ACT research report series: Estimation of item response models using the EM algorithm for finite mixtures

Woodruff, David J.; Hanson, Bradley A.

doi:10.1037/e427312008-001

Cited by 20 publications

(33 citation statements)

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“…As shown by Woodruff and Hanson (1996), the ML estimation for at iteration s can be simplified by expressing the equation as a function of two additive terms, φ( Δ )+ψ( π ), as follows: and where is the conditional likelihood (often referred to as “posterior” likelihood) for examinee i where Θ i = q k , given the fixed known values y i , , and , and computed by Notice here that the first term φ( Δ ) depends only on Δ and the second term ψ( π ) depends only on π . Thus, the M step at iteration s finds the ML estimates and that maximize φ( Δ ) and ψ( π ), respectively.…”

Section: Fixed Parameter Calibration Using Mmle Via the Em Algorithmmentioning

confidence: 84%

“…The MMLE‐EM method for the usual test data in which all items are “free” and need to be calibrated has been well established in the literature (e.g., Bock & Aitkin, 1981; Mislevy, 1984; Mislevy & Bock, 1985; Woodruff & Hanson, 1996). This section describes the essential elements of the MMLE‐EM approach for FPC, which can be viewed as an adapted version of the usual MMLE‐EM method.…”

Section: Fixed Parameter Calibration Using Mmle Via the Em Algorithmmentioning

confidence: 99%

“…The EM algorithm uses the complete data likelihood to find values of the parameters Δ and π which maximize the observed data likelihood (Woodruff & Hanson, 1996). This approach is based on the fact that maximizing the observed data likelihood is equivalent to maximizing the conditional expectation of the complete data likelihood, where the expectation is taken with respect to the conditional distribution of the missing data given the observed data and some fixed known values of the parameters (Dempster, Laird, & Rubin, 1977; McLachlan & Krishnan, 1997).…”

Section: Fixed Parameter Calibration Using Mmle Via the Em Algorithmmentioning

confidence: 99%

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A Comparative Study of IRT Fixed Parameter Calibration Methods

Kim

2006

J Educational Measurement

118

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This article provides technical descriptions of five fixed parameter calibration (FPC) methods, which were based on marginal maximum likelihood estimation via the EM algorithm, and evaluates them through simulation. The five FPC methods described are distinguished from each other by how many times they update the prior ability distribution and by how many EM cycles they use. Specifically, the five FPC methods included no prior weights updating and one EM cycle (NWU‐OEM) or multiple EM cycles (NWU‐MEM), one prior weights updating and one EM cycle (OWU‐OEM) or multiple EM cycles (OWU‐MEM), and multiple weights updating and multiple EM cycles (MWU‐MEM) methods. All the five FPC methods were evaluated in terms of recovery of the underlying ability distribution and item parameters. An important factor in the simulation was three different ability (normal) distributions—N(0, 1), N(0.5, 1.22), and N(1, 1.42)—for FPC groups, with the fixed item parameters obtained with a reference N(0, 1) group. Only the MWU‐MEM method appeared to perform properly under all the three distributions. Under the N(0, 1) distribution, the NWU‐MEM and OWU‐MEM methods also appeared to perform properly. Under the N(0.5, 1.22), and N(1, 1.42) distributions, however, the four methods other than the MWU‐MEM method resulted in some or severe under‐estimation in the recovery.

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Section: Fixed Parameter Calibration Using Mmle Via the Em Algorithmmentioning

confidence: 84%

Section: Fixed Parameter Calibration Using Mmle Via the Em Algorithmmentioning

confidence: 99%

Section: Fixed Parameter Calibration Using Mmle Via the Em Algorithmmentioning

confidence: 99%

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A Comparative Study of IRT Fixed Parameter Calibration Methods

Kim

2006

J Educational Measurement

118

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“…Therefore, they must be estimated from sample data using, for example, marginal maximum likelihood estimation via the EM algorithm (Bock and Aitkin 1981;Harwell et al 1988;Thissen 1982;Woodruff and Hanson 1996) or Bayes modal estimation via the EM algorithm (Harwell and Baker 1991;Mislevy 1986;Tsutakawa and Lin 1986). The ability distribution is often taken a priori as a unit normal distribution [denoted N(0, 1)], but can be estimated from sample data parametrically (Mislevy 1984) or discretely (Bock and Aitkin 1981;Woodruff and Hanson 1996) using the EM algorithm. The maximum likelihood estimates of relative weights (i.e., probabilities) at discrete ability points are often labeled ''posterior'' weights in the literature (Mislevy and Bock 1990).…”

Section: Practical Issuesmentioning

confidence: 99%

The estimation of the IRT reliability coefficient and its lower and upper bounds, with comparisons to CTT reliability statistics

Kim

Feldt

2010

Asia Pacific Educ. Rev.

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The primary purpose of this study is to investigate the mathematical characteristics of the test reliability coefficient q XX 0 as a function of item response theory (IRT) parameters and present the lower and upper bounds of the coefficient. Another purpose is to examine relative performances of the IRT reliability statistics and two classical test theory (CTT) reliability statistics (Cronbach's alpha and Feldt-Gilmer congeneric coefficients) under various testing conditions that result from manipulating large-scale real data. For the first purpose, two alternative ways of exactly quantifying q XX 0 are compared in terms of computational efficiency and statistical usefulness. In addition, the lower and upper bounds for q XX 0 are presented in line with the assumptions of essential tau-equivalence and congeneric similarity, respectively. Empirical studies conducted for the second purpose showed across all testing conditions that (1) the IRT reliability coefficient was higher than the CTT reliability statistics; (2) the IRT reliability coefficient was closer to the Feldt-Gilmer coefficient than to the Cronbach's alpha coefficient; and (3) the alpha coefficient was close to the lower bound of IRT reliability. Some advantages of the IRT approach to estimating test-score reliability over the CTT approaches are discussed in the end.Keywords Test reliability Á Item response theory (IRT) Á Lower and upper bounds of reliability coefficient Á Test score metric versus ability score metric Even when a test form is developed using item response theory (IRT), practitioners often use the test score (X) metric as in classical test theory (CTT) rather than the ability score (h) metric as a basis for reporting examinees' scores. This preference is related to the practical problem that the h metric is often not easily understood by examinees. In this paper, the test score metric refers to the one that is developed by summing item scores, regardless of whether dichotomous or polytomous.Although not all assumed this context, many studies (e.g., Dimitrov 2003;Kolen et al. 1996; Lord 1980, p. 52;May and Nicewander 1994;Shojima and Toyoda 2002) have presented formulas for computing the IRT counterpart of test score reliability in CTT as a function of known item parameters and some distribution of ability. However, as described in detail later, the formulas shown for quantifying IRT-based test score reliability (simply referred to as IRT test reliability) are not based on the same assumptions and thus do not lead to the same reliability coefficient. In fact, some of the formulas (e.g., Kolen et al. 1996;May and Nicewander 1994) are for exactly quantifying the IRT test reliability coefficient (denoted q XX 0) and others (e.g., Dimitrov 2003;Shojima and Toyoda 2002) are for approximating the exact coefficient.The primary purpose of this study is to investigate the mathematical characteristics of the IRT test reliability coefficient and present its lower and upper bounds. For this, two alternative ways of exactly quantifying q XX 0 are comp...

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“…Expectation maximization algorithm is useful within both supervised & semi-supervised methods. [1] Several techniques for learning statistical models have been developed recently by researchers in machine learning & data mining. All of these secrets must address a similar set of representational algorithmic choices & must face a set of statistical challenges unique to learning from relational data.…”

Section: Literature Review a Bhawna Nigam (2011) Document Classimentioning

confidence: 99%

Enhance Clustering Mechanism using Expectation Maximization Algorithm for Gaussian Model

Suthar¹

2018

IJRASET

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ACT research report series: Estimation of item response models using the EM algorithm for finite mixtures

Cited by 20 publications

References 20 publications

A Comparative Study of IRT Fixed Parameter Calibration Methods

A Comparative Study of IRT Fixed Parameter Calibration Methods

The estimation of the IRT reliability coefficient and its lower and upper bounds, with comparisons to CTT reliability statistics

Enhance Clustering Mechanism using Expectation Maximization Algorithm for Gaussian Model

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