The frequencies of m independent p-way contingency tables are analyzed by a model that assumes that the ordinal categorical data in each of m groups are generated from a latent continuous multivariate normal distribution. The parameters of these multivariate distributions and of the relations between the ordinal and latent variables are estimated by maximum likelihood. Goodness-of-fit statistics based on the likelihood ratio criterion and the Pearsonian chisquare are provided to test the hypothesis that the proposed model is correct, that is, it fits the observed sample data. Hypotheses on the invariance of means, variances, and polychoric correlations of the latent variables across populations are tested by Wald statistics. The method is illustrated on an example involving data on three five-point ordinal scales obtained from male and female samples.

The polychoric correlation is discussed as a generalization of the tetrachoric correlation coefficient to more than two classes. Two estimation methods are discussed: Maximum likelihood estimation, and what may be called “two-step maximum likelihood” estimation. For the latter method, the thresholds are estimated in the first step. For both methods, asymptotic covariance matrices for estimates are derived, and the methods are illustrated and compared with artificial and real data.

Previous influential simulation studies investigate the effect of underlying non-normality in ordinal data using the Vale–Maurelli (VM) simulation method. We show that discretized data stemming from the VM method with a prescribed target covariance matrix are usually numerically equal to data stemming from discretizing a multivariate normal vector. This normal vector has, however, a different covariance matrix than the target. It follows that these simulation studies have in fact studied data stemming from normal data with a possibly misspecified covariance structure. This observation affects the interpretation of previous simulation studies.

The polychoric correlation is a popular measure of association for ordinal data. It estimates a latent correlation, i.e., the correlation of a latent vector. This vector is assumed to be bivariate normal, an assumption that cannot always be justified. When bivariate normality does not hold, the polychoric correlation will not necessarily approximate the true latent correlation, even when the observed variables have many categories. We calculate the sets of possible values of the latent correlation when latent bivariate normality is not necessarily true, but at least the latent marginals are known. The resulting sets are called partial identification sets, and are shown to shrink to the true latent correlation as the number of categories increase. Moreover, we investigate partial identification under the additional assumption that the latent copula is symmetric, and calculate the partial identification set when one variable is ordinal and another is continuous. We show that little can be said about latent correlations, unless we have impractically many categories or we know a great deal about the distribution of the latent vector. An open-source R package is available for applying our results.

Although there has been significant research on the relationship between alcohol consumption and demographic and psychological influences, this does not consider the effect of social influence among older drinkers and if these effects differ between men and women. One aspect of social influence is social capital. The aim of this paper is to examine whether relational and cognitive social capital are associated with higher or lower risk of alcohol use among adults aged 50 years or older and to assess the extent to which this relationship differs between men and women. To investigate this, data were collected from a cross-sectional questionnaire survey of adults over the age of 50 in the United Kingdom who were recruited from general practitioners. The sample consisted of 9,984 individuals whose mean age was 63.87 years. From these data, we developed proxy measures of social capital and associate these with the respondent's level of alcohol consumption as measured on the Alcohol Use Disorders Identification Test (AUDIT-10) scale. In the sample, just over 20 per cent reported an increasing risk or dependency on alcohol. Using two expressions of social capital – relational (social relationships) and cognitive (knowledge acquisition and understanding) – we found that greater levels of both are associated with a reduced risk of higher drinking risk. Being female had no significant effect when combined with relational capital but did have a significant effect when combined with cognitive capital. It is argued that interventions to enhance social relations among older people and education to help understand alcohol risks would be helpful to protect older people from the damaging effects of excessive alcohol consumption.