Power analysis to detect misfit in SEMs with many items: Resolving unrecognized problems, relating old and new approaches, and "matching" power analysis approach to data analysis approach.

It is unappreciated that there are four different approaches to power analysis for detecting misspecification by testing overall fit of structural equation models (SEMs) and, moreover, that common approaches can yield radically diverging results for SEMs with many items (high ). Here we newly relate these four approaches. Analytical power analysis methods using theoretical null and theoretical alternative distributions (Approach 1) have a long history, are widespread, and are often contrasted with "the" Monte Carlo method-which is an oversimplification.

On the Common but Problematic Specification of Conflated Random Slopes in Multilevel Models.

For multilevel models (MLMs) with fixed slopes, it has been widely recognized that a level-1 variable can have distinct between-cluster and within-cluster fixed effects, and that failing to disaggregate these effects yields a conflated, uninterpretable fixed effect. For MLMs with random slopes, however, we clarify that two different types of slope conflation can occur: that of the fixed component (termed fixed conflation) and that of the random component (termed random conflation). The latter is rarely recognized and not well understood.

R-squared Measures for Multilevel Models with Three or More Levels.

Applications of multilevel models (MLMs) with three or more levels have increased alongside expanding software capability and dataset availability. Though researchers often express interest in R-squared measures as effect sizes for MLMs, R-squareds previously proposed for MLMs with three or more levels cover a limited subset of choices for how to quantify explained variance in these models. Additionally, analytic relationships between total and level-specific versions of MLM R-squared measures have not been clarified, despite such relationships becoming increasingly important to understand when there are more levels.

Closing the Word-Problem Achievement Gap in First Grade: Schema-Based Word-Problem Intervention with Embedded Language Comprehension Instruction.

The main purpose of this study was to test the effects of word-problem intervention, with versus without embedded language comprehension instruction, on at-risk 1 graders' word-problem performance. We also isolated the need for a structured approach to word-problem intervention and tested the efficacy of schema-based instruction at 1 grade. Children (n=391; mean age = 6.

Effect size measures for longitudinal growth analyses: Extending a framework of multilevel model R-squareds to accommodate heteroscedasticity, autocorrelation, nonlinearity, and alternative centering strategies.

Developmental researchers commonly utilize multilevel models (MLMs) to describe and predict individual differences in change over time. In such growth model applications, researchers have been widely encouraged to supplement reporting of statistical significance with measures of effect size, such as R-squareds (R ) that convey variance explained by terms in the model. An integrative framework for computing R-squareds in MLMs with random intercepts and/or slopes was recently introduced by Rights and Sterba and it subsumed pre-existing MLM R-squareds as special cases.

New Recommendations on the Use of R-Squared Differences in Multilevel Model Comparisons.

When comparing multilevel models (MLMs) differing in fixed and/or random effects, researchers have had continuing interest in using R-squared differences to communicate effect size and importance of included terms. However, there has been longstanding confusion regarding which R-squared difference measures should be used for which kind of MLM comparisons. Furthermore, several limitations of recent studies on R-squared differences in MLM have led to misleading or incomplete recommendations for practice.

Problems with Rationales for Parceling that Fail to Consider Parcel-Allocation Variability.

In structural equation modeling applications, parcels-averages or sums of subsets of item scores-are often used as indicators of latent constructs. Parcel-allocation variability (PAV) is variability in results that arises sample alternative item-to-parcel allocations. PAV can manifest in all results of a parcel-level model (e.

Time of day effects on the relationship between daily sleep and anxiety: An ecological momentary assessment approach.

Previous research has linked sleep disturbance to anxiety. However, evidence for this relation has been inconsistent, largely limited to retrospective reports that do not account for daily variability, and silent on when the association is most pronounced. Thus, the present study utilized ecological momentary assessment (EMA) to examine the effects of daily deviations in total sleep time (TST) and person-average TST on anxiety and whether these effects varied as a function of time of day in a sample of unselected adults (N = 138).

Quantifying explained variance in multilevel models: An integrative framework for defining R-squared measures.

Researchers often mention the utility and need for R-squared measures of explained variance for multilevel models (MLMs). Although this topic has been addressed by methodologists, the MLM R-squared literature suffers from several shortcomings: (a) analytic relationships among existing measures have not been established so measures equivalent in the population have been redeveloped 2 or 3 times; (b) a completely full partitioning of variance has not been used to create measures, leading to gaps in the availability of measures to address key substantive questions; (c) a unifying approach to interpreting and choosing among measures has not been provided, leading to researchers' difficulty with implementation; and (d) software has inconsistently and infrequently incorporated available measures. We address these issues with the following contributions.

Ignoring a Multilevel Structure in Mixture Item Response Models: Impact on Parameter Recovery and Model Selection.

The current study investigated the consequences of ignoring a multilevel structure for a mixture item response model to show when a multilevel mixture item response model is needed. Study 1 focused on examining the consequence of ignoring dependency for within-level latent classes. Simulation conditions that may affect model selection and parameter recovery in the context of a multilevel data structure were manipulated: class-specific ICC, cluster size, and number of clusters.

A framework of R-squared measures for single-level and multilevel regression mixture models.

Psychologists commonly apply regression mixture models in single-level (i.e., unclustered) and multilevel (i.

Explicating the Conditions Under Which Multilevel Multiple Imputation Mitigates Bias Resulting from Random Coefficient-Dependent Missing Longitudinal Data.

Random coefficient-dependent (RCD) missingness is a non-ignorable mechanism through which missing data can arise in longitudinal designs. RCD, for which we cannot test, is a problematic form of missingness that occurs if subject-specific random effects correlate with propensity for missingness or dropout. Particularly when covariate missingness is a problem, investigators typically handle missing longitudinal data by using single-level multiple imputation procedures implemented with long-format data, which ignores within-person dependency entirely, or implemented with wide-format (i.

Anxiety and Depression in Children of Depressed Parents: Dynamics of Change in a Preventive Intervention.

The current study examined effects of a preventive intervention on patterns of change in symptoms of anxiety and depression in a sample of children of depressed parents. Parents with a history of depression (N = 180) and their children (N = 242; 50% female; M = 11.38; 74% Euro-American) enrolled in an intervention to prevent psychopathology in youth.

The relationship between multilevel models and non-parametric multilevel mixture models: Discrete approximation of intraclass correlation, random coefficient distributions, and residual heteroscedasticity.

Multilevel data structures are common in the social sciences. Often, such nested data are analysed with multilevel models (MLMs) in which heterogeneity between clusters is modelled by continuously distributed random intercepts and/or slopes. Alternatively, the non-parametric multilevel regression mixture model (NPMM) can accommodate the same nested data structures through discrete latent class variation.

Accounting for Parcel-Allocation Variability in Practice: Combining Sources of Uncertainty and Choosing the Number of Allocations.

Item parceling remains widely used under conditions that can lead to parcel-allocation variability in results. Hence, researchers may be interested in quantifying and accounting for parcel-allocation variability within sample. To do so in practice, three key issues need to be addressed.

Effects of parceling on model selection: Parcel-allocation variability in model ranking.

Research interest often lies in comparing structural model specifications implying different relationships among latent factors. In this context parceling is commonly accepted, assuming the item-level measurement structure is well known and, conservatively, assuming items are unidimensional in the population. Under these assumptions, researchers compare competing structural models, each specified using the same parcel-level measurement model.

Partially nested designs in psychotherapy trials: A review of modeling developments.

Objective: Individually-randomized psychotherapy trials are often partially nested. For instance, individuals assigned to a treatment arm may be clustered into therapy groups for purposes of treatment administration, whereas individuals assigned to a wait-list control are unclustered. The past several years have seen rapid expansion and investigation of methods for analyzing partially nested data.

A Latent Transition Analysis Model for Latent-State-Dependent Nonignorable Missingness.

Psychologists often use latent transition analysis (LTA) to investigate state-to-state change in discrete latent constructs involving delinquent or risky behaviors. In this setting, latent-state-dependent nonignorable missingness is a potential concern. For some longitudinal models (e.

Cautions on the Use of Multiple Imputation When Selecting Between Latent Categorical versus Continuous Models for Psychological Constructs.

Clinical psychology researchers studying adolescents and young adults long have been interested in characterizing the latent categorical (classes/profiles) versus continuous (factors) nature of psychological syndromes. To inform this debate, researchers sometimes compare the fit of finite mixture versus factor analysis models to symptom data. This study explains and evaluates how missing data handling methods can impact results of this important model fit comparison.

Handling Missing Covariates in Conditional Mixture Models Under Missing at Random Assumptions.

Mixture modeling is a popular method that accounts for unobserved population heterogeneity using multiple latent classes that differ in response patterns. Psychologists use conditional mixture models to incorporate covariates into between-class and/or within-class regressions. Although psychologists often have missing covariate data, conditional mixtures are currently fit with a conditional likelihood, treating covariates as fixed and fully observed.

Peer victimization (and harsh parenting) as developmental correlates of cognitive reactivity, a diathesis for depression.

Prior research has shown cognitive reactivity to be a diathesis for depression. Seeking evidence for the developmental origins of such diatheses, the current study examined peer victimization and harsh parenting as developmental correlates of cognitive reactivity in 571 children and adolescents (ages 8-13 years). Four major findings emerged.

Structural Equation Modeling Approaches for Analyzing Partially Nested Data.

Study designs involving clustering in some study arms, but not all study arms, are common in clinical treatment-outcome and educational settings. For instance, in a treatment arm, persons may be nested in therapy groups, whereas in a control arm there are no groups. Methodological approaches for handling such partially nested designs have recently been developed in a multilevel modeling framework (MLM-PN) and have proved very useful.

Understanding Linkages Among Mixture Models.

The methodological literature on mixture modeling has rapidly expanded in the past 15 years, and mixture models are increasingly applied in practice. Nonetheless, this literature has historically been diffuse, with different notations, motivations, and parameterizations making mixture models appear disconnected. This pedagogical review facilitates an integrative understanding of mixture models.

Individual influence on model selection.

Researchers in psychology are increasingly using model selection strategies to decide among competing models, rather than evaluating the fit of a given model in isolation. However, such interest in model selection outpaces an awareness that one or a few cases can have disproportionate impact on the model ranking. Though case influence on the fit of a single model in isolation has been often studied, case influence on model selection results is greatly underappreciated in psychology.