After presenting the BLIM, naive tests of invariance are presented along with theoretical results showing that also the general version of the test suffers for the same problems.
The BLIM is defined as a quadruple (Q, K, [pi], r), in which:
A way to assess violations of the parameter invariance assumption of the BLIM would be to partition the whole data set into two independent groups, to fit the BLIM in each of them (say, Group 1 and Group 2), and to apply some suitable statistical test of the difference between the parameter estimates in the two groups.
Then, according to the BLIM, the conditional probability that in a randomly sampled response pattern R, an item q is failed by careless error, given that the size of R is below the cutoff is
Thus, when the parameters of the BLIM are estimated from only a part of the data set (below/ above), one obtains biased parameter estimates.
For this reason it is recommended not using methods like the one described in this section for testing parameter invariance of the BLIM.
The remaining parameters of the BLIM were fixed to constant values, and the restriction [p.
2], leads to reject the error parameter invariance assumption of the BLIM even when it is respected by the data, irrespectively of the values of the proportions [p.
The BLIM was then estimated to both groups, in each of the 9 pairs, and the means of the parameter estimates were compared to those computed by applying Equation (8) for [[beta].
Subsequently, in each of the 9 conditions, the BLIM was fitted to the data in each of the two groups below and above the cutoff, and the means of the [[beta].