MCMC estimation in MlwiN

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MCMC estimation in MlwiN
MCMC estimation is a big topic and is given a
pragmatic and cursory treatment here. Interested
students are referred to the manual MCMC
estimation in MLwiN available from http//multile
vel.ioe.ac.uk/beta/index.html
In the workshop so far you have been using IGLS
(Iterative Generalised Least Squares) algorithm
to estimate the models using MQL and PQL
approximations to handle discrete responses.
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IGLS versus MCMC
IGLS
MCMC
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Bayesian framework
MCMC estimation operates in a Bayesian framework.
A bayesian framework requires one to think about
prior information we have on the parameters we
are estimating and to formally include that
information in the model. We may make the
decision that we are in a state of complete
ignorance about the parameters we are estimating
in which case we must specify a so called
uninformative prior. The posterior
distribution for a paremeter ? given that we have
observed y is subject to the following rule
p(?y)? p(y ?)p(?)
Where p(?y) is the posterior distribution for ?
given we have observed y p(y ?) is the
likelihood of observing y given ? p(?) is the
probability distribution arising from some
statement of prior belief such as we believe
?N(1,0.01). Note that we believe ?N(1,1) is
a much weaker and therefore less influential
statement of prior belief.
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Applying MCMC to multilevel models
Lets start with a ML Normal response
We have the following unknowns
There joint posterior is
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Gibbs sampling
Evaluating the expression for the joint posterior
with all the parameters unknown is for most
models, virtually impossible. However, if we take
each unknown parameter in turn and temporarily
assume we know the values of the other
parameters, then we can simulate directly from
the so called conditional posterior
distribution. The Gibbs sampling algorithm cycles
through the following simulation steps. First we
assume some starting values for our unknown
parameters
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Gibbs sampling cntd
We now have updated all the unknowns in the
model. This process is repeated many times until
eventually we converge on the distribution of
each of the unknown parameters.
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IGLS vs MCMC convergence
IGLS algorithm converges, deterministically to a
distribution.
MCMC algorithm converges on a distribution.
Parameter estimates and intervals are then
calculated from the simulation chains.
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MCMC for discrete response models
GIBBS sampling relies on being able to sample
from the conditional posterior directly, in some
models for some parameters the conditional
posterior can not be arranged into a form that
corresponds to a known distribution we can sample
from directly. This is the case for
In such cases we need to use another type of MCMC
sampling known as Metropolis-Hastings sampling
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Metropolis-Hastings Sampling
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DIC and model comparison
Deviance Information Criterion
DIC is sum of two terms fit complexity or
deviance effective number of parameters
We want to maximise fit and minimize model
complexity
This corresponds to lower deviance and lower
effective number of parameters
So smaller DIC correspond to better models
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To illustrate lets take a simple model
Deviance4553.96, Effective number of params
1.97, DIC4553.961.974555.93
Actually effective number of parameters is really
2, but our estimate of effective number of
parameters used in the DIC is very close.
Why estimate the effective number of parameters?
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Comparison of SLML with DIC
Students are nested within 65 schools. If we fit
a multilevel model
What is the effective number of parameters now?
66(J-1) interceptslope?
No. because uj are assumed to come from a
distribution which places constraints on the
values they can take, this means the effective
number of parameters(number of independent
parameters) will be less than 66.
ML Deviance4257.85, Effective number of params
53.96, DIC4311.81
SL Deviance4553.96, Effective number of params
1.97, DIC4555.93
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Fitting schools with fixed effects
True effective number of params is now 66 and
estimated number is very close.
ML(fixed effects) Deviance4252.73, Effective
number of params 65.5, DIC4318.81
ML(random effects) Deviance4257.85, Effective
number of params 53.96, DIC4311.81
SL Deviance4553.96, Effective number of params
1.97, DIC4555.93
In terms of DIC ML(random effects) is best model
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Other MCMC issues
By default MLwiN uses flat, uniformative priors
see page 5 of MCMC estimation in MLwiN (MEM) For
specifying informative priors see chapter 6 of
MEM. For model comparison in MCMC using the DIC
statistic see chapters 3 and 4 MEM. For
description of MCMC algorithms used in MLwiN see
chapter 2 of MEM.
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When to consider using MCMC in MLwiN
If you have discrete response data binary,
binomial, multinomial or Poisson (chapters 11,
12, 20 and 21). Often PQL gives quick and
accurate estimates for these models. However, it
is a good idea to check against MCMC to test for
bias in the PQL estimates.
If you have few level 2 units and you want to
make accurate inferences about the distribution
of higher level variances.
Some of the more advanced models in MLwiN are
only available in MCMC. For example, factor
analysis (chapter 19), measurement error in
predictor variables (chapter 14) and CAR spatial
models (chapter 16)
Other models, can be fitted in IGLS but are
handled more easily in MCMC such as multiple
imputation (chapter 17), cross-classified(chapter
14) and multiple membership models (chapter 15).
All chapter references to MCMC estimation in
MLwiN.