Questions: Dimensionality Assessment and Bifactor Models

Question 1

A psychologist administers a depression questionnaire with 20 items and obtains Cronbach's alpha = .91. She concludes the test is unidimensional and that only a total score should be reported. What is the critical problem with this reasoning?

• A. Cronbach's alpha must reach .95 or higher before a total score is justified
• B. Subscale scores should always be reported regardless of the underlying factor structure
• C. High alpha reflects internal consistency, not unidimensionality — strong group factors can produce high alpha even in a clearly multidimensional test
• D. She should have used omega-total instead, which would definitively confirm unidimensionality

Answer: (C)

Alpha is sensitive to the average inter-item correlation and scale length, not to whether items measure one latent trait or several. A test with two or three strong correlated subfactors will yield high alpha while being clearly multidimensional — bifactor analysis might reveal that affective, somatic, and cognitive symptom clusters all carry substantial specific variance beyond a general depression factor. The correct tool for assessing unidimensionality is exploratory or confirmatory factor analysis, not alpha.

Question 2

A bifactor model fitted to an intelligence battery yields omega-hierarchical = .83 and omega-subscale for verbal ability = .61. What does this pattern most clearly support?

• A. The verbal subscale is unreliable and subscale scores should not be reported
• B. Both a total score and verbal subscale scores carry meaningful information and can both be legitimately reported
• C. The general factor explains all meaningful variance; the verbal subscale adds nothing beyond the total score
• D. The bifactor model is misspecified because omega-subscale should always exceed omega-hierarchical

Answer: (B)

Omega-hierarchical of .83 indicates the general factor accounts for substantial reliable variance in the total score, justifying a full-scale score. Omega-subscale of .61 for verbal ability indicates the verbal subscale has meaningful reliable variance *beyond* the general factor — not captured by the total score. Both numbers being substantial is exactly the pattern that justifies reporting both levels. Reporting only the total score in this situation discards clinically relevant information.

Question 3

A bifactor model simultaneously models a general factor (which all items load on) and group-specific factors (which subsets of items load on), allowing the two levels of structure to be estimated at once.

• T. True
• F. False

Answer: True

This is the defining feature of bifactor models. Every item receives two loadings: one on the general factor and one on its group factor. This is what makes bifactor models more informative than either a pure unidimensional model (which ignores the group structure) or a standard correlated-factors model (which doesn't cleanly separate general from specific variance). The simultaneous estimation allows omega coefficients to be computed that attribute variance to each level.

Question 4

A Cronbach's alpha of .90 provides strong evidence that a psychological test measures a single latent trait.

• T. True
• F. False

Answer: False

Alpha measures internal consistency — the degree to which items correlate with each other — not dimensionality. A test can achieve high alpha through several correlated subfactors without being unidimensional. For example, an anxiety measure with distinct cognitive, somatic, and behavioral facets that all correlate with each other could easily yield alpha = .90 while a bifactor model shows significant specific variance in each facet. Unidimensionality requires factor analysis, not alpha.

Question 5

What is the key advantage of omega-hierarchical over Cronbach's alpha when deciding whether to justify reporting a single total score from a multidimensional psychological test?

Model answer: Omega-hierarchical specifically estimates what proportion of total score variance is attributable to the general factor alone, after accounting for group factors. If it is high, the general factor dominates the total score and reporting a single score is defensible. Cronbach's alpha inflates when items form correlated clusters, so it can be high even when group factors carry substantial specific variance — it cannot distinguish between 'one strong general factor' and 'several correlated specific factors.' Omega-hierarchical makes that distinction directly.

The practical decision about what scores to report turns on whether the general factor or the group factors are doing more work. Omega-hierarchical answers that question. A low omega-hierarchical (say .50) alongside high alpha (.90) would signal that alpha is being inflated by group structure, and that subscale scores would be more interpretable than a total. A high omega-hierarchical means the total score reflects mostly the general factor — the variance that subscales share — which is what makes a single total score scientifically defensible.