Differential Privacy Applications to Bayesian and Linear Mixed Model Estimation

We consider a particular maximum likelihood estimator (MLE) and a computationally-intensive Bayesian method for differentially private estimation of the linear mixed-effects model (LMM) with normal random errors. The LMM is important because it is used in small area estimation and detailed industry tabulations that present significant challenges for confidentiality protection of the underlying data. The differentially private MLE performs well compared to the regular MLE, and deteriorates as the protection increases for a problem in which the small-area variation is at the county level. More dimensions of random effects are needed to adequately represent the time- dimension of the data, and for these cases the differentially private MLE cannot be computed. The direct Bayesian approach for the same model uses an informative, but reasonably diffuse, prior to compute the posterior predictive distribution for the random effects. The differential privacy of this approach is estimated by direct computation of the relevant odds ratios after deleting influential observations according to various criteria

Stop or Continue Data Collection: A Nonignorable Missing Data Approach for Continuous Variables

We present an approach to inform decisions about nonresponse follow-up sampling. The basic idea is (i) to create completed samples by imputing nonrespondents' data under various assumptions about the nonresponse mechanisms, (ii) take hypothetical samples of varying sizes from the completed samples, and (iii) compute and compare measures of accuracy and cost for different proposed sample sizes. As part of the methodology, we present a new approach for generating imputations for multivariate continuous data with nonignorable unit nonresponse. We fit mixtures of multivariate normal distributions to the respondents' data, and adjust the probabilities of the mixture components to generate nonrespondents' distributions with desired features. We illustrate the approaches using data from the 2007 U. S. Census of Manufactures