A General Procedure for Parameter Estimation for the Law of Comparative Judgement

Abstract

This paper gives a numerical procedure for estimating the parameters of the law of comparative judgement under a variety of conditions. A broad class of goodnessof-fit functions is treated, including weighted least-squares and maximumlikelihood criteria. Any response function that has the necessary derivatives can be employed. Various kinds of constraints may be imposed on the parameters of the model. Special attention is given to linear constraints; and to equality constraints, where any of the parameters may be assumed to be known in advance and any subset of the parameters may be assumed to take on either a specified or an unspecified common value. In the maximum-likelihood case, for large samples, approximate standard errors and confidence intervals for the estimated parameters may be obtained, and goodness of fit may be tested by the likelihoodratio technique. Numerical examples are provided. the symbol, vs,, the law of comparative judgement states that Letting n,, = P(X, < X,).pc=B(X,) ( l < i < n ) , ada = uq2 rsj = raq ( l < j d i < n ) ,the random variable, Xi -X,, has an expected value of prpi and a variance of d,a = uia + u,a -2ri, ui 0 , . Consequently,Z d 5 = (4 -4 -Pi + P,)/dd$ 1973,26, 240-2601

Limited information estimation and testing of Thurstonian models for paired comparison data under multiple judgment sampling

“…In closing, it should be noted that the ULS estimator employed here is closely related to the classical least squares procedures proposed for estimating Thurstonian models (see Arbuckle & Nugent, 1973;Bock & Jones, 1968, and references therein). However, these use first order information only (i.e., binary choices), which are assumed to be independent.…”

Bayesian analysis of order-statistics models for ranking data

“…In the previous example, it can be seen that VA can be transformed to VB and Vc by applying (11) This identifiability problem is well-known in the context of Thurstone's order statistics models and multinomial probit models (Arbuckle & Nugent, 1973;Bunch, 1991;Dansie, 1985;and Yai et al, 1997). Various solutions by imposing constraints to the covariance matrix V has been proposed in the literature and are summarized in Table 1.…”

“…for any constant vector c (Arbuckle & Nugent, 1973), V can never be identified unless it is structured. In the previous example, it can be seen that VA can be transformed to VB and Vc by applying (11) This identifiability problem is well-known in the context of Thurstone's order statistics models and multinomial probit models (Arbuckle & Nugent, 1973;Bunch, 1991;Dansie, 1985;and Yai et al, 1997).…”

Multilevel logistic regression for polytomous data and rankings

“…Bloxom (1972) and Arbuckle and Nugent (1973) are seminal contributions, introducing common factor models in the pair-comparisons literature. Probit versions for rankings, with the restriction ⌿ (1) = I, have been discussed in a series of papers (Böckenholt, 1993;Brady, 1989;Chan & Bentler, 1998).…”