[Paper Review] On the dimensional indeterminacy of one-wave factor analysis under causal effects

Tyler J. VanderWeele, C. J. K. Batty|arXiv (Cornell University)|Jan 28, 2020

Mental Health Research Topics4 citations

TL;DR

This paper demonstrates that one-wave factor analysis cannot reliably distinguish between a single-factor model and a two-factor model with causal effects between factors, due to convergence of covariance structures. When one factor causally influences another, the resulting data pattern may be equally well explained by a single factor, undermining the validity of single-factor interpretations in psychological scales without prior causal assumptions.

ABSTRACT

It is shown, with two sets of indicators that separately load on two distinct factors, independent of one another conditional on the past, that if it is the case that at least one of the factors causally affects the other, then, in many settings, the process will converge to a factor model in which a single factor will suffice to capture the covariance structure among the indicators. Factor analysis with one wave of data can then not distinguish between factor models with a single factor versus those with two factors that are causally related. Therefore, unless causal relations between factors can be ruled out a priori, alleged empirical evidence from one-wave factor analysis for a single factor still leaves open the possibilities of a single factor or of two factors that causally affect one another. The implications for interpreting the factor structure of psychological scales, such as self-report scales for anxiety and depression, or for happiness and purpose, are discussed. The results are further illustrated through simulations to gain insight into the practical implications of the results in more realistic settings prior to the convergence of the processes. Some further generalizations to an arbitrary number of underlying factors are noted.

Motivation & Objective

• To investigate the limitations of one-wave factor analysis in identifying true factor structures when causal effects exist between latent factors.
• To demonstrate that causal relationships between factors can produce covariance structures indistinguishable from those of a single factor.
• To challenge the validity of interpreting psychological scales (e.g., anxiety, depression) as measuring a single underlying dimension without ruling out causal interdependencies.
• To illustrate through simulations how convergence to a single-factor solution occurs even when two causally related factors underlie the data.
• To extend the findings to models with more than two latent factors, highlighting broader implications for factor model interpretation.

Proposed method

• The authors model two latent factors that are conditionally independent given past variables, then introduce a causal effect from one factor to the other.
• They derive the resulting covariance structure of observed indicators under such a causal model, showing convergence toward a single-factor solution in the limit.
• The analysis uses structural equation modeling (SEM) frameworks to formalize the relationships between latent factors and observed indicators.
• Simulations are conducted to explore the practical implications before asymptotic convergence, using realistic parameter values.
• Theoretical derivations are extended to models with an arbitrary number of latent factors, showing that causal chains can similarly induce dimensional indeterminacy.
• The paper relies on mathematical derivation and simulation to demonstrate that the same observed covariance matrix can be generated by both a single-factor model and a causally related two-factor model.

Experimental results

Research questions

• RQ1 Can one-wave factor analysis reliably distinguish between a single-factor model and a two-factor model with causal effects between factors?
• RQ2 To what extent do causal relationships between latent factors produce covariance structures that mimic those of a single factor?
• RQ3 How do the results of one-wave factor analysis change when causal effects are present but unaccounted for in the model?
• RQ4 What are the practical implications of this indeterminacy for interpreting psychological scales such as those measuring anxiety, depression, or happiness?
• RQ5 Can the dimensional indeterminacy be resolved without prior knowledge of causal structure?

Key findings

• When one factor causally affects another, the covariance structure among observed indicators converges to that of a single-factor model, making the two models statistically indistinguishable in one-wave data.
• Empirical evidence from one-wave factor analysis for a single factor cannot rule out the possibility of two causally related factors.
• Simulations show that even before full convergence, the data may already exhibit strong single-factor patterns, reducing the power to detect multiple factors.
• The indeterminacy is not limited to two factors; it generalizes to models with an arbitrary number of latent factors connected via causal chains.
• Without a priori assumptions about causal structure, factor analysis with one wave of data cannot uniquely determine the true factor dimensionality.