethnic groups. It provided a reasonable approximation to the data for Black and White women but not for Hispanic women. Tests for differential item functioning (DIF) were therefore conducted for only Black and White women. Analyses revealed no more than trivial instances of nonuniform DIF but more substantial evidence of uniform DIF for 3 of the 18 items. After having established partial strong factorial invariance of the instrument, latent factor means were found to be significantly higher for Black than for White women on all 3 subscales (somatization, depression, anxiety). In conclusion, the instrument may be used for mean comparisons between Black and White women.     

Testing the co-existence of two timing strategies for motor control in a unique task: The synchronisation spatial-tapping task

The control of rhythmic action sequences may involve two distinct timing strategies, i.e., event-based and emergent timing, which are usually revealed through finger-tapping and circle-drawing tasks, respectively. There is a lively debate concerning the possibility of coexistence of the two modes of timing for the execution of a single task. If one considers emergent timing as simply an absence of explicit representation of a time interval, then by definition, the two modes of timing cannot coexist. However, if one considers that emergent timing engages control of another motor parameter, e.g., a control of movement through space rather than time, then the possibility of coexistence needs to be reassessed. In the present study, we designed a hybrid of finger-tapping and circle-drawing tasks for which the demands for space and time control were present at the same time in order to reassess the coexistence hypothesis. Seventy-eight participants performed a spatial-tapping task in which finger taps were to be produced in synchrony with a regular metronome to 6 visual targets presented around a virtual circle. The metronome set ten distinct tempi (1100–300 ms). Using autocorrelation analyses on timing variables, we show that motor timing was event-based at slow tempi and emergent at faster tempi. Through an analysis of the trajectory, we confirm that an increase in the spatial control of movement took place congruently with a switch from event-based to emergent timing modes. At these fast tempi, timing and spatial errors were correlated but only at the specific target location for which a dynamical anchor point was revealed. Hence, we conclude that the coding of emergent timing has a spatial nature from which emerge timing regularities. This spatio-temporal strategy insures the performance of sequential motor actions when cognitive effort is too high for the use of pure event-based timing strategies.     

The effect of bilingualism on lexical learning and memory across two language modalities: some evidence for a domain-specific,but not general,advantage

ABSTRACT

The present study was conducted to replicate bilingual advantages in short-term memory for language-like material and word learning in young adults and extend this research to the sign domain, ultimately with the goal of investigating the domain specificity of bilingual advantages in cognition. Data from 112 monolingual hearing non-signers and 78 bilingual hearing non-signers were analysed for this study. Participants completed a battery of tasks assessing sign and word learning, short-term memory, working memory capacity, intelligence, and a language and demographic questionnaire. Overall, the results of this study suggested a bilingual advantage in memory for speech-like material – no other advantage (or disadvantage) was found. Results are discussed within the context of recent large-scale experimental and meta-analytic studies that have failed to find bilingual advantages in domain-general abilities such as attention control and working memory capacity in young adults.