Numerous methods have been invented to estimate the parameters of Rasch measurement models. Wright and Masters (1982) describe the Conditional (FCON), Unconditional (JMLE UCON), Pair-wise (PAIR) and Normal Approximation (PROX) algorithms. When any of these methods is applied with strict Rasch model specifications, the parameters corresponding to extreme score vectors in the empirical data are inestimable. In practice, observations forming part of extreme score vectors are eliminated from the data set before the remaining parameters are estimated.
The fact that there is always some probability of the occurrence of extreme score vectors, regardless of the parameter values, has proven difficult to manage. Even FCON, which specifically prohibits the possibility of extreme person scores, includes in the sample space of its likelihood equations the response vectors corresponding to extreme item scores. Unfortunately, including the likelihood of extreme vectors in the estimation procedure introduces bias into the estimates, as Haberman (1977) observed for the UCON algorithm.
Estimation bias, however, is not inevitably statistical inconsistency, because as the number of parameters becomes large in a reasonable way, the probability of extreme vectors becomes vanishingly small. Haberman (1977) gives the consistency conditions for UCON. The existence of bias is of no great concern so long as its nature and magnitude is understood. Thus Wright and Douglas (1977) propose a correction for bias in UCON which has proved useful in conventional testing. For FCON, the bias has generally been negligible.
Extreme High Person Vector | ||
Extreme Low Item Vector | Vectors of Non-Extreme Observations | Extreme High Item Vector |
Extreme Low Person Vector | ||
Figure 1. Sample space of all possible vectors of observations for estimation algorithms applied to two-facet data. |
The way in which the probability of extreme score vectors is included in the sample space for each of the estimation methods is diagrammed in Figure 1 for a conventional two-facet test. The sample space of the UCON and PROX algorithms corresponds to the entire figure.
The sample space for FCON is the horizontal sub-rectangle, without the extreme person vectors, when conditioning is on person scores. When conditioning is on item scores, then the sample space for FCON is the vertical sub-rectangle, without the extreme item vectors. This lack of congruity between the sample spaces for the two FCON formulations shows that their resulting estimates can not be exactly compatible. Log-linear models can be formulated to parallel either the FCON or UCON approaches.
The PAIR algorithm (Choppin, 1968) does not use entire response vectors in its estimation, but rather employs a comparison of the responses made by persons to pairs of items. It takes no account, however, of the fact that a response could form part of an extreme score vector, and so bias due to the possibility of extreme vectors remains.
For two-facet tests with complete dichotomous data, bias in estimation has been shown to be either negligible or removable by a simple correction. The application of Rasch techniques to rating scales, incomplete data sets (e.g., CAT responses) and more complex situations raises the bias question anew. Either the particular bias for each new situation requires investigation, or a new estimation technique needs to be developed which produces unbiased estimates.
The Extra-Conditional (XCON) estimation procedure, also called XMLE, Exclusory Maximum Likelihood Estimation CMLE, described here, eliminates the possibility of most extreme score vector combinations from its sample space, and so eliminates this source of bias from Rasch measurement estimations. XCON is similar to UCON in derivation, except that an extra condition is added so that extreme score vectors do not form part of the estimation sample space.
UCON | XCON | ||
Extreme high 1 | P_{ni1} | MAX | |
Non-extreme 1 | P_{ni1}-MAX | E_{ni1} | |
Non-extreme 0 | P_{ni0} | P_{ni0}-MIN | E_{ni0} |
Extreme low 0 | MIN | ||
Figure 2. Sample space of the possible (0, 1) responses to dichotomous item i by person n. |
In Figure 2, the left rectangle comprises the sample space for a response to a dichotomous item. (A similar figure can be constructed for a rating scale item). UCON, following the Rasch model directly, models the top half of the rectangle, the probability of a success, to be
P_{ni }= exp(B_{n}-D_{i}) / ( 1 + exp(B_{n}- D_{i}) )
with the usual conventions. Based on this equation, UCON then provides parameter estimates which maximize the likelihood of the empirical data.
Bias is introduced because the sample space in Figure 2 includes two areas in which the empirical response would form part of an extreme score vector. These are represented in Figure 2 by an area, MAX, in which a success on the item forms part of an extreme high score vector (a maximum score on the test), and by an area, MIN, in which a failure on the item would form part of an extreme low score vector (a zero score on the test).
If we consider only estimable data sets, the probability of scoring a one, E_{ni1}, becomes,
E_{ni1} = (P_{ni1} - MAX) / (1 - MIN - MAX)
and this equation is the basis of the XCON estimation algorithm. For a two-facet test, MAX is given by
MAX = (Probability of extreme high item score vector for item i) + (Probability of extreme high person score vector for person n) - (Probability of both vectors simultaneously).
MIN is defined in a similar way for low score vectors.
Referring back to Figure 1, removing MIN and MAX is equivalent to removing the outside sub-rectangles and leaving just the central sub-rectangle.
Simulation studies indicate that, both for Atwo items taken by many persons@ (Andersen 1973) and for Athree items taken by many persons@, XCON is almost indistinguishable from FCON. In larger data sets, XCON has the virtues of UCON which are joint estimation of all parameters and computational economy. XCON's further statistical properties are under investigation.
The precise estimation equations for XCON are set forth in Linacre (1989).
John M. Linacre
Andersen, E. B. Conditional inference for multiple-choice questionnaires. British Journal of Mathematical and Statistical Psychology, 1973, 26, 31-44.
Choppin, B. H. Item banking using sample-free calibration. Nature, 1968, 219, p. 870-872, reprinted in Evaluation in Education, 1985, 9:1 p. 81-85.
Haberman, S.J. Maximum likelihood Estimates in Exponential Response Models. The Annals of Statistics, 1977, 5, p. 815- 841.
Linacre, J.M. Many-Facet Rasch Measurement. Chicago: MESA Press. 1989.
Wright, B.D. & Douglas, G.A. Best procedures for sample-free item analysis. Applied Psychological Measurement, 1977, 1, 281- 294.
Wright, B.D. & Masters, G.N. Rating Scale Analysis: Rasch Measurement. Chicago: MESA Press. 1982.
Extra-conditional (XCON) algorithm [XMLE: Exclusory Maximum Likelihood Estimation]. Linacre JM. … Rasch Measurement Transactions, 1989, 3:1 p.47-48
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