Item Analysis by the Hierarchical Generalized Linear Model
Author(s): Akihito Kamata
Journal of Educational Measurement, Vol. 38, No. 1 (Spring, 2001)
This paper provides an example of an explicit alternative formulation of a multilevel item response model, both as a two-level model and, as an extension, a three-level model.
HGLM is an extension of the generalized linear model (GLM) (McCullagh & Nelder, 1989) to hierarchical data that enables HLM to deal with models having non-normal errors.
Four perspectives on multilevel formulation of IRT models are represented as background information：
l This first perspective is characterized by the treatment of person ability parameters as random parameters in an IRT model, a treatment originally intended to facilitate MMLE item parameter estimation.
l A second perspective on multilevel formulation of IRT models is represented by the multiple-group IRT model, which assumes individuals are grouped by a common characteristic, such as ethnic group or school attended.
l The third perspective is the decomposition of an item parameter into more than one parameter in IRT modeling.
l The fourth perspective on multilevel formulation of IRT models relates to attempts to investigate rater effects in rating scales, such as performance assessments.
l First, the two-level item analysis model is formulated and shown to be algebraically equivalent to the Rasch model.
The two-level item analysis model is formulated following the GLM framework：
Level-1 is the item-model
The linear predictor model of a sampling distribution of item responses is considered to be the level-1 model.
log(pij/(1- pij))= hij=b0j+b1j X1ij + b2j X2ij +…+bkj Xkij
= b0j+
Pij: the probability that j gets item i correct
i: item (1,…, k )
j: person (1,…, n )
q: dummy variable q=1 when q=i or q=0
b0j : intercept
bqj: coefficient associated with Xqij
Level-2 is the person-model, in which b0j is assumed to be a random effect across persons.
b0j=g00 +m0j
b1j=g10
…
b (k-1)j=g(k-1)0
m0j: random component of b0j
when level-l and level-2 models are combined, the linear predictor model is as follows:
When i=q, the equation is algebraically equivalent to Rasch Model.
l Next, a two-level item analysis model with person-characteristic variables is presented, offering a means of analyzing the effects of the person variables alone. Level-1 is the same, while a predictor variable is added in level-2.
b0j=g00 +g01W1j +…+ g0pWpj+ m0j*
b1j=g10
…
b (k-1)j=g(k-1)0
Wsj are person-level predictor measures for predictor s and person j.
l Lastly, a three-level item analysis is presented that can provide estimates of group-level abilities as well as person-level abilities, quantify the variation of person-characteristic variable effects across groups, and reveal any interaction effect between a group-characteristic variable and a person-characteristic variable.
For example, a new level that represents school is added to the model.
Level-1——item-level model：
i=1, .. , k-1
j=1, .. ,n
m=1, .. ,r
: the qth dummy variable for the ith item for person j in school m
: the effect of the reference item
: the effect of the qth item compared to the reference item
The level-2 models for the parameters assumed to be constant across people are person-level models.
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