Spurious Latent Classes in the Mixture Rasch Model
Natalia Alexeev, Jonathan Templin and Allan S. Cohen
In this study, a mixture Rasch model was applied to detect spurious latent classes. Test data from the 2003 administration of a statewide Grade 8 mathematics test from a southeastern state were used for this example. The results of the empirical study suggest that underfit items appear to be involved in creating extra classes in MRMs.
The following three simulation studies were used to investigate the phenomenon. the model was applied to data from items that initially were selected based on fit to a one-class 3-parameter IRT model. The assumptions of the one-class Rasch model, clearly were violated by data that fit the 3-parameter model Results suggested how latent classes could be explained and also suggested that these latent classes might be due to applying a mixture Rasch model to 3PL data.
The first simulation study intended to investigate the impact of fitting an incorrect model and data were simulated as from a one-class 2PL model. These results suggest that the one-class Rasch model did not fit the 2PL data well. The Rasch model required more than one class when the distribution for discrimination parameters violated Rasch data even moderately.
The second simulation study examined how many items were needed, how high the item discrimination needed to be, and how large a sample size was needed to cause an extra latent class to form. The result shows the lower the discrimination index, the larger the number of items required to trigger the second class.
The third simulation study applied a mixture 2PL model to two-class MRM, serving as a guiding example. For all conditions and all replications, two-class solutions were a better fit according to the BIC and its sample-adjusted version for both mixture Rasch and mixture 2PL models.
The BIC information index was used to identify the number of classes.
Some key concepts:
Traditional IRT models are defined by several assumptions，one of which is invariance. If this assumption is violated, mixture IRT models may be applied.
Mixture Rasch model as a special case of finite mixture distribution models with multivariate, categorical observed variables can simultaneously estimate a latent ability and latent class membership, whose assumption is that the data were drawn from an observable mixture of populations.
Simulation study1:
Item length: 10 or 30
Sample size: 600,1500,3000
Strong violation of the Rasch assumption
Log(ai)~N(0,0.25)
bi~N(0,0.25)
Moderate violation of the Rasch assumption
Log(ai)~N(0,0.09)
bi~N(0,0.1)
Simulation study2:
q~N(0,1)
Item length: 30
Non-Rasch item: 1-10
Sample size: 4000, 6000, 8000
ai: 1.25-2.75
bi~N(0,1)
10 replications
Simulation study 3:
First class
Item difficulties: -2.5-2.5
(–2.5, –2, –1.5, –1, –.75, –.5, –.25, 0, .25, .5, .75, 1, 1.5, 2, 2.5.)
Second class
The last block of five items was moved to be the first five items.
Sample size: 1500, 3000
Item length: 15, 30
30 replications
Comments
1 The paper is interesting. The logic is very easier to understand though some sentences are a bit confused for me.
2 So I wonder when the sample size is small, whether will the result still be ok. If the initial model for one data is not fit the Rasch model, only several items with high discrimination parameter in the model will have a great impact on results when we apply MRM to do the analysis. And the second simulation is for 2PL model, whether the result can be generalized to 3PL.
3 I am not familiar to mixture IRT model. According to the paper, if the invariance assumption of IRT is violated, then we can use the MIRT model. It is a sufficient condition not a necessary condition. So it can be used to study DIF.