ERIC EJ844431: An Introduction to the DA-T Gibbs Sampler for the Two-Parameter Logistic (2PL) Model and beyond

2005·0.25 MB·English

by  ERIC

Psicolo´gica(2005),26,327-352
An Introduction to the DA-T Gibbs Sampler for
the Two-Parameter Logistic (2PL) Model and
Beyond
GunterMaris&TimoM.Bechger
Cito(TheNetherlands)
The DA-T Gibbs sampler is proposed by Maris and Maris (2002) as a
Bayesian estimation method for a wide variety of Item Response Theory
(IRT)models. ThepresentpaperprovidesanexpositoryaccountoftheDA-
T Gibbs sampler for the 2PL model. However, the scope is not limited to
the2PLmodel. ItisdemonstratedhowtheDA-TGibbssamplerforthe2PL
may be used to build, quite easily, Gibbs samplers for other IRT models.
Furthermore, the paper contains a novel, intuitive derivation of the Gibbs
samplerandcouldbereadforagraduatecourseonsampling.
Introduction
Let Y = 1 denote the event that person p gives the correct answer to
pi
item i, and θ his ability. Assume that there exists a latent response variable,
p
X , such that person p solves item i if X is larger than a threshold δ . That
pi pi i
is,
P(Y = 1|θ ) = P(X > δ |θ ) .
pi p pi i p
Itisseenthattheprobabilityofacorrectresponsedependsonthethresholdof
the item as well as the ability of the respondent. The probability of a correct
responseasafunctionofabilityiscalledtheItemResponseFunction(IRF).
Addresscorrespondenceto: GunterMaris,Cito,P.O.Box1034,NL-6801MG,Arnhem,
TheNetherlands. E-mail: gunter.maris[at]citogroep.nl;Tel:[phone].
328 G.MarisandT.M.Bechger
1
0.9
0.8
0.7
0.6
θ1|)
Y=pi 0.5
Pr(
0.4
0.3
0.2
0.1
0
−5 −4 −3 −2 −1 0 1 2 3 4 5
θ
Figure1. IRFsfortwo2PLitemswithdifferentparameters.
UndertheTwo-ParameterLogistic(2PL)model(Birnbaum,1968),X
pi
isassumedtofollowalogisticdistributionwithmeanα θ andscaleparame-
i p
terβ = 1sothat
∞
P(X > δ |θ ,α ,δ ) = (x > δ )f(x |θ ,α )dx
pi i p i i pi i pi p i pi
−∞
∞ exp(x −α θ )
pi i p
= (x > δ ) dx
Z−∞ pi i [1+exp(xpi −αiθp)]2 pi
exp(α θ −δ )
= Z i p i ,
1+exp(α θ −δ )
i p i
where(x > δ )denotesanindicatorvariablethatisoneifx > δ ,andzero
pi i pi i
otherwise. TheTwo-parameterNormalOgive(2NO)model(Birnbaum,1968)
isobtainedwhenthedistributionofthelatentresponsevariablesisnormal.
The discrimination parameter α determines how fast the IRF changes
i
withability. Ifα ispositive(negative),theprobabilityofansweringcorrectly
i
is an increasing (decreasing) function of ability. The Rasch model (Rasch,
1980) is a special case of the 2PL where all items have a discrimination pa-
rameterequaltoone.
AbouttheDA-TGibbssamplerforthe2PL 329
Asitstands,the2PLisunidentifiable. Specifically,
exp(α∗θ∗ −δ∗)
P(Y = 1|θ ,α ,δ ) = i p i
pi p i i 1+exp(α∗θ∗ −δ∗)
i p i
where
θ −c
α∗ = α d, δ∗ = δ −α c, θ∗ = p ,
i i i i i p d
and c and d are arbitrary constants. To deal with this indeterminacy we arbi-
trarily set α = 1, and δ = 0. This means that the item parameters must be
1 1
interpretedrelativetothefirstitem.
The purpose of this paper is to provide an expository account of
Bayesian estimation of the 2PL focussing on the DA-T Gibbs sampler devel-
oped by Maris and Maris (2002). In addition, we offer an intuitive derivation
of the Gibbs sampler and demonstrate that the DA-T Gibbs sampler for the
2PL can be used to build Gibbs samplers for many other Item Response The-
ory(IRT)models. Amongothers,weconsidertheLinearLogisticTestModel
(LLTM) (Fischer, 1995), the 3PL (Birnbaum, 1968), and the Nedelsky model
formultiplechoiceitems(Bechger,Maris,Verstralen&Verhelst,2005).
Gibbs Sampling
Let Λ = (Λ ,...,Λ ), m ≥ 2, denote a vector of parameters.1 In
1 m
Bayesianstatistics,theunknownparametersareconsideredrandomvariables.
Bayes theorem states that the posterior density (the posterior, for short) of Λ
giventheobserveddatay isgivenby
f(y|λ)f(λ)
f(λ|y) = ,
f(y)
where f(y|λ) denotes the likelihood function, and f(y) the marginal likeli-
hood function. The prior density f(λ) (prior, for short) expresses substantive
knowledge concerning the parameters prior to data collection. In Bayesian
statisticsallinferencesabouttheparametersarebasedupontheposterior.
1Weusesubscriptstodistinguishparametervectorsfromscalars.
330 G.MarisandT.M.Bechger
Second
parameter
First
parameter
Second
parameter
First
parameter
Figure2. SchematicrepresentationoftwoiterationsoftheGibbssamplerwithtwo
parameters. Theplotmustbereadfromupperlefttolowerright.
The Gibbs sampler is an iterative procedure to generate parameter val-
uesλ(0),λ(1),...fromtheposterior. Thefirstngeneratedvaluesarediscarded
and the rest is considered to be a dependent and identically distributed (did)
samplefromtheposterior. Thismeansthat
1. ThedistributionofΛ(n+j) giventhedataistheposteriorforallj > 0.
2. Conditionaluponthedata,Λ(n+j) isnot independentofΛ(n+i).
In this section we discuss how the Gibbs sampler works, why it works,
and when it works. Alternative explanations can be found, for instance, in
Casella and George (1992), Tanner (1996), or Ross (2003). The reader is
referredtoTierney(1994)foramorerigoroustreatment.
AbouttheDA-TGibbssamplerforthe2PL 331
How
The procedure starts by choosing an initial value λ(0). Then, in each
successive iteration, individual parameters are sampled independently from
their so-called full conditional distributions. The order in which the parame-
tersaresampledisarbitrary.
The full conditional density (the full conditional, for short) is the den-
sity given the observed data and the current value of all other parameters. In
the sequel, we will often use the shorthand notation f(λ |...) for the full
k
conditional of parameter λ . To determine, up to a constant, the full condi-
k
tional of a parameter λ we write down the density f(λ,y) and remove all
k
factorsthatareunrelatedtoλ . Specificexampleswillbegivenbelow.
k
Figure2representstwoiterationsofafictitiousGibbssampler,withtwo
parameters being sampled at each iteration. The closed curve represent the
support of a two-dimensional posterior. The solid lines indicate the support
ofthefullconditionalsandthecrossesdenotearbitraryvaluessimulatedfrom
the different full conditional distributions. Observe that the Gibbs sampler
“travels” through the support of the posterior along horizontal and vertical
paths. Note that the support of the posterior must be such that every region
canbereachedbytheGibbssampler,irrespectiveofthepointofdeparture.
With a did sample from the posterior we may use the Monte Carlo
method to calculate an unbiased estimate of the posterior expectation of any
functiong(λ,y):
1
g(λ,y)f(λ|y)dλ ≈ g(λ(j),y) ,
n
s j
where n denotesZthe number of sampled values λ(j). That is, we approxi-
s X
mate the expectation by the sample mean. The posterior probability that a
parameterissmallerorequaltoaconstantt,forexample,isestimatedby
1
λ(j) ≤ t .
n k
s j
Thevarianceoftheestimatoroftheposteriorexpectationcanbeestimatedby
X(cid:16) (cid:17)
thevarianceoverindependentreplicationsoftheGibbssampler.
332 G.MarisandT.M.Bechger
3
2
1
0
−1
−2
−3
−4
−5
0 200 400 600 800 1000 1200 1400 1600 1800 2000
3
2
1
0
−1
−2
−3
−4
−5
2000 2500 3000 3500 4000 4500 5000
iterations
Figure3. Plotofsampledvaluesagainstiterations.
Unfortunately, there is no established way to determine an appropriate
value for n. One option is to look at plots of λ(1),λ(2),... against iterations
for a number of independent replications. An illustration with four indepen-
dentreplicationsisgiveninFigure3. If,afterniterations,thevaluesappearto
fluctuatearoundacommonstationaryvalue,thismaybetakenascircumstan-
tial evidence that n is large enough. In Figure 3, the plots appear to stabilize
after about 1200 iterations. However, there is no way to be sure since we do
not know what will happen after 5000 iterations. Other ad hoc methods to
assess the required number of iterations are surveyed by Gelman and Rubin
(1992)orGill(2002,chapter11).
AbouttheDA-TGibbssamplerforthe2PL 333
Why
Let Λ(n),n ≥ 0 denoteaMarkovchain. AMarkovchainisastochas-
tic process such that each value depends only on its immediate predecessor;
thatis,forƒn > 0, '
f(λ(n)|λ(n−1),...,λ(0)) = f(λ(n)|λ(n−1)) .
TheGibbssamplerisaproceduretosimulateaMarkovchainsuchthat
the marginal distribution of Λ(n) converges to the posterior if n increases.
Convergence to the posterior is guaranteed if the following conditions are
satisfied:
1. Theposterioristheinvariantdistribution.
2. Thechainisirreducible.
Invariance means that if λ(0) is drawn from the posterior, then all sub-
sequentvaluesarealsodrawsfromtheposterior. Suppose,foreaseofpresen-
tation,thattherearetwoparameters.2 Theirposteriordensityis
f(λ ,λ |y) = f(λ |λ ,y)f(λ |y) .
1 2 1 2 2
To sample from this posterior, we draw λ(1) from the marginal posterior dis-
2
tribution and then λ(1) from the distribution conditional upon λ(1). Note that
1 2
thelatterisafullconditionalasdefinedinthepreviousparagraph.
SupposewesetupaMarkovchaintodrawλ(1)fromthemarginalposte-
2
riordistribution. Convergenceisfasterifthedependencebetweensubsequent
values is weaker. Thus we aim for a weak degree of dependence. Specifi-
cally,weensurethatΛ(1) andΛ(0) areindependentandidenticallydistributed
2 2
conditionaluponΛ(0). Thatis,
1
f(λ(0),λ(1)|y) = f(λ(0)|λ(0),y)f(λ(1)|λ(0),y)f(λ(0)|y)dλ(0) .
2 2 2 1 2 1 1 1
If we integrate f(λ(0),λZ(1)|y) with respect to λ(0) (or λ(1)), we see that Λ(0)
2 2 2 2 2
andΛ(1) havethesamemarginaldistribution. Thisdistributionisthemarginal
2
2Theargumentforthegeneralcasefollowsbymathematicalinduction.
334 G.MarisandT.M.Bechger
posterior. Itfollowsthat
f(λ(0),λ(1)|y)
f(λ(1)|λ(0),y) = 2 2
2 2 f(λ(0)|y)
2
f(λ(0)|λ(0),y)f(λ(0)|y)
= f(λ(1)|λ(0),y) 2 1 1 dλ(0)
2 1 f(λ(0)|y) 1
2
= Z f(λ(1)|λ(0),y)f(λ(0)|λ(0),y)dλ(0) .
2 1 1 2 1
Toproduceavalueλ(1) froZmtheposteriordistributionwemayusethemethod
2
ofcomposition(Tanner,1996,section3.3.2)asfollows:
1. Drawλ(0) fromtheposterior.
2
2. Drawλ(0) fromthefullconditionalf(λ |λ(0),y).
1 1 2
3. Drawλ(1) fromthefullconditionalf(λ |λ(0),y).
2 2 1
ThisprocedureisaGibbssamplerstartingwithadrawfromtheposterior.
λ(0)
2
λ(0)
1
λ(1) λ(1)
1 2
λ(1)
1
λ(12) λ(22)
etc.
Figure4. Schematicpictureofthesamplingprocedure. Withintherectangleisthe
Gibbssampler.
AbouttheDA-TGibbssamplerforthe2PL 335
Withλ(1)drawnfromthemarginalposterior,wethendrawλ(0)fromthe
2 1
full conditional f(λ |λ(1),y) and repeat the process with λ(1) replacing λ(0),
1 2 2 2
etc. Schematically, the sampling procedure may be depicted as in Figure 4
wherethevaluesgeneratedbytheGibbssampleraredrawninsidearectangle.
ItcanbeshownthatthesevaluesaretherealizationofaMarkovchainwhose
invariantdistributionis,byconstruction,theposterior. Thevaluesoutsidethe
rectangleneednotbegeneratedinwhichcaseweobtaintheGibbssampleras
describedintheprevioussection.
Irreducibility refers to the fact that it must be possible to reach each
region in the support of the posterior (see e.g., Figure 2). This is true for the
majorityofapplications.
When
Gibbs sampling is useful when the full conditionals are tractable. If
so, it provides an estimation procedure that can be implemented relatively
quickly. We call a distribution tractable if there is a simple and efficient
method to generate a sample from it. Methods for stochastic simulation can
befound,forinstance,inDevroye(1986),Ripley(1997),orRoss(2001).
There are many situations where the full conditionals are not tractable.
Thisis,infact,thecaseofthe2PL(Maris&Maris,2002). Inthenextsection,
we will demonstrate that the DA-T Gibbs sampler is a variant of the Gibbs
samplerdesignedtogivetractablefullconditionals.
The DA-T Gibbs Sampler for the 2PL
ThePrior
We conveniently assume that the parameters are a priori independent.
Thatis,
f(θ,δ,α) = f(θ ) f(δ )f(α ) . (1)
p i i
p i
We also assume that all prior distributions are tractable. Note that the priors
Y Y
must be chosen relative to the item whose parameters are arbitrarily fixed to
identifythemodel.
336 G.MarisandT.M.Bechger
TheFullConditionals
DA stands for Data Augmentation which entails adding latent data as
(auxiliary) parameters (Tanner & Wong, 1987). The principle of DA may be
statedasfollows:
Augmenttheobserveddatawithlatentdatasothattheaugmented
posteriordistributionis“simple”. (e.g.,Tanner,1996,p. 38)
Here, the continuous latent responses are added as parameters. Our hope is
thatDAwillresultintractablefullconditionals. Let’ssee!
TheDAposteriorofthe2PLisproportionalto
f(θ,δ,α,x,y) = f(y|x,θ,δ,α)f(x|θ,δ,α)f(θ,δ,α)
= f(y|x,δ)f(x|θ,α)f(θ,δ,α) .
Personsareassumedtobeindependentofoneanother,sothat
1ifx > δ andy = 1
pi i pi
f(y|x,δ) = f(y |x ,δ ) = 1ifx ≤ δ andy = 0
pi pi i pi i pi
p i (cid:134) 8 0otherwise (cid:144)
>
= YY(x > δ )ypi(x ≤<δ )1−ypi ,
pi i pi i
p i >
:
and
YY
f(x|θ,α) = f(x |θ ,α )
pi p i
p i
exp(x −α θ )
pi i p
= YY .
[1+exp(x −α θ )]2
p i pi i p
Thusf(θ,δ,α,x,y)equals YY
exp(x −α θ )
(x > δ )ypi(x ≤ δ )1−ypi pi i p f(θ,δ,α) . (2)
pi i pi i [1+exp(x −α θ )]2
p i pi i p
The next step is to derive the full conditionals, including those for the
YY
latent responses. Let us first consider the full conditional distribution of δ ,
i