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We describe the Bedside Patient Rescue (BPR) project, the goal of which is risk prediction of adverse events for non-ICU patients using ~200 variables (vitals, lab results, assessments, ...). There are several missing predictor values for most patients, which in the health sciences is the norm, rather than the exception. A Bayesian approach is presented that addresses many of the shortcomings to standard approaches to missing predictors: (i) treatment of the uncertainty due to imputation is straight-forward in the Bayesian paradigm, (ii) the predictor distribution is flexibly modeled as an infinite normal mixture with latent variables to explicitly account for discrete predictors (i.e., as in multivariate probit regression models), and (iii) certain missing not at random situations can be handled effectively by allowing the indicator of missingness into the predictor distribution only to inform the distribution of the missing variables. The proposed approach also has the benefit of providing a distribution for the prediction, including the uncertainty inherent in the imputation. Therefore, we can ask questions such as: is it possible this individual is at high risk but we are missing too much information to know for sure? How much would we reduce the uncertainty in our risk prediction by obtaining a particular missing value? This approach is applied to the BPR problem resulting in excellent predictive capability to identify deteriorating patients.

A difficulty in joint modeling of continuous and discrete response variables is the lack of a natural multivariate distribution. For joint modeling of clustered observations on binary and continuous responses, we study a correlated probit model that has an underlying normal latent variable for the binary responses. Catalano and Ryan have factored the model into a marginal and a conditional component and used generalized estimating equations methodology to estimate the effects. We propose a Monte Carlo expectation–conditional maximization algorithm for finding maximum likelihood estimates of the mixed model itself, extending and accelerating an algorithm for models with binary responses. We demonstrate the methodology with a developmental toxicity study measuring fetal weight and a binary malformation status for several litters of mice. A simulation study suggests that efficiency gains of joint fittings over separate fittings of the response variables occur mainly for small datasets with strong correlation...

A computationally practical form of probit analysis for multiple response variables was applied to data on health service utilization as it relates to type of supplemental insurance coverage and other demographic covariates in a sample of 4,658 individuals over age 65 on Medicare. The quantal response variables comprised use of five health services: Medical provider visit, hospital outpatient visit, emergency room visit, hospital inpatient stay, and home health care visit. An individual could use any or all of these services. Supplemental insurance coverage was categorized as Medicaid, employer-related private insurance, and privately purchased insurance, and any combination of these insurance coverages was possible. Covariates included health status, education, urban location, race, sex, marital status, employment status, age, and income. The multivariate probit model estimates association between insurance coverage variables and health service utilization patterns jointly for the five health se...

The multivariate probit model (MPM) is a particular case of the class of correlated prediction models. A correlated prediction model is especially useful when prediction or classification is envisaged into diagnostic classes that are combinations of binary responses. The parameter vector consists of a “location” part and an “association” part. The location part accounts for the effect the regressors have on the marginal probabilities of the binary responses. The association part corrects these probabilities, taking into account that the responses are related. This article investigates conditions for the existence and unicity of the maximum likelihood estimator (MLE) of the parameter vector. It turns out that the existence and uniqueness of the MLE for the location parameters when the association parameters are known are related to those of the multigroup logistic model. Necessary and sufficient conditions are given for the existence of the MLE of the association part. On the other hand the condit...

(2003). Logit and Probit: Ordered and Multinomial Models. Journal of the American Statistical Association: Vol. 98, No. 463, pp. 775-776.

In this article we propose two minimum chi-square estimators for a bivariate probit model. We call one estimator the Full Information and the other Limited Information Minimum Chi-Square because the first takes account of all the a priori information while the second does not. Both estimators are shown to be consistent. Moreover, the first is shown to be asymptotically as efficient as the maximum likelihood estimator and yet is computationally much simpler. For illustration, estimates are computed for the data used by Ashford and Sowden [1970.]