Abstract
A review of various priors used in Bayesian estimation under the two-parameter logistic model is presented together with clear mathematical definitions of the prior distributions. Examples that compared Bayesian estimation methods are presented using empirical data. The effects of the priors and their specifications on both item and ability parameter estimates are demonstrated. The computer program OpenBUGS that implements the rejection sampling method is the main program employed in the study. Issues in Bayesian estimation and use of priors in item response theory are discussed.
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Appendices
Appendix A: OpenBUGS Code
model { # 2PL model for (i in 1:I) { for (j in 1:J) { logit(p[i, j]) <- alpha[j] ∗ (theta[i] - beta[j]) x[i, j] ~ dbern(p[i, j]) } # ability prior theta[i] ~ dnorm(0, 1) } # item Priors for (j in 1:J) { a[j] ~ dchisqr(10) alpha[j] <- sqrt(a[j] ∗ 0.1) beta[j] ~ dunif(-5, 5) # beta[j] ~ dnorm(mub, taub) # GS2 # alpha[j] ~ dlnorm(0, 2) # GS3 # beta[j] ~ dnorm(0, 0.5) # GS3 } # hyperpriors # mub ~ dflat() # GS2 # taub ~ dgamma(2.5, 5) # GS2 } # kct data list(I = 35, J = 18, x = structure(.Data = c( 1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0, 1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0, 1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0, 1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0, 1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0, 1,1,1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0, 1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,1,0,1,0,0,1,1,0, 1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0, 1,1,1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0, 1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0, 1,1,1,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0, 1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 ), .Dim = c(35, 18)) ) # initial values (e.g., GS1) list( a = c( 10,10,10,10,10,10, 10,10,10,10,10,10, 10,10,10,10,10,10 ), beta = c( 0,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0 ),
# mub = 0, taub = 1, theta = c( -0.4519851, 0.2231436, 0.2231436, -0.6931472, 0.2231436, 0.2231436, 1.2527630, 0.2231436, 0.2231436, 0.4519851, -0.2231436, -0.2231436, 0.2231436, 0.4519851, 0.9555114, 0.2231436, 0.0000000, 0.4519851, 0.0000000, 0.4519851, 0.6931472, 0.6931472, 0.6931472, 1.2527630, -0.9555114, 0.2231436, -0.4519851, 0.2231436, 0.2231436, 0.0000000, 0.2231436, 0.4519851, -0.6931472, 0.6931472, -1.6094379 ) )
Appendix B: Summary of Priors and Specifications
Papers in Tables 4, 5 and 6 are not exhaustive. Estimation techniques in the tables include JBME, MCMC, and MBE. The acronym BME designates Bayes modal estimation, BE designates Bayes estimation (i.e., posterior mean), EAP designates expected a posteriori (i.e., posterior mean via quadratures), and MAP designates maximum a posteriori (i.e., posterior mode with known item parameters). The types of priors can be classified into two; one without any hierarchical structure and the other with some hierarchical structure for which parameters are modeled with hyperpriors and hyperparameters (i.e., Hierarchical). Priors can also be differentiated as ones with exchangeability for which the same prior will be applied to all items in a test or a subtest (i.e., Exchangeable), others with capability of assigning an individual prior on each parameter (i.e., Individual), and also others obtained with information from the current data (i.e., Empirical). It should be noted that in the tables, the names of the distributions might sound the same but could be mathematically, trivially different. Each paper should be consulted and carefully read before employing the priors in one’s research. Also note that several keywords from the computer programs (e.g., SPR, TPR, FLO, AJ, BJ, PA, etc.) are used without any explications.
There are more than six additional, relevant papers that could be included in Tables 4, 5 and 6. The relevant papers are as follows (but without full references): Spiegelhalter et al.’s (1996) “BUGS 0.5 Examples Volume 1”; Johnson and Albert’s (1999) “Ordinal Data Modeling”; Curtis’s (2010) “Journal of Statistical Software, 36”; Nathesan et al.’s (2016) “Frontiers in Psychology, 7”; Luo and Ziao’s (2017) “Educational and Psychological Measurement, Febuary 1”; and Parchev et al.’s (2017) “CRAN Package irtoys”. The six papers mentioned in Tables 4, 5 and 6 are representative ones.
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Kim, SH. et al. (2021). Priors in Bayesian Estimation Under the Two-Parameter Logistic Model. In: Wiberg, M., Molenaar, D., González, J., Böckenholt, U., Kim, JS. (eds) Quantitative Psychology. IMPS 2020. Springer Proceedings in Mathematics & Statistics, vol 353. Springer, Cham. https://doi.org/10.1007/978-3-030-74772-5_28
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