Maydeu-Olivares and Joe (J. Am. Stat. Assoc. 100:1009–1020, 2005; Psychometrika 71:713–732, 2006) introduced classes of chi-square tests for (sparse) multidimensional multinomial data based on low-order marginal proportions. Our extension provides general conditions under which quadratic forms in linear functions of cell residuals are asymptotically chi-square. The new statistics need not be based on margins, and can be used for one-dimensional multinomials. We also provide theory that explains why limited information statistics have good power, regardless of sparseness. We show how quadratic-form statistics can be constructed that are more powerful than X2 and yet, have approximate chi-square null distribution in finite samples with large models. Examples with models for truncated count data and binary item response data are used to illustrate the theory.

Factor or conditional independence models based on copulas are proposed for multivariate discrete data such as item responses. The factor copula models have interpretations of latent maxima/minima (in comparison with latent means) and can lead to more probability in the joint upper or lower tail compared with factor models based on the discretized multivariate normal distribution (or multidimensional normal ogive model). Details on maximum likelihood estimation of parameters for the factor copula model are given, as well as analysis of the behavior of the log-likelihood. Our general methodology is illustrated with several item response data sets, and it is shown that there is a substantial improvement on existing models both conceptually and in fit to data.

In categorical data analysis, two-sample cross-validation is used not only for model selection but also to obtain a realistic impression of the overall predictive effectiveness of the model. The latter is of particular importance in the case of highly parametrized models capable of capturing every idiosyncracy of the calibrating sample. We show that for maximum likelihood estimators or other asymptotically efficient estimators Pearson’s X2 is not asymptotically chi-square in the two-sample cross-validation framework due to extra variability induced by using different samples for estimation and goodness-of-fit testing. We propose an alternative test statistic, X2xval, obtained as a modification of X2 which is asymptotically chi-square with C - 1 degrees of freedom in cross-validation samples. Stochastically, X2xval≤ X2. Furthermore, the use of X2 instead of X2xval with a χ2C - 1 reference distribution may provide an unduly poor impression of fit of the model in the cross-validation sample.

We introduce a family of goodness-of-fit statistics for testing composite null hypotheses in multidimensional contingency tables. These statistics are quadratic forms in marginal residuals up to order r. They are asymptotically chi-square under the null hypothesis when parameters are estimated using any asymptotically normal consistent estimator. For a widely used item response model, when r is small and multidimensional tables are sparse, the proposed statistics have accurate empirical Type I errors, unlike Pearson’s X2. For this model in nonsparse situations, the proposed statistics are also more powerful than X2. In addition, the proposed statistics are asymptotically chi-square when applied to subtables, and can be used for a piecewise goodness-of-fit assessment to determine the source of misfit in poorly fitting models.

The provision of pensions for Civil Servants and other employees in public office, such as the police, as well as in large private businesses, became more widespread in the second half of the nineteenth century. Such pensions, and other non-pay benefits, including sick pay, not only helped with recruitment but also provided a means of managing the retirement of workers who were deemed to be incapable of performing their roles. The rules governing eligibility to receive a pension in the Metropolitan Police in London were closely linked to the certification of poor health. Police doctors restricted the certification of sickness as a reason for retirement because it impacted the size of the force, resulted in the loss of more experienced men, and added to the cost of the pension fund. This strategy generated conflict with the workforce, resulting in industrial unrest. Piecemeal reforms failed to address workers’ concerns until 1890, when the rights to receive a pension were improved. These reforms, rather than stricter vigilance by police doctors, were an effective way of retaining experienced officers in the police force.