Modelovn vazeb mezi asymetri motorickch symptom Parkinsonovy
nemoci a lateralitou tla
Comparison of results from General health questionnaire
Comparison of maximum likelihood and bayesian estimation of
Rasch model: What we gain by using bayesian approach?
Jan tochl, Ph.D.Department of PsychiatryUniversity of
CambridgeEmail: [email protected] of the presentation
Brief introduction to the concept of bayesian statisticsUsing R
and Winbugs for estimation of bayesian Rasch modelAnalysis and
comparison of both methodologies in General health
questionnaireGeneral ideas and introduction to bayesian statisticsA
bit of theoryWhat 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(|X)
We have a model, which gives us the likelihood P(X|)
independenceWe need to use P(X|) to find P(|X) i.e. to invert
the probability
Bayes theorem
Published in 1763
Allows to go from P(X|) to P(|X)
Prior distribution of parametersIts a constant!Posterior
distributionBayes 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
12OpenbugsCan 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)
Item fit of Rasch (1PLM) model and Mokken model
itemDifficultyDiscriminationChi-squarep-valueGHQ151.723.5730.02
