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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

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

A bit of theory……

- 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)

- 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

- 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

- 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

- Published in 1763
- Allows to go from P(X|θ) to
- P(θ|X)

Prior distribution of parameters

It´s a constant!

Posterior distribution

- 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

- 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 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

- 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“)

General Health Questionnaire, items 1-7

- 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)