Shrinkage Estimation of Log-odds Ratios for Comparing Mobility Tables

Abstract

Statistical analysis of mobility tables has long played a pivotal role in comparative stratification research. This article proposes a shrinkage estimator of the log-odds ratio for comparing mobility tables. Building on an empirical Bayes framework, the shrinkage estimator improves estimation efficiency by “borrowing strength” across multiple tables while placing no restrictions on the pattern of association within tables. Numerical simulation shows that the shrinkage estimator outperforms the usual maximum likelihood estimator (MLE) in both the total squared error and the correlation with the true values. Moreover, the benefits of the shrinkage estimator relative to the MLE depend on both the variation in the true log-odds ratio and the variation in sample size among mobility regimes. To illustrate the effects of shrinkage, the author contrasts the shrinkage estimates with the usual estimates for the mobility data assembled by Hazelrigg and Garnier for 16 countries in the 1960s and 1970s. For mobility tables with more than two categories, the shrinkage estimates of log-odds ratios can also be used to calculate summary measures of association that are based on aggregations of log-odds ratios. Specifically, the author constructs an adjusted estimator of the Altham index and, with a set of calibrated simulations, demonstrates its usefulness in enhancing both the precision of individual estimates and the accuracy of cross-table comparisons. Finally, using two real data sets, the author shows that in gauging the overall degree of social fluidity, the adjusted estimator of the Altham index agrees more closely with results from the Unidiff model than does the direct estimator of the Altham index.