Biplots in Reduced - Rank Regression

2 SUMMARY Regression problems with a number of related response variables are typically analyzed by separate multiple regressions. This paper shows how these regressions can be visualized jointly in a biplot based on reduced-rank regression. Reduced-rank regression combines multiple regression and principal components analysis and can therefore be carried out with standard statistical packages. The proposed biplot highlights the major aspects of the regressions by displaying the least-squares approximation of fitted values, regression coefficients and associated t-ratio's. The utility and interpretation of the reduced-rank regression biplot is demonstrated with an example using public health data that were previously analyzed by separate multiple regressions. 1. Introduction In comparison with multiple regression, multivariate regression is rarely used by applied statisticians. When a number of response variates is of interest, each response variate is usually analyzed in a separate multiple regression. This practice is justified by the Gauss-Markoff setup of regression theory. In this setup, estimation in multivariate regression reduces to a series of multiple regressions (e.g. RAO, 1973: section 8c2). This holds true also for maximum likelihood estimation (e.g. MARDIA et al. 1979: section 6.2). Things change if a restriction is imposed on the rank of the matrix of regression coefficients (ANDERSON, 1951, 1984). This yields a more parsimonious model in which response variates react to the regressor variables only through a restricted number of 'latent variables'. The latent variables can be estimated by canonical variates. IZENMAN (1975) introduced the apt name reduced-rank regression. Despite further work by, reduced-rank regression has found few applications. This is unfortunate. In our view, reduced-rank regression is particularly useful, because it offers the possibility to produce a plot that easily summarises the major aspects of the regressions. This possibility has so far not been exploited. In this paper we show how the biplot graphic display (GABRIEL, 1971; 1982) can help to visualize the reduced-rank model. The proposed biplot represents geometrically the fitted values of the regressions, the estimated regression coefficients and their associated t-ratios. The two-dimensional biplot is exact for the rank 2 model. For higher ranks, it forms a least-squares approximation. We demonstrate the utility of the reduced-rank regression biplot with an example using public health data that has previously been analyzed by separate multiple regressions (KUNST et al., 1990). Reduced-rank regression can be carried out by standard computer programmes for multivariate analysis. There exist essentially two different methods of estimation. Depending on the assumptions …