This document discusses some aspects of modeling multivariate data in the context of regression analysis, in particular when outcomes are of mixed type (continuous, linear, count, etc.) Many common methods - some dating back more than 80 years [1] - deal with multiple outcomes of the same type in the context of hypothesis testing, especially for normally distributed outcomes and the extension to outcomes arising from generalized linear models. Various regression models for multivariate outcomes with different marginal distributions have been developed more recently [2], often tailored to quite specific circumstances.

Here, we discuss and explore properties of some methods which attempt to be slightly more general in that they may be applied to a wide range of mixed types of data. Naturally, there is a trade-off to this apparent flexibility, namely interpretability and power. By reducing the possibility of model mis-specification, we make it somewhat harder to ensure that the interpretation of the results is as unambiguous and straightforward as in simpler, less flexible models. More bluntly, as in the famous tale of the miller and his son [3], by trying to accommodate too many data-generating mechanisms we risk failing to effectively analyze any of them. We attempt to show via simulation studies and analysis of data that this risk isn't too great.

Chapter 1 reviews the so-called density ratio model, a semi-parametric regression model for outcomes arising from an exponential family of distributions. We explore the extension of this model to multivariate endpoints, examine its performance under various data-generating mechanisms, and present an application to a health care data set. Implementation details and a description of the user-friendly interface are also provided.

Chapter 2 begins with a detailed overview and literature review of secondary data analysis models, since this topic may be quite unfamiliar even to practicing statisticians. Next, we apply the (multivariate) density ratio model of Chapter 1 to generalize an existing, popular secondary data analysis model. Simulation studies present the performance of this model, and we discuss some theoretical properties of the proposed estimator.

The third chapter charts a different course by focusing on a particular kind of hypothesis test for mixed-type outcomes. A common reason for performing joint analysis of several outcomes is the possibility of conducting a test of `joint effect', which may be of inherent interest, of utility with small samples and correspondingly low power, or undertaken with respect to multiple testing considerations. The methodology presented there does not attempt to construct a joint model which combines each outcome, adding a great degree of flexibility and avoiding many pitfalls (in particular, mis-specification and interpretation issues) of mixed-data models.

Finally, the Appendices provide supplementary details, proofs, and longer formulae as appropriate.