베이지안 분석에서 사전분포의 이해와 적용 = Understanding and applying prior distributions in Bayesian analyses

다국어 초록 (Multilingual Abstract)

The Bayesian estimation method has recently received a lot of attention in the social sciences. The Bayesian method has a special factor of prior distribution that can reflect researchers’ background knowledge in the estimation process. The specification of the prior distribution affects the overall estimation. Despite prior distribution being the most important factor in Bayesian analysis, there is a lack of methodological research for understanding and appropriately specifying the prior distribution. Therefore, the present study tries to help researchers to apply the prior distribution to their estimation by addressing the importance of the prior distribution and the overall content of the prior specification. First, we explore the method that researchers do not directly specify the prior distribution. This method means selecting the default prior distribution automatically provided by the program, and if you want to use this option, you must know exactly what kind of the default prior distribution is actually provided. For this, we discuss the default priors of frequently used programs, as well as the known problem of the default priors. Second, we address the method that researchers do specify the prior distribution by themseleves. The prior distributions that can be directly specified include noninformative prior distributions and informative prior distributions. Which prior distribution to use is determined by the presence of prior information on parameters. This study deals with the necessity of noninformative prior distributions and the proposed method when specifying them, provides studies that can be referenced when specifying informative prior distributions, and explores criteria that can be referenced for the select of informativeness by synthesizing the criteria across many studies. We provide practical help through data examples applying the methods discussed in the text, and finally discuss the significance and limitations of the present study.

국문 초록 (Abstract)

최근 베이지안 추정 방법이 사회과학 분야에서 많은 관심을 받고 있다. 베이지안 방법에는 연구자의 배경지식을 추정에 반영할 수 있는 사전분포라는 특별한 요소가 있으며, 이를 어떻게 지정하는지가 추정 전반에 영향을 미친다. 사전분포는 베이지안 분석에서 가장 중요한 요소임에도 불구하고, 사전분포를 이해하고 적절히 지정하기 위해 참고할 수 있는 방법론적 연구는 부족한 상황이다. 본 연구는 사전분포의 중요성과 사전분포 지정에 대한 전반적인 내용을 다룬다. 먼저, 연구자가 사전분포를 직접 지정하지 않는, 즉 프로그램이 제공하는 디폴트 사전분포 방법을 알아본다. 자주 사용되는 프로그램들의 디폴트 사전분포를 알아봄과 더불어 디폴트 사전분포의 알려진 문제점도 확인한다. 다음으로는 연구자가 사전분포를 직접 지정하는 방법에 대해 다룬다. 직접 지정할 수 있는 사전분포에는 무정보 사전분포와 정보 사전분포가 있으며, 어떤 사전분포를 이용할지는 모수에 대한 사전 정보의 유무에 따라 결정된다. 무정보 사전분포의 필요성과 이를 지정할 때 참고할 수 있도록 제안된 방법을 다루고, 정보 사전분포를 지정할 때 참고할 수 있는 연구들을 제공하며, 여러 연구의 기준을 종합해 연구자의 정보성 선택에 참고할 수 있는 기준을 탐색한다. 이후 본문에서 논의한 방법들을 적용한 자료 예시를 통해 실질적 도움을 제공하고자 하였으며, 마지막으로 본 연구의 의의와 한계에 대해 논의한다.

참고문헌 (Reference)

1 최유정, "사회부과 완벽주의와 사회불안의 관계에서 대인존재감과 자기수용의 매개효과" 한국콘텐츠학회 20 (20): 259-270, 2020

2 이지은, "대학생의 평가염려 완벽주의와 사회불안의 관계에서 무조건적 자기수용의 매개효과와 성별의 조절효과" 학생생활상담연구소 39 (39): 25-45, 2018

3 최재광, "대학생의 사회부과적 완벽주의가 우울에 미치는 영향: 불확실성에 대한 인내력 부족과 무조건적 자기수용의 매개효과를 중심으로" 중소기업융합학회 8 (8): 183-191, 2018

4 서영임, "대학생의 다차원적 완벽주의가 사회불안 및 우울에 미치는 영향: 무조건적 자기수용의 매개효과를 중심으로" 한국정서행동장애학회 34 (34): 197-215, 2018

5 Merkle, E. C., "blavaan : Bayesian structural equation models via parameter expansion" 85 (85): 1-30, 2015

6 Lunn, D. J., "WinBUGS-a Bayesian modelling framework : concepts, structure, and extensibility" 10 (10): 325-337, 2000

7 Spiegelhalter, D. J, "WinBUGS user manual, Version 1.4" MRC Biostatistics Unit

8 Zondervan-Zwijnenburg, M., "Where do priors come from? Applying guidelines to construct informative priors in small sample research" 14 (14): 305-320, 2017

9 McNeish, D., "Two-Level Dynamic Structural Equation Models with Small Samples" 26 (26): 948-966, 2019

10 Baron, R. M., "The moderator–mediator variable distinction in social psychological research : Conceptual, strategic, and statistical considerations" 51 (51): 1173-1182, 1986

11 Depaoli, S., "The impact of inaccurate"informative"priors for growth parameters in Bayesian growth mixture modeling" 21 (21): 239-252, 2014

12 Hobert, J. P., "The effect of improper priors on Gibbs sampling in hierarchical linear mixed models" 91 (91): 1461-1473, 1996

13 Lee, D. Y., "The Effects of adolescent’s Perfectionistic Self-Presentation on Social anxiety: focusing on the mediation effects of unconditional self-acceptance" The Catholic University of Korea 2020

14 Lunn, D., "The BUGS book: A practical introduction to Bayesian analysis" CRC Press 2012

15 Carpenter, B., "Stan : A probabilistic programming language" 76 (76): 1-32, 2017

16 Hox, J., "Small sample size solutions: A guide for applied researchers and practitioners" Taylor & Francis 255-265, 2020

17 Kim, S. Y., "Single and multiple ability estimation in the SEM framework : A noninformative Bayesian estimation approach" 48 : 563-591, 2013

18 Cortopassi, A. C., "Self-concealment, ego depletion, and drug dependence among young sexual minority men who use substances" 4 (4): 272-281, 2017

19 R Core Team, "R: A language and environment for statistical computing [Computer software manual]" R Foundation for Statistical Computing

20 Steer, R. A., "Psychometric characteristics of the Beck Depression Inventory-II with college students" 30 (30): 128-136, 1997

21 van Erp, S., "Prior sensitivity analysis in default Bayesian structural equation modeling" 23 (23): 363-388, 2018

22 Gelman, A., "Prior distributions for variance parameters in hierarchical models(comment on article by Browne and Draper)" 1 (1): 515-534, 2006

23 Smid, S. C., "Predicting a distal outcome variable from a latent growth model : ML versus Bayesian estimation" 27 (27): 169-191, 2020

24 Miočević, M., "Power in Bayesian mediation analysis for small sample research" 24 (24): 666-683, 2017

25 Kaplan, D., "Oxford handbook of quantitative methods" Oxford University Press 407-437, 2013

26 McNeish, D., "On using Bayesian methods to address small sample problems" 23 (23): 750-773, 2016

27 Polson, N. G., "On the half-Cauchy prior for a global scale parameter" 7 (7): 887-902, 2012

28 Muthen, L. K., "Mplus user’s guide. (8th ed.) [Computer software manual]" Muthen & Muthen 2019

29 Hagger, M. S., "Motivational predictors of students’ participation in out-of-school learning activities and academic attainment in science : An application of the trans-contextual model using Bayesian path analysis" 67 : 232-244, 2018

30 Barnard, J., "Modeling covariance matrices in terms of standard deviations and correlations, with application to shrinkage" 10 (10): 1281-1311, 2000

31 McNeish, D., "Modeling clustered data with very few clusters" 51 (51): 495-518, 2016

32 Fang, J., "Mediation effects in 2-1-1 multilevel model : evaluation of alternative estimation methods" 26 (26): 591-606, 2019

33 Depaoli, S., "Measurement and structural model class separation in mixture CFA : ML/EM versus MCMC" 19 (19): 178-203, 2012

34 Plummer, M., "JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling" Technische Universität Wien 124 : 2003

35 Yu, R., "Investigating different approaches to develop informative priors in hierarchical Bayesian safety performance functions" 56 : 51-58, 2013

36 Kruschke, J. K., "Introduction to special section on Bayesian data analysis" 6 (6): 272-273, 2011

37 Kelly, B. C., "Influences of motivational contexts on prescription drug misuse and related drug problems" 48 : 49-55, 2015

38 Hoyle, R. H., "Handbook of Applied Multivariate Statistics and Mathematical Modeling" Academic Press 465-497, 2000

39 Lee, S. Y., "Evaluation of the Bayesian and maximum likelihood approaches in analyzing structural equation models with small sample sizes" 39 (39): 653-686, 2004

40 Miočević, M., "Different Roles of Prior Distributions in the Single Mediator Model with Latent Variables" 56 (56): 20-40, 2021

41 Mohajerin, B., "Differences between expressive suppression and cognitive reappraisal in opioids and stimulant dependent patients" 2 : 8-14, 2013

42 Falkenström, F., "Development and validation of a 6-item working alliance questionnaire for repeated administrations during psychotherapy" 27 (27): 169-183, 2015

43 Smid, S. C., "Dangers of the Defaults : A Tutorial on the Impact of Default Priors When Using Bayesian SEM With Small Samples" 11 : 3536-, 2020

44 Falkenström, F., "Confirmatory Factor Analysis of the Patient Version of the Working Alliance Inventory–Short Form Revised" 22 (22): 581-593, 2015

45 Can, S., "Collinear latent variables in multilevel confirmatory factor analysis : A comparison of maximum likelihood and Bayesian estimations" 75 (75): 406-427, 2015

46 Kelly, B. C., "Chasing the bean : Prescription drug smoking among socially active youth" 56 : 632-638, 2015

47 Xu, C., "Calibration of crash risk models on freeways with limited real-time traffic data using Bayesian meta-analysis and Bayesian inference approach" 85 : 207-218, 2015

48 Muthén, B., "Bayesian structural equation modeling : a more flexible representation of substantive theory" 17 (17): 313-335, 2012

49 Baldwin, S. A., "Bayesian methods for the analysis of small sample multilevel data with a complex variance structure" 18 (18): 151-164, 2013

50 Yuan, Y., "Bayesian mediation analysis" 14 (14): 301-322, 2009

51 Asparouhov, T., "Bayesian analysis of latent variable models using Mplus"

52 Smid, S. C., "Bayesian Versus Frequentist estimation for Structural Equation Models in Small Sample Contexts : A Systematic Review" 27 (27): 131-161, 2020

53 Gelman, A., "Bayesian Data Analysis" CRC Press 2013

54 van de Schoot, R., "Analyzing small data sets using Bayesian estimation: The case of posttraumatic stress symptoms following mechanical ventilation in burn survivors" 6 (6): 25216-, 2015

55 Jeffreys, H., "An invariant form for the prior probability in estimation problems" 186 (186): 453-461, 1946

56 Hox, J. J., "An introduction to structural equation modeling" 11 : 354-373, 1998

57 Chiorri, C., "An approximate measurement invariance approach to within-couple relationship quality" 5 : 983-, 2014

58 Preacher, K. J., "Addressing moderated mediation hypotheses : Theory, methods, and prescriptions" 42 (42): 185-227, 2007

59 Gelman, A., "A weakly informative default prior distribution for logistic and other regression models" 2 (2): 1360-1383, 2008

60 van de Schoot, R., "A systematic review of Bayesian articles in psychology : The last 25 years" 22 (22): 217-239, 2017

61 van de Schoot, R., "A gentle introduction to Bayesian analysis : Applications to developmental research" 85 (85): 842-860, 2014

62 Holtmann, J., "A comparison of ML, WLSMV, and Bayesian methods for multilevel structural equation models in small samples : A simulation study" 51 (51): 661-680, 2016

63 Kim, S. Y., "A comparison of Bayesian to maximum likelihood estimation for latent growth models in the presence of a binary outcome" 44 (44): 447-457, 2020

64 Browne, W. J., "A comparison of Bayesian and likelihood-based methods for fitting multilevel models" 1 (1): 473-514, 2006

65 Depaoli, S., "A Bayesian approach to multilevel structural equation modeling with continuous and dichotomous outcomes" 22 (22): 327-351, 2015

66 Lüdtke, O., "A 2×2 taxonomy of multilevel latent contextual models : Accuracy–bias trade-offs in full and partial error correction models" 16 (16): 444-467, 2011