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J an Štochl, Ph.D. Department of Psychiatry University of Cambridge Email: js883@cam.ac.uk. Comparison of maximum likelihood and bayesian estimation of Rasch model: What we gain by using bayesian approach? . Comparison of results from General health questionnaire.
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Jan Štochl, Ph.D. • Department of Psychiatry • University of Cambridge • Email: js883@cam.ac.uk Comparison of maximum likelihood and bayesian estimation of Rasch model: What we gain by using bayesian approach? Comparison of results from General health questionnaire
Content of the presentation Brief introduction to the concept of bayesian statistics Using R and Winbugs for estimation of bayesian Rasch model Analysis and comparison of both methodologies in General health questionnaire
General ideas and introduction to bayesian statistics A bit of theory……
What is Bayesian statistics? • It is an alternative to the classical statistical inference (classical statisticians are called „frequentist“) • Bayesians view the probability as a statement of uncertainty. In other words, probability can be defined as the degree to which a person (or community) believes that a proposition is true. • This uncertainty is subjective (differs across researchers)
Bayesians versus frequentists • A frequentist is a person whose long-run ambition is to be wrong 5% of the time • A Bayesian is one who, vaguely expecting a horse, and catching a glimpse of a donkey, strongly believes he has seen a mule
Bayes theorem and modeling • Our situation – fit the model to the observed data • Models give the probability of obtaining the data, given some parameters: • This is called the likelihood • We want to use this to learn about the parameters
Inference • We observe some data, X, and want to make inferences about the parameters from the data – i.e. find out about P(θ|X) • We have a model, which gives us the likelihood P(X|θ) • independenceWe need to use P(X|θ) to find P(θ|X) – i.e. to invert the probability
Bayes theorem • Published in 1763 • Allows to go from P(X|θ) to • P(θ|X) Prior distribution of parameters It´s a constant! Posterior distribution
Bayes theorem and adding more data • Suppose we observe some data, X1, and get a posterior distribution: • What if we later observe more data, X2? If this is independent of X1, then so that • i.e. the first posterior is used as the prior to get the second posterior
Features of Bayesian approach • Flexibility to incorporate your expert opinion on the parameters • Although this concept is easy to understand, it is not easy to compute. Fortunately, MCMC methods have been developed • Finding prior distribution can be difficult • Misspecification of priors can be dangerous • The less data you have the higher is the influence of priors • The more informative are priors the more they influence the final estimates
When to use Bayesian approach? • When the sample size is small • When the researcher has knowledge about the parameter values (e.g. from previous research) • When there are lots of missing data • When some respondents have too few responses to estimate their ability • Can be useful for test equating • Item banking
Openbugs • Can handle many types of data (including polytomous) • Can handle many types of models (SEM, IRT, Multilevel……) • Possibility to use syntax language or special graphical interface to introduce the model (doodles) • Provides standard errors of the estimates • Provides fit statistics (bayesian ones) • Can be remotely used from R (packages „R2Winbugs“, „R2Openbugs“, „Brugs“, „Rbugs“…) • Results from Openbugs can be exported to R and further analyzed (packages „coda“, „boa“)
Practical comparison of maximum likelihood and bayesian estimation of Rasch model General Health Questionnaire, items 1-7
General Health Questionnaire (GHQ) • 28 items, scored dichotomously (0 and 1), 4 unidimensional subscales (7 items each) • Only one subscale is analyzed (items 1-7) • Rasch model is used, maximum likelihood estimates are obtained in R (package „ltm“), bayesian estimates in Openbugs (and analyzed in R) • 2 runs in Openbugs : • - first one with vague (uninformative) priors for difficulty parameters (normal distibution with mean=0 and sd=10) • - second one with mix of informative and uninformative priors for difficulty parameters (to demonstrate the influence of priors)
Further reading and software • General literature on bayesian IRT analysis • Congdon, P (2006). Bayesian Statistical Modelling, 2nd edition. Wiley. • Congdon, P. (2005). Bayesian Methods for Categorical Data, Wiley. • Congdon, P. (2003). Applied Bayesian Modelling, Wiley. • Winbugs User Manual (available online) from • http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/manual14.pdf • Winbugs discussion archive http://www.jiscmail.ac.uk/lists/bugs.html • Lee, S.Y. (2007). Structural Equation Modelling: A Bayesian Approach, Wiley. • Iversen, G. R. (1984). Bayesian Statistical Inference: Sage. Available software • Winbugs, Openbugs, Jags (freely available) • R (freely available) - package „mokken“ • Mplus (commercial)