# Reduced rank regression for mixed predictor and response variables

## TL;DR

This paper introduces a new regression method that handles mixed types of predictor and response variables using optimal scaling and a majorization-minimization algorithm.

## Contribution

The novel contribution is the generalized mixed reduced rank regression (GMR3) method for mixed variable types with optimal scaling and maximum likelihood estimation.

## Key findings

- GMR3 effectively handles mixed predictor and response variable types using optimal scaling.
- Simulation studies demonstrate the algorithm's performance across different variable types and sample sizes.
- The method is applied to the Eurobarometer Surveys dataset to showcase its practical utility.

## Abstract

In this paper, we propose the generalized mixed reduced rank regression method, GMR3 for short. GMR3 is a regression method for a mix of numeric, binary and ordinal response variables. The predictor variables can be a mix of binary, nominal, ordinal and numeric variables. For dealing with the categorical predictors we use optimal scaling. A majorization‐minimization algorithm is derived for maximum likelihood estimation. A series of simulation studies is shown (Section 4) to evaluate the performance of the algorithm with different types of predictor and response variables. In Section 5, we briefly discuss the choices to make when applying the model the empirical data and give suggestions for supporting such choices. In a second simulation study (Section 6), we further study the behaviour of the model and algorithm in different scenarios for the true rank in relation to sample size. In Section 7, we show an application of GMR3 using the Eurobarometer Surveys data set of 2023.