Abstract
One important tool for assessing whether a data set can be described equally well with a Rasch Model (RM) or a Linear Logistic Test Model (LLTM) is the Likelihood Ratio Test (LRT). In practical applications this test seems to overly reject the null hypothesis, even when the null hypothesis is true. Aside from obvious reasons like inadequate restrictiveness of linear restrictions formulated in the LLTM or the RM not being true, doubts have arisen whether the test holds the nominal type-I error risk, that is whether its theoretically derived sampling distribution applies. Therefore, the present contribution explores the sampling distribution of the likelihood ratio test comparing a Rasch model with a Linear Logistic Test Model. Particular attention is put on the issue of similar columns in the weight matrixW of the LLTM: Although full column rank of this matrix is a technical requirement, columns can differ in only a few entries, what in turn might have an impact on the sampling distribution of the test statistic. Therefore, a system of how to generate weight matrices with similar columns has been established and tested in a simulation study. The results were twofold: In general, the matricesconsidered in the study showed LRT results where the empirical alpha showed only spurious deviations from the nominal alpha. Hence the theoretically chosen alpha seems maintained up to random variation. Yet, one specific matrix clearly indicated a highly increased type-I error risk: The empirical alpha was at least twice the nominal alpha when using this weight matrix. This shows that we have to indeed consider the internal structure of the weight matrix when applying the LRT for testing the LLTM. Best practice would be to perform a simulation or bootstrap/re-sampling study for the weight matrix under consideration in order to rule out a misleadingly significant result due to reasons other than true model misfit.
Highlights
G. Rasch (1960) introduced a statistical model that allows for describing the probability of a positive response to a dichotomous item by means of two real-valued parameters, βi (i = 1, . . . , k) covering the difficulty of the item i and θv (v = 1, . . . , n) characterizing person v in terms of the ability to solve this item or proneness to endorse a statement: p(+|θv, βi) =eθv −βi + eθv−βi (1)Austrian Journal of Statistics, Vol 38 (2009), No 4, 221–230The Linear Logistic Test Model (LLTM; Fischer, 1972, 1973, 1983, 1995; Scheiblechner, 1971, 1972) is an extension of the Rasch Model (RM)
A system was established which allows for the generation of an indefinite number of weight matrices with similar columns, a selection of which has been analysed
The empirical alpha was evaluated using two tolerance levels, 10% and 20%, the more stricter of which (10%) will be used for our decision concerning the correctness of the Likelihood Ratio Test (LRT)
Summary
The Linear Logistic Test Model (LLTM; Fischer, 1972, 1973, 1983, 1995; Scheiblechner, 1971, 1972) is an extension of the Rasch Model (RM) It decomposes each item difficulty parameter βi into a weighted sum of basic parameters ηj The normalizing constant c is required to compensate for an (admissible) shift of the βi (for instance for norming purposes) without affecting the ηj (Fischer, 1995). It can be eliminated by setting c = −1/k i j wijηj, wi∗j = wij −1/k i wij (Fischer, 1983). Applying the LLTM when the RM does not hold would be of little interest, as it would boil down to the decomposition of an item parameter that innately has not adequately described the data
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