Mark G. Haviland

Bifactor Models and Rotations: Exploring the Extent to which Multidimensional Data Yield Univocal Scale Scores

The application of psychological measures often results in item response data that arguably are consistent with both unidimensional (a single common factor) and multidimensional latent structures (typically caused by parcels of items that tap similar content domains). As such, structural ambiguity leads to seemingly endless “confirmatory” factor analytic studies in which the research question is whether scale scores can be interpreted as reflecting variation on a single trait. An alternative to the more commonly observed unidimensional, correlated traits, or second-order representations of a measures latent structure is a bifactor model. Bifactor structures, however, are not well understood in the personality assessment community and thus rarely are applied. To address this, herein we (a) describe issues that arise in conceptualizing and modeling multidimensionality, (b) describe exploratory (including Schmid–Leiman [Schmid & Leiman, 1957] and target bifactor rotations) and confirmatory bifactor modeling, (c) differentiate between bifactor and second-order models, and (d) suggest contexts where bifactor analysis is particularly valuable (e.g., for evaluating the plausibility of subscales, determining the extent to which scores reflect a single variable even when the data are multidimensional, and evaluating the feasibility of applying a unidimensional item response theory (IRT) measurement model). We emphasize that the determination of dimensionality is a related but distinct question from either determining the extent to which scores reflect a single individual difference variable or determining the effect of multidimensionality on IRT item parameter estimates. Indeed, we suggest that in many contexts, multidimensional data can yield interpretable scale scores and be appropriately fitted to unidimensional IRT models.

Scoring and modeling psychological measures in the presence of multidimensionality.

Confirmatory factor analytic studies of psychological measures showing item responses to be multidimensional do not provide sufficient guidance for applied work. Demonstrating that item response data are multifactorial in this way does not necessarily (a) mean that a total scale score is an inadequate indicator of the intended construct, (b) demand creating and scoring subscales, or (c) require specifying a multidimensional measurement model in research using structural equation modeling (SEM). To better inform these important decisions, more fine-grained psychometric analyses are necessary. We describe 3 established, but seldom used, psychometric approaches that address 4 distinct questions: (a) To what degree do total scale scores reflect reliable variation on a single construct? (b) Is the scoring and reporting of subscale scores justified? (c) If justified, how much reliable variance do subscale scores provide after controlling for a general factor? and (d) Can multidimensional item response data be represented by a unidimensional measurement model in SEM, or are multidimensional measurement models (e.g., second-order, bifactor) necessary to achieve unbiased structural coefficients? In the discussion, we provide guidance for applied researchers on how best to interpret the results from applying these methods and review their limitations.

Item Response Theory

Item response theory (IRT) is an increasingly popular approach to the development, evaluation, and administration of psychological measures. We introduce, first, three IRT fundamentals: (a) item response functions, (b) information functions, and (c) invariance. We next illustrate how IRT modeling can improve the quality of psychological measurement. Available evidence suggests that the differences between IRT and traditional psychometric methods are not trivial; IRT applications can improve the precision and validity of psychological research across a wide range of subjects.

Alexithymia in women and men hospitalized for psychoactive substance dependence.

Self-report alexithymia, depression, and anxiety inventories were completed by 204 (84 women and 120 men) psychoactive substance-dependent patients during their first week of hospitalization. Eighty-five of the 204 patients (41.7%) scored in the alexithymic range on the revised Toronto Alexithymia Scale (TAS-20). Womens average alexithymia, depression (Beck Depression Inventory [BDI]), and anxiety (State-Trait Anxiety Inventory-State [STAI-S]) scores were higher than mens average scores. Ethnic (Hispanic whites v non-Hispanic whites) and diagnostic (alcohol v drug v mixed-substance dependence) group differences were not significant. To examine the interrelationships among alexithymia, depression, and anxiety, a causal model confirmed in medical students was tested. The model was reconfirmed; state anxiety predicted depression and alexithymia, and depression predicted alexithymia. These findings are consistent with previous research and compatible with the view that a state of alexithymia can result from severe anxiety and depression.

Evaluating bifactor models: Calculating and interpreting statistical indices.

Bifactor measurement models are increasingly being applied to personality and psychopathology measures (Reise, 2012). In this work, authors generally have emphasized model fit, and their typical conclusion is that a bifactor model provides a superior fit relative to alternative subordinate models. Often unexplored, however, are important statistical indices that can substantially improve the psychometric analysis of a measure. We provide a review of the particularly valuable statistical indices one can derive from bifactor models. They include omega reliability coefficients, factor determinacy, construct reliability, explained common variance, and percentage of uncontaminated correlations. We describe how these indices can be calculated and used to inform: (a) the quality of unit-weighted total and subscale score composites, as well as factor score estimates, and (b) the specification and quality of a measurement model in structural equation modeling. (PsycINFO Database Record

Multidimensionality and Structural Coefficient Bias in Structural Equation Modeling: A Bifactor Perspective

In this study, the authors consider several indices to indicate whether multidimensional data are “unidimensional enough” to fit with a unidimensional measurement model, especially when the goal is to avoid excessive bias in structural parameter estimates. They examine two factor strength indices (the explained common variance and omega hierarchical) and several model fit indices (root mean square error of approximation, comparative fit index, and standardized root mean square residual). These statistics are compared in population correlation matrices determined by known bifactor structures that vary on the (a) relative strength of general and group factor loadings, (b) number of group factors, and (c) number of items or indicators. When fit with a unidimensional measurement model, the degree of structural coefficient bias depends strongly and inversely on explained common variance, but its effects are moderated by the percentage of correlations uncontaminated by multidimensionality, a statistic that rises combinatorially with the number of group factors. When the percentage of uncontaminated correlations is high, structural coefficients are relatively unbiased even when general factor strength is low relative to group factor strength. On the other hand, popular structural equation modeling fit indices such as comparative fit index or standardized root mean square residual routinely reject unidimensional measurement models even in contexts in which the structural coefficient bias is low. In general, such statistics cannot be used to predict the magnitude of structural coefficient bias.

Applying Bifactor Statistical Indices in the Evaluation of Psychological Measures

ABSTRACT The purpose of this study was to apply a set of rarely reported psychometric indices that, nevertheless, are important to consider when evaluating psychological measures. All can be derived from a standardized loading matrix in a confirmatory bifactor model: omega reliability coefficients, factor determinacy, construct replicability, explained common variance, and percentage of uncontaminated correlations. We calculated these indices and extended the findings of 50 recent bifactor model estimation studies published in psychopathology, personality, and assessment journals. These bifactor derived indices (most not presented in the articles) provided a clearer and more complete picture of the psychometric properties of the assessment instruments. We reached 2 firm conclusions. First, although all measures had been tagged “multidimensional,” unit-weighted total scores overwhelmingly reflected variance due to a single latent variable. Second, unit-weighted subscale scores often have ambiguous interpretations because their variance mostly reflects the general, not the specific, trait. Finally, we review the implications of our evaluations and consider the limits of inferences drawn from a bifactor modeling approach.

Item Response Theory and the Measurement of Clinical Change

An instruments sensitivity to detect individual-level change is an important consideration for both psychometric and clinical researchers. In this article, we develop a cognitive problems measure and evaluate its sensitivity to detect change from an item response theory (IRT) perspective. After illustrating assumption checking and model fit assessment, we detail 4 features of IRT modeling: (a) the scale information curve and its relation to the bandwidth of measurement precision, (b) the scale response curve and how it is used to link the latent trait metric with the raw score metric, (c) content-based versus norm-based score referencing, and (d) the level of measurement of the latent trait scale. We conclude that IRT offers an informative, alternative framework for understanding an instruments psychometric properties and recommend that IRT analyses be considered prior to investigations of change, growth, or the effectiveness of clinical interventions.

A California Q-set alexithymia prototype and its relationship to ego-control and ego-resiliency

The primary purposes of the present study were to use the Q-sort method to develop a measure of alexithymia and to locate the construct within a two-dimensional (ego-control and ego-resiliency) model of personality. Thirteen professional judges described the characteristics of the alexithymic personality with the 100-item California Q-set. Scores from the sorts were aggregated to form the Alexithymia Prototype, which had a Spearman-Brown reliability of 0.99. Alexithymic people were described as having difficulties experiencing and expressing emotion, lacking imagination, and being literal, socially conforming, and utilitarian; they lack insight, are humorless, and experience meaninglessness; and anxiety and tension find outlet in bodily symptoms. This description is consistent, for the most part, with modern formulations of the alexithymia construct. In the language of the two-dimensional personality model, alexithymic individuals appear to be overcontrolling and lacking ego-resiliency (i.e., constricted, anxious, rigid, and withdrawn). We, therefore, compared the Alexithymia Prototype with two independently developed prototypes, Overcontrol and Ego-Resiliency. The Q-correlations between alexithymia and overcontrol and between alexithymia and ego-resiliency were 0.45 and -0.70, respectively. Although item analyses confirmed moderate overlap between alexithymia and overcontrol and considerable overlap between alexithymia and lacking ego-resiliency (ego-brittle), item differences suggest that alexithymia, indeed, is a unique personality construct.

MULTIDIMENSIONALITY AND STATE DEPENDENCY OF ALEXITHYMIA IN RECENTLY SOBER ALCOHOLICS

In this study, we a) examined the appropriateness of using a single global score to represent alexithymia and b) constructed a model to examine the relationship between alexithymia and depression in recently sober alcoholics applying for inpatient care. To measure alexithymia, we used the Toronto Alexithymia Scale (TAS); to measure depression, we used the revised Beck Depression Inventory (BDI). Factor analyses identified three alexithymia factors (Feelings, Daydreaming, and External Thinking) and two depression factors (Somatic-Performance and Cognitive-Affective). The three TAS factors were not positively related to each other; the two BDI factors were. We used LISREL software to examine the relationships between the TAS factors and the BDI factors. The only two significant unidirectional coefficients were between the TAS-Feelings factor and the two BDI factors. Our results suggest that in recently sober alcoholics, alexithymia, as measured by the TAS, consists of three independent, unrelated dimensions. Moreover, only the dimension associated with an inability to identify feelings and to distinguish them from bodily sensations is related to depressive symptoms. To determine whether this alexithymia feelings dimension actually is dependent on situational depression and/or anxiety will require confirmation in additional samples, such as primary alexithymics and patients with major depressive disorders.