Robust tests of equivalence for k independent groups

A common question of interest to researchers in psychology is the equivalence of two or more groups. Failure to reject the null hypothesis of traditional hypothesis tests such as the ANOVA F‐test (i.e., H₀: μ₁ = … = μ_k) does not imply the equivalence of the population means. Researchers interested in determining the equivalence of k independent groups should apply a one‐way test of equivalence (e.g., Wellek, 2003). The goals of this study were to investigate the robustness of the one‐way Wellek test of equivalence to violations of homogeneity of variance assumption, and compare the Type I error rates and power of the Wellek test with a heteroscedastic version which was based on the logic of the one‐way Welch (1951) F‐test. The results indicate that the proposed Wellek–Welch test was insensitive to violations of the homogeneity of variance assumption, whereas the original Wellek test was not appropriate when the population variances were not equal.     

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.     

Four improved statistics for contrasting means by correcting skewness and kurtosis

This paper is concerned with removing the influence of non‐normality in the classical t‐statistic for contrasting means. Using higher‐order expansion to quantify the effect of non‐normality, four corrected statistics are provided. Two aim to correct the mean bias and two to correct the overall distribution. The classical t‐statistic is also robust against non‐normality when the observed variables satisfy certain structures. A special case is when the marginal distributions of the contrast are independent and identically distributed.     

A Method for the Quantitative Recording of Eye Movements

Many writers have implicitly or explicitly stated that nonparametric tests are free from the assumption of homogeneity of variance. Nonparametric tests for difference in central tendencies generally involve the assumption of homogeneity of variance. The assumption of homogeneity of variance for the t test and for nonparametric tests serves the same purpose: it allows the user to draw more specific inferences when the null hypothesis is rejected.     

Effect of non-normality on test statistics for one-way independent groups designs

The data obtained from one‐way independent groups designs is typically non‐normal in form and rarely equally variable across treatment populations (i.e. population variances are heterogeneous). Consequently, the classical test statistic that is used to assess statistical significance (i.e. the analysis of variance F test) typically provides invalid results (e.g. too many Type I errors, reduced power). For this reason, there has been considerable interest in finding a test statistic that is appropriate under conditions of non‐normality and variance heterogeneity. Previously recommended procedures for analysing such data include the James test, the Welch test applied either to the usual least squares estimators of central tendency and variability, or the Welch test with robust estimators (i.e. trimmed means and Winsorized variances). A new statistic proposed by Krishnamoorthy, Lu, and Mathew, intended to deal with heterogeneous variances, though not non‐normality, uses a parametric bootstrap procedure. In their investigation of the parametric bootstrap test, the authors examined its operating characteristics under limited conditions and did not compare it to the Welch test based on robust estimators. Thus, we investigated how the parametric bootstrap procedure and a modified parametric bootstrap procedure based on trimmed means perform relative to previously recommended procedures when data are non‐normal and heterogeneous. The results indicated that the tests based on trimmed means offer the best Type I error control and power when variances are unequal and at least some of the distribution shapes are non‐normal.     

Adaptive robust estimation and testing

We examined nine adaptive methods of trimming, that is, methods that empirically determine when data should be trimmed and the amount to be trimmed from the tails of the empirical distribution. Over the 240 empirical values collected for each method investigated, in which we varied the total percentage of data trimmed, sample size, degree of variance heterogeneity, pairing of variances and group sizes, and population shape, one method resulted in exceptionally good control of Type I errors. However, under less extreme cases of non‐normality and variance heterogeneity a number of methods exhibited reasonably good Type I error control. With regard to the power to detect non‐null treatment effects, we found that the choice among the methods depended on the degree of non‐normality and variance heterogeneity. Recommendations are offered.     

From understanding others' needs to prosocial action: Motor and social abilities promote infants' helping

In the present study, we test the main hypothesis that infants' understanding of others' needs translates into helping behavior, when critical motor and social competencies have emerged, early in the second year. We assessed the understanding of others' needs in an eye‐tracking paradigm and the helping behavior of 10‐ (n = 41) and 16‐month‐olds (n = 37). Furthermore, we assessed the motor and social abilities of 16‐month‐olds. Critically, while infants understood others' needs already at 10 months, fine motor and social interaction skills moderated the link between infants' prosocial understanding and helping behavior at 16 months. This provides first evidence that infants' helping behavior relates to their understanding of others' needs. Furthermore, we found that fine motor, gross motor, and social interaction skills predicted early helping behavior by themselves. These findings highlight that the emergence of infants' helping behavior is the result of a developmental system that includes infants' understanding of others' needs and also their motor and social competencies. The link between infants' understanding of others' needs and their early helpful actions provide further support for the prosocial nature of early helping behavior.     

The impact of sample non‐normality on ANOVA and alternative methods

In this journal, Zimmerman (2004, 2011) has discussed preliminary tests that researchers often use to choose an appropriate method for comparing locations when the assumption of normality is doubtful. The conceptual problem with this approach is that such a two‐stage process makes both the power and the significance of the entire procedure uncertain, as type I and type II errors are possible at both stages. A type I error at the first stage, for example, will obviously increase the probability of a type II error at the second stage. Based on the idea of Schmider et al. (2010) , which proposes that simulated sets of sample data be ranked with respect to their degree of normality, this paper investigates the relationship between population non‐normality and sample non‐normality with respect to the performance of the ANOVA, Brown–Forsythe test, Welch test, and Kruskal–Wallis test when used with different distributions, sample sizes, and effect sizes. The overall conclusion is that the Kruskal–Wallis test is considerably less sensitive to the degree of sample normality when populations are distinctly non‐normal and should therefore be the primary tool used to compare locations when it is known that populations are not at least approximately normal.     

How satisfied are you with your job? Estimating the reliability of scores on a single‐item job satisfaction measure

Estimating the reliability of scores on single‐item measures can be difficult because commonly used internal consistency estimates of reliability cannot be calculated. When longitudinal data is available, statistical models can be used to decompose the variability in the latent variable at each wave into trait versus state variance. Then, reliability can be estimated as a ratio of the sum of the trait variance that is captured in repeated assessments over the total variance. The current study used latent trait‐state‐error models on a nine‐year longitudinal data (N = 5,003) to estimate the test–retest reliability of scores on a single‐item measure of job satisfaction. Results showed that job satisfaction scores were somewhat unreliable (r_xx = .49–.59) and amenable to change.     

Comparing Job Applicants to Non‐applicants Using an Item‐level Bifactor Model on the HEXACO Personality Inventory

The present study evaluated the ability of item‐level bifactor models (a) to provide an alternative explanation to current theories of higher order factors of personality and (b) to explain socially desirable responding in both job applicant and non‐applicant contexts. Participants (46% male; mean age = 42 years, SD = 11) completed the 200‐item HEXACO Personality Inventory‐Revised either as part of a job application (n = 1613) or as part of low‐stakes research (n = 1613). A comprehensive set of invariance tests were performed. Applicants scored higher than non‐applicants on honesty‐humility (d = 0.86), extraversion (d = 0.73), agreeableness (d = 1.06), and conscientiousness (d = 0.77). The bifactor model provided improved model fit relative to a standard correlated factor model, and loadings on the evaluative factor of the bifactor model were highly correlated with other indicators of item social desirability. The bifactor model explained approximately two‐thirds of the differences between applicants and non‐applicants. Results suggest that rather than being a higher order construct, the general factor of personality may be caused by an item‐level evaluative process. Results highlight the importance of modelling data at the item‐level. Implications for conceptualizing social desirability, higher order structures in personality, test development, and job applicant faking are discussed. Copyright © 2017 European Association of Personality Psychology     

Imagination links with schizotypal beliefs,not with creativity or learning

Imagination refers to creating mental representations of concepts, ideas, and sensations that are not contemporaneously perceived by the senses. Although it is key to human individuality, research on imagination is scarce. To address this gap, we developed here a new psychometric test to assess individual differences in imagination and explored the role of imagination for learning, creativity, and schizotypal beliefs. In a laboratory‐based (N = 180) and an online study (N = 128), we found that imagination is only weakly associated with learning achievement and creativity, accounting for 2–8% of the variance. By contrast, imagination accounted for 22.5% of the variance in schizotypal beliefs, suggesting overall that imagination may be more indicative of cognitive eccentricities rather than benefit the accumulation of knowledge or production of novel and useful ideas.     

Mother‐child interaction is associated with neurocognitive outcome in extremely low gestational age children

Early mother‐child interaction is one of the factors suggested to have an impact on neurocognitive development of extremely low gestational age (ELGA) children. Our aim was to examine associations of mother‐child interaction with neurocognitive outcome, neurological impairments and neonatal brain injuries in ELGA children. A prospective study of 48 ELGA children, born before 28 gestational weeks (26.3 ± 1.2 weeks, birth weight 876 g ± 194 g), and 16 term controls. Brain MRI was performed at term‐equivalent age. At two years of corrected age, the mother‐child interaction was assessed in a structured play situation using the Erickson Scales and Mutually Responsive Orientation Scales. Neurocognitive outcome was assessed with Griffiths Mental Developmental Scales (GMDS) and Bayley Scales of Infant and Toddler Development ‐ Third Edition (BSID‐III) and with Hempel neurological examination. Among ELGA children, higher quality of dyadic relationship and maternal sensitivity, responsiveness, and supportiveness were associated with positive neurocognitive outcome measured both with GMDS and BSID‐III (adjusted p < 0.05). This association remained after adjusting for mother's educational level. Neurological impairments at two years, white matter or gray matter abnormalities in MRI at term‐equivalent age, and grade III‐IV intraventricular hemorrhage during the neonatal period were not associated with mother‐child interaction. This study emphasizes the importance of the quality of mother‐child interaction after extremely preterm birth for neurocognitive development. Neonatal brain injury and neurological impairments were not associated with worse parent‐child interaction after two years.     

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.     

A note on statistical power in multi‐site randomized trials with multiple treatments at each site

We derive the statistical power functions in multi‐site randomized trials with multiple treatments at each site, using multi‐level modelling. An F statistic is used to test multiple parameters in the multi‐level model instead of the Wald chi square test as suggested in the current literature. The F statistic is shown to be more conservative than the Wald statistic in testing any overall treatment effect among the multiple study conditions. In addition, we improvise an easy way to estimate the non‐centrality parameters for the means comparison t‐tests and the F test, using Helmert contrast coding in the multi‐level model. The variance of treatment means, which is difficult to fathom but necessary for power analysis, is decomposed into intuitive simple effect sizes in the contrast tests. The method is exemplified by a multi‐site evaluation study of the behavioural interventions for cannabis dependence.     

A note on consistency of non-parametric rank tests and related rank transformations

The extent to which rank transformations result in the same statistical decisions as their non‐parametric counterparts is investigated. Simulations are presented using the Wilcoxon–Mann–Whitney test, the Wilcoxon signed‐rank test and the Kruskal–Wallis test, together with the rank transformations and t and F tests corresponding to each of those non‐parametric methods. In addition to Type I errors and power over all simulations, the study examines the consistency of the outcomes of the two methods on each individual sample. The results show how acceptance or rejection of the null hypothesis and differences in p‐values of the test statistics depend in a regular and predictable way on sample size, significance level, and differences between means, for normal and various non‐normal distributions.     

A Two Factor ANOVA-like Test for Correlated Correlations: CORANOVA

Testing homogeneity of correlations with Fisher's Z is inappropriate when correlations are themselves correlated. Suppose measurements of brain activation and performance are taken before and during a verbal memory task. Of interest are changes in activity gradients in specific regions, R₁, R₂, R₃, and performance, V. The "correlated correlations" of interest ρ_V,R1 , ρ_V,R2 , and ρ_V,R3 , have a single variable, V, in common. We wish to compare these correlations between males and females, across regions, and to assess an interaction of the correlation. Fisher's Z can compare pairs of correlations, and Olkin and Finn's (1990) method can test homogeneity of correlated correlations across a single within factor (based on asymptotic normality), but no current procedure can test a region by gender (within by between) interaction of correlations. We propose a nonparametric method for testing this interaction and both main effects. The procedure is analogous to two-way ANOVA, but hypotheses test homogeneity of correlations, not means. The null distributions are estimated with permutations, avoiding asymptotic distributional assumptions and enhancing applicability to smaller samples and non-normal data. Simulations demonstrated maintenance of correct level (power = alpha level under the null) for normal and non-normal data and small samples. The Olkin-Finn test had inflated level for non-normal data or small samples. The Fisher's Z had inflated level for non-normal data, but not for small samples. Our method had better efficiency across contrasts and data types and sizes. Applied to correlations between regional laterality of blood flow and verbal memory performance, the method showed sensitivity to a biologically meaningful sex by region interaction in these correlations. A SAS macro for CORANOVA is available.     

Omnibus hypothesis testing in dominance-based ordinal multiple regression

Often quantitative data in the social sciences have only ordinal justification. Problems of interpretation can arise when least squares multiple regression (LSMR) is used with ordinal data. Two ordinal alternatives are discussed, dominance-based ordinal multiple regression (DOMR) and proportional odds multiple regression. The Q2 statistic is introduced for testing the omnibus null hypothesis in DOMR. A simulation study is discussed that examines the actual Type I error rate and power of Q2 in comparison to the LSMR omnibus F test under normality and non-normality. Results suggest that Q2 has favorable sampling properties as long as the sample size-to-predictors ratio is not too small, and Q2 can be a good alternative to the omnibus F test when the response variable is non-normal.     

A robust approach for analyzing unbalanced factorial designs with fixed levels

The goal of this study was to investigate the performance of Hall’s transformation of the Brunner-Dette-Munk (BDM) and Welch-James (WJ) test statistics and Box-Cox’s data transformation in factorial designs when normality and variance homogeneity assumptions were violated separately and jointly. On the basis of unweighted marginal means, we performed a simulation study to explore the operating characteristics of the methods proposed for a variety of distributions with small sample sizes. Monte Carlo simulation results showed that when data were sampled from symmetric distributions, the error rates of the original BDM and WJ tests were scarcely affected by the lack of normality and homogeneity of variance. In contrast, when data were sampled from skewed distributions, the original BDM and WJ rates were not well controlled. Under such circumstances, the results clearly revealed that Hall’s transformation of the BDM and WJ tests provided generally better control of Type I error rates than did the same tests based on Box-Cox’s data transformation. Among all the methods considered in this study, we also found that Hall’s transformation of the BDM test yielded the best control of Type I errors, although it was often less powerful than either of the WJ tests when both approaches reasonably controlled the error rates.     

A little help from my friends: Lack of social interaction predicts greater boredom during the COVID-19 pandemic

Were people bored during the pandemic, and if so why? One possibility is lack of social interaction due to restrictions on social activity intended to slow the spread of communicable disease. In a 3-week daily diary study (n = 438; international community sample) social interaction predicted boredom and its consequences. People felt more bored on days when they engaged in less social interaction than usual (in-person or virtually), largely driven by a lack of meaning. In turn, boredom predicted lower well-being concurrently, and more virtual interaction the next day; people dispositionally higher in trait boredom also reported more solitary (but not partnered) sexual activity. In conclusion, this study suggests that maintaining social connections, even during a pandemic, may be important to mitigate boredom and improve overall well-being.