Testing and modelling non‐normality within the one‐factor model

Maximum likelihood estimation in the one‐factor model is based on the assumption of multivariate normality for the observed data. This general distributional assumption implies three specific assumptions for the parameters in the one‐factor model: the common factor has a normal distribution; the residuals are homoscedastic; and the factor loadings do not vary across the common factor scale. When any of these assumptions is violated, non‐normality arises in the observed data. In this paper, a model is presented based on marginal maximum likelihood to enable explicit tests of these assumptions. In addition, the model is suitable to incorporate the detected violations, to enable statistical modelling of these effects. Two simulation studies are reported in which the viability of the model is investigated. Finally, the model is applied to IQ data to demonstrate its practical utility as a means to investigate ability differentiation.     

A short note on the maximal point‐biserial correlation under non‐normality

The aim of this paper is to derive the maximal point‐biserial correlation under non‐normality. Several widely used non‐normal distributions are considered, namely the uniform distribution, t‐distribution, exponential distribution, and a mixture of two normal distributions. Results show that the maximal point‐biserial correlation, depending on the non‐normal continuous variable underlying the binary manifest variable, may not be a function of p (the probability that the dichotomous variable takes the value 1), can be symmetric or non‐symmetric around p = .5, and may still lie in the range from ?1.0 to 1.0. Therefore researchers should exercise caution when they interpret their sample point‐biserial correlation coefficients based on popular beliefs that the maximal point‐biserial correlation is always smaller than 1, and that the size of the correlation is always further restricted as p deviates from .5.     

Improved statistics for contrasting means of two samples under non‐normality

This paper presents the asymptotic expansions of the distributions of the two‐sample t‐statistic and the Welch statistic, for testing the equality of the means of two independent populations under non‐normality. Unlike other approaches, we obtain the null distributions in terms of the distribution and density functions of the standard normal variable up to n^?1, where n is the pooled sample size. Based on these expansions, monotone transformations are employed to remove the higher‐order cumulant effect. We show that the new statistics can improve the precision of statistical inference to the level of o (n^?1). Numerical studies are carried out to demonstrate the performance of the improved statistics. Some general rules for practitioners are also recommended.     

Asymptotic expansion of the distributions of the estimators in factor analysis under non‐normality

Equations for the Edgeworth expansion of the distributions of the estimators in exploratory factor analysis and structural equation modeling are given. The equations cover the cases of non‐normal data, as well as normal ones with and without known first‐order asymptotic standard errors. When the standard errors are unknown, the distributions of the Studentized statistics are expanded. Methods of constructing confidence intervals of population parameters with arbitrary asymptotic confidence coefficients are given using the Cornish‐Fisher expansion. Simulations are performed to see the usefulness of the asymptotic expansions in exploratory factor analysis with rotated solutions and confirmatory factor analysis. The results show that asymptotic expansion gives substantial improvement of approximation to the exact distribution constructed by simulations over the usual normal approximation.     

The impact of experienced versus non‐experienced suggestions on children's recall of repeated events

Three experiments were conducted to test the proposition that children's suggestibility about an occurrence of a repeated event is heightened when an interviewer suggests false details that were experienced in non‐target occurrences of the event as opposed to new details that never occurred. In each experiment, children participated in a repeated event during which specific items varied each time (e.g. the children always got a sticker but the theme of the sticker was different in each occurrence). Separate biasing and memory interviews were then conducted. In Experiment 1, the interviewer merely suggested that the false details might have occurred in the event. In the remaining experiments, the suggested details were clearly linked to the target occurrence with either a contextual or temporal cue. The potential moderating effect of the child's age (Experiment 1) and the retention interval (Experiments 1 and 2) were also examined. Consistent with the initial hypothesis, suggestions about experienced (non‐target) details were more likely to be repeated by the children compared to suggestions about non‐experienced details. In Experiments 2 and 3, experienced suggestions were also more likely to inhibit children's recall of the target occurrence. The relevance and generalizability of these findings to the legal setting are discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

The impact of mothers' trait anxiety on their expectations of approach‐avoidance behaviour in anxious and non‐anxious children

This study investigated estimates of approach and avoidance behaviour in clinically anxious and non‐anxious children, and whether mothers' expectations of their children's avoidance differed as a function of high trait anxiety (HTA) versus low trait anxiety (LTA). Participants were 62 clinically anxious and 60 non‐anxious children aged 7–12 years and their mothers. Estimates of avoidance were obtained using an analogue task in which children and mothers were given threat and pleasant information about two novel animals and were asked to estimate children's avoidance of the threat animal's habitat when the threat animal was present (threat condition) and absent (safe condition) from the habitat and when its presence was uncertain (ambiguous condition). Contrary to expectation, anxious children did not differ from controls in estimates of avoidance in any condition. However, relative to HTA mothers of anxious children and LTA mothers of non‐anxious children, HTA mothers estimated greater approach behaviour by their non‐anxious children in the threat condition. Findings suggest that mothers' expectations of children's approach‐avoidance behaviour is influenced by both maternal and child factors.     

Childhood maltreatment and adult personality disorder symptoms in a non‐clinical sample

It can be argued that the well‐substantiated relationship between childhood maltreatment and adult personality disorder (PD) symptoms may be confounded by comorbid symptoms of depression, anxiety or dysfunctional childhood family environments. Therefore, the current study was designed to test the hypothesis that retrospective reports of childhood maltreatment would still be significantly related to reports of more PD symptoms when statistically controlling for these factors. One hundred and seventy‐eight non‐clinical participants were divided into groups reporting childhood maltreatment (n = 54) or not (n = 124) according to scores on the Childhood Trauma Questionnaire. Participants also completed questionnaires measuring current depression, anxiety, and PD symptoms as well as retrospective reports of their childhood environment. Results showed that individuals reporting childhood maltreatment reported more symptoms of PD than those not reporting childhood maltreatment, even when statistically controlling for depression, anxiety and retrospective reports of dysfunctional family environment. These findings underscore the relevance and independent contribution of childhood maltreatment to the development of PDs, with important implications for further research and clinical practice.     

Gender differences on the Five to Fifteen questionnaire in a non‐referred sample with inattention and hyperactivity‐impulsivity and a clinic‐referred sample with hyperkinetic disorder

Lambek, R., Trillingsgaard, A., Kadesjö, B., Damm, D. & Thomsen, P. H. (2010). Gender differences on the Five to Fifteen questionnaire in a non‐referred sample with inattention and hyperactivity‐impulsivity and a clinic‐referred sample with hyperkinetic disorder. Scandinavian Journal of Psychology 51, 540–447. The aim of the present study was to examine gender differences in children with inattention, hyperactivity, and impulsivity on the Five to Fifteen (FTF) parent questionnaire. First, non‐referred girls (n = 43) and boys (n = 51) with problems of attention and hyperactivity‐impulsivity and then clinic‐referred girls (n = 35) and boys (n = 66) with hyperkinetic disorder (HKD) were compared on the FTF. Results suggested that non‐referred boys were more hyperactive‐impulsive than non‐referred girls, whereas clinic‐referred boys and girl with HKD were more similar than dissimilar on the FTF questionnaire. Secondly, it was examined whether the application of gender mixed norms versus gender specific norms would result in varying proportions of clinic‐referred children with HKD being identified as impaired on the subdomains of the FTF questionnaire. Based on results it was concluded that the use of a gender mixed normative sample may lead to overestimation of impairment in boys with HKD, but the type of sample applied to define impairment on the FTF should depend on the purpose for applying the questionnaire.     

On sample size calculation for 2×2 fixed‐effect ANOVA when variances are unknown and possibly unequal

The factorial 2 × 2 fixed‐effect ANOVA is a procedure used frequently in scientific research to test mean differences between‐subjects in all of the groups. But if the assumption of homogeneity is violated, the test for the row, column, and the interaction effect might be invalid or less powerful. Therefore, for planning research in the case of unknown and possibly unequal variances, it is worth developing a sample size formula to obtain the desired power. This article suggests a simple formula to determine the sample size for 2 × 2 fixed‐effect ANOVA for heterogeneous variances across groups. We use the approximate Welch t test and consider the variance ratio to derive the formula. The sample size determination requires two‐step iterations but the approximate sample sizes needed for the main effect and the interaction effect can be determined separately with the specified power. The present study also provides an example and a SAS program to facilitate the calculation process.     

Repeated measures one‐way ANOVA based on a modified one‐step M‐estimator

Wilcox, Keselman, Muska and Cribbie (2000) found a method for comparing the trimmed means of dependent groups that performed well in simulations, in terms of Type I errors, with a sample size as small as 21. Theory and simulations indicate that little power is lost under normality when using trimmed means rather than untrimmed means, and trimmed means can result in substantially higher power when sampling from a heavy‐tailed distribution. However, trimmed means suffer from two practical concerns described in this paper. Replacing trimmed means with a robust M‐estimator addresses these concerns, but control over the probability of a Type I error can be unsatisfactory when the sample size is small. Methods based on a simple modification of a one‐step M‐estimator that address the problems with trimmed means are examined. Several omnibus tests are compared, one of which performed well in simulations, even with a sample size of 11.     

Personality dimensions and emotional problems: The mediating role of irrational beliefs in Pakistani adult non‐clinical sample

This study presents the first examination of the relation between the Big Five personality traits, irrational beliefs and emotional problems in Pakistan, which is an understudied country in the psychological distress literature. A total of 195 participants (aged 25–60 years), employees at COMSATS University, completed a demographic information sheet, the Big Five Personality Questionnaire, the Irrational Belief Inventory and two subscales of the Brief Symptom Inventory including depression and anxiety. Direct effects of neuroticism, openness and conscientiousness were also observed for depression and anxiety. Structural Equation Modelling demonstrated that irrational beliefs played a significant mediating role in the relationship between neuroticism and anxiety and neuroticism and depression. The results highlight the importance of cognitive beliefs in functionally linking personality traits and emotional problems.     

Relationships between magical thinking,obsessive‐compulsiveness and other forms of anxiety in a sample of non‐clinical children

Despite the obvious phenomenological similarities between magical thinking and obsessive‐compulsiveness, the relationship between them has been the subject of few empirical investigations in samples of children. The present study aimed to examine the relationship between a general epistemic stance towards magical causation and tendencies towards obsessive‐compulsiveness in a non‐clinical sample of schoolchildren. One‐hundred and two children, aged between 5 and 10 years (48 boys and 54 girls), completed questionnaire measures designed to assess magical thinking, obsessive‐compulsiveness, and other forms of anxiety. School teachers completed a measure of strengths and difficulties for each child. General belief in magical causation was correlated with all types of anxiety, not just obsessive‐compulsiveness, with significant correlations shown for boys in the sample, but not girls. General belief in magical causation contributed little to the prediction of obsessive‐compulsiveness beyond general anxiety. In this study, a general epistemic stance towards magical causation did not differentiate obsessive‐compulsiveness from other anxiety dimensions. The findings are considered in the context of developmental theories of magical and scientific causal reasoning.     

Non‐parametric three‐way mixed ANOVA with aligned rank tests

Research problems that require a non‐parametric analysis of multifactor designs with repeated measures arise in the behavioural sciences. There is, however, a lack of available procedures in commonly used statistical packages. In the present study, a generalization of the aligned rank test for the two‐way interaction is proposed for the analysis of the typical sources of variation in a three‐way analysis of variance (ANOVA) with repeated measures. It can be implemented in the usual statistical packages. Its statistical properties are tested by using simulation methods with two sample sizes (n = 30 and n = 10) and three distributions (normal, exponential and double exponential). Results indicate substantial increases in power for non‐normal distributions in comparison with the usual parametric tests. Similar levels of Type I error for both parametric and aligned rank ANOVA were obtained with non‐normal distributions and large sample sizes. Degrees‐of‐freedom adjustments for Type I error control in small samples are proposed. The procedure is applied to a case study with 30 participants per group where it detects gender differences in linguistic abilities in blind children not shown previously by other methods.     

Randomization tests as alternative analysis methods for behavior‐analytic data

Randomization statistics offer alternatives to many of the statistical methods commonly used in behavior analysis and the psychological sciences, more generally. These methods are more flexible than conventional parametric and nonparametric statistical techniques in that they make no assumptions about the underlying distribution of outcome variables, are relatively robust when applied to small‐n data sets, and are generally applicable to between‐groups, within‐subjects, mixed, and single‐case research designs. In the present article, we first will provide a historical overview of randomization methods. Next, we will discuss the properties of randomization statistics that may make them particularly well suited for analysis of behavior‐analytic data. We will introduce readers to the major assumptions that undergird randomization methods, as well as some practical and computational considerations for their application. Finally, we will demonstrate how randomization statistics may be calculated for mixed and single‐case research designs. Throughout, we will direct readers toward resources that they may find useful in developing randomization tests for their own data.     

Beta‐binomial ANOVA for multivariate randomized response data

There is much empirical evidence that randomized response methods improve the cooperation of the respondents when asking sensitive questions. The traditional methods for analysing randomized response data are restricted to univariate data and only allow inferences at the group level due to the randomized response sampling design. Here, a novel beta‐binomial model is proposed for analysing multivariate individual count data observed via a randomized response sampling design. This new model allows for the estimation of individual response probabilities (response rates) for multivariate randomized response data utilizing an empirical Bayes approach. A common beta prior specifies that individuals in a group are tied together and the beta prior parameters are allowed to be cluster‐dependent. A Bayes factor is proposed to test for group differences in response rates. An analysis of a cheating study, where 10 items measure cheating or academic dishonesty, is used to illustrate application of the proposed model.     

The relationship between cognitive emotion regulation strategies and emotional problems: comparison between a clinical and a non‐clinical sample

This study focuses on the relationship between the use of specific cognitive emotion regulation strategies and emotional problems. Two samples were included: 99 adults from a clinical population and 99 matched non‐clinical adults. Data was obtained in both groups on the use of nine cognitive emotion regulation strategies: self‐blame, other‐blame, rumination, catastrophizing, putting into perspective, positive refocusing, positive reappraisal, acceptance, and refocus on planning. Logistic regression analyses show that self‐blame, catastrophizing, and positive reappraisal were, relative to the other strategies, the most important variables for distinguishing between the two samples. While the first two strategies were reported significantly more often by the clinical than by the non‐clinical sample, positive reappraisal was reported significantly more often by the non‐clinical sample. The results suggest that cognitive emotion regulation strategies may be a useful target for prevention and intervention. Copyright © 2002 John Wiley & Sons, Ltd.