Ordinal data, ordered scale points, and order statistics
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 \(\xi\).
ordered Dirichlet prior, ordered cumulative probabilities, ordered scales
scientific article; zbMATH DE number 3844898
scientific article