Ordinal data, ordered scale points, and order statistics
type: Item
instance of: scholarly article
title: Ordinal data, ordered scale points, and order statistics
zbMATH Open document ID: 0532.62095
DOI: 10.1007/BF02293685
author: Pieter Vijn
published in: Psychometrika
publication date: 1983-01-01T00:00:00Z
review text: The numbers of individuals \(n_ 1,...,n_{k+1}\) in \(k+1\) response categories follow a multinomial distribution with sample size \(N=n_ 1+...+n_{k+1}\) and probabilities \(\pi_ 1,...,\pi_{k+1}\). The response categories are labeled \(1,...,k+1\) and considered as ordinal responses corresponding to an underlying latent continuous variable \(\xi\) with ordered unknown cutpoints \(\theta_ 1<...<\theta_ k\). For \(\xi \leq \theta_ 1\) the response category is 1, for \(\theta_ 1<\xi \leq \theta_ 2\) the response category is 2, etc. Let F be the cumulative distribution function of \(\xi\). Then \(\gamma_ j=F(\theta_ j)=\pi_ 1+...+\pi_ j\) is the probability of a response category less than or equal to j for \(j=1,...,k.\) An ordered Dirichlet distribution is used as a Bayesian prior of \(\gamma_ 1,...,\gamma_ k\). The posterior distribution of \(\gamma_ 1,...,\gamma_ k\) is also a Dirichlet distribution, and from this the induced posterior distribution of \(\theta_ 1,...,\theta_ k\) can be deduced. Since it is complicated to evaluate, normal approximations are used to approximate it. Illustrations are provided for logistic scales and for some more general scales on \(\xi\).
Mathematics Subject Classification ID: 62P15
Mathematics Subject Classification ID: 62F15
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 3844898
zbMATH Keywords: ordinal data
zbMATH Keywords: order statistics
zbMATH Keywords: ordered Dirichlet prior, ordered cumulative probabilities, ordered scales
zbMATH Keywords: latent population
zbMATH Keywords: coverages
zbMATH Keywords: log odds ratio
zbMATH Keywords: stochastic ordering
zbMATH Keywords: complementary log-log scale
zbMATH Keywords: logistic scale
zbMATH Keywords: normal approximations
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02293685
OpenAlex ID: W2056594759
description: scientific article; zbMATH DE number 3844898