**Research**

Previous Bayesian multinomial probit models have required the analyst to choose a base or reference category that identifies the location of the latent utilities that are commonly used to define the model. Previous work has shown that the choice of which category serves as the base category can impact the posterior predictions. We describe a Bayesian multinomial probit model that requires no base category and therefore is symmetric with respect to relabeling the outcome categories. We achieve this through a series of sum-to-zero identifying restrictions on the latent utilities and the regression parameters. We propose an efficient marginal data augmentation Gibbs sampler to estimate the model.

Burgette and Nordheim (2012). "The trace restriction: An alternative identification strategy for the Bayesian multinomial probit model." Journal of Business and Economic Statistics, 30(3): 404-410.

Previously, multinomial probit (MNP) models have been
identified by fixing one of the variance parameters at one. In
a consumer choice dataset, we demonstrate that posterior
predictions can be sensitive to the choice of which element we
fix. To avoid this arbitrary choice, we propose a model that
instead restricts the trace of the covariance matrix. In
simulated data, we find that this results in more reliable
predictions. Further, the trace restriction can provide
stronger identification, yielding more meaningful marginal
posterior distributions. The trace restriction is now
the default behavior of the **MNP** package of
Kosuke Imai and David van Dyk.

Public release of spatially-referenced microdata can entail significant risk that motivated intruders will be able to learn the identities of respondents who provide sensitive data. To mitigate this risk, it is standard to aggregate data over large geographic areas, which can degrade the utility of the data for legitimate researchers. As an alternative, we propose methods to produce synthetic sets of areal identifiers. Our goal is to simulate multiple sets of data that--on average--retain the statistical properties of the observed data, while protecting respondents' anonymity. We propose methods to simulate areal identifiers using a multinomial probit model. Because this results in a model that (in typical applications) will have hundreds or even thousands of response categories, we propose a sparse structure for the multinomial model. Further, we suggest a simplified, latent Potts model structure for the regression coefficients, which can help to preserve spatial relationships. We demonstrate our methods on simulated and genuine data.

Burgette and Nordheim (2010). "A full Gibbs sampler for the Bayesian multinomial probit switching model." Manuscript available here.

In this paper, we propose a model for a selection or
switching model with a multinomial response. These models are
useful when respondents self-select the level of a treatment,
or select themselves into or out of the sample. Unlike related
work, our algorithm only requires Gibbs steps. C code and an R
interface are available in the **endogMNP**
package, available from CRAN.

In an ongoing study of adverse birth outcomes, the study team switched from one analytical lab to another when measuring levels of contaminants like lead in the mothers' blood. Inspection makes it clear that the marginal distributions of contaminant measurements are very different for the two labs. We describe three Bayesian nonparametric approaches for flexibly combining the observations into a single lab's scale. Through a series of simulation studies, we provide guidelines for when each method is most appropriate. We then apply our methods to the birth data.

Burgette and Reiter (2010). "Multiple imputation for missing
data via sequential regression trees," American Journal of Epidemiology,
170(9): 1070-1076. Available here.

Burgette and Reiter (2011). "Modeling
adverse birth outcomes via confirmatory factor quantile
regression." Forthcoming at Biometrics.

Burgette, Reiter and Miranda (2011). "Exploratory
quantile regression with many covariates: An application
to adverse birth outcomes." Epidemiology
22(6): 859-866.

We propose a Bayesian growth mixture model to jointly examine the associations between longitudinal blood pressure trajectories, preterm birth (PTB), and low birth weight (LBW). The model partitions women into distinct classes characterized by a longitudinal mean arterial pressure curve and joint probabilities of PTB and LBW.