The polyserial correlation coefficient
type: Item
instance of: scholarly article
title: The polyserial correlation coefficient
zbMATH Open document ID: 0536.62045
DOI: 10.1007/BF02294164
author: Fritz Drasgow
author: Neil J. Dorans
author: Ulf Henning Olsson
published in: Psychometrika
publication date: 1982-01-01T00:00:00Z
review text: Suppose the joint probability distribution of the variables X with \(E(X)=\mu\), Var X\(=\sigma^ 2\), and \(\eta\) with \(E(\eta)=0\), Var \(\eta =I\), is binormal with correlation \(\rho_{X_{\eta}}=\rho\). Instead of the underlying continuous variable \(\eta\), the authors consider the ordinal categorical variable Y defined by a monotonic step function \(Y=y_ j\) if \(\tau_{j-1}\leq \eta<\tau_ j (j=1,2,...,r)\), with \(y_{j-1}<y_ j\) and \(\tau_ 0=-\infty\), \(\tau_{j-1}<\tau_ j\), \(\tau_ r=\infty\), whose probabilities are obviously \(p_ j=P(Y=y_ j)=\Phi(\tau_ j)-\Phi(\tau_{j-1})\) with \(\Phi(\tau)=\int^{\tau}_{-\infty}\phi(t)dt\), \(\phi(t)=\exp(-t^ 2/2)/\sqrt{2\pi}\), and thence derive the ''point-polyserial'' correlation between X and Y \[ {\tilde \rho}=\rho \sum^{r-1}_{j=1}\phi(\tau_ j)\cdot(y_{j+1}-y_ j)/\sigma_ y. \] This most general relation depends on r, on the threshold values \(\tau_ j\), and on the scoring ones \(y_ j\). It generalizes known results on biserial correlation \((r=2)\), and about other special scoring systems, as those studied by \textit{N. R. Cox} [Biometrics 30, 171-178 (1974; Zbl 0292.62022)] and \textit{N. Jaspen} [Serial correlation. Psychometrika 11, 23-30 (1946)]. The relation is used in estimating the polyserial correlation \(\rho\) from a sample of N observations \((x_ i,y_ i)\), \(i=1,...,N\), on the variable (X,Y). Assuming a scoring system with \(y_ j=\) consecutive entire numbers, there are to be estimated the unknown model parameters \(\rho,\mu,\sigma,\tau_ 1,...,\tau_{r-1}.\) The authors study three methods: 1) simultaneous estimation of all parameters by maximum likelihood, solving a complicated non-linear equation system; 2) the two-step method in which, after having estimated \(\mu\) and \(\sigma^ 2\) by the sample statistics \(\bar x\), \(s^ 2_ x\), and \(\tau_ 1,...,\tau_{r-1}\) by the inverse of a normal distribution function applied to the observed marginal distribution of Y, a conditional maximum likelihood estima…
Mathematics Subject Classification ID: 62H20
Mathematics Subject Classification ID: 65C05
zbMATH DE Number: 3852231
zbMATH Keywords: dichotomous variables
zbMATH Keywords: polychotomous variables
zbMATH Keywords: latent variables
zbMATH Keywords: ordinal categorical variable
zbMATH Keywords: monotonic step function
zbMATH Keywords: polyserial correlation
zbMATH Keywords: simultaneous estimation
zbMATH Keywords: maximum likelihood
zbMATH Keywords: non-linear equation system
zbMATH Keywords: two-step method
zbMATH Keywords: conditional maximum likelihood estimate
zbMATH Keywords: ad hoc estimator
MaRDI profile type: Publication
cites work: Q3667792
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description: scientific article; zbMATH DE number 3852231