Comparison of results from General health questionnaire

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# Comparison of results from General health questionnaire - PowerPoint PPT Presentation

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

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)
• 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)