Ordinal analysis

Abstract: Much behavioral rescarch involves comparing the central tendencies of different groups, or of the same subjects under different conditions, and the usual analysis is some form of mean comparison. This article suggests that an ordinal statistic, d, is often more appropriate. d compares the number of times a score from one group or condition is higher than one from the other, compared with the reverse. Compared to mean comparisons, d is more robust and equally or more powerful; it is invariant under transformation; and it often conforms more closely to the experimenter's research hypothesis. It is suggested that inferences from d be based on sample estimates of its variance rather than on the more traditional assumption of identical distributions

Abstract: Background Epidemiologists are often interested in estimating the risk of several related diseases as well as adverse outcomes, which have a natural ordering of severity or certainty. While most investigators choose to model several dichotomous outcomes (such as very low birthweight versus normal and moderately low birthweight versus normal), this approach does not fully utilize the available information. Several statistical models for ordinal responses have been proposed, but have been underutilized. In this paper, we describe statistical methods for modelling ordinal response data, and illustrate the fit of these models to a large database from a perinatal health programme. Methods Models considered here include (1) the cumulative logit model, (2) continuation-ratio model, (3) constrained and unconstrained partial proportional odds models, (4) adjacent-category logit model, (5) polytomous logistic model, and (6) stereotype logistic model. We illustrate and compare the fit of these models on a perinatal database, to study the impact of midline episiotomy procedure on perineal lacerations during labour and delivery. Finally, we provide a discussion on graphical methods for the assessment of model assumptions and model constraints, and conclude with a discussion on the choice of an ordinal model. The primary focus in this paper is the formulation of ordinal models, interpretation of model parameters, and their implications for epidemiological research. Conclusions This paper presents a synthesized review of generalized linear regression models for analysing ordered responses. We recommend that the analyst performs (i) goodness-of-fit tests and an analysis of residuals, (ii) sensitivity analysis by fitting and comparing different models, and (iii) by graphically examining the model assumptions.

Abstract: We propose a technique that we call HodgeRank for ranking data that may be incomplete and imbalanced, characteristics common in modern datasets coming from e-commerce and internet applications. We are primarily interested in cardinal data based on scores or ratings though our methods also give specific insights on ordinal data. From raw ranking data, we construct pairwise rankings, represented as edge flows on an appropriate graph. Our statistical ranking method exploits the graph Helmholtzian, which is the graph theoretic analogue of the Helmholtz operator or vector Laplacian, in much the same way the graph Laplacian is an analogue of the Laplace operator or scalar Laplacian. We shall study the graph Helmholtzian using combinatorial Hodge theory, which provides a way to unravel ranking information from edge flows. In particular, we show that every edge flow representing pairwise ranking can be resolved into two orthogonal components, a gradient flow that represents the l 2-optimal global ranking and a divergence-free flow (cyclic) that measures the validity of the global ranking obtained—if this is large, then it indicates that the data does not have a good global ranking. This divergence-free flow can be further decomposed orthogonally into a curl flow (locally cyclic) and a harmonic flow (locally acyclic but globally cyclic); these provides information on whether inconsistency in the ranking data arises locally or globally. When applied to statistical ranking problems, Hodge decomposition sheds light on whether a given dataset may be globally ranked in a meaningful way or if the data is inherently inconsistent and thus could not have any reasonable global ranking; in the latter case it provides information on the nature of the inconsistencies. An obvious advantage over the NP-hardness of Kemeny optimization is that HodgeRank may be easily computed via a linear least squares regression. We also discuss connections with well-known ordinal ranking techniques such as Kemeny optimization and Borda count from social choice theory.