Estimation of Item Response Models

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Estimation of Item Response Models

• Mister Ibik
• Division of Psychology in Education
• Arizona State University
• EDP 691 Advanced Topics in Item Response Theory

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Motivation and Objectives

• Why estimate?
• Distinguishing feature of IRT modeling as
  compared to classical techniques is the presence
  of parameters
• These parameters characterize and guide inference
  regarding entities of interest (i.e., examinees,
  items)
• We will think through
• Different estimation situations
• Alternative estimation techniques
• The logic and mathematics underpinning these
  techniques
• Various strengths and weaknesses
• What you will have
• A detailed introduction to principles and
  mathematics
• A resource to be revisitedand revisitedand
  revisited

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Outline

• Some Necessary Mathematical Background
• Maximum Likelihood and Bayesian Theory
• Estimation of Person Parameters When Item
  Parameters are Known
• ML
• MAP
• EAP
• Estimation of Item Parameters When Person
  Parameters are Known
• ML
• Simultaneous Estimation of Item and Person
  Parameters
• JML
• CML
• MML
• Other Approaches

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Background Finding the Root of an Equation

• Newton-Raphson Algorithm
• Finds the root of an equation
• Example the function f(x) x2
• Has a root (where f(x) 0) at x 0

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Newton-Raphson

• Newton-Raphson takes a given point, x0, and
  systematically progresses to find the root of the
  equation
• Utilizes the slope of the function to find where
  the root may be
• The slope of the function is given by the
  derivative
• Denoted
• Gives the slope of the straight line that is
  tangent to f(x) at x
• Tangent best linear prediction of how the
  function is changing
• For x0, the best guess for the root is the point
  where f'(x) 0
• This occurs at
• So the next candidate point for the root is

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Newton-Raphson Updating (1)

• Suppose x0 1.5

f'(x0) 3
f(x0) 2.25
x1 0.75
x0 1.5
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Newton-Raphson Updating (2)

• Now x1 0.75

f'(x1) 1.5
f(x1) 0.5625
x2 0.375
x1 0.75
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Newton-Raphson Updating (3)

• Now x2 0.375

f'(x2) 0.75
f(x2) 0.1406
x3 0.1875
x2 0.375
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Newton-Raphson Updating (4)

• Now x3 0.1875

f'(x3) 0.375
f(x3) 0.0352
x4 0.0938
x3 0.1875
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Newton-Raphson Example
Iteration Value f(x)
0 1.5000 2.2500 3.0000 0.7500 0.7500
1 0.7500 0.5625 1.5000 0.3750 0.3750
2 0.3750 0.1406 0.7500 0.1875 0.1875
3 0.1875 0.0352 0.3750 0.0938 0.0938
4 0.0938 0.0088 0.1875 0.0469 0.0469
5 0.0469 0.0022 0.0938 0.0234 0.0234
6 0.0234 0.0005 0.0469 0.0117 0.0117
7 0.0117 0.0001 0.0234 0.0059 0.0059
8 0.0059 0.0000 0.0117 0.0029 0.0029
9 0.0029 0.0000 0.0059 0.0015 0.0015
10 0.0015 0.0000 0.0029 0.0007 0.0007
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Newton-Raphson Summary

• Iterative algorithm for finding the root of an
  equation
• Takes a starting point and systematically
  progresses to find the root of the function
• Requires the derivative of the function
• Each successive point is given by
• The process continues until we get arbitrarily
  close, as usually measured by the change in some
  function

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Difficulties With Newton-Raphson

• Some functions have multiple roots
• Which root is found often depends on the start
  value

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Difficulties With Newton-Raphson

• Numerical complications can arise
• When the derivative is relatively small in
  magnitude, the algorithm shoots into outer space

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Logic of Maximum Likelihood

• A general approach to parameter estimation
• The use of a model implies that the data may be
  sufficiently characterized by the features of the
  model, including the unknown parameters
• Parameters govern the data in the sense that the
  data depend on the parameters
• Given values of the parameters we can calculate
  the (conditional) probability of the data
• P(Xij 1 ?i, bj) exp(?i bj)/(1 exp(?i
  bj))
• Maximum likelihood (ML) estimation asks What
  are the values of the parameters that make the
  data most probable?

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Example Series of Bernoulli Variables With
Unknown Probability

• Bernoulli variable P(X 1) p
• The probability of the data is given by pX
  (1-p)(1-X)
• Suppose we have two random variables X1 and X2
• When taken as a function of the parameters, it is
  called the likelihood
• Suppose X1 1, X2 0
• P(X1 1, X2 0p) L(pX1 1, X2 0) p
  (1-p)
• Choose p to maximize the conditional probability
  of the data
• For p 0.1, L 0.1 (1-0.1) 0.09
• For p 0.2, L 0.2 (1-0.2) 0.16
• For p 0.3, L 0.3 (1-0.3) 0.21

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Example Likelihood Function
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The Likelihood Function in IRT

• The Likelihood may be thought of as the
  conditional probability, where the data are known
  and the parameters vary
• Let Pij P(Xij 1 ?i, ?j)
• The goal is to maximize this function what
  values of the parameters yield the highest value?

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Log-Likelihood Functions

• It is numerically easier to maximize the natural
  logarithm of the likelihood
• The log-likelihood has the same maximum as the
  likelihood

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Maximizing the Log-Likelihood

• Note that at the maximum of the function, the
  slope of the tangent line equals 0
• The slope of the tangent is given by the first
  derivative
• If we can find the point at which the first
  derivative equals 0, we will have also found the
  point at which the function is maximized

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Overview of Numerical Techniques

• One can maximize the lnL function by finding a
  point where its derivative is 0
• A variety of methods are available for maximizing
  L, or lnL
• Newton-Raphson
• Fisher Scoring
• Estimation-Maximization (EM)
• The generality of ML estimation and these
  numerical techniques results in the same concepts
  and estimation routines being employed across
  modeling situations
• Logistic regression, log-linear modeling, FA,
  SEM, LCA

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ML Estimation of Person Parameters When Item
Parameters Are Known

• Assume item parameters bj, aj, and cj, are known
• Assume unidimensionality, local and respondent
  independence

• Conditional probability now depends on person
  parameter only

• Likelihood function for the person parameters
  only

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ML Estimation of Person Parameters When Item
Parameters Are Known

• Choose each ?i such that L or lnL is maximized
• Lets suppose we have one examinee
• Maximize this function using any of several
  methods
• Well use Newton-Raphson

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Newton-Raphson Estimation Recap

• Recall NR seeks to find the root of a function
  (where 0)
• NR updates follow the general structure

What is the derivative of this function?
What is our function of interest?
Current value

• Derivative of the function of interest

• Updated value

Function of interest
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Newton-Raphson Estimation of Person Parameters

• Newton-Raphson uses the derivative of the
  function of interest
• Our function is itself a derivative, the first
  derivative of lnL with respect to ?i
• Well need the second derivative as well as the
  first derivative
• Updates given by

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ML Estimation of Person Parameters When Item
Parameters Are Known The Log-Likelihood

• The log-likelihood to be maximized
• Select a start value and iterate towards a
  solution using Newton-Raphson
• A hill-climbing sequence

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ML Estimation of Person Parameters When Item
Parameters Are Known Newton-Raphson

• Start at -1.0

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ML Estimation of Person Parameters When Item
Parameters Are Known Newton-Raphson

• Move to 0.09

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ML Estimation of Person Parameters When Item
Parameters Are Known Newton-Raphson

• Move to -0.0001
• When the change in ?i is arbitrarily small (e.g.,
  less than 0.001), stop estimation
• No meaningful change in next step
• The key is that the tangent is 0

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Newton-Raphson Estimation of Multiple Person
Parameters

• But we have N examinees each with a ?i to be
  estimated
• We need a multivariate version of the
  Newton-Raphson algorithm

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First Order Derivatives

• First order derivatives of the log-likelihood
• ?lnL/??i only involves terms corresponding to
  subject i

Why???
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Second Order Derivatives

• Hessian second order partial derivatives of the
  log-likelihood
• This matrix needs to be inverted
• In the current context, this matrix is diagonal

Why???
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Second Order Derivatives

• The inverse of the Hessian is diagonal with
  elements that are the reciprocals of the diagonal
  of the Hessian
• Updates for each ?i do not depend on any other
  subjects ?

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Second Order Derivatives

• The updates for each ?i are independent of one
  another
• The procedure can be performed one examinee at a
  time

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ML Estimation of Person Parameters When Item
Parameters Are Known Standard Errors

• The approximate, asymptotic standard error of the
  ML estimate of ?i is
• where I(?i) is the information function
• Standard errors are
• asymptotic with respect to the number of items
• approximate because only an estimate of ?i is
  employed
• asymptotically approximately unbiased

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ML Estimation of Person Parameters When Item
Parameters Are Known Strengths

• ML estimates have some desirable qualities
• They are consistent
• If a sufficient statistic exists, then the MLE is
  a function of that statistic (Rasch models)
• Asymptotically normally distributed
• Asymptotically most efficient (least variable)
  estimator among the class of normally distributed
  unbiased estimators
• Asymptotically with respect to what?

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ML Estimation of Person Parameters When Item
Parameters Are Known Weaknesses

• ML estimates have some undesirable qualities
• Estimates may fly off into outer space
• They do not exist for so called perfect scores
  (all 1s or 0s)
• Can be difficult to compute or verify when the
  likelihood function is not single peaked (may
  occur with 3-PLM or more complex IRT models)

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ML Estimation of Person Parameters When Item
Parameters Are Known Weaknesses

• Strategies to handle wayward solutions
• Bound the amount of change at any one iteration
• Atheoretical
• No longer common
• Use an alternative estimation framework (Fisher,
  Bayesian)
• Strategies to handle perfect scores
• Do not estimate ?i
• Use an alternative estimation framework
  (Bayesian)
• Strategies to handle local maxima
• Re-estimate the parameters using different
  starting points and look for agreement

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ML Estimation of Person Parameters When Item
Parameters Are Known Weaknesses

• An alternative to the Newton-Raphson technique is
  Fishers method of scoring
• Instead of the Hessian, it uses the information
  matrix (based on the Hessian)
• This usually leads to quicker convergence
• Often is more stable than Newton-Raphson
• But what about those perfect scores?

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Bayes Theorem

• We can avoid some of the problems that occur in
  ML estimation by employing a Bayesian approach
• All entities treated as random variables
• Bayes Theorem for random variables A and B

Posterior distribution of A, given B The
probability of A, given B.

• Conditional probability of B, given A

• Prior probability of A

• Marginal probability of B

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Bayes Theorem

• If A is discrete
• If A is continuous
• Note that P(BA) L(AB)

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Bayesian Estimation of Person Parameters The
Posterior

• Select a prior distribution for ?i denoted P(?i)
• Recall the likelihood function takes on the form
  P(Xi ?i)
• The posterior density of ?i given Xi is
• Since P(Xi) is a constant

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Bayesian Estimation of Person Parameters The
Posterior

• The Likelilhood

• The Prior

• The Posterior

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Maximum A Posteriori Estimation of Person
Parameters

• The Maximum A Posteriori (MAP) estimate is
  the maximum of the posterior density of ?i
• Computed by maximizing the posterior density, or
  its log
• Find ?i such that
• Use Newton-Raphson or Fisher scoring
• Max of lnP(?i Xi) occurs at max of lnP(Xi
  ?i) lnP(?i)
• This can be thought of as augmenting the
  likelihood with prior information

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Choice of Prior Distribution

• Choosing P(?i) U(-8, 8) yields the posterior to
  be proportional to the likelihood
• In this case, the MAP is very similar to the ML
  estimate
• The prior distribution P(?i) is often assumed to
  be N(0, 1)
• The normal distribution commonly justified by
  appeal to CLT
• Choice of mean and variance identifies the scale
  of the latent continuum

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MAP Estimation of Person Parameters Features

• The approximate, asymptotic standard error of the
  MAP is
• where I(?i) is the information from the posterior
  density
• Advantages of the MAP estimator
• Exists for every response pattern why?
• Generally leads to a reduced tendency for local
  extrema
• Disadvantages of the MAP estimator
• Must specify a prior
• Exhibits shrinkage in that it is biased towards
  the mean May need lots of items to swamp the
  prior if its misspecified
• Calculations are iterative and may take a long
  time
• May result in local extrema

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Expected A Posteriori (EAP) Estimation of Person
Parameters

• The Expected A Posteriori (EAP) estimator is the
  mean of the posterior distribution
• Exact computations are often intractable
• We approximate the integral using numerical
  techniques
• Essentially, we take a weighted average of the
  values, where the weights are determined by the
  posterior distribution
• Recall that the posterior distribution is itself
  determined by the prior and the likelihood

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Numerical Integration Via Quadrature

• The Posterior Distribution
• With quadrature points
• Evaluate the heights of the distribution at each
  point
• Use the relative heights as the weights

? .165
.021 / .165 .127
.002 / .165 .015
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EAP Estimation of via Quadrature

• The Expected A Posteriori (EAP) is estimated by a
  weighted average
• where H(Qr) is weight of point Qr in the
  posterior (compare Embretson Reise, 2000 p.
  177)
• The standard error is the standard deviation in
  the posterior and may also be approximated via
  quadrature

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EAP Estimation of via Quadrature

• Advantages
• Exists for all possible response patterns
• Non-iterative solution strategy
• Not a maximum, therefore no local extrema
• Has smallest MSE in the population
• Disadvantages
• Must specify a prior
• Exhibits shrinkage to the prior mean If the
  prior is misspecified, may need lots of items to
  swamp the prior

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ML Estimation of Item Parameters When Person
Parameters Are Known Assumptions

• Assume
• person parameters ?i are known
• respondent and local independence
• Choose values for item parameters that maximize
  lnL

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Newton-Raphson Estimation

• What is the structure of this matrix?

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ML Estimation of Item Parameters When Person
Parameters Are Known

• Just as we could estimate subjects one at a time
  thanks to respondent independence, we can
  estimate items one at time thanks to local
  independence
• Multivariate Newton-Raphson

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ML Estimation of Item Parameters When Person
Parameters Are Known Standard Errors

• To obtain the approximate, asymptotic standard
  errors
• Invert the associated information matrix, which
  yields the variance-covariance matrix
• Take the square root of the elements of the
  diagonal
• Asymptotic w.r.t. sample size and approximate
  because we only have estimates of the parameters
• This is conceptually similar to those for the
  estimation of ?
• But why do we need a matrix approach?

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ML Estimation of Item Parameters When Person
Parameters Are Known Standard Errors

• ML estimates of item parameters have same
  properties as those for person parameters
  consistent, efficient, asymptotic (w.r.t.
  subjects)
• aj parameters can be difficult to estimate, tend
  to get inflated with small sample sizes
• cj parameters are often difficult to estimate
  well
• Usually because theres not a lot of information
  in the data about the asymptote
• Especially true when items are easy
• Generally need larger and more heterogeneous
  samples to estimate 2-PL and 3-PL
• Can employ Bayesian estimation (more on this
  later)