Bayesian Estimation for Item Factor Analysis Models with Sparse Categorical Indicators

ABSTRACT

Psychometric models for item-level data are broadly useful in psychology. A recurring issue for estimating item factor analysis (IFA) models is low-item endorsement (item sparseness), due to limited sample sizes or extreme items such as rare symptoms or behaviors. In this paper, I demonstrate that under conditions characterized by sparseness, currently available estimation methods, including maximum likelihood (ML), are likely to fail to converge or lead to extreme estimates and low empirical power. Bayesian estimation incorporating prior information is a promising alternative to ML estimation for IFA models with item sparseness. In this article, I use a simulation study to demonstrate that Bayesian estimation incorporating general prior information improves parameter estimate stability, overall variability in estimates, and power for IFA models with sparse, categorical indicators. Importantly, the priors proposed here can be generally applied to many research contexts in psychology, and they do not impact results compared to ML when indicators are not sparse. I then apply this method to examine the relationship between suicide ideation and insomnia in a sample of first-year college students. This provides an important alternative for researchers who may need to model items with sparse endorsement.

KEYWORDS:

• Bayesian estimation
• item factor analysis
• sparse categorical indicators

Article information

Conflict of interest disclosures: The author signed a form for the disclosure of potential conflicts of interest. The author reported no financial or other conflicts of interest in relation to the work described.

Ethical principles: The author affirms having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respecting the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.

Funding: This work was partially supported by Grant F31DA035523 from the National Institute on Drug Abuse. This research was supported in part by the National Institute on Drug Abuse of the National Institutes of Health under award number F31DA035523. The content is solely the responsibility of the author and does not necessarily represent the official views of the National Institutes of Health.

Role of the funders/sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.

Acknowledgments: The author is grateful to Kiara Timpano and Brad Schmidt for providing the data used in the empirical example. The author also wants to thank Patrick Curran, Amy Herring, David Thissen, Kenneth Bollen, Daniel Bauer, and Andrea Hussong for their feedback on this research and two anonymous reviewers for their comments on prior versions of this article.

Notes

1 Specifically, the polychoric correlation coefficients that are used in limited-information estimation approaches are sensitive to low frequencies in bivariate contingency tables (Olsson, 1979; Savalei, 2011).

2 The reverse is also true; for example, very low intercepts could lead to an item that is almost always endorsed and non-endorsement is sparse.

3 The Bayesian framework I focus on is not the only possible approach. Maximum a posteriori (MAP or modal Bayes) estimation pairs prior distributions from Bayesian statistics with a method of estimation similar to ML estimation (Mislevy, 1986). I focus on “full” Bayesian inference and MCMC to describe the posterior distribution for its generality and potential to scale to higher dimensional problems.

4 The labeling of informative versus uninformative for peaked and diffuse priors, respectively, is in widespread use but can be misleading as a flat prior may be highly informative for some purposes, and the level of information in a particular prior varies case by case (see Zhu & Lu, 2004). I avoid referring to flat prior distributions as “uninformative” for this reason.

5 Many flat priors do not have “proper” probability distributions, which means that they do not integrate to 1. For example, a uniform distribution on the real line (U(−∞, ∞)) is improper. The use of improper priors can lead to an improper posterior distribution, invalidating inference; therefore, using improper priors requires care to ensure that the posterior distribution is proper.

6 The influence of informative prior distributions, accurate or inaccurate, for misspecified IFA models is an important topic for future research.

7 Because many concepts of Hamiltonian dynamics and HMC are unfamiliar to non-physicists, a detailed description of HMC is beyond the scope of this paper. I refer interested readers to Neal (2011) and Gelman et al. (2013, pp. 300–308) for more details; however, note that this material is necessarily technical.

8 In this study, 500 replications per condition were sufficient for Monte Carlo convergence. This was checked by examining running average plots for key estimates across replications.

9 This model specification is only locally identified (Bollen & Bauldry, 2010; Loken, 2005), as there is a sign indeterminacy for the factor loadings on one or both factors. For the estimation routines used in Mplus for these models and data, the sign indeterminacy is not an issue and leads to solutions with a majority of positive factor loadings.

10 Specifically, for the baseline condition with no sparse items and moderately informative priors, scaling to a latent factor resulted in small estimated effective sample size (e.g., less than 100) for multiple parameters in approximately 10% of the replications after 10,000 replications (half-burn-in). Scaling by setting the factor variances to 1, however, resulted in higher estimated effective sample size (e.g., minimum 569) and sampling was twice as fast.

11 As noted by a reviewer, the prior variance used here may simply be too large for models with categorical data. Convergence may be achievable under a condition with “flat” priors over a smaller region.

12 For the conditions with sparseness, I separately examined results for all replications, including replications with effective sample size below the cutoff. The results did not differ meaningfully for any outcome.

13 The sampling efficiency for these conditions could be further improved using suggestions in the User's Guide (see Ch. 21, Stan Development Team, 2015), but the syntax becomes less intuitive and general.