The polyserial correlation coefficient
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 estimate of \(\rho\) is computed; 3) an ad hoc estimator \({\hat \rho}=r_{xy}\cdot s_ y/\sum_{j}\phi({\hat \tau}_ j)\) of \(\rho\) is determined by inserting in the above-mentioned relation the sample estimates \(r_{xy}\) for \({\tilde \rho}\), \(s_ y\) for \(\sigma_ y\), \({\hat \tau}{}_ j\) for \(\tau_ j.\) The three methods are compared by Monte Carlo simulation (four-way 2\(\cdot 2\cdot 3\cdot 2\) factorial design with factors N, symmetry or asymmetry of threshold system (\(\tau)\), \(\rho\), r, and with 50 replications in each cell). All three methods perform well, whereas the direct use of \(r_{xy}\) would be rather misleading.
scientific article; zbMATH DE number 3852231