The Effect of Item Response Dependency on Trait or Ability Dimensionality
ACT Research Report Series87-10The Effect of Item Response Dependency on Trait or Ability DimensionalityJudith A Spray Terry AckermanSeptember 1987 For additional copies write: ACT Research Report Series P.O. Box 168 Iowa City, Iowa 52243©1988 by The American College Testing Program. All rights reserved. THE EFFECT OF ITEM RESPONSE DEPENDENCY ON TRAIT OR ABILITY DIMENSIONALITYJudith A. Spray Terry A. Ackerman {I ABSTRACTThe purpose of this paper was to investigate various levels of item response dependency using principal component analyses. Item response data were simulated using an IRT-based dependency model which describes a two-state Markov process. Results indicated that when the IRT assumption of local independence was violated, items within a dependent sequence were clearly identified by their loadings on a second principal component, in addition to the common first principal component shared by all of the items. Under "realistic conditions" of local dependence, the response data retained their unidimensional characteristics. Concern for the effect that item dependency may have on ability estimation is also discussed. a& The Effect of Item Response Dependency on Trait or Ability DimensionalityThe assumption of local independence allows the joint probability distri*ibution of observing a response vector, U = (u1u 2...u^) given ability 0^ to be written as a product of k marginal probability functions orKP(U, = U,t u2 = U2, uk = uk|e.) = .MrtUj = Uj|e.)| .A particular violation of this assumption can arise in the following way. Suppose that, for a k-item test, m of the items form a subset such thatP(U. — u . , U, - u », . .. , U — u , U . ■" u . j U. " u 10. )1 i 7 2 2 m m m+1 m+1 k k' lkp(u, = u , 10. )p(u, = u J o . , u .).,.p(u = u |e.( u .) n [p(u. = u ,10.)] .1 11 1 2 21 1 m m' 1 m-1 . 1 J ij=m+l JFor example, the three geometry items pictured in Figure 1 could represent the first three items on a 20-item geometry test. The joint probabilitydensity function for an examinee with ability 0. could be written asP( U 1 - Uj, U 2 - u 2, U 3 — u 3,...U20 - u2oI®£) -20Hvl = u,|e.)p(u2 = uJe., u^PCUjle., u2) n [p(u. = u.|e.))j=4 Jin order to account for item response dependence between items 1 and 2 and between items 2 and 3.Does the joint probability density in equation (2) imply that the dimensionality of the space defined by the item responses as greater than one, even 2though only a scalar value of fh is assumed? To investigate this question, we have used a finite, two-state (0 or 1, incorrect or correct) Markov chain or process to model the dependence within the m-item sequence.Let Pj(0.) represent the probability of an examinee with trait measure 0^ answering test item j correctly, independently of any other test item.Then define a transition matrix between any adjacent items, j-1 and j in the k-item test as specified below.0jth-1 item1In this model, 0,.^ represents the probability that an examinee with trait 0^ will move from an incorrect response on item j-1 (state 0) to a correct response on item j (state 1). Similarly, B.j represents a transition probability from a correct response on item j-1 to an incorrect response on* * ,item j. The probabilities, 1 - cu ^ and 1 - 0 ^ imply state consistency between items.We note that items j and j-1 are assumed to be adjacent test items only for the purpose of discussion in this paper. This is not a requirement, however, and in fact all discussion may be generalized to any two test items, j and j-T, where t = l,2,...,k-l and j = t+1,t+2 ,...,m.jth item0 1**1 - a. .a. .ij**S. .1 - B. .ijij 3These four cell probabilities are functions of (1) the jth item-by-ith person interaction, as given by P^(0^), and (2) the amount and direction of any item dependency. This definition of the transition probabilities is similar in structure to the latent Markov chain model described by Lazarsfeld and Henry (1968). These probabilities are defined as follows.a. . = aP .(0.) ij J iand8tj = 6V9i>whereQ .(0- ) = 1 - P.(0.) . j i J iThe parameters, a and 0, are weights that describe the dependency rela- tionship with 0 < o < 1 and 0 < 0 < 1. For the purpose of the simple examples provided in this paper, the weights are assumed to be constant but not necessarily equal to each other for all adjacent pairs of items in the m-item sequence. This is not re
