Abstract
A novel exploratory approach is developed to the analysis of a large table of counts. It uses random-effects models where the cells of the table (representing types of individuals) form the higher level in a multilevel model. The model includes Poisson variation and an offset to model the ratio of observed to expected values thereby permitting the analysis of relative rates. The model is estimated as a Bayesian model through MCMC procedures and the estimates are precision-weighted so that unreliable rates are down-weighted in the analysis. Once reliable rates have been obtained graphical and tabular analysis can be deployed. The analysis is illustrated through a study of the occupational class distribution for people of different age, birthplace-origin and generation in Australia. The case is also made that even where there is a full census there is a need to move beyond a descriptive analysis to a proper inferential and modelling framework. We also discuss the relative merits of Full and Empirical Bayes approaches to model estimation.
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Notes
For details on the TableBuilder facility see http://www.abs.gov.au/websitedbs/censushome.nsf/home/tablebuilder—accessed July 29 2014.
The Normality assumption of the cell differentials is obviously a key assumption for the validity of the variance in summarising the differences in the relative risk. This can be informally assessed with a Normal probability plot. In practice we have found that this assumption is generally met; no doubt due to using the log transform. Moreover, McCulloch and Neuhaus (2011) have found model results are generally robust to the shape of the random-effects distribution. An exception to this would be marked outliers for particular cells which could be accommodated by specifying separate fixed effects for these cells which would make them immune to shrinkage.
For ease of exposition (and as per normal practice) the imprecision in the ratio is dependent only on the imprecision of the observed count. It is being assumed that the expected count is precise. A more realistic formulation is given in Talbot et al. (2011). The specific nature of the weighting for this log-Normal model is considered by Papageorgiou and Ghosh (2012, Eqs. 1–3); albeit in an empirical Bayes formulation.
The weight is a form of interclass correlation coefficient for each cell that measures the amount of true variability (the level 2 variance) in the underling rates relative to the total observed variability. In the measurement literature, the reliability \(w_j \) is often symbolised by \(\rho _{yy} \) to convey the internal dependency of a measured y variable.
The between cell variance at level 2 summarizes the differences between cells, but usefully it is not the variance of the shrunken differentials, but the variance of the raw differentials. Consequently it is not the estimated between group variance of the sample, but the estimated between-group variance in the population.
For a more general discussion of these advantageous properties see the classic papers of James and Stein (1961), and Lindley and Smith (1972). Their benefits are extolled in Kendall (1959) ‘song’, and in the expository paper of Efron and Morris (1977) which studies baseball averages and disease distributions.
The estimate of pD is given by the difference between the average deviance and the deviance at the expected value of the unknown parameters.
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Jones, K., Owen, D., Johnston, R. et al. Modelling the occupational assimilation of immigrants by ancestry, age group and generational differences in Australia: a random effects approach to a large table of counts. Qual Quant 49, 2595–2615 (2015). https://doi.org/10.1007/s11135-014-0130-8
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DOI: https://doi.org/10.1007/s11135-014-0130-8