Modelling Comparison Data with Ordinal Response

Giuseppe Casalicchio
In this thesis, we present several extensions of the Bradley-Terry-Luce model, which is known as a pair comparison model. The aim of pair comparisons is to elicit an overall ranking for a set of objects that are compared pairwise by judges, or, as we will call them, 'subjects'. The presented extensions allow for ordinal responses and the inclusion of subject-specific covariates as well as order effects (or object-specific order effects). If an order effect is present, there is an advantage or
more » ... sadvantage in a pair comparison for the object that is presented first. In the context of sport competitions, this effect is equivalent to having a home advantage (or a team-specific home advantage). The inclusion of subject-specific covariates in a Bradley-Terry-Luce model may lead to an over-parameterized model. This is because subject-specific covariates are included in the model as subject-object interactions, and for each additional subject-specific covariate, we need to estimate as many subject-object parameters as there are object parameters. To tackle this problem, we suggest a component-wise boosting algorithm that is able to select the most important subject-object interactions. The performance of this algorithm is investigated through a simulation study. This thesis comes along with an R-Package called ordBTL that can be used to fit the proposed models and extensions. Lastly, we will illustrate some extensions by applying them to different datasets. τ Kendall's τ rank correlation coefficient Φ dispersion parameter iii Nomenclature vectors and matrices x i = (x i1 , . . . , x iP ) vector of subject-specific covariates for subject i x (r,s) = x (r,s) 1 , . . . , x (r,s) M −1 vector of object indicators when the pair (r, s) is compared x (r,s) iC = x (r,s) i1 , . . . , x (r,s) iC vector of candidate covariates for subject i when the pair (r, s) is compared (x (r,s) ic = x ip x (r,s) m ) h (r,s) = h (r,s) 1 , . . . , h (r,s) M vector of dummy variables indicating objectspecific order effects (or team-specific home advantages) Q (K−1)×q constraint matrix (for the thresholds) X (r,s) O comparison-specific design matrix when comparing the pair (r, s) once X iO design matrix containing all X (r,s) O for all pairs (r, s) that are compared by subject i X O design matrix containing X iO , ∀i (contains all comparisons made by all subjects) X (r,s) iC = 1 K−1 ⊗ x (r,s) iC matrix of candidate covariates for subject i when comparing the pair (r, s) X C = [x ·1 , . . . , x ·C ] matrix containing candidate covariates (contains all X (r,s) iC ∀r = s, i) X (r,s) i = X (r,s) O , X (r,s) iC design matrix containing object indicators and candidate covariates for a comparison of the pair (r, s) made by subject i X = [X O , X C ] complete design matrix with object indicators and candidate covariates for all comparisons and all subjects miscellaneous X m latent variable referring to object m m ∼ iid random variable referring to object m x = max{z ∈ Z | z ≤ x} floor function
doi:10.5282/ubm/epub.21746 fatcat:dvlugcbj5vhi7kmnn36fdjdyry