Please cite or reference this post if you use it for publication purposes: Wiegand, J.P. (October 9, 2015). Bayesian SEM (BSEM) Application and Example [Blog post]. Retrieved from https://justinwiegand.com/blog/?p=331
Sounds good, right?
Bayesian priors allow cross-loadings and residual covariances of SEM’s to vary a small degree (i.e., replace exact zeros with approximate zeros from informative, small-variance priors) and be evaluated (see Asparouhov, Muthén, & Morin, 2015; Muthén, & Asparouhov 2012). The researcher can thereby discover whether 0 cross-loadings likely exist in the “true” population model, given their data, and refine their model accordingly.
I recently had the opportunity to apply this technique to a bi-factor model for a new scale that had previously only been subjected to a traditional CFA. The model had strong theoretical support, but fit indices and some loadings were not supportive of the general factor as theorized. These conditions provide a perfect opportunity to use Bayesian CFA (BCFA) to refine the model for cross-validation.
MPlus 7 was used to specify and test a model where cross-loadings were assigned normally distributed priors with 0 means and variances of .01. The technique is not difficult for those familiar with CFA in MPlus. Working from a typical CFA, the researcher need only:
- Specify “Bayes” as your estimator (shown with option recommendations):
ANALYSIS: ESTIMATOR = BAYES; !Uses two independent MCMC chains
PROCESSORS = 2; !To speed up computations if you have 2 processors
FBITERATIONS=15000; !Minimum number of Markov Chain iterations
- Include all cross-loadings for a factor below the actual loadings (in our case, the specific factors in a bi-factor model).
- Thus for a data set with 10 items (y1-y10), and two factors (f1 and f2) with loadings y1-y3 and y4-y10 respectively, go from:
f1 BY y1-y3;
f2 BY y4-y10;
f1 BY y1-y3
f2 BY y4-y10
- Next, assign labels to the cross-loadings so a Bayesian prior can be specified for each. To do so, expand the syntax with cross-loadings as follows (cross-loadings can be labeled however desired–I am using “xlam” to refer to lamda’s [loadings] of the cross-loadings [“x” for cross]):
f1 BY y1-y3
y4-y10 (f1xlam1-f1xlam7); !Cross-loadings (with assigned labels)
f2 BY y4-y10
y1-y3(f2xlam1-f2xlam3); !Cross-loadings (with assigned labels)
- Finally, model priors are specified by defining all the cross-loadings’ distribution, mean, and variance. A statement is simply added at the end of the “Model” command as follows (for cross-loading priors normally distributed with mean 0 and variances of 0.01):
f1xlam1-f2xlam3~N(0,0.01); !You can list across factors if ordered
Naturally, identification and metric setting needs to be addressed first (for example, MPlus sets the first loading of a factor to one by default, but you may want to free that and instead set the variance of each factor to 1 instead). Yet, hopefully this illustrates the ease at which cross-loadings can be assigned Bayesian priors. For those interested in assigning residual covariances prior distributions the concept is very similar. A full example is given in the Asparouhov, Muthén, and Morin (2015) article.
Now for the beauty (i.e., application) of BCFA. Cross-loadings were assigned a normal prior distribution with mean 0 and variance of 0.01. Were the actual loadings 0 given our data? The prior and posterior distributions for each loading can be viewed to verify this (choose “Plot” > “View Plots” in MPlus) or the confidence intervals for the cross-loadings can be examined in the MPlus output. Below is an example of a cross-loading for an item I analyzed:
First, examine the prior distribution (made up of 15,000 iterations):
This looks right, the mean is essentially 0 and variance (Std Dev squared) 0.01 as specified.
Now, did the distribution change given our data? Check the posterior distribution to find out (or the confidence intervals for the cross-loadings in the MPlus output):
It appears the distribution did change. Given our data, the posterior distribution does not include 0 (for a 95% confidence interval). This suggests that the specific cross-loading should be freed.
This simple illustration is just one way BCFA (and BSEM) adds value for scale and model development. I refer you to the linked articles for more. Let me know if you’ve had a chance to apply the technique and what you learned.
Pervin, L. A. (1968). Performance and satisfaction as a function of individual-environment fit. Psychological Bulletin
(1), 56–68. https://doi.org/10.1037/h0025271
Aafaqi, R., Ansari, M., Ahmad, Z., & Lee, G. (2005). Work Values Fit and Organizational Commitment Among Medical Doctors. Presented at the SIOP 2005 Conference Program. Retrieved from http://www.siop.org/ProgramOnWeb/Default.asp?year=2005
Aarons, G. A., Ehrhart, M. G., Farahnak, L. R., Sklar, M., & Horowitz, J. (2015). Discrepancies in Leader and Follower Ratings of Transformational Leadership: Relationship with Organizational Culture in Mental Health. Administration and Policy in Mental Health and Mental Health Services Research
, 1–12. https://doi.org/10.1007/s10488-015-0672-7
Abosch, K. S. (1995). The promise of broadbanding. Compensation and Benefits Review
(1), 54. Retrieved from http://search.proquest.com/docview/213674624?accountid=14553
Abrahams, N. M., Neumann, I., & Githens, W. H. (1971). Faking Vocational Interests: Simulated Versus Real Life Motivation. Personnel Psychology
(1), 5–12. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=6264085&site=ehost-live
Academic Senate For California Community Colleges. (2008). Promoting Thoughtful Faculty Conversations about Grade Distributions. Academic Senate for California Community Colleges.
Ackerman, P. L., & Beier, M. E. (2003). Intelligence, Personality, and Interests in the Career Choice Process. Journal of Career Assessment
(2), 205–218. https://doi.org/10.1177/1069072703011002006
Ackerman, P. L., & Heggestad, E. D. (1997). Intelligence, personality, and interests: Evidence for overlapping traits. Psychological Bulletin
(2), 219–245. https://doi.org/10.1037/0033-2909.121.2.219
Ackerman, R. A., Donnellan, M. B., & Robins, R. W. (2012). An Item Response Theory Analysis of the Narcissistic Personality Inventory. Journal of Personality Assessment
(2), 141–155. https://doi.org/10.1080/00223891.2011.645934
Ackerman, R. A., Witt, E. A., Donnellan, M. B., Trzesniewski, K. H., Robins, R. W., & Kashy, D. A. (2010). What Does the Narcissistic Personality Inventory Really Measure? Assessment
(1), 67–87. https://doi.org/10.1177/1073191110382845
Adkins, C. L. (1995). Previous Work Experience And Organizational Socialization: A Longitudinal Examination. Academy of Management Journal
(3), 839–862. https://doi.org/10.2307/256748
Adler, P. S., & Kwon, S.-W. (2002). Social capital: Prospects for a new concept. Academy of Management Review
(1), 17–40. Retrieved from http://amr.aom.org/content/27/1/17.short
Aguinis, H., & Gottfredson, R. K. (2010). Best-practice recommendations for estimating interaction effects using moderated multiple regression. Journal of Organizational Behavior
(6), 776–786. https://doi.org/10.1002/job.686
Aguinis, H., & Kraiger, K. (2009). Benefits of Training and Development for Individuals and Teams, Organizations, and Society. Annual Review of Psychology
(1), 451–474. https://doi.org/10.1146/annurev.psych.60.110707.163505
Aguinis, H., & Smith, M. A. (2007). Understanding the impact of test validity and bias on selection errors and adverse impact in human resource selection. Personnel Psychology
(1), 165–199. https://doi.org/10.1111/j.1744-6570.2007.00069.x
Aguinis, H., Dalton, D. R., Bosco, F. A., Pierce, C. A., & Dalton, C. M. (2011). Meta-Analytic Choices and Judgment Calls: Implications for Theory Building and Testing, Obtained Effect Sizes, and Scholarly Impact. Journal of Management
(1), 5–38. https://doi.org/10.1177/0149206310377113
Aguinis, H., O’Boyle, E., Gonzalez-Mulé, E., & Joo, H. (2016). Cumulative Advantage: Conductors and Insulators of Heavy-Tailed Productivity Distributions and Productivity Stars. Personnel Psychology
(1), 3–66. https://doi.org/10.1111/peps.12095
Aguinis, H., Edwards, J. R., & Bradley, K. J. (2017). Improving Our Understanding of Moderation and Mediation in Strategic Management Research. Organizational Research Methods
(4), 665–685. https://doi.org/10.1177/1094428115627498
Ahearne, M., Mathieu, J., & Rapp, A. (2005). To Empower or Not to Empower Your Sales Force? An Empirical Examination of the Influence of Leadership Empowerment Behavior on Customer Satisfaction and Performance. Journal of Applied Psychology
(5), 945–955. https://doi.org/http://dx.doi.org/10.1037/0021-9010.90.5.945
Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Thousand Oaks, CA: SAGE Publications.
Aiken, L. H., Clarke, S. P., Sloane, D. M., Sochalski, J., & Silber, J. H. (2002). Hospital Nurse Staffing and Patient Mortality, Nurse Burnout, and Job Dissatisfaction. JAMA: Journal of the American Medical Association, 288(16), 1987.
Al-Aiban, K. M., & Pearce, J. L. (1993). The Influence of Values on Management Practices. International Studies of Management & Organization
(3), 35. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=9408010026&site=ehost-live
Albert, J. H. (1992). Bayesian Estimation of Normal Ogive Item Response Curves Using Gibbs Sampling. Journal of Educational and Behavioral Statistics
(3), 251–269. https://doi.org/10.3102/10769986017003251
Alessandri, G., & Vecchione, M. (2012). The higher-order factors of the Big Five as predictors of job performance. Personality and Individual Differences
(6), 779–784. https://doi.org/10.1016/j.paid.2012.05.037
Alge, B. J., Ballinger, G. A., Tangirala, S., & Oakley, J. L. (2006). Information privacy in organizations: Empowering creative and extrarole performance. Journal of Applied Psychology
(1), 221–232. https://doi.org/http://dx.doi.org/10.1037/0021-9010.91.1.221
Allen, D. (2008). Retaining Talent: A Guide to Analyzing and Managing Employee Turnover (p. 57). SHRM.
Allen, A. (2016). Curvilinearity of the relationships between selected personality variables and leadership effectiveness
(Ph.D.). Alliant International University, United States -- California. Retrieved from http://search.proquest.com.proxy2.library.illinois.edu/docview/1762748836/abstract/CF26457CD3564377PQ/43
Allen, J., & Robbins, S. (2010). Effects of interest–major congruence, motivation, and academic performance on timely degree attainment. Journal of Counseling Psychology
(1), 23–35. https://doi.org/10.1037/a0017267
Allen, M. J., & Yen, W. M. (1979). Introduction to Measurement Theory. Monterey, CA: Brooks/Cole Publishing Company.
Allen, T. D., Johnson, H.-A. M., Xian Xu, Biga, A., Rodopman, O. B., & Ottinot, R. C. (2008). Mentoring and Protege Narcissistic Entitlement. Journal of Career Development
(4), 385–405. https://doi.org/10.1177/0894845308327735
Allison, C. J., & Cossette, I. (2007). Theory and practice in recruiting women for STEM careers. In Women in Engineering Programs and Advocates Network
. Retrieved from http://journals.psu.edu/wepan/article/view/58487
Allport, F. H. (1933). Institutional behavior; essays toward a re-interpreting of contemporary social organization. Chapel Hill: University of North Carolina Press.
Alter, C. (1990). An Exploratory Study of Conflict and Coordination in Interorganizational Service Delivery System. Academy of Management Journal
(3), 478–502. https://doi.org/10.2307/256577
American Psychiatric Association. (2013). Diagnostic and statistical manual of mental disorders: DSM-5 (Fifth edition). Arlington, VA: American Psychiatric Association.
Ames, D., Maissen, L. B., & Brockner, J. (2012). The role of listening in interpersonal influence. Journal of Research in Personality
(3), 345–349. https://doi.org/10.1016/j.jrp.2012.01.010
Ames, D. R., Rose, P., & Anderson, C. P. (2006). The NPI-16 as a short measure of narcissism. Journal of Research in Personality
(4), 440–450. Retrieved from http://www.sciencedirect.com/science/article/pii/S0092656605000504
Aminzade, R. (1984). Capitalist Industrialization and Patterns of Industrial Protest: a Comparative Urban Study of Nineteenth-Century France. American Sociological Review
(4), 437–453. https://doi.org/10.2307/2095461
Anand, S., Meuser, J. D., Vidyarthi, P. R., & Ekkirala, S. (2013). Leader Fairness and Employee I-Deals: Coworkers as the Enablers. Academy of Management Proceedings
(1), 16962. https://doi.org/10.5465/AMBPP.2013.16962abstract
Anderson, J. C. (1979). Local union participation: A re-examination. Industrial Relations
(1), 18–31. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&db=bth&AN=4551979&site=ehost-live
Anderson, C., Ames, D. R., & Gosling, S. D. (2008). Punishing Hubris: The Perils of Overestimating One’s Status in a Group. Personality and Social Psychology Bulletin
(1), 90–101. https://doi.org/10.1177/0146167207307489
Anderson, N., Ones, D. S., Sinangil, H. K., & Viswesvaran, C. (Eds.). (2001). Handbook of Industrial, Work & Organizational Psychology: Volume 1: Personnel Psychology (Vol. 1). SAGE.
Angrist, J. D., & Pischke, J.-S. (2009). Mostly harmless econometrics: an empiricist’s companion. Princeton: Princeton University Press.
Antonakis, J., Bendahan, S., Jacquart, P., & Lalive, R. (2010). On making causal claims: A review and recommendations. The Leadership Quarterly
(6), 1086–1120. https://doi.org/10.1016/j.leaqua.2010.10.010
Antonioni, D., & Park, H. (2001). The effects of personality similarity on peer ratings of contextual work behaviors. Personnel Psychology
(2), 331–360. Retrieved from http://search.proquest.com.proxy2.library.illinois.edu/docview/220137692/abstract/B17B4BEA38704446PQ/1
Appelbaum, E., Bailey, T., Berg, P., & Kalleberg, A. L. (2000). Manufacturing Advantage: Why High Performance Work Systems Pay Off. Ithaca, NY: Cornell University Press.
Aquino, K., Stewart, M. M., & Reed, A. (2005). How Social Dominance Orientation and Job Status Influence Perceptions of African-American Affirmative Action Beneficiaries. Personnel Psychology
(3), 703–744. https://doi.org/10.1111/j.1744-6570.2005.681.x
Araujo, A., Gottlieb, D., & Moreira, H. (2007). A model of mixed signals with applications to countersignalling. RAND Journal of Economics (Wiley-Blackwell)
(4), 1020–1043. https://doi.org/10.1111/j.0741-6261.2007.00124.x
Arbour, S. (2008). Using person-environment fit and careers stage to examine satisfaction, commitment and work strain in Canadian nurses
(Ph.D.). University of Windsor (Canada), Canada. Retrieved from http://search.proquest.com.proxy2.library.illinois.edu/docview/304557527/abstract/F911A3AFBC76400BPQ/980
Argote, L., McEvily, B., & Reagans, R. (2003). Managing knowledge in organizations: An integrative framework and review of emerging themes. Management Science
(4), 571–582. Retrieved from http://pubsonline.informs.org/doi/abs/10.1287/mnsc.49.4.571.14424
Armstrong, P. I., & Rounds, J. (2010). Integrating Individual Differences in Career Assessment: The Atlas Model of Individual Differences and the Strong Ring. Career Development Quarterly
(2), 143–153. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=EJ912669&site=ehost-live