The Comparative Effects of Compensatory and Noncompensatory Two-Dimensional Data on Unidimensional IRT Estimates

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Abstract
This study compared the effects of using a unidi mensional IRT model with two-dimensional data gener ated by noncompensatory and compensatory multidi mensional IRT models. Within each model, simulated datasets differed according to the degree of correlation between two vectors of θ parameters, ranging from 0 to .95. Results showed that the number-correct distri butions for each group of datasets were generally comparable, although factor analyses of tetrachoric correlations suggested that differences existed in the structure of the data from the two models. For the uni dimensional parameter estimates, it was found that the â values from the noncompensatory model appeared to be averages of the a1 and a2 values, while the â values from the compensatory model were best considered as an estimate of the sum of the a1 and a2 values. Con versely, the b values for the noncompensatory data were consistently greater than the b1 values, while the b values from the compensatory model were best con sidered as the average of the b1 and b2 values. For both models the θ estimates were most highly related to the average of the two θ parameters. However, for the noncompensatory model there was a general in crease in the strength of this relationship with in creases in ρ(θ1,θ 2). For the compensatory model, the strength of this relationship did not show a great deal of change with differences in ρ(θ1,θ 2). Index terms: Compensatory multidimensional IRT models, Item response theory, Multidimensional IRT models, Noncompensatory multidimensional IRT models, Pa rameter estimation, Violations of unidimensionality.