Effects of varying magnitude and patterns of response dependence in the unidimensional Rasch model

By adding items with responses identical to a selected item, Smith (2005) investigated the effect of the response dependence on person and item parameter estimates in the dichotomous Rasch model. By varying the magnitude of response dependence among selected items, rather than their having perfect dependence, this paper provides additional insights into the effects of response dependence on the same estimates in the same model. Two sets of simulations are reported. In the first set, responses to all items except the first were dependent on either the first item or on the immediately preceding item; in the second set, subsets of items were formed first, and then within each of these subsets, responses to all items in a subset except the first were dependent on either the first item or on the immediately preceding item. The effects of dependence were noticeable in all of the statistics reported. In particular, the fit statistics and the parameter estimates showed increasing discrepancies from their theoretical values as a function of the magnitude of the dependence. In some cases, however, two related statistics gave the impression of improvement as a function of increased dependency; first the standard deviation of person estimates showed an increase, and second the index analogous to traditional reliability showed relative increase. In addition to the estimates and depending on the structure and magnitude of the dependence, the person distribution was affected systematically, ranging from becoming skewed to becoming bimodal. The effects on the distribution help explain some of the effects on the statistics reported. In the case of the second set of simulations in which the dependence is within subsets of items, it is possible to take account of the response dependence. This is done by summing the responses of the items within each subset to form a polytomous item and then analyzing the data in terms of a smaller number of polytomous items. This way of accounting for dependence, in which the maximum score for the test as a whole remains the same, gives a more accurate value of the reliability and a more realistic distribution of the person estimates than when the dependence within subsets of items is not taken into account.