# Warm's Mean Weighted Likelihood Estimates (WLE) of Rasch Measures

For a mathematical description of WLE, see RMT (2009), 23:1, 1188-9

Ronald A. Fisher (1922) formulated the concept of the likelihood of the data given a statistical model that is hypothesized to have generated those data and a set of parameter estimates. Maximum likelihood estimates are the parameters values which maximize the likelihood that the observed data would have been generated. Thus MLE values correspond to the mode of the likelihood function.

Thomas Warm (1989) points out the that those modal estimates are biased when viewed from the likelihood function as whole. He suggests that, rather than the mode of the likelihood function, estimates should be based on its mean. These estimates have come to be called "Warm estimates", and his approach is Warm (or Weighted) (Mean) Likelihood Estimation (WLE or WMLE).

In the Rasch model, the estimation of MLE and WLE require iteration. WLE is more computationally intensive. Warm demonstrates that the asymptotic variance of MLE and WLE estimates are the same, meaning that the estimates have the same model standard errors.

WLE estimates are generally slightly more central than MLE estimates, though the implications of this for practical applications are not clear, because the difference is usually less than the standard error of the estimates.

Fisher R.A., (1922) "On the mathematical foundations of theoretical statistics" Philosophical Transactions of the Royal Society of London (A) 222, 1922, p. 309-368.

Warm T.A. (1989) "Weighted Likelihood Estimation of Ability in Item Response Theory." Psychometrika, 54, 427-450.

John Michael Linacre

Note: Winsteps and Facets do not use MMLE or WLE, because Winsteps and Facets embody the philosophy that:
1. transposing the data matrix (exchanging the persons and items) should impact estimation as little as possible.
2. estimating the person and item measures from a free analysis should produce the same results as (i) anchoring the items at their free estimates, and estimating the person measures, or (ii) anchoring the persons at their free estimates and estimating the item measures.

Warm's Mean Weighted Likelihood Estimates (WLE) of Rasch Measures. Warm T.A. … Rasch Measurement Transactions, 2007, 21:1 p. 1094

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
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