The effect of multiple indicators on the power to detect inter‐individual differences in change

Hertzog et al. evaluated the statistical power of linear latent growth curve models (LGCMs) to detect individual differences in change, i.e., variances of latent slopes, as a function of sample size, number of longitudinal measurement occasions, and growth curve reliability. We extend this work by investigating the effect of the number of indicators per measurement occasion on power. We analytically demonstrate that the positive effect of multiple indicators on statistical power is inversely related to the relative magnitude of occasion‐specific latent residual variance and is independent of the specific model that constitutes the observed variables, in particular of other parameters in the LGCM. When designing a study, researchers have to consider trade‐offs of costs and benefits of different design features. We demonstrate how knowledge about power equivalent transformations between indicator measurement designs allows researchers to identify the most cost‐efficient research design for detecting parameters of interest. Finally, we integrate different formal results to exhibit the trade‐off between the number of measurement occasions and number of indicators per occasion for constant power in LGCMs.     

On the power of multivariate latent growth curve models to detect correlated change

We evaluated the statistical power of single-indicator latent growth curve models (LGCMs) to detect correlated change between two variables (covariance of slopes) as a function of sample size, number of longitudinal measurement occasions, and reliability (measurement error variance). Power approximations following the method of Satorra and Saris (1985) were used to evaluate the power to detect slope covariances. Even with large samples (N = 500) and several longitudinal occasions (4 or 5), statistical power to detect covariance of slopes was moderate to low unless growth curve reliability at study onset was above .90. Studies using LGCMs may fail to detect slope correlations because of low power rather than a lack of relationship of change between variables. The present findings allow researchers to make more informed design decisions when planning a longitudinal study and aid in interpreting LGCM results regarding correlated interindividual differences in rates of development.     

Equivalence tests for comparing correlation and regression coefficients

Equivalence tests are an alternative to traditional difference‐based tests for demonstrating a lack of association between two variables. While there are several recent studies investigating equivalence tests for comparing means, little research has been conducted on equivalence methods for evaluating the equivalence or similarity of two correlation coefficients or two regression coefficients. The current project proposes novel tests for evaluating the equivalence of two regression or correlation coefficients derived from the two one‐sided tests (TOST) method (Schuirmann, 1987, J. Pharmacokinet. Biopharm, 15, 657) and an equivalence test by Anderson and Hauck (1983, Stat. Commun., 12, 2663). A simulation study was used to evaluate the performance of these tests and compare them with the common, yet inappropriate, method of assessing equivalence using non‐rejection of the null hypothesis in difference‐based tests. Results demonstrate that equivalence tests have more accurate probabilities of declaring equivalence than difference‐based tests. However, equivalence tests require large sample sizes to ensure adequate power. We recommend the Anderson–Hauck equivalence test over the TOST method for comparing correlation or regression coefficients.     

Bayes factor design analysis: Planning for compelling evidence

A sizeable literature exists on the use of frequentist power analysis in the null-hypothesis significance testing (NHST) paradigm to facilitate the design of informative experiments. In contrast, there is almost no literature that discusses the design of experiments when Bayes factors (BFs) are used as a measure of evidence. Here we explore Bayes Factor Design Analysis (BFDA) as a useful tool to design studies for maximum efficiency and informativeness. We elaborate on three possible BF designs, (a) a fixed-n design, (b) an open-ended Sequential Bayes Factor (SBF) design, where researchers can test after each participant and can stop data collection whenever there is strong evidence for either \(\mathcal {H}_{1}\) or \(\mathcal {H}_{0}\), and (c) a modified SBF design that defines a maximal sample size where data collection is stopped regardless of the current state of evidence. We demonstrate how the properties of each design (i.e., expected strength of evidence, expected sample size, expected probability of misleading evidence, expected probability of weak evidence) can be evaluated using Monte Carlo simulations and equip researchers with the necessary information to compute their own Bayesian design analyses.     

Consistent estimation in the rasch model based on nonparametric margins

Consider the class of two parameter marginal logistic (Rasch) models, for a test ofm True-False items, where the latent ability is assumed to be bounded. Using results of Karlin and Studen, we show that this class of nonparametric marginal logistic (NML) models is equivalent to the class of marginal logistic models where the latent ability assumes at most (m + 2)/2 values. This equivalence has two implications. First, estimation for the NML model is accomplished by estimating the parameters of a discrete marginal logistic model. Second, consistency for the maximum likelihood estimates of the NML model can be shown (whenm is odd) using the results of Kiefer and Wolfowitz. An example is presented which demonstrates the estimation strategy and contrasts the NML model with a normal marginal logistic model.This research was supported by NIMH traning grant, 2 T32 MH 15758-06 and by ONR contract N00014-84-K-0588. The author would like to thank Diane Lambert, John Rolph, and Stephen Fienberg for their assistance. Also, the comments of the referees helped to substantially improve the final version of this paper.     

Generalization of equivalence relations from photos to objects by preschool children

Generalized equivalence classes are stimulus classes that consist of equivalent stimuli and other physically similar class‐member stimuli. The present study evaluated whether preschool children would form equivalence classes among photos of abstract objects (2D) and show equivalence generalization to the corresponding objects (3D), printed photos (2D stimuli), and to black‐and‐white drawn pictures (2D stimuli). Six typically developing children were taught arbitrary relations to establish three 3‐member equivalence classes with 2D stimuli presented on a computer screen. AB‐AC baseline relations (for half of the participants) and AB‐BC relations (for the other half) were taught using a multiple‐probe design to assess taught and tested relations. After class formation, three types of generalization probes were conducted: generalization to 3D stimuli, generalization between 2D (printed photos) and 3D stimuli, and generalization to drawn pictures (2D). All of the participants formed the equivalence classes. Two participants met the criterion for all three generalization probe types. Two participants presented mixed results across tests, and two participants did not exhibit equivalence generalization. The results demonstrated equivalence generalization from 2D to 3D stimuli in preschool children, although the variability across participants suggests that such generalization cannot be assumed a priori.     

Examining Factor Score Distributions to Determine the Nature of Latent Spaces

Similarities between latent class models with K classes and linear factor models with K ? 1 factors are investigated. Specifically, the mathematical equivalence between the covariance structure of the two models is discussed, and a Monte Carlo simulation is performed using generated data that represents both latent factors and latent classes with known amounts of overlap. It is shown that, under certain conditions, the distribution of factor scores can be related to the continuity of the latent space via tests of multimodality as suggested by McDonald (1967) McDonald, R. P. 1967. Non-linear factor analysis. Psychometric Monographs, 15 Psychometric Society [Google Scholar].     

An investigation of the psychometric evidence of CBM-R passage equivalence: Utility of readability statistics and equating for alternate forms

Claims have been made that grade appropriate curriculum-based measurement of reading (CBM-R) passages are of comparable difficulty and can be used interchangeably to monitor student progress. Empirical evidence to support claims of equivalence has been lacking. This research investigated the basis for making claims of equivalence. The use of readability statistics to justify passage equivalence was found to be lacking. Using a general measurement model for congeneric tests, CBM-R passages were found to measure a single latent factor with a high degree of reliability. However, evidence indicated that the raw scores, words read correctly per minute, across passages did not provide equivalent measurements. Statistical equating was investigated as an approach to overcome the lack of equivalence with promising results.     

Constructive Logic with Strong Negation is a Substructural Logic. II

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL _ew of the substructural logic FL _ew . The main result of Part I of this series [41] shows that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL _ew (namely, a certain variety of FL _ew -algebras) are term equivalent. In this paper, the term equivalence result of Part I [41] is lifted to the setting of deductive systems to establish the definitional equivalence of the logics N and NFL _ew . It follows from the definitional equivalence of these systems that constructive logic with strong negation is a substructural logic. Presented by Heinrich Wansing     

The Role of Response Effort and Motion Study in Functionally Equivalent Task Designs and Alternatives

We evaluated the effects of functionally equivalent task designs and alternatives, as validated by motion study procedures, on dependent variables (nonadaptive responses, use of alternative, attempts at task, and completed attempts at task) relevant to performing a selected task. First, we evaluated the effects of functionally equivalent task designs on the dependent variables. Second, we evaluated the effects of an efficient functionally equivalent alternative on the variables. Third, we compared the effects of the efficient functionally equivalent alternative with a less efficient functionally equivalent alternative on the same variables. The results showed that the inefficient functionally equivalent task design occasioned higher rates of nonadaptive responses than the efficient functionally equivalent task design. The results also showed that the functionally equivalent task designs and alternatives competed within and across response classes to reduce nonadaptive responses. Mixed results were obtained in comparing the effects of the efficient versus the less efficient functionally equivalent alternatives. We provide evidence for extending the current concept of functional equivalence to include task design responses as well as alternative responses in functional equivalence training.     

Measurement Equivalence/Invariance of the Abusive Supervision Measure Across Workers From Taiwan and the United States

Growing international research interest in negative-leadership behaviors prompts the need to examine whether measures of ineffective leadership developed in the United States are equivalent across countries outside the United States. B. J. Tepper's (2000) abusive supervision measure has been used widely inside and outside the United States and merits research attention on its construct equivalence across different cultural settings. The authors conducted a series of multigroup confirmatory factor analyses to investigate the measurement equivalence of this measure across Taiwan (N = 256) and the United States (N = 389). Configural invariance was established, suggesting that both U.S. and Taiwanese samples perceive abusive supervision as a single-factor concept. Furthermore, the establishment of partial metric invariance and partial scalar invariance suggests that the abusive supervision measure is applicable to crosscultural comparisons in latent means, construct variance, construct covariances, and unstandardized path coefficients with the caution that workers from different cultures calibrate their responses differently when answering some items.     

Higher Order Factor Structure of the WISC-IV in a Clinical Neuropsychological Sample

A confirmatory factor analysis was conducted examining the higher order factor structure of the WISC-IV scores for 344 children who participated in neuropsychological evaluations at a large children's hospital. The WISC-IV factor structure mirrored that of the standardization sample. The second order general intelligence factor (g) accounted for the largest proportion of variance in the first-order latent factors and in the individual subtests, especially for the working memory index. The first-order processing speed factor exhibited the most unique variance beyond the influence of g. The results suggest that clinicians should not ignore the contribution of g when interpreting the first-order factors.     

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.     

The cost of closure: logical realism,anti-exceptionalism,and theoretical equivalence

Philosophers of science often assume that logically equivalent theories are theoretically equivalent. I argue that two theses, anti-exceptionalism about logic (which says, roughly, that logic is not a priori, that it is revisable, and that it is not special or set apart from other human inquiry) and logical realism (which says, roughly, that differences in logic reflect genuine metaphysical differences in the world) make trouble for this commitment, as well as a closely related commitment to theories being closed under logical consequence. I provide three arguments. The first two show that anti-exceptionalism about logic provides an epistemic challenge to both the closure and the equivalence claims; the third shows that logical realism provides a metaphysical challenge to both the closure and the equivalence claims. Along the way, I show that there are important methodological upshots for metaphysicians and philosophers of logic, in particular, lessons about certain conceptions of naturalism as constraining the possibilities for metaphysics and the philosophy of logic.

     

Internet versus paper‐and‐pencil survey methods in psychological experiments: Equivalence testing of participant responses to health‐related messages

Despite experiments being increasingly conducted over the Internet, few studies have tested whether such experiments yield data equivalent to traditional methods' data. In the current study, data obtained via a traditional sampling method of undergraduate psychology students completing a paper‐and‐pencil survey (N = 107) were compared with data obtained from an Internet‐administered survey to a sample of self‐selected Internet‐users (N = 94). The data examined were from a previous study that had examined the persuasiveness of health‐related messages. To the extent that Internet data would be based on a sample at least as representative as data derived from a traditional student sample, it was expected that the two methodologies would yield equivalent data. Using formal tests of equivalence on persuasion outcomes, hypotheses of equivalence were generally supported. Additionally, the Internet sample was more diverse demographically than the student sample, identifying Internet samples as a valid alternative for future experimental research.     

A direct proof of the equivalence of free categorial grammars and simple phrase structure grammars

In [2], Bar-Hillel, Gaifman, and Shamir prove that the simple phrase structure grammars (SPGs) defined by Chomsky are equivalent in a certain sense to Bar-Hillel's bidirectional categorial grammars (BCGs). On the other hand, Cohen [3] proves the equivalence of the latter ones to what the calls free categorial grammars (FCGs). They are closely related to Lambek's syntactic calculus which, in turn, is based on the idea due to Ajdukiewicz [1]. For the reasons which will be discussed in the last section, Cohen's proof seems to be at least incomplete. This paper yields a direct proof of the equivalence ofFCGs andSPGs.Allatum est die S Marlii 1976     

Executive function and psychosocial adjustment in healthy children and adolescents: A latent variable modelling investigation

The objective of this study was to establish latent executive function (EF) and psychosocial adjustment factor structure, to examine associations between EF and psychosocial adjustment, and to explore potential development differences in EF-psychosocial adjustment associations in healthy children and adolescents. Using data from the multisite National Institutes of Health (NIH) magnetic resonance imaging (MRI) Study of Normal Brain Development, the current investigation examined latent associations between theoretically and empirically derived EF factors and emotional and behavioral adjustment measures in a large, nationally representative sample of children and adolescents (7–18 years old; N = 352). Confirmatory factor analysis (CFA) was the primary method of data analysis. CFA results revealed that, in the whole sample, the proposed five-factor model (Working Memory, Shifting, Verbal Fluency, Externalizing, and Internalizing) provided a close fit to the data, χ²(66) = 114.48, p < .001; RMSEA = .046; NNFI = .973; CFI = .980. Significant negative associations were demonstrated between Externalizing and both Working Memory and Verbal Fluency (p < .01) factors. A series of increasingly restrictive tests led to the rejection of the hypothesis of invariance, thereby precluding formal statistical examination of age-related differences in latent EF-psychosocial adjustment associations. Findings indicate that childhood EF skills are best conceptualized as a constellation of interconnected yet distinguishable cognitive self-regulatory skills. Individual differences in certain domains of EF track meaningfully and in expected directions with emotional and behavioral adjustment indices. Externalizing behaviors, in particular, are associated with latent Working Memory and Verbal Fluency factors.     

Multigroup confirmatory factor analysis and structural invariance with age of the Behavior Rating Inventory of Executive Function (BRIEF)—French version

The parent and teacher forms of the French version of the Behavioral Rating Inventory of Executive Function (BRIEF) were used to evaluate executive function in everyday life in a large sample of healthy children (N = 951) aged between 5 and 18.

Several psychometric methods were applied, with a view to providing clinicians with tools for score interpretation. The parent and teacher forms of the BRIEF were acceptably reliable. Demographic variables (such as age and gender) were found to influence the BRIEF scores. Confirmatory factor analysis was then used to test five competing models of the BRIEF's latent structure. Two of these models (a three-factor model and a two-factor model, both based on a nine-scale structure) had a good fit. However, structural invariance with age was only obtained with the two-factor model.

The French version of the BRIEF provides a useful measure of everyday executive function and can be recommended for use in clinical research and practice.     

Meaning in life of young and elderly adults: An examination of the factorial validity and invariance of the Life Regard Index

The main question of the present investigation was whether the Life Regard Index (LRI) is an adequate instrument to study possible differences between young and elderly adults with regard to experienced meaning in life. Participants in this study were a group of 206 young adults (M= 17.8 years, 49.0% female) and a group of 373 elderly adults (M = 65.90 years, 58.8% female). Respondents completed a Dutch paper-and-pencil version of the LRI. The LISREL confirmatory factor analytic model was used to test for the equivalence of measurement and of structure of the instrument. Results show that in both age groups the items of the LRI were found to be distributed to the two a priori dimensions of meaning in life, Framework and Fulfillment. Only the factor loadings of the Framework items were invariant across both groups. Neither the error of measurement, nor the structure of the underlying concept were equivalent for young and elderly adults. Young adults were found to experience less meaning in life than the elderly.     

On Generalization of Definitional Equivalence to Non-Disjoint Languages

For simplicity, most of the literature introduces the concept of definitional equivalence only for disjoint languages. In a recent paper, Barrett and Halvorson introduce a straightforward generalization to non-disjoint languages and they show that their generalization is not equivalent to intertranslatability in general. In this paper, we show that their generalization is not transitive and hence it is not an equivalence relation. Then we introduce another formalization of definitional equivalence due to Andréka and Németi which is equivalent to the Barrett–Halvorson generalization in the case of disjoint languages. We show that the Andréka–Németi generalization is the smallest equivalence relation containing the Barrett–Halvorson generalization and it is equivalent to intertranslatability, which is another definition for definitional equivalence, even for non-disjoint languages. Finally, we investigate which definitions for definitional equivalences remain equivalent when we generalize them for theories in non-disjoint languages.